PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINE return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DAIE PPEU A DATE DUE DATE DUE
fig 3 MN,

59V.

 

.A

A
(“a

g; I 1

mm M003

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1100 claim“

 

 

PREPARTUM SELECTION OF HOLSTEIN HEIFERS TO
MAXIMIZE PROFIT OF MICHIGAN DAIRY PRODUCERS

By

Brian R. Radke

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1998

 

ABSTRACT

PREPARTUM SELECTION OF HOLSTEIN HEIFERS TO
MAXIMIZE PROFIT OF MICHIGAN DAIRY PRODUCERS

By

Brian R Radke

Based on the culling optimization and simulation literature, it can be concluded that the
Michigan dairy industry has an excessive cull rate. The corollary is that producers have an
excess of heifers, dictating the need for a method of prepartum heifer selection. Selecting
heifers on the basis of lifetime profit corrected for the opportunity cost of postponed
replacement will maximize producer profit. While this profit measure can not be calculated
prior to death, it has moderate correlation with first lactation milk production. Estimated
brwding values (EBV) which are fimctions of the heifers’ parents’ predicted transmitting
abilities (PTA) are the only widely available data on which to base heifer selection
decisions. While theoretically and intuitively parents’ PTA should predict subsequent
daughter milk production, and therefore, lifetime profit corrected for the opportunity cost
of postponed replacement, empirical support for this relationship was lacking. Based on
the data of5, 123 Michigan Holstein heifers, parents’ PTA of milk, fat, and protein
available just prior to the heifers’ calvings account for under 7% of the within herd
variation in milk production. While diffuse, the distribution of within herd rank
correlations between predicted mature equivalent milk production, based on genetic
evaluations, and the actual mature equivalent milk production, was skewed to the left
suggesting the evaluations may be useful in prepartum heifer selection. The omission

oflleifers Which were culled prior to first lactation, did not appear to cause selection bias in

"a.

.5.

.2. .

X

wx

- \

}\

the sample data. It also appeared herd could be modeled as a random effect. Despite
evidence that first lactation milk production did influence producers’ culling decisions,
parents’ PTA of milk, fat and protein estimated after the birth of 5,619 Michigan Holstein
heifers were unable to predict heifer culling during rearing or first lactation. This also
suggested Michigan producers were not basing their culling decisions on genetics. There
appeared to be little difi‘erence in the voluntary nature of voluntary versus involuntary
wllings casting doubt on the existence or utility of such a culling classification scheme.
Modified Box-Complex, a search algorithm, was used to determine optimal heifer
selection rules under three scenarios. Separate data sets of 58 and 57 Michigan herds were
used to derive and test the rules, respectively. The objective of the rule was to maximize
average profit per heifer, summed over all herds in the data set, subject to a given
proportion of heifers being selected. Profit was a function of the returns to heifer sales
which varied across the four raising periods the rule operated over and lifetime profit
corrected for opportunity cost of each heifer estimated from first lactation milk
production. Genetic data consisted of PTA milk, fat, protein, and associated reliabilities of
each parent in each of the four periods. Considerable sampling error was encountered in
terms of rule performance between the samples used to derive and test the selection rules.
Regardless of scenario, selection based on EBV of milk in the most profitable period to
sell heifers performed comparably to the rule developed by Complex. Use of either of
these rules resulted in a $20 per heifer improvement in profit over random selection. While
heifer selection based on EBV milk is more profitable than random selection, only a

portion of the profit available in heifer selection is being captured.

 

This research is dedicated to my family who, although I’m not sure they understood,
supported my leaving practice to return to academia.

Specifically, this is for Natalie, who understands all too well why I left practice. Who,

during the last four and a half years, endured my worst -— studying nights and weekends,
and my best- windsurfing, while she toiled away. Your turn?

iv

 

inn

a:

£7

In?

/r

ACKNOWLEDGEMENTS

I am indebted to my major professor James W. Lloyd, who treated me as a
respected colleague. Our relationship was the envy of many of the other Agricultural
Economics graduate students. J'un supported me throughout my studies and research. He
further contributed to my development by opening doors for my participation in a number
of presentations and workshops to varied audiences, as well as providing consulting
opportunities. And when the time was right, Jim gave the nudge out of the nest to
encourage me to stretch my wings. On a personal note, Natalie and I are gratefiil to the
entire Lloyd family for their support and fiiendship.

I also thank my guidance committee: J. Roy Black, Stephen Harsh, Jefi‘ M.
Wooldridge, John B. Kaneene, and Robert J. Tempelman for providing guidance in their
respective areas of expertise.

Finally, the financial support of the Department of Agricultural Economics through
the Animal Industry Initiative funds provided by the Agricultural Experiment Station at
Michigan State University while the majority of this work was conducted is gratefully

acknowledged.

t"... A

0.. .O-o-

.
b

r t,
'.
-9

W';

"l '
Y

5*.- r
‘l‘ .I

I 'I

TABLE OF CONTENTS

List of Tables

List of Figures

Chapter 1 - Statement of the Research Issue and Introduction

Chapter 2 - Present Value Models

Chapter 3 - Dynamic Optimization and Culling Models

Chapter 4 - Heifer Selection

Chapter 5 -— Selection Index Theory

Chapter 6 - Summary of Literature Review and Hypotheses

Chapter 7 — Prediction and Ranking of First Lactation Milk Using Parent PTA

Chapter 8 - Prediction of Culling Prior to Second Lactation based on Parents’
Predicted Transmitting Abilities

Chapter 9 - Complex Solution to the Value of Genetic Information in Prepartum
Holstein Heifer Selection

Chapter 10 - Conclusions

Appendix A - Fixed Effect Logit (Conditional Logistic) Proof for a Cluster of 3
with One Success

Appendix B - Modified Box-Complex FORTRAN Code for the Heifer Selection
Problem

Appendix C - FORTRAN Code to Simulate Herd Profits Given a Heifer
Selection Rule

vii

33

88

101

145

148

179

218

259

273

279

300

LIST OF TABLES

Table 3.1. Production, as a percent of the mean production for that parity, below
which insemination would not be optimum by parity for the base scenario (8000

kg ME).

Table 3.2. Selected results of sensitivity analysis of altering basel scenario
assumptions on the optimum culling policy.

Table 3.3. Resulting yearly annuity and culling rates when average ME was 8000
kg, including and not including the financial lossesl associated with cows
involuntarily culled.

Table 3.4. Simulated level of total culling to maximize objectives under varying
conditions.

Table 3.5. Results of Dutch simulation of four selection policies during first
lactation.

Table 3.6. Results of US simulation of four selection policies.

Table 3.7. Selected results of culling optimization studies.

Table 3.8. Selected results of culling simulation studies.

Table 4.1. Approximate reliabilities of RNIOC fi'om combination of mature
equivalent yield of milk and fat during first lactation (yield), 14 linear type traits
and final score (type), and total months in milk to 84 months of age (MIM), and
MIM predicted from 36 months of age (PMIM).

Table 5.1. Estimated breeding value, and first lactation production 95%
confidence intervals for the range of parent reliabilities of milk production
genetic value.

Table 5.2. Heritabilities (on diagonal), phenotypic (above diagonal) and genetic
(below diagonal) correlations among first lactation yield traits (Jairath et al.,
1995).

Table 7.1. Parent PTA of milk, fat and protein, reliabilities for milk-fat, and first
lactation mature equivalent milk production for 5,123 animals.

vii

49

49

53

62

84

86

91

126

131

173

Table 7.2. Regression coefficients, and standard errors for each of the three
regressions of standardized first lactation mature equivalent milk production on
standardized parent PTA of milk, fat and protein for 5,123 animals.

Table 7.3. Frequency distributions of within herd Spearman’s rank correlations
comparing predictions and actual first lactation milk production for 576 herds.

Table 8.1. Parent PTA of milk, fat and protein for 5,619 animals.

Table 8.2. Coeficients and their standard errors from the FE probit and logit
models of regressing culling on parent’s PTA.

Table 8.3. Mean effect of a 100 kg change in PTA on probability of being culled,
and associated 95% confidence intervals (CI).

Table 8.4. Predictive ability of FE probit model.

Table 8.5. Predictive ability of FE logit model for the 174 herds with a single
animal culled.

Table 8.6. Coeficients and their standard errors fi'om the ordered FE probit
model of regressing culling classification on parent’s PTA.

Table 8.7. Mean efl‘ect and standard errors (SE) of a 100 kg change in PTA on
the probability of being culled for each culling classification.

Table 8.8. Predictive ability of ordered FE probit model for those animals which
were actually culled.

Table 8.9. Coefiicients and their standard errors from the FE probit and logit
models of regressing culling on MEM.

Table 8.10. Predictive ability of FE probit model of regressing culling on MEM.

Table 8.11. Culling predictive ability of FE logit model using MEM for the 238
herds with a single animal culled.

Table 9.1. Genetic information for the first and fourth periods, along with first
lactation milk production and lifetime profit corrected for the opportunity cost
of postponed replacement.

Table 9.2. Scenarios used to develop and test heifer selection rules.

Table 9.3. Data utilizing heifer selection rules.

viii

174

175

213

213

214

214

214

215

215

216

216

216

217

252

253

254

Table 9.4. Fourth period weights, and their average, for two scenario 1 rules 255
with identical selection decisions for the 58 rule deriving herds.

Table 9.5. Means and variances of the increase in herd average heifer profit of 256

the selection rules utilizing genetic infomation over random selection for the
three scenarios.

ix

LIST OF FIGURES

Figure 7.1. Spearman’s rank correlations between RE MEM and FE MEM
predictions for 576 herds.

Figure 7.2. Spearman’s rank correlations between actual MEM and RE MEM
predictions for 576 herds.

Figure 7.3. Spearman’s rank correlations between actual MEM and FE MEM
predictions for 576 herds.

Figure 9.1. Average profit per heifer from Complex and theoretical genetic
selection rules relative to random selection under scenario 1 for 57 herds.

Figure 9.2. Histogram of difference in average profit per heifer between
Complex and theoretical genetic selection rules under scenario 1 for 57 herds

176

177

178

257

258

.n.

.n\

D‘

t.
0‘.

.2.

5,.

Chapter 1

STATEMENT OF THE RESEARCH ISSUE AND INTRODUCTION

Statement of the Research Issue

There is a lack of thought, much less knowledge, in the area of prepartum Holstein
heifer selection. Numerous optimization and simulation models suggest the culling rate of
cows in Michigan dairy herds is excessive, resulting in a 2% to 40% decrease in producer
profit. These studies indicate a method of heifer selection is required.

Given the need to select only a portion of the available heifers, it would be
desirable to select those animals which will subsequently be most profitable. This contrasts
the naive method of random heifer selection employed in culling optimization studies.
Genetic information, including milk, fat, and protein estimated breeding values (EBV) and
their associated reliabilities, are the most viable data on which to base heifer selection
decisions. An EBV, the estimation of an animal’s genetic merit for the respective trait, is
the sum of its parents’ predicted transmitting abilities (PTA) for the trait.

However, the utility of a profit maximizing decision rule for heifer selection based
on genetic information is not clear for a number of reasons. Firstly, the relationship
between profitability and the genetic information is not known. Secondly, it is not clear
how the value of the genetic information updated every six months over the two year
period of heifer rearing will compare to the rearing costs of those periods. Finally, the

decision rule is complicated by its dynamic nature - heifer selection in previous rearing

periods will alter the subsequent heifer population on which selection is practiced, and this
must be considered by the decision rule.

Random heifer selection should improve producer profitability and cash flow
versus a scenario where heifers are not selected, but may not be intuitively appealing to
producers. A profit maximizing method of heifer selection will not only fiirther improve
producer profitability, but may also be more acceptable to managers thereby encouraging

them to utilize the management technique of heifer selection.

Introduction

Michigan milk producers face an increasingly competitive market, not only
nationally but as well internationally. Least cost and efficient production are necessary to
survive in commodity markets. The call for attention to the economically important issue
of culling has been made (Nelson, 1994). As well “additional research is required to
determine how dairy replacement systems reflect the lifetime productivity and profitability
of the dairy cow” (Heinrichs, 1993).

Briefly, Chapters 2 through 5 of this dissertation consist of literature review.
Chapter 6 contains a summary of the literature review and the three hypotheses that were
generated as a result. The following three chapters (7, 8, and 9) each evaluate one of the
hypotheses. Chapter 10 summarizes the dissertation and discusses potential areas of
further research.

While much of the literature review uses mathematical explanations, intuitive

explanations of the concepts are also provided. Chapters 2 and 3 consider the economics

of asset replacement theory, also known as capital budgeting. While capital budgeting is
not directly applied in this dissertation, the current research is highly predicated upon
studies which have applied this theory to dairy herds. As this wealth of work on optimal
culling rates in dairy herds has been largely overlooked, it is important to review this
literature along with its associated methods. Given an understanding and acceptance of the
results from this large body of literature, the logic of the necessity of a method of heifer
selection will become clear.

Following a brief introduction to neoclassical production economics, Chapter 2
reviews standard replacement theory using Net Present Value analysis. While some models
of dairy herd culling have employed this technique, it is not particularly appropriate to the
current problem for various reasons including the replacements are not identical to the
animals they replace. These models are however useful for gaining an intuitive
understanding of the replacement issue, as well as, stressing the importance of time
horizon, model specification, and opportunity costs. Despite the lack of sophistication,
these models suggest dairy cow culling, at least in the Netherlands, may be greater than
that to maximize profit.

Chapter 3 considers dynamic programming optimization and simulations of dairy
herd culling. A number of studies will be considered to stress the diversity of these
models, in terms of modeling approach, and assumptions. Despite the diversity of these
models (including the country modeled), the results are remarkably similar - culling rates

to optimize profitability are less than those currently employed by the dairy industry. This

Du

.nh

‘7.

I

use

literature is tediouSIY I'eviewed because despite the consensus, it is unfortunately not well
understood by the dairy industry or its advisors.

The simulation studies help to provide some understanding of why maximum
culling rates do not maximize income. The excessive culling rates used by the dairy
industry may maximize milk production and genetic gain. However, the industry and its
advisors have failed to consider the cost of these gains, namely the opportunity cost of the
heifers.

Chapter 3 concludes with a discussion of why the dairy industry as a whole has too
high a culling rate. This is not a trivial concern as the research presented in Chapters 2 and
3 suggests the industry is acting inefficiently and irrationally fiom an economic
perspective. Included in the discussion of the potential motivations of the dairy industry is
my argument that involuntary culls simply rarely exist.

The conclusion to be drawn from Chapters 2 and 3 is that many Michigan dairy
producers have too high a culling rate. This directly suggests these producers have an
excess of heifers, necessitating a method of heifer selection.

The weakness of these culling models is their modeling of heifer selection, which
was often random or arbitrarily based on some genetic parameter. Given that all
replacement heifers are not required, Chapter 4 considers the potential methods of ranking
heifers for maximum profit. Lifetime profit corrected for the opportunity cost of
postponed replacement is revealed to be the economically correct objective in heifer
selection. This value is shown to be moderately correlated with first lactation milk

production.

.-

Ni

ht

L59

L7

(I!

Hi

Currently estimated breeding value, based on parents’ PTA, is one of the few
indicators of first lactation milk production, and by association, lifetime profit corrected
for the opportunity cost of postponed replacement. Chapter 5 critically evaluates the
Selection Index estimation of genetic value.

Chapter 6 summarizes the major points from the literature review. It is apparent
that estimates of genetic value currently represent the only readily available means of
hdfer selection. Chapter 6 then presents the three hypotheses which were evaluated in this
research

While parents’ PTA should have some predictive ability for subsequent first
lactation milk production and therefore be usefiil in heifer selection for maximum profit,
surprisingly this theoretical relationship has not been empirically evaluated to any degree.
In keeping not only with the standard methodology of economics, but as well, the
scientific method of logical positivism it was deemed important to empirically examine the
theory. So prior to generating a heifer selection rule based on genetic estimates it was
deemed important to return to the basics of evaluating how well genetics predicted
subsequent milk production. This research is presented in Chapter 7.

With some indication that genetics may be useful in heifer selection, the research
turned to exploring a method of heifer selection. Producers are likely making the correct
decisions as to which animals they are culling, the problem is the late timing of the
decision. By waiting until after fi'eshening to decide which animals to cull, producers are
paying too high a price to make this correct decision - the value of the decision is less than

its cost. If genetics could predict which animals the producers would subsequently cull, a

5 ii}:

lii

u b

;
h a”,

‘2...

M...

P“ .I

I‘i'

a1"

very good heifer selection rule with a powerfirl motivational influence to lower dairy
producers’ culling rates would result. Chapter 8 considers the ability of genetic indices to
predict culling during rearing or the first lactation. As well, some light is shed on the
voluntary nature of what has traditionally been deemed voluntary versus involuntary culls.

Following the use of econometric methodology to elicit a method of selecting
heifers using parents’ genetic estimates, Chapter 9 turns to an operations research
approach. Modified Box-Complex, a search algorithm, was used to derive a potentially
dynamic method of heifer selection to maximize producer profit. This approach
incorporated the multi-period nature of heifer selection, where every six months during the
rearing process new improved genetic estimates were produced and there was an
economic consequence, either positive or negative, associated with raising the heifers for
that period. Given the proportion of heifers to be selected, the rules consisted of the
proportion of heifers and the weightings for the genetic information in each period. Using
a sample of herds which was used solely for testing the rules, Complex’s rule was tested
against the profits associated with random heifer selection and a rule based on genetic
theory.

Chapter 10 provides a summary of this dissertation, and discusses potential

avenues of further research.

‘t
r |
‘1‘. -

riff

all

awn"

.4...

BIBLIOGRAPHY

Heinrichs, A. J. 1993. Raising dairy replacements to meet the needs of the let century. J.
Dairy Sci. 76:3179-3187.

Nelson, A J. 1994. Information needs of the dairy industry for health and nutrition
management. J. Dairy Sci. 77:1984-1991.

 

"'p’n
‘k-«d.

Chapter 2

PRESENT VALUE MODELS

livestock culling is fundamentally a replacement issue. This chapter will briefly
introduce the standard neoclassical production economics approach to input selection.
Finding this technique inadequate in the assessment of durable assets, the Net Present
Value (NPV) method of capital budgeting will be employed to optimize increasingly
complex replacement models. The flexibility of these models will be showcased by
addressing the issue of the optimal calving interval, before turning to the animal
replacement decision. It will become apparent that the dynamic and complex nature of
replacement in general, and dairy cattle in particular, is frequently underestimated. For
example, these results highlight that the approach of collecting milk production data on
dairy cows with the goal of assessing the most profitable time for insemination is so
simplistic as to be meritless.

While even the moderately complex analytical models presented in this chapter are
not fully adequate to address replacement decisions in dairy herds these models and their
applications do provide some insight into the problem. These models provide intuition into
the understanding of capital asset replacement theory decisions including the importance
of choosing the appropriate model and time horizon for a given problem, as well as,
considering the opportunity costs of replacement. Applying standard replacement theory, I
also explain why if genetics is useful in heifer selection, such a selection scheme should be

associated with a higher culling rate among the cows than a herd which randomly selects

.a
m a“
ll- ‘ '

.' o";
,3 ‘1. fil

.4. «r
u “bo-

F; an:

I' on
4

ifi‘u.
. ~;“~d

heifers. Turning to applications of these NPV models in the literature, numerical versions
of the models which suggest that as of the mid-1980s, culling strategies in the Netherlands
were undesirable and associated with excessive culling rates are reviewed. Applying an
understanding of NPV models and replacement economics, the result that the profit
maximizing culling rates were most sensitive to the price of the replacement relative to the
cow and minimally influenced by milk price or rate of genetic improvement are explained.
Finally, Van Arendonk’s (1991) application of replacement theory revealing that neither
lifetime profit nor profit/day correctly rank animals based on profit is reviewed along with
the economically correct method of ranking cows.

Production economics is the area of neoclassical economies which considers the
selection and application of constrained inputs to maximize an objective, commonly profit.
Inputs used in production can be classified according to their functional life span:
expendables are instantaneously and completely consumed in the production process,
while durable inputs or assets have the potential to supply services for multiple periods
(Robison and Barry, 1996). Production economics tends to focus on the comparative
static analysis of instantaneous production firnctions utilizing expendable inputs; all
dynamic or intertemporal components are ignored (Beattie and Taylor, 1993). Feeding a
dairy cow to produce milk is an example of such a production function, and can be

mathematically represented as
m = f a (2. 1)

where m = milk production

f = level of feed supplied

8'!

1-

'v.

9c

5-
0

I

If

or = the productivity of feed in milk production.

The profit (1:) fi'om such a production fiinction is a fiinction of the prices of milk (P) and
feed (A).

1t=Pf°‘—Af (2.2)
Equating the marginal cost of the feed with the marginal value product of the milk reveals
the profit maximizing level of the feed input. As we will see in the next chapter, failure to
understand or use this simple static profit maximizing decision rule has resulted in poor
advice on optimal culling rates fi'om dairy industry advisors.

Given the optimal input level, static analysis can then be conducted to assess the
efi‘ects of changes in any of the parameters on the optimal level of feed.

However, even simple extensions of the above problem can not be adequately dealt
within a standard production economics fi'amework. Consider a situation where the
amount of feed available is limited for a given period of time, for example the inventory of
feed is fixed at level F until time period T when the next harvest will occur. Intuitively it
can be reasoned that the production functions are now temporally related. Decisions made
in any time period will have ramifications on subsequent periods. So the feed available in a

given period (t) is a function of the total amount available and the amount previously fed.

t-1

05f,sF-§fl 15,3T (2.3)

Equations (2.1) and (2.2) then become

m, = f, (2. 4)

10

I

I.

H.

'1

9::

H’

'L

In

I-I'

r
2*.

1:1

1'
20f,“ -Af,) s.t. Equation (2.3) (2-5)
r=1

The solution will then consist of a time path of optimum feed levels in each period. Static
analysis of this problem is rarely appropriate given that changes in f, will have
ramifications on the amount fed in time periods greater than 1. Comparative static analysis
is apropos in a dynamic system only in the special situation in which the time periods are
uncoupled. This requires, at a minimum, a nonbinding feed constraint which reduces the
problem to solving Equation (2.2) T times.

Given that static analysis may not be suitable for solving situations involving
expendable inputs, it is not surprising that this technique is inadequate for dealing with
production using durable assets, which by definition have multiple period lives. Consider a
feed wagon purchase decision, the initial cost of the wagon will be home when purchased
in the first period, while returns and expenses associated with operation of the wagon will
be realized in subsequent periods. And finally, at the end of its usefiil life, a return or cost
will likely be associated with retirement of the asset.

Capital budgeting is one technique used to analyze the economics of assets or
inputs which generate costs and returns over multiple periods. While payback, accounting
return on investment (i.e., return on equity, or return on assets), benefit-cost analysis (or
profitability index), and internal rate of return have all been used to assess the profitability
of capital assets, net present value is the method of choice. NPV is the only method which
unambiguously considers all the cash flows generated by an asset, the time value of
money, does not make overly rigid reinvestment assumptions, and provides a unique

solution (Robison and Barry, 1996). The formula for NPV in discrete time is

11

pf

(5

JV

 

NPV=—Vo +2 12* + V‘")

2. 6
,.1(1+r)’ (1+r)" ( )

where Vo = the initial investment in the asset
R, = is the net cash flow in time period t
n = the life span of the asset
V(n) = the (salvage) value of the asset in time period n

r = the discount rate of the challenger.
The challenger is the next best alternative use of the investment, while the defender is the
asset explicitly being considered and for which the NPV is being calculated. The discount
rate which is associated with a particular challenger and should be adjusted for risk and
taxes (Robison and Barry, 1996). While the challenger may be a interest bearing
investment, it is incorrect to generally define r as the interest rate. This incorrect definition
of r in not uncommon. Ifnothing else such a definition begs the question which interest
rate - that associated with a chequing account, savings account, certificate of deposit,

treasury bill, grade of private or public bond?

Equation (2.6) can be equivalently expressed in continuous time as

NPV = -V0 + }R(t)e'"dt + e‘”V(n) (2. 7)
0

Now consider the simple decision of when to sell a cull dairy cow, assuming no
replacement (Perrin, 1972). The decision mle is derived by setting the derivative of the
objective function (NPV in Equation (2.7)) with respect to the choice variable (time) equal

to zero (Beattie and Taylor, 1993).

12

rflVPV

dn = R(n)e"" + V '(n)e’”' — re'"'V(n) = O (2. 8)

Equation (2.8) can be simplified to

R(n) + V'(n) = rV(n) (2. 9)
Negativity of the second derivative is required for a global maximum. Equation (2.9)
reveals that the optimal time to sell is the period in which the current period return, R(n),
plus the change in salvage value, V’(n), are equal to the opportunity cost of the salvage
value, rV(n). The original cost of the animal does not influence the decision of when to sell
the animal. This is an application of the economic principle of ignoring fixed cost in
subsequent decision making. Clearly then, optimizing the decision of when to cull the
animal does not necessarily result in a non-negative net present value.

More complex replacement problems will now be considered. The vehicle for this
analysis will be the selection of the optimal lactation length of a dairy cow. This problem is
appropriate to consider not only because it is the focus of a large body of contemporary
research, but as well, it will serve to highlight some other issues. Namely, that the solution
technique must be customized to the problem being considered —- static analysis is not
necessarily appropriate for all expendable inputs, and capital budgeting is not limited to
only strict capital assets. The lactation length decision also makes apparent the complexity
of even a seemingly simple problem, and the necessity of the optimal lactation length as an
input to optimal culling decisions.

Consider first that, assuming a constant dry period length, the optimal lactation
length can be equivalently stated as the optimal calving interval or time of conception.

This problem then is one of optimal replacement, but the factor being considered for

13

 

 

 

. w

1

‘3‘

5‘

replacement, the lactation, does not fit the strict definition of a capital asset, since it is not
a separate physical entity with investment and salvage values. In contrast, the animal itself,
is a strictly defined capital asset. As this optimal calving interval problem is explored it
should become apparent that solving this problem is integral to solving the problem of
actual cow replacement.

Consider a simple model of a lactation which accounts only for the daily net

returns (1(1)) of a single lactation.

NPV = Tl(t)e'"dr (2. 10)
O

For a global maximum, strict concavity(NPV' > 0, and NPV" < 0), is required in the
neighborhood of the solution to the first order condition (Beattie and Taylor, 1993). Ifit

can be assumed that daily returns, which are equivalent to net marginal profit, follow the

same general shape as the lactation curve, a global maximum is assured.

The profit maximizing decision of when to end the only lactation will obviously be

the equimarginal rule of static production -- continue milking the cow until the net return

is zero (Equation (2.12)).

Lg" =1(m)e-- = o (2.11)

I(m)=0 (2. 12)

This simple single lactation model overlooks the reality that a lactation is often

ended in preparation for a subsequent lactation. The problem is truly one of optimal

l4

 

replacement of a cyclical production function, namely lactations. A more realistic NPV
model can then be respecified in Equation (2.13). It includes a constant dry period cost
and arbitrarily assumes culling after the third lactation, and, similar to the problem above,
assumes all lactations are identical. Notice that while multiple lactations of a single animal
are considered, the fundamental decision is lactation length, as animal replacement is not

considered.

NPV = --Vo + Erma-"d: + 5"" [:I(t)e'"dt -— c] + (“at”) [I](t)e‘"dt - c]

+e"("‘+"2+"3)SV

(2. 13)

where V0 = the initial purchase price of the animal

[(1) = the net lactational returns

c = the cost associated with the constant length dry period

SV = the salvage value of the cow at the end of the third lactation

n, = the length of lactation i.
The cost of the dry period is accrued at the beginning of the subsequent lactation. Note
that the cost of the dry period in the third lactation had been omitted since with a constant
salvage value the animal would be sold prior to the dry period. Making 6 a firnction oft
would permit the cost of the dry period to vary with the length of the lactation. Taking the

derivative of NPV with respect to each n,

(Hi/PV

d" = (“mallow-"'3 - re"‘"1+"2+"3’SV = o (2.14)
3

 

WV
412

 

= e"""l(n2 )e’m2 — re"("‘*"2)[?l(t)e’"dt - c] - re"("‘+"2+"3)SV = O
o

15

 

cu

(2.15)

"3
M = «n, )e'"'1 — re'”" [imam — c] — re"‘"‘”’“[ 1 l(t)e’"dr — c]
(1711 o o (2. 16)
—re"("‘+"2+"3)S V = 0
These first order conditions for a maximum can be restated as
[(713) = rSV (2. 17)
n:
[(112) = r [1(r)e'"dr + e'"'3SV — c (2. 18)
o

101,) = r[?l(t)e'"dt - c + e'"'2 {Tl(t)e'"dt - c + e'"’3 SVB (2. 19)
o 0

Equation (2.17) suggests that the animal should be sold in that period of the third lactation
in which the net profit is equal to the opportunity cost of salvage. Comparing the decision
rules for the third lactation (Equation (2.17)) and the single lactation (Equation (2.12))
reveals that since the right hand side (RHS) of Equation (2.17) is positive, I(n;)>1(m).
Recall that in the neighborhood of the optimal length of lactation the profit function must
be increasing at a decreasing rate (i.e., the function must be strictly concave) so the
marginal profit is decreasing. Therefore, a higher net marginal profit function (1013))
corresponds to a shorter lactation length and unambiguously n 3<m. Intuitively this makes
sense since the RHS is the opportunity cost of postponed culling in the last lactation. Once

sold the salvage value could be invested in the challenger at rate r. So once marginal profit

16

 

it!

‘u.

drops to this level it is optimal to sell. Obviously this will occur prior to the net lactational
returns dropping to zero. There is a cost associated with not culling the animal, and as a
result the level of profit at which the animal is sold will be higher than if this cost is not

considered.

Solving for SV in Equation (2.17) and substituting into Equation (2.18) results in

Equation (2.20).

1(n2) = r[?l(t)e'"dt - c] + e""3 [(113 ) (2. 20)
0

From Equation (2.20) it can be deduced that n2< n3, for the RHS of Equation (2.20)
includes not only the discounted salvage value opportunity cost of Equation (2.17), but
also the opportunity cost of the third lactation less the cost of the dry period. Since the
marginal profit of the last day of the second lactation must be greater than that of the third
lactation, the second lactation must be shorter than the third.

Dividing Equation (2.18) by r and substituting into Equation (2.19) results in
Equation (2.21) which reveals that m< n2, since again, the first lactation confronts all
those opportunity costs that the second faced, discounted back another lactation, as well

as the opportunity cost of the second lactation itself.
I(n,) = r[?l(t)e'"dt - c) + e’"'21(n2) (2. 21)
o

The optimal lactation length is then a fimction of the number of subsequent lactations with

n1<n2<n3 <m. Clearly then, the simple lactation decision rule of Equation (2.12), by

17

ignoring the cost of the dry period and the opportunity cost of future lactations,
overestimates the optimal length of all lactations, and therefore, calving intervals.

Even intuitively, it is apparent that the calving interval decision is dynamic. The
rationale behind shorter calving intervals is to forego the lower returns associated with the
end of lactation, in order to return the animal to peak production when returns are
supposedly higher, ignoring the peri-parturient risks for the animal. Using this reasoning, it
should be apparent that the consideration of a single lactation is an inappropriate method
of problem formulation.

This simple model can provide insight into what factors should be considered when
assessing when to breed a cow. Given identical milk and profit functions, which would
approximate production of mature cows, the optimal calving interval is a function of the
rate of decline of milk production and profit, in other words, persistency of milk
production. Peak level of milk production, which is felt to be a good indicator of the area
under the lactation curve, is relevant only when levels of production will vary in
subsequent lactations as occurs with animals in their first or second lactation. In terms of
first lactation animals the breeding decision is now a function of both persistency and total
production - when to end the generally highly persistent, but lower yielding first lactation
to enter the higher production, lower persistency mature lactations. Notice that assuming a
constant length dry period, the shorter the calving interval the greater the proportion of
time spent in the dry period. So even with poorly persistent mature cows, very short
lactations which maximize the amount of time the animal spends in or near peak

production may not be economically optimal. Thus, the rationale for the argument that

18

given the ability to maintain persistency, such as with bovine somatotropin, the optimal
calving interval will be extended. (Of course this assumes that the use of bST is
economically sound, and ignores issues such as the shift in the proportion of the herd
milking and dry at any point.) The simple model indicates as the cost of the dry period
increases, ceteris paribus, the lactation length increases. This cost could included not only
obvious items such as feed, but as well, the potential cost of periparturient disease. It
should also be apparent that the suggestion of a single optimal calving interval for all the

animals in a herd is inappropriate, as is the “cost of a day open” concept.

Given this analysis, the inappropriateness of collecting empirical milk production
data on cows to then determine the optimal insemination period is obvious. At a minimum
for such an analysis, as the previous models showed, all future lactations would need to be
considered. Practically such an analysis would need to account for the censored nature of

the data due to culling.

Also, consideration of only a single animal’s lactations is only appropriate if that
animal is not replaced. In reality, most dairy herds operate under a binding herd size
constraint, and the sale of a current asset is required for the introduction of a new asset.
So ultimately the insemination decision includes not simply when to inseminate but as well
who to breed, and at what point breeding should no longer be continued. To address the
issue of asset replacement, infinite replacement models will now be considered. While
these models can apply to any asset, the finite lives and therefore lactations of individual

cows, are not particularly suitable for such examples, so attention will focus on animal

19

 

replacement. Following the review of infinite models, the role of individual animal

lactations in these models will the broached.

Assuming an infinite time horizon of identical replacement, this problem can be

modeled as
"I "3
NPV = [ g(t)e‘"d1 + e""" Tam-"dz + exists) I g(t)e’"dt + (2. 22)
O 0 O
Whereg(t)=netcashflowinperiodt
g(n) = -Vo + Ig(t)e’"dt + e'"'V(n)
0

andn,-=thelife span ofanimalj, j=1,...oo.

The optimal life span of each asset is again determined by taking the derivative of NPV

 

withrespect to eachnj.
dNPV _ - "2 _,, _, , "3 -
—dn—1—=g(n,)e "" —re "'1 Ig(t)e dl-re ("1 ”“IgMe "d1+”°=0 (2°23)
0 O
"3 "4
“E, = g(n2)e‘"" —re-'"2 {grower - re""'=*"3’ {nae-"d: + = 0 .

(2. 24)

Due to the infinite time horizon all the first derivatives are the same and therefore n, = n;

= "3 = = na. Equation (2.22) can then be restated as

20

NPV = (1 +e'"' ”"2" +e"3" )Ig(t)e'"dt (2.25)
0

 

 

1 " -,,
'(1-e'”)ig(t)e d: (2.26)
with first order condition
My ( 1 j -n, '9‘". " -rr
__ = m g(n)e — —— g(r)e d1 = 0 (2. 27)
dn 1-e (1_e-r~)2i

g(n) = ——r—— ? g(r)e"'dt (2. 28)
(1-e‘")o

The integral in Equation (2.28) is simply the present value of a replacement, while

(7%,?)- is the annuity factor. As a result, the right-hand side is the annuity or average
- e

return fiom future replacements. Equation (2.28) then represents the standard decision
rule of asset replacement - replace when the marginal (annual) value of the defender (the
LHS of Equation (2.28)) equals the average (annual) present value of the challenger
(Paris, 1960). The maximum average value of a subsequent replacement can be considered
the opportunity cost of postponed replacement. As alluded to in the lactation replacement
model, as the number of future replacements increases so does the opportunity cost of
replacement, so the optimal life span of an asset with infinite replacement will be shorter

than if a finite number of replacements were available.

21

 

fur

Finally consider the case of continuous improvement of the asset (Robison and

Barry, 1996) as is expected in livestock due to genetic improvement.

NPV, = Tam-"d: +(1+h)e'""" Tam-"dz +(1+h)2e*'lw+m)Tg(t)e-~d, +
O 0 O

(2. 29)

Where 11 represents the rate of improvement between successive assets. The optimal life
span of each asset is again determined by taking the first derivative of NPV with respect to
each 71,), where the i subscripting simply serves to distinguish this NPV model with

improvement fiom that of Equation (2.22) without improvement of replacements.

 

 

my "21‘ "31'
' = g(n1,. )e‘mv -(1+h)re'""" Ig(t)e'"dt—(1+h)2re"(""'+"2‘) jg(z)e'"dz +.-. = o
dnu o o
(2.30)
M” =(1+h)g<n2.)e'"'2‘ -(1+h)2re‘""" "liaise”:
2‘ ° (2. 31)

Mr
- (1 + h)3m-'('m+"3‘) j g(t)e’"dt + = o
O

Dividing both sides of Equation (2.31) by (1+h),

(0va
4'12:

 

"31 "41'
= g(n2, )e'W —(1+h)re'"’2" jg(r)e'"dz —(1+h)2re“'<"2i*"=i’ Ig(t)e'"dt + = o
O 0
(2.32)

22

 

 

Comparing Equations (2.30) and (2.32), it is again apparent that the first order derivatives
are identicaland nu =n2, =n3, =---.

Equation (2.29) can then be restated as

NPV, =[1+(1+h)e'""' +(1+h)2e"2"" +(1+h)3e"3"" +---]Tg(t)e‘"dt (2.33)
O

 

 

 

 

 

1 “1 .,
=[1—(1+h)e""’]£gme d! (2'34)
withfirstordercondition
WVr=[ 1 J ' —n.,._ “kW-m . t '"dt=0 235
d». 1-<1+h)e-'~ 3"” (1-(1+h)e""')2]:g()e ‘ ’
gin,)= "1”) lane“ (2.36)

(1-(1+h)e""")o

This first order condition assumes the function is well behaved (h>0, g’(t) > 0

and g" < O ). By comparing Equations (2.28) and (2.36), it is apparent that n)<n. Having
previously seen the efi‘ect of opportunity costs on the optimal length of asset life, it is not
surprising that the length of each assets’ life is shorter with improvement than without.
And that the greater the rate of improvement the greater the decrease in length of asset

life.

 

 

dg(n ) r "‘ - d"
’ = g(t)e "dt > 0 -—'- < 0 (2. 37)
‘fl’ (1-(1+h)e""')2ii db

23

 

A constant optimal asset life (e.g., n“. = 21,, = 21,, = -- -) with constant improvement is not
as intuitive a result.

The inverse relationship between the rate of improvement of the asset and the
optimal asset lifetime bears emphasizing and will be used in Chapter 3. In Chapter 3,1
review herd culling models which employ two different methods of heifer selection:
random and based on genetics. Now due to genetic improvement fiom generation to
generation even with random selection the potential replacement heifers are improving.
However, if genetic indices are usefirl in heifer selection, then by definition the rate of
improvement under this selection scheme should be greater than that under random
selection. Ifthis is the case, then the optimal length of the cows’ lives should be shorter
than under random selection. Ceteris paribus, failure to find a shorter cow life (i.e., higher
culling rate) with genetic selection as opposed to random selection would suggest genetic
indices are not usefirl in heifer selection.

Burt (1965) showed that under conditions of uncertain replacement (i.e.,
involuntary culling) and risk neutrality, the decision rule for the infinite horizon was simply
to replace when the expected net current period return equaled the expected weighted
average net return fiom the potential replacement.

While the preceding results have provided analytical results which aid in
understanding the culling problem and solutions, they present limited application to
solving actual culling decisions for two reasons. First, recall that the opportunity cost of
postponed replacement is the average present value of potential replacements. This then

requires optimization of all future replacements with respect to all choice variables. For

24

 

H

’s
“I

up

. ,{v

[-1

I:

As.

example Equation (2.13), the NPV for a single animal derived from it’s three potential
lactations, could then be substituted for g(t) in Equation (2.25). Even with identical
replacement, solving this problem would then require simultaneous consideration of four
first order conditions, one each for n, m, rig, and n3. Analytical solutions then become
tedious due to the large number of simultaneous equations. Second, the net revenue flows
of a dairy cow are very variable throughout a given lactation and across lactations. At the
extreme, consider the negative returns during the dry period, which are in fact preparatory
for the subsequent lactation. This lack of identical net return or profit firnctions over time
and between challenger and defender realistically removes the possibility of analytical
sohrtions. As we shall see this also presents difficulties in numerical analysis of these
problems.

The numerical technique of comparing the marginal profit of the defender versus
the average annual profit of the challenger, also known as the marginal net revenue
approach, has been modeled for replacement decisions. Renkema and Stelwagen (1979)
developed the primary model used for NPV analysis of the culling decision.

In their model the earned income, under Dutch conditions, for a lactation was
calculated as gross revenue from milk and calves, less cost of feed and sundries, the
change in slaughter value was also considered. Assuming a 12 month calving interval and
a constant mature equivalent milk production, the earned income was calculated for
lactations 1 through 14. Incorporating the probability of disposal at the end of the next
lactation, the expected marginal profit associated with the next year was calculated. With

the cost of the replacement heifer and the marginal expected earned income for every

25

 

«L

M.

rd

15 H

N.\

lactation, the non-discounted average annual income of the replacement heifer associated
with successive lactations could have been calculated, permitting the determination of the
optimal lifespan of the replacement. However, instead the replacement’s lifespan was
assumed to be 4.2 lactations, approximating the average productive life of the typical
Dutch Friesian. The analysis was repeated with a higher level of milk, and as well, the
assumption of identical replacement was relaxed by repeating the analysis using a 1% rate
of genetic improvement in milk production.

Many of the assumptions of the model are unsatisfying and potentially biasing. For
example, the assumption of culling at the end of the lactation versus at day 205 was
subsequently found to overestimate profitability by 30% (Van Arendonk, 1984). Care is
needed in extrapolating this model of a single challenger and defender to a herd situation.
To make this extension all animals in the herd are required to be homogeneous with
respect to the 12 month calving interval and mature equivalent milk production. While the
model can compare a genetically superior replacement to a cow, it cannot represent a herd
with multiple levels of genetic capability. This is a drawback of all such marginal revenue
models (Van Arendonk, 1984).

Although the decision rule approximated that of the marginal revenue optimization
model, note the lack of discounting, without having optimized the length of the
replacement’s life, it could easily be argued that this model is actually a simulation rather
than an optimization. Simulation versus optimization will be more fully discussed in

Chapter 3.

26

Given the shortcomings of this model, the result nonetheless are interesting. It was
determined that increasing herdlife from an average productive life of 3.3 to 5.3 lactations
increased annual earned income by 20%. This simplified model suggests the benefit of
prolonged productive life, this benefit was minimally affected by genetic improvement.
The economic return fi'om longer life was much more sensitive to the net cost of
replacement (i.e. cost of replacement heifer less salvage value of the culled cow) than
changes in the milk price.

The realism of this model was increased by assuming culling at day 245 of lactation
and used in a traditional simulation model (Korver and Renkema, 1979) which will be
discussed under simulations in Chapter 3.

The model was also further updated and expanded to include the time value of
money and optimal insemination decisions (Dijkhuizen, Renkema and Stelwagen, 1984).
The potential calving interval was 345 to 525 days, and at 20 day intervals the replacement
decision as well as the insemination decision for the current lactation was made.
Replacement was possible in future lactations only at day 225. At each heat (20 day
interval) the animal was to be inseminated if the sum of her expected future marginal
revenues in the case of pregnancy, less the annuity of the average replacement, was
greater than zero. The correlation between length of successive calving intervals was 0.6.
This cumulative future difference between the defender and challenger is actually the
decision rule used in the original model, in place of that stated. Full numeration of

expected returns for each animal are then required.

27

 

I

r

s‘

r.

ail"

7«

P

a

NH

>

x

a.)

\.N-

in

it.

Ifit was deemed uneconomical to inseminate, the animal was not culled until the
point in her lactation where the expected returns for the remainder of the lactation were
less than the annuity of the replacement. A 1% annual genetic increase in milk production
was assumed.

While it is not clear, it appears that similar to the original model, the lifespan of the
replacement has not been optimized, but rather arbitrarily chosen.

The results were in the form of critical production levels by lactation, and days at
insemination below which it is not optimal to inseminate a cow. These guides indicate that
a young cow which is average or above average in level of production can profitably be
inseminated up to 8 to 9 months post partum, suggesting the culling strategy in the
Netherlands was not optimal. While the results were not sensitive to the price of milk,
since this afi‘ected the returns of both replacements and defenders, changes in the cost of a
replacement did afi‘ect the critical production level. As expected from theory, the model
found the greater the persistence of milk production, the longer it was profitable to
inseminate open cows. The authors never calculated the distribution of productive lives in
the herd to determine whether rational expectations were firlfilled (i.e., if the assumed
length of the replacement’s productive life equaled that of a herd resulting fiom following
the guide). Failure to fulfill the self-expectation nullifies any apparent solution. This may
have been what the authors were referring to as “more insight is desired into the economic
consequences of making replacement decisions over a number of years with different herds
on the above mentioned criteria”. Also note that future replacements are a fimction of the

animals currently in the herd and management, including culling. So the simple static

28

 

/\

./;‘.

'I75/

4-

£17

“I

I“.

i

comparison of a defender with a genetically superior heifer is over simplified. The dynamic
nature of the problem is clear - decisions made today affect future decisions.

While the realism of the model has been increased through the modifications some
unrealistic assumptions, such as the timing of firture culling still remain. These
simplifications are a direct function of this marginal method of analysis. As was discussed
previously, permitting net return or profit firnctions to vary over time and between
challenger and defender would require calculation of all expected net revenue for all
potential future decisions. As the number of potential decisions increases, the complexity
and number of calculations quickly become overwhelming.

Kuipers (1980) also produced a marginal net revenue model to evaluate the
expected marginal profit of a challenger over the remainder of the current lactation and
first six months of the subsequent one. This poor choice in time horizon for consideration
of the marginal value of the challenger underestimated the value of younger animals, since,
as noted earlier, production and net revenue increase over the initial lactations.

It is only recently that the concepts fiom replacement theory models have been
applied to other avenues such as cow ranking for profitability (Van Arendonk, 1991). A
large amount of literature has focused on the contributions of such factors as increased
length of herdlife and milk yield on cow profitability. The most common profit firnctions
used to determine the dependent proxy variable for profit have been lifetime profit and
profit per day of life. Van Arendonk realized that the former implies no replacement or
implicitly a non-binding herd size constraint. Profit per day of life implies identical

replacement, which is unrealistic due to genetic improvement and the lack of systematic

29

difi‘erences between heifers available for replacement of different cows. The failure to
account for the opportunity cost of future replacement has resulted in the possible
millification of a large amount of previous work. This area will be considered further in
Chapter 4.

In summary, this chapter focused on the optimal calving interval decision, as well
as, animal replacement. It was apparent that even these simple problems are complex and
increasingly dificult to solve due to the number of choice variables, the intertemporal and
often dynamic nature of the problem. The importance of considering model specification
and time horizon also were evident. While these models provided some intuitive insight
into some of the underlying principles of these solutions, such as the importance of
opportunity costs, ultimately they were unable to answer the complex question of the
optimal herd level culling and replacement strategies. Simplified numerical models
suggested the potential for improvement in culling strategy, at least under Dutch
conditions. The difiiculty with numerical analysis was the requirement to consider all
potential firture decisions and returns. A subset of these dynamic problems, namely those
that can be recursively stated, can be solved with greater ease through dynamic
programming. This technique and the results of replacement models using this technique

will be addressed in Chapter 3.

30

BIBLIOGRAPHY

Beattie, B. R, and C. R Taylor. 1993. Page 1-8 in the Economics of Production. Kreiger
Publishing Company, Malabar, Florida.

Burt, O. R 1965. Optimal replacement under risk. J. Farm Econ. 47:324-346.

Dijkhuizen, A A, J. A Renkema, and J. Stelwagen. 1984/85. Economic aspects of
reproductive failure in dairy cattle. II The decision to replace animals. Prev. Vet. Med.
3:265-276.

Faris, J. E. 1960. Analytical techniques used in determining the optimum replacement
pattern. J. Farm Econ. 44:755-766.

Korver, S. and J. A Renkema. 1979. Economic evaluation of replacement rates in dairy
herds 1]. Selection of cows during first lactation. Livest. Prod. Sci. 6229-3 7.

Kuipers, A. 1980. Development and comparison of selection criteria for cows and bulls
with a dairy herd simulation model. PhD. Thesis, Univ. Wisconsin, Madison.

Perrin, R K. 1972. Asset replacement principles. Am. J. Agr. Econ. 54:60-67.

Renkema, J. A, and J. Stelwagen. 1979. Economic evaluation of replacement rates in
dairy herds 1. Reduction of replacement rates through improved health. Livest. Prod. Sci.
6: 15-27.

Robison, L. J., and P. J. Barry. 1996. Ch. 1, 2, 8, 11, 21 in Present Value Models and
Investment Analysis. lst ed. The Academic Page, Northport, Alabama.

Van Arendonk, J. A. M. 1984. Studies on the replacement policies in dairy cattle. Z.
Tierzuchtg. Zuchtgsbiol. 1012330-340.

31

 

i-s

1‘3.

u
vAOF‘
vol

it”

pal.
v

Van Arendonk, J. A. M. 1992. Use of profit equations to determine relative economic
value of dairy herd life and production from field data. J. Dairy Sci. 74:1101-1107.

32

Chapter 3

DYNAMIC OPTIMIZATION AND CULLING MODELS

From Chapter 2 it was evident that while the traditional NPV approach to animal
replacement lent some intuition to the understanding of the problem, it was not
particularly well suited to actually solving these problems. In this chapter another solution
technique, dynamic programming, will be reviewed. Afier defining the model through an
example, analytical dynamic programming will be covered to gain familiarity with the
model and its methods. Numerical dynamic programming solutions, a much more practical
approach to complex dynamic problems such as livestock replacement will follow. Given
this foundation the results of dairy cattle culling studies utilizing dynamic programming
and optimization will be reviewed. While tedious, consideration of these models, in at least
a superficial way, is necessary for two reasons. First, in order to draw proper conclusions
about the culling issue based on the results of these models, the diversity of the models
must be realized. And second, these models should be reviewed for this area of
management tends to be overlooked by the industry despite the wealth of knowledge in
this area. These studies are summarized in Tables 3.7 and 3.8 at the end of the chapter.
The issues of voluntary and involuntary culling and simulation versus optimization will
also be considered. The conclusion will be that the market is operating inefficiently
through excessive culling, corroborating evidence of this inefficiency is presented. I then

try to resolve why the market is operating in an economically irrational manner.

33

 

Id

'6‘

3.x

To introduce dynamic programming an analytical solution to a simple replacement
model will be used. Analytical models tend to be quite general, utilizing symbols rather
than numbers. The advantage of this method is that it will provide some insight into the
solution, as the models in Chapter 2 did. Numerical analysis, where the values of the
parameters are used, is easier to Operationalize to get actual results especially in complex
problems.

Consider the problem of when, in its lifetime, to sell an animal without

replacement. This is an example of a simple optimal st0pping problem. Reconsider

 

 

Equation (2.6) from Chapter 2
NPV = -V(, + Z R' + V‘") (3. 1)
,.1 (1+ r)‘ (1+ r)"
The objective is to maximize the profit given that the animal is present so the initial
investment may be omitted
R' + V‘") (3. 2)
,.1 (1+ r)‘ (1+ r)"
The net cash flow in any lactation (m) can be denoted as
1r,=R(n,)+d,V, 1sr.<_T (3.3)

The notation has been altered slightly so R(nJ explicitly recognizes that the returns in a
lactation fi'om milk production are a fimction of the lactation number m. As was discussed

in Chapter 2, in order to optimize the culling decision, the length of the lactation needs

34

 

 

I»

,3

Th

\rsl

11‘

\l\

also to be optimized. To maintain simplicity assume that the R(nJare already optimized
with respect to lactation length.
Let d. be a dichotomous choice or control variable with

(3. 4)

0 if the animal is not culled at t
' - 1 ifthe animal is culled at r

V, remains the salvage value of the animal in period t, and T is the maximum potential
lifespan of the animal, after which the animal has no value, Vm =0.
Equation (3.2), in dynamic programming termed the objective functional, can then

be generically restated

maxBn,(n,,d,)+ B'*’1t,+1(n,+,,d,+1 ) + + Warmer” (3.5)
wherefl= 1/1+r, and [:1 Sn.

In traditional dynamic programming terminology, It, is the return function
representing the immediate period return associated with n,, the state variable, and d,, the
control variable. The state variable or vector, if more than one state variable, define the
state of the system in period t. In period t, the decision maker cannot affect the state
variables, they are taken as given. The decision maker has direct control over the control
variable or vector, (1,, in period t. The transition equation, which links the state and control

variables, for this problem is
"1+1 = "1+ ] lfdg=0

n...= 0(151’5 T-t) ifd,=l (3.6)

35

This simple transition equation states that if the animal has not been culled, it will proceed
to the next lactation. Obviously once the animal has been culled, there can be no
subsequent returns. Notice that the current state variable is a function of the previous

control variable

"1+1 = 8101:1613) (3. 7)

This then defines the problem as being recursive (i.e., the current controls are only a

function of previous, not future, controls). To see this note that

"r = gl-1(nI-1rd(-1) (3. 8)
".4 = g.-2(n.-2.d,-z) (3. 9)
"2 = 81011.0’1) (3. 10)

Sequential substitution of Equations 3.8 through 3.10 into 3.7 reveals
n,+1 = §,(n,,d1,d2,...,d,_,,d,) (3. ll)

Lagging Equation (3.11) one period and substituting into Equation (3.5) reveals

r
maxZB'i,(n1,d1.d2,...,d,_1,d,) (3. l2)

t-l

36

 

9rd

. 1‘0“

1’1
r2.

' '1
1’
is:

5 0:1

’0
,5 ”5‘
“ Wm.

|.. .
‘A

'S

“A

The return firnction in any period is then only a firnction of the current and previous
controls. Since dynamic programming solves problems recursively, this specialized
recursive problem structure is required for this technique. Calculus of variations and the
maximum principle are other techniques appropriate for solving recursive dynamic
optimintion problems. The former requires the transition equation have the restrictive
form of depending solely on the current control and not the current state (i.e.,
nm=g.(d.)). The maximum principle is usefirl in continuous time formulations, where it is
also known as control theory. Control theory utilizes the Hamiltonian as a method to solve
difi‘erential equations. Solving stochastic infinite horizon problems using control theory is
quite complex and not well suited to numerical analysis. In discrete time the Hamiltonian

reduces to the Lagrangian, which gives the same results as dynamic programming.

Dynamic programming then reduces to selecting the optimum set of policy
fiinctions from a space of fiinctions, d,=h{nJ. This formidable task is made easier through

the use of the principle of optimality.
Suppose {In L; is a set of policy functions that solve the recursive problem

Equation (3.5). Now consider the modified problem

maxiB'rrAnndfl (3. 13)

[-1

Let {h,' }T lbe the optimal policy functions for the modified problem. The principle
(-

of optimality states that for the periods 1' to T, h,= f. Notice that it is the optimal policy

function that is being chosen, not the optimal control choice. The principle of optimality

37

 

then suggests that the optimal policy functions are invariant to the earlier time periods.
The optimum firnction is selected such that it is unaffected by changing circumstances.
Notice that as the state variables change the control variables will change but not the
policy function. Also the principle of optimality does not necessarily suggest the policy
function is time invariant (i.e., h,=h.+,). These points will become more clear with an
example.

Based on the principle of optimality Equation (3.5) can be restated in Bellman’s

current period value function.

V. (n. l = mflwnnd. ) + 13214., (n... l} (3. 14)

subject to the transition Equations (3.6), [St S T, and given m.

The principle of optimality and the recursive nature of the problem pemrit a simple
technique to solve for the optimal policy functions over a space of functions. Dynamic
programming starts at the terminal period and solves for the optimal policy function for
each time period in a sequentially recursive fashion. The result will be a time path of

control variables given an initial value of the state variable.

Substituting the specific form of the return function (Equation (3.3)) into Equation

(3.14) reveals
m». > = my{BR(nr 1+ 134V, + Bszl'hn )} (3. 15)

subject to the transition Equations (3.6), 15.1 _< T, and given 11,.

The terminal period value function is denoted

38

iv

['7

If

I

Vr ("r) = "2"{BRlnrl1’ BdrVr + BZVT+1(nT+1)} (3. 15)

T denotes the end of the animal’s life, and as a result Vrt 1(nrt 1)=0. So the decision rule
will be to cull the animal at T, if it has not occurred previously. The terminal period policy

fimctions are:

(3. 17)

O ifnT =0
dT =

1 if», = T

Due to the categorical nature of the control variable, the solution technique to this
problem does not involve the standard technique of differentiation, since due to its
discontinuous nature the equation is not differentiable. This then presents another problem
with analytical results, including NPV techniques, which was not addressed in the previous
chapter.

Notice that the solution is a policy fimction; the control variable being a firnction
of the state variable. As the state variable changes the optimal control variable will
correspondingly change. This optimal policy function for the terminal period will,
however, never change regardless of how far back into time the problem is worked.

Substituting Equation (3.17) into (3.16), the terminal period value function is

BR(T)+BVT ifn, = T
V1011): , (3. 18)
if n, = 0
At r=T-1, Bellman’s equation becomes
Vanni-m ax{sRm+Bdr-iVT-. «Liszt/Tim} (3.19)

T-1

39

 

Ifculling has not already occurred there are two possible options for the period T -1. Ifthe

animal is not to be culled, d1.) =0, and culling occurs in the final period, so

Vr.r("r-I)=flR(77+BilRm+Vr] (3.20)
Or the animal may be culled, d“ =1, and V’(nr) =0. The returns generated would then be

Vr-I("r-1)= 13R” '1)+ 5V“-

The decision rule for T-l, then is

o if V,_, < spam + V,]
d,_1 = (3. 21)

1 if VH 2 [411(7) +VT]

This decision rule is similar to those from Chapter 2, cull the animal now if the salvage
value today is greater than or equal to the discounted returns from lactation and salvage

value in the next lactation. Culling when the terms are equal has arbitrarily been assumed.

BRU’) + smax{V,_,,p[R(7) + V,]} if n,_, = T-1
0 if "M = 0

19.101“) = { (3. 22)

Notice from Equation (3.18) that fiRfl)+ fiVT = V’(nr) so all terms from Equation (3.19)
are represented in Equation (3.22).

Similarly, for period T -2, the animal can be culled in which case the returns are
16R” '2) + flVr-z (3. 23)

or alternatively, culled later with returns

fiRfl-2)+fi’(R(T—1)+max{VT_,,B[R(T)+VT]}) (3.24)

40

which can be restated using Equation (3.22) as

5R” '2)+fl2 Va!” '1) (3. 25)
The decision rule is then
0 if Vr-z < BVT-1inT-1)
d,_2 = . (3. 26)
1 ‘f Vr-z 2 BVr-r ("r-1)
With the current period value function being
BMW-2) + Bmax{V7--2, BVT-t ("r-1 )} if "r-z = T ’ 2
Vr-z ("r—2) = (3- 27)
Ifn7-_2 = O
In general then, Bellman’s equation becomes
BR“) + BmaX{Vr rBVr+1 ("1+1 )} if "r = t
V102. ) = (3. 23)
if n, = 0
The optimal decision rule or policy function is characterized as
0 if Vt < BVr+1("r+1)
d, = . (3. 29)
1 if V: 2 BVr+1(nr+1)

The rule is then to cull now if the optimized discounted returns from all firture periods are
less than or equal to the current salvage value. Despite the recursive nature of the solution

technique the results are forward looking considering the possible future returns.

41

This analytical approach can be easily modified to include replacement and efi‘ects
of uncertainty of the state variables. Uncertainty is introduced to dynamic programming
through making the transition equation dependent on an identically and independently
distributed random variable with a mean of zero, 6.1,. In the context of the present
example this error term could include the probability of unplanned death. Equation (3.7)

would be modified to
"1+1 = gr(nrvdrrer+1) (3° 30)

Since the random shock is unknown when the control variable is chosen, the future path of
the state variables is uncertain. The stochastic version of Equation (3.15), the current

period value fimction with the specific return function is

14(n.)= mJX{BR(ni)+ 134V. + Eamon, )} (3. 31)

where E. is the expectation operator conditional on information available at time I. At 1,
R011) and V, are known with certainty. Since the expectation of am is zero, the stochastic
aspect of the problem is eliminated (Bellman, 1957). The approach to solving the problem
is again one of determining the set of optimal policy firnctions, d,=h,(n.). Notice however
that since n, is a stochastic variable and d, is a fimction of the state variable, the time path
of the control variables is itself a stochastic process.

Substituting the policy function into the transition equation reveals

"1+1 = gr ("1 1hr ("1 )1814-1) (3' 32)

42

The sequence of state variables then follows a Markov process, in that the current state
contains all the relevant information for predicting future states. While this may seem quite
restrictive, using vectors, autoregressive processes can also be represented as Markov
processes.

It is a simple matter to introduce the returns fiom multiple animals into the
objective functional to represent replacement. Iteration of Bellman’s equation would
simply continue for more periods. Solving dynamic programming problems through
iterations requires some consideration for the time horizon of the problem. As the example
illustrates the farther back in time one goes the less critical the generally arbitrary value of
the terminal state becomes. As iterations continue into the past it is possible for the system
to enter a steady state, where the policy firnctions are time invariant (i.e., h,= h" 1).

Analytical problems are not easily adjusted to solve for multiple state and control
variables. Rather numerical analysis is more amicable to such complex problems. There are
a number of methods for numerical analysis of dynamic programming problems. The firll

enumeration of firture possible states will be examined here.

The terminal value function is a vector with the number of elements equal to the
number of possible states. The state and control variables are members of a finite defined
set. For numerical analysis, a finite state Markov chain is used. A probability matrix
contains the probabilities of being in a certain vector of the state variables in H 1 given a
present vector of the state variables. For a given state vector the terminal period value
fimction is calculated through consideration of the return function and terminal value

function for every possible control vector. The control vector associated with the

43

 

maximum value of the terminal period value firnction is the optimal control policy for the
given state vector at T -1. This is repeated for every value of the state vector, resulting in

an optimal control vector for every possible state vector for the terminal period.

In this full enumeration approach iteration occurs backwards in time. The period
value function for T -1 , is calculated similarly to the original iteration. The terminal period
(T) value fimction, is determined by using the finite state Markov chain to weight the
possible value functions in T by the probability of the respective state vector occurring
given the state vector in T -I . (This is simply an application of Burt’s (1965) idea fi'om
Chapter 2 that the expected value is a fi1nction of the summation of each possible value
weighted by it’s probability of occurrence.) Summing the current return firnction, and the
terminal value function gives the current period value firnction in T -l. The optimal policy
fimction for period T -1, is a vector of state variables and the associated control variables
which maximize the respective state variable’s current period value function. The policy

firnction from the first iteration is then the Optimal policy function for the next period (i.e.,

I).

By assuming identical future replacement the marginal net revenue approach from
Chapter 2 is a special case of dynamic programming (Burt, 1965).

The advantages of numerical solutions are the relaxation of the model
requirements such as continuous difi‘erentiability. Also there is no need to solve numerous
analytical simultaneous equations. The disadvantages are the lack of intuition provided by
the model, which is important not only in gaining insight into the general or specific

problem, but as well, greater difficulty in realizing errors in the problem formulation or

 

solution technique. These models also require a large body of information which is
commonly not available. For example transition probabilities between all possible
combinations of states are required.

Highly complex dynamic optimization programs are available to solve for
management decisions in dairy herds. Van Arendonk’s model (Van Arendonk, 1985a; Van
Arendonk and Dijkhuizen, 1985) will the be primary focus as it is the most sophisticated
and widely used model. While components of the model have been validated at least for
Dutch conditions (Van Arendonk, 1985), in general the complexity of dynamic
optimization models make their evaluation quite difficult, for due to their large size firll
disclosure of the model is not possible. Furthermore, the accuracy of those parts presented
is often not obvious, for example, there is little data on which to base evaluation of
probabilities of conception for each combination of days in milk, and animal age. Model
evaluation will then be based, not only on the inputs used in the model, but also on how
well its description of the herd matches generally accepted parameters (i.e., income over
fwd cost). Some of the inputs and assumptions, as well as, outputs not necessarily directly
related to the culling issue will then be reported. Sensitivity analysis will reflect the
robustness of model results to input changes. Following this the last source of model
validation - comparison of the results with other optimization and simulation models will
be considered. Tables 3.7 and 3.8 contain selected results of some of the optimization and
simulation models, respectively, that will be reviewed in this chapter.

The objective of Van Arendonk’s stochastic dynamic programming model is to

maximize the expected present value of the monthly net revenue stream for a replacement

45

 

heifer and her replacements over a 15 year planning horizon. The control variables are the
replacement and insemination decisions. The model operates recursively, starting in the
final month of the planning horizon, where the terminal value of animals was carcass
(salvage) value. The optimization then proceeds backwards, at each month making the
decision to replace or keep the animal based on which decision coupled with the
respective subsequent optimum decisions yielded the greater expected value of net
revenue. In considering the insemination decision, the probability of conception, cost of
insemination, and expected net revenues calculated from milk and fat yields, calf values, .
carcass values, feed costs, health costs and housing costs (Van Arendonk, 1985) were
considered. The model has been updated numerous times.

One collaboration was with Rogers and McDaniel (1988) to adjust the model to

US dairy conditions. Briefly, the state variables and the number of levels of each variable

 

 

 

were:
lactation number 12
stage of lactation 16
milk yield in past lactations 15
milk yield in present lactations 15
time of concergion 6

 

Cows were considered for insemination in months 2 through 7 of lactation.
V°luntary replacement was defined as replacement due to the decision making of the
model. Involuntary culling reasons included mastitis, other health disorders, milking speed,
tempefllment, pendulous udders, other husbandry related disorders, and death. The
monthly probabilities of involuntary culling and associated economic costs by parity were

ex0861mm to the model. The proportion of involuntary culling ranged from 20% in the

46

first month of lactation to 4% in the last month. First lactation animals had the lowest
probability of involuntary culling at 10%, while the parity 12 animals had a 30% chance.
Cows were not considered for voluntary culling prior to the second month of lactation,
and sale of an animal was not necessarily concurrent with the decision to cull the animal.

The repeatability of milk yield was assumed to be 0.55 for adjacent records and
0.50 for those two years apart. Conception rates were assumed to be independent of the
previous calving interval and milk yield in the previous and present lactation. The marginal
probabilities of conception decreased with increased parity and increased from month two
to five of a given parity before declining slightly in the sixth and seventh months of
lactation. For example the marginal probabilities of conception in the second and fifth
month for a first parity animal are respectively 0.30 and 0.52, whereas the comparable
numbers for a fifth parity animal are 0.29 and 0.49. Genetic improvement in first lactation
cows was assumed to be SlO/year. Improvement in older cows was calculated based on a
0.9 genetic correlation between first and later lactations, and age factors for milk yield.

The annual discount rate of 4% was used in determination of the expected net
present value fi'om a replacement heifer over the 15 year planning horizon, in earlier use of
the model the optimum replacement policy was not influenced by a reduction in the
planning horizon from 20 to 15 years (Van Arendonk, and Dijkhuizen, 1985). This present
value was then converted to a yearly annuity for purposes of comparison among various
management alternatives. Strictly speaking, reinvestment at the discount rate and an
infinite time horizon are required for an annuity to be an appropriate measure of

profitability (Robison, and Barry, 1996). Prices and parameters were chosen to be

47

.e

a

'\

£7

/"

A"

I

representative of American Holstein dairy farms in 1986. Milk price was $24.64! 100 kg
with a 80.3696 fat difi‘erential. The base price of a mixed ration with 16% crude protein
and 1.72 Mcal NEJkg DM was $0.084/Mcal NE.. Energy requirement, after accounting
for mobilization of body reserves, limited DMI. Average 305 d mature equivalent milk
yield was 8000 kg with 3.6% fat test. Price per unit of semen was $13.

All calves were sold at one week of age, with heifers being valued at $75 more
than bull calves. Carcass values were calculated from body weight, dressing percentage
(46.4 to 49.2%), and price per kilogram of carcass weight ($1.793 to $3.032) and varied
fiom $571 to 3521. An unlimited supply of freshening 24 month old heifers, at a real cost
of$1100 each, was assumed.

Of the 259,200 potential combination of state variables, profit after conception
was calculated for 14,940 combinations and fiiture profitability for 174,300 (Van
Arendonk, 1988). Cost of management and labor for milk harvest were not included.

The optimum average culling rate was 25.1% (8.6% voluntary and 16.5%
involuntary), which was lower than the estimated national average of 3 1%. Following first
lactation, where 35% of replacements were voluntary, the marginal voluntary culling rate
declined until the sixth lactation. Only high producing cows were kept for many lactations
because otherwise the net revenues fi'om the lower producing cows were sufficient to
compensate for the decline in fertility and increased probability of involuntary culling. The
optimum policy resulted in an average calving interval of 377 days, with 38% of the cows

having intervals of greater than 13 months. Table 3.1 is a reproduction from the paper.

48

 

.Q .\s"

5.x.

Table 3.1. Production, as a percent of the mean production for that parity, below which
insemination would not be optimum by parity for the base scenario (8000 kg ME).
Month in lactation

 

 

Parity 3 5 7
1 78 82 86
2 78 82 86
3 78 82 90
4 82 86 90
5 86 90 94
6 9O 94 102
7 94 98 106
8 98 106 110
9 106 1 10 1 18
10 1 10 1 18 126
11 122 126
12

 

Feed costs represented 42% of the value of milk yield for an average cow, and
marginal feed costs represented about 24% of any additional amount of milk yield for a
cow.

Table 3.2. Selected results of sensitivity analysis of altering basel scenario assumptions on
the optimum culling policy.

 

 

 

Yearly Culling Rates (%)
Scenario Annuity (S) Overall Involuntary Voluntary
Base 443 25.1 16.5 8.6
9600 kg ME 732 26.8 16.3 10.5
$19.71/100 kg milk 73 23.5 16.8 6.7
”957/100 kg milk 814 27.6 16.2 11.4
$0.067/Mcal NE. 599 25.8 16.4 9.4
$0.101/Mcal NE. 286 24.8 16.6 8.2
SlOOO/freshening heifer 476 28.0 16.0 13.0
$1200/fi'eshening heifer 412 23.7 16.8 6.9

 

1. 8000 kg ME, $24.64/ 100 kg milk, $0.084/Mcal NE), 31 IOO/freshening heifer.
The efi‘ect of the sensitivity analysis on the yearly annuity is as expected with

increased output or output price, or decreased input prices being associated with increased

49

profitability and vice versa. The overall optimum culling rates maintain a narrow range and
never attain the estimated current national average.

The unexpected increased voluntary culling rate associated with increased milk
production is due to an increased absolute difference in production between younger and
more mature animals as a result of estimating production on a proportional basis relative
to the mature equivalent (Rogers, Van Arendonk, et al., 1988b). Nevertheless, the
difi‘erence in the optimal policy is minimal. This insensitivity of optimal policy to herd level
of production is expected, contrary to conventional wisdom, by simply reflecting on the
earlier replacement models. The choice is between the current animal and the replacement,
increasing the production and profit of each has no effect on the relative difl‘erence and
therefore little effect on the replacement policy.

Similarly, altering milk price or feed cost had little effect on the optimal policy. In
contrast, the culling rates were highly sensitive to the cost of the replacement. Lowering
bdfer costs by 9% increased voluntary culling in all lactations, but none more so than the
doubling in first lactation.

The authors concluded “culling rates in the US may be higher than optimum. Our
model suggests they should be near 25% in most situations. Perhaps farmers who manage
well could eliminate rearing of replacements with undesirable pedigrees or purchase less
replacements and improve their profits....increased culling to improve yield per cow may
not be justified due to an increased number of replacements needed.”

These results are consistent with the optimum policies generated by the same

model under Dutch conditions (Van Arendonk, and Dijkhuizen, 1985) which tend to have

50

 

lower beefprices, calf values and feed costs than the US. The model suggested the
optimal Dutch culling rate was 28%, and sensitive to a change in the difference between
heifer price and carcass value, but insensitive to changes in milk, feed or calf prices or
conception rate. This study compared modeling animals with 15 levels of milk production
on a mature equivalent basis versus simply assuming average production, corrected for
age, of all animals. The variation in production between present and replacement cows
was important when considering the effects of changes in production or price situations on
optimum replacement policies. Doubling then quadrupling the rate of genetic improvement
in milk revenues minus feed costs resulted in minimal effects on culling rate. This failure
for genetic progress to affect the optimal policy is not surprising given Equation (2.37).
For given a low rate of improvement (i.e., h 5 1%) it can be seen that the efi‘ect on the
optimal life will be minimal. Also, as expected based on the explanation in Chapter 2,
insemination continued longer with higher persistency of milk production but again the
efi‘ect on herd life was slight.

A sister paper provides some interesting insight into the dynamics of total culling
rate, involuntary and voluntary culling (Rogers, Van Arendonk, et al., 1988b). As is
evident in Table 3.3, increased voluntary culling partially compensated for the lowered
involuntary culling. The increase in yearly annuity associated with a lowered involuntary
culling rate was due to lower net replacement costs (cost of heifer less carcass value),
decreased frequency of low yielding cows and increased productive life of high yielding
cows. The latter two efi‘ects were responsible for over 50% of the increase in the armuity

indicating that forced reduction in the intensity of voluntary culling is a major cost

51

associated with excessive involuntary culling. This fact is firrther highlighted by the last
two rows of Table 3.3 where the economic loss associated with involuntary cows was
removed. Of the $113 gain in yearly annuity associated with reducing the base level of
involuntary culling to only infertile cows, only $32 was associated with the direct financial
losses of those involuntarily culled, while the remainder was associated with reductions in
net revenues due to limited opportunities for voluntary culling. Once the coeficient of
variation for milk yield was made consistent, altering the involuntary culling probabilities
using the 9600 ME level of production as the base had similar results as the 8000 kg ME
base. Lowering involuntary culling by one cow per year in a 100-cow herd improves net
revenue by approximately $750 to $900/yr (a 1.6% to 1.3% increase). This finding that
lowering the involuntary culling rate resulted in a considerable increase in the annuity, and
a lesser decrease in the overall culling rate due to a somewhat compensatory rise in
voluntary culls, was also produced under Dutch conditions (Van Arendonk, 1985a). This
suggests that management should concentrate on the lowering of involuntary culls as a
method of achieving lower culling rates. As with any model, prior to embracing
recommendations the model and its assumptions must be compared with reality. The
evaluation of whether management should focus on decreasing involuntary culls will be
postponed until a general discussion of voluntary and involuntary culling which follows

later in the chapter.

52

 

“fl ..

‘1'

' f

“PW

sex.

3‘

u I...

Table 3.3. Resulting yearly annuity and culling rates when average ME was 8000 kg,
including and not including the financial losses1 associated with cows involuntarily culled.

 

 

 

Percent of base involuntary Yearly Culling Rates (VQ

culling probabilities Annuity (8) Overall Involuntary Voluntary
120 420 27.2 19.4 7.8
100 443 25.1 16.5 8.6

so 465 23.4 13.6 9.8

o 556 16.8 1.62 15.2
100‘ 475 24.5 16.8 7.7
80‘ 492 23.9 13.8 9.1

 

1. All financial losses fiom reduced carcass value, idle production factors, lowered milk
yield prior to disposal, and health costs associated with involuntary culling.
2. Cows still not pregnant at the eighth month of lactation.

The results of the model appear valid, and the general conclusion that the optimal
culling rate should be under 30% for US conditions can be made, and the potential for
sizable economic loss is present due to non-optimal culling policy.

Van Arendonk (1987) subsequently investigated the effects of ignoring variation of
variables in the optimal culling rate as determined by his dynamic programming model.
The base culling rate was 23.8%. The resulting optimal culling rates, in parentheses,
follow the respective omitted variables. It was determined that excluding the variation in
carcass value due to age, stage of lactation, and reason for replacement (25.1%), or
ignoring the feed cost (27.8%) variation increased culling especially among the first
lactation animals. The results were similar when the objective was maximization of milk
revenues (26%), or the scenario where all cows were inseminated and kept if they
conceived (16.8%).

Congleton and King (1985) estimated future incomes from an extremely

sophisticated biological simulation model (Congleton, 1984) which only assumed culling

at 180 days if open and a maximum of 4 services. All other culling reasons were simulated.

53

 

Voluntary culling occurred if the NPV of a heifer was greater than that of the cow
currently in the herd. The NPV of a cow was the sum of her own discounted cash flows
for the time horizon weighted by their probability of occurrence, and the sum of her
potential replacements’ discounted cash flows for the time horizon weighted by their
probability of occurrence. Similarly, the NPV of the potential replacement heifer included
her and all her potential replacements’ weighted discounted cash flows. Using a dynamic
program they found relatively few cows (21%) were culled compared to most commercial
dairy herds. Their results suggest culling of first lactation animals should be limited to that
due to involuntary culling. They also found that increasing the time horizon beyond two
calving intervals increased the culling rate to its maximum of 21% at the maximum six
lactation time horizon.

A simple dynamic program of Irish conditions (Killen and Kearney, 1978) included
a twenty-year time horizon, no variation in production within a lactation was modeled and
culling was assumed to occur prior to the initiation of a lactation. The optimal culling rate
was found to vary from 1/5 to 1/6 as the price of milk, level of herd production, and
involuntary culling rates were varied to represent conditions in the industry over the
previous two decades.

A Canadian dynamic programming culling model also illustrated the efl‘ects of
model time horizon, as well as, modeling animal weight (Stewart, Burnside et al., 1977).
The state variables consisted of lactation number (7 levels), estimated 305-day nrilk yield
(1 1 levels), average 305-day milk fat % (7 levels), and body weight (5 levels). A 12 month

calving interval was used, culling occurred at 60 days post partum. The probability of

54

mun

Wm.

\\

I: d-

e a

culling and some of the transition probabilities were based on Canadian cow records.
Annual genetic improvement of milk production of 0.9% per year was included.

Heavier cows required higher milk production to be kept. Increasing milk price or
decreasing feed price resulted in more intensive culling of lighter cows, although
subsequently this efi‘ect was deemed to have such a minor impact on profit that it could be
omitted from the model. In contrast to most other studies the optimal policies were
insensitive to net cost of replacement (replacement cost less salvage value). Use of a 5
year rather than 10 year time horizon resulted in more intensive culling. Altering the
feeding regimen from corn silage and hay providing the roughage to haylage as the sole
source altered the optimal policy for only 6% of the states. The actual optimal culling rates
were not mentioned.

A subsequent paper (Stewart, Burnside et al., 1978) also found that the policies
did not change when the efi’ect of genetic improvement in production was removed. The
simplification of an average replacement heifer was overcome by incorporating a
probability distribution of possible replacement heifers into the model. The probabilities of
weight, milk production and fat test of the heifer were included. The authors found that
the specification of an average replacement tends to underestimate the replacement value,
but the effect appeared minor as the optimal policies changed for only 3.3% of the states.

This issue of a standard replacement is not new. Ben-Ari et a1. (1983) recognized
that unlike traditional replacement theory heifers cannot be considered standard
replacement items. Kristensen (1987), in his dynamic programming replacement model for

dairy cows, incorporated this concept by including the breeding value of the sire as a state

55

variable. The advantage of using each parents’ genetic information was suggested. The
model predicted heavier culling among animals of lower genetic value.

There is general agreement among the optimization models that the optimal culling
rate is between 20 to 30% of the herd. This assumes that culls are sold for beefand not
dairy. This suggests that most herds will not need all their potential replacement heifers.
The optimal culling rates were sensitive to the relative cost of a replacement versus that of
the culled animal, but was not sensitive to most other parameters including genetic trend
or herd level of milk production. The greatest economic gain from reduced culling is
lowering that due to involuntary reasons.

Initially it may be curious that all the models, except for Congleton and King
(1985), assumed a base rate of involuntary culling when in fact the purpose of the models
was to optimize culling decisions. This assumption is pragmatic. While all culling has an
economic component, in order to optimize involuntary culling it must be modeled. The
modeling of the reasons for involuntary culling require a very detailed, data intensive
model. For example, consider the disease of mastitis. To incorporate this disease
occurrence one would need to know the transition probability from each state vector to all
the potential state vectors. The later would reflect the varying severity and duration of
potential mastitis episodes. While survival analysis can assist with determination of the
transition probabilities the economic impacts in terms of treatment costs are also required.
While dynamic optimization models including the stochastic occurrence and optimal
culling decisions related to single diseases such as mastitis are starting to be developed

(Houben, Huirne, et al., 1994) they are not very prevalent. The biological and economic

56

 

data required for these models are generally not available. The issue of voluntary versus
involuntary culling as it applies to actual dairy famrs will be addressed further in the
chapter.

The model of Rogers et al. did not include the effect of seasonality. Two models
suggest the potential impact of season on the optimal culling policies. Under Dutch
conditions (Van Arendonk, 1986) the optimum policy for inseminating and replacing cows
was greatly afi‘ected by the seasonal variation in production, and to a much smaller extent,
in calf prices and feed costs. Variation in prices of replacement heifers and salvage
animals, as well as, conception rate played a minimal role in causing the new policies
under the efi‘ect of seasonality. Under southern US conditions (DeLorenzo, Spreen et al.,
1992) seasonality of milk price, replacement cost, and conception rate were included in
the model. The resulting decision rules were significantly different than those generated
without seasonality. Use of the new policies resulted in monthly milk over and
underproduction that mimicked that actually found in Florida. The seasonal efi‘ects on
conception rates and lactational performance are the main factors determining the seasonal
pattern and optimal culling rate nearer 40% (McCullough and DeLorenzo, 1996).

Another management problem which should ideally be solved with the replacement
problem is the optimal feeding regime (Kennedy, 1986). If feed prices are not stochastic,
the optimum ration need only be optimized once for each possible state vector.

The dairy dynamic programming models only considered optimization of
insemination and culling policy ignoring the effects of other management decisions. Similar

simplified models are the norm in beef cow replacement as well. A recent beefcow

57

replacement paper (Frasier and Pfeiffer, 1994), however, also included the decisions of
breeding season length and winter nutrition. A sensitivity analysis of long-term capital
gains exclusions, feed costs, cost of replacement females and returns from weaned calves
and cull cows, showed very stable feeding and breeding policies. Similar to the results
fiom the dairy models, the optimal culling policy varied drastically with a unilateral change
in the cost of replacement, while simultaneous changes in the prices for both cows and
calves had little efi’ect on the policy. The study also highlighted the effects the relative
easing of capital gains taxes can have on optimal replacement, since the returns fi'om
home-raised replacements culled from the beef herd are considered capital gains.

Tronstad and Gum (1994) in their study, cited the use of stochastic rather than
deterministic livestock prices as the reason that their model of beef cow culling found it
may be profitable to keep nonpregnant cows in contrast to previous models.

Congleton (1988b) considered the effect of uncertainty of future production and
milk price state variables in estimation of future income. It is possible that the optimal
decisions will vary with the realized future state. The example used is that with a drop in
future milk price the mature cow, with higher production in the next few lactations, would
be preferred over the younger animal. In a highly simplified analysis the advantage of
increased predictive value of animal performance far outweighed the benefit of more
accurate prediction of firture milk prices.

Another problem with dynamic programming is that unless producers have the
capacity to run these programs for their specific situation, the ability to apply the results is

questionable. For McCullough and DeLorenzo (1996) showed that while varying

58

 

individual inputs rarely caused meaningful changes in optimal policy, varying multiple
inputs consistently resulted in meaningful changes. While with the advancement of
computer technology, repeating the computer runs is not as great an impediment as it was
a decade ago, this approach still provides little into insight into the reasons for the culling
decisions or the cost of not following the recommendations. Tronstad and Gum (1994)
used Classification and Regression Trees (CART), to provide advice on culling decisions.
The process generates two binary decision trees, one each for open and pregnant cows,
which provides an easy to understand set of culling rules, with the associated cost of
ignoring the recommended action. While the percent of animals correctly classified by
CART, as compared to the DP, at the terminal node of the decision tree was as low as
64%, the overall accuracy of CART was 95% and 99.5% of correctly classified “open”
and “pregnant” cows respectively. Using the decision rule of CART yielded over 99% of
the expected returns from the DP. The use of CART in conjunction with spring and fall
calving increased expected wealth by 10% over the simple rule of thumb of culling all
open cows in the fall with spring-only calving.

Finally, a difiiculty with optimization models is that only a point estimate of the
retums fi'om the policy is given, with no indication of the variation in the potential returns.
Repeated simulation is a method to evaluate the distribution.

This is an appropriate place to make the distinction between simulation and
optimization. Notice that all Optimization models contain simulations. Generally a
mathematical model is being used to simulate some phenomenon (i.e., the profits

generated by a Holstein cow in her third lactation.) However not all simulation models are

59

 

able to optimize. While this may seem so obvious as to be trivial, it is often overlooked.
Some of the confusion may be due simply to that fact that computers are often used to
solve both types of problems.

The advantages of simulations are their ability to compare results between optimal
policies and with those currently used, validate dynamic models and describe the transition
to optimized structure of the herd (DeLorenzo, Spreen et al., 1992).

Consider a recent (1996) paper by Galton, Everett, Van Amburgh and Dauman
concerning the economics of extended calving intervals with use of bST. The result of this
study is “a 18.0 month calving interval is recommended for improved animal performance
and herd profitability”. The simple fact that this was a simulation, not an optimization
study, appears to have escaped the authors. At best, the correct conclusion fiom this study
is that under the conditions and assumptions of the simulation the bST cows with an
extended calving interval had greater returns than the traditional management group with a
13.2 month calving interval. (Even this result is suspect due to a number of firndamental
flaws in the study design, including, but not limited to, ignoring the increased physical and
economic pressure bST treatment places on the milking cow facilities, using a single point
estimate for length of herd life rather than a distribution, assuming different culling rates
between treatments, and a time horizon bias that favors the bST animals.) Assuming that
bST does impart an economic benefit, it is entirely possible that a 17, 19, 20, or even a
13.2 month calving interval is superior to one of 18 months. Consider that if the goal of
the study was not simply to promote bST but rather increase producer profitability there

are a myriad of other potential management changes that could be considered, some of

60

a“; :1} have
21533111111 1
11.11:!) c
:1» inpiis:
1:") 5.11 1
1.1511
011:: $1
121: L: 10%
"Pf-Z among
13491: prod
31:00) Wt
3:213: lile
(“Elan be
2'1. {50, 7"
millior
133101 me
Firm 191
.3: .fi

“4%

‘i‘ 7:918?

Kid .
41:;k‘ ’_
It 1713121“

vi‘ib

s...

,s
\"mbr'i'i

h
U‘u‘l

which may have less associated risk. Such a biased and misinterpreted study only provides
disinformation and misleads those without the expertise or time to assess the study.
Unfortunately confusion over simulation and optimization and the correct interpretation of
equally simplistic studies are not uncommon. Obviously the appeal of such models is their
simplicity, but it is not clear what is being accomplished by oversimplifying a complex
problem.

Other simulations have been more fi'uitful. Dijkhuizen and Stelwagen (1988)
revealed the robustness of the culling guide generated by Van Arendonk’s dynamic
program among varied reproductive efficiencies. Their simulation model included
stochastic production, heat detection, conception rate and involuntary disposal. Feed
intake, body weight and prices were deterministic. The culling guide produced a higher
return over the policies of retaining all cows which conceive within 240 days, or no
insemination beyond 160 days of lactation, under all combinations of heat detection rate
(HDR) (50, 70, 90%) and conception rate (40, 60, 80%). But the difference was small
under conditions of poor reproductive performance due to the limited ability to make
culling decisions. Updating the guide which was created assuming 70% HDR and average
conception rates per lactation that varied from 0.47 to 0.63 (Van Arendonk and

Dijkhuizen, 1985) to reflect the poor reproductive performance had minimal improvement
on the returns.

A deterministic model of Irish conditions (Allaire, and Cunningham, 1980)
provides insight into the effects of voluntary culling based on level of production. Three

levels of involuntary culling (15, 20, 25%), and three levels of replacement cost relative to

61

 

beefprice (0.94, 1.26, 1.57) were simulated for 16 years. Maximum allowable levels of
voluntary culling were 26, 19, and 11%, for involuntary levels of 15, 20, and 25%,
respectively. Replacement occurred at the beginning of the following year. Results
presented are for the final year. The greater the involuntary culling level the lower the
annual milk production per cow, as mature high producing cows are replaced with lower
producing heifers. For a given level of involuntary culling the increase in milk production
per cow rises then plateaus and may then decrease as voluntary culling increased. Also as
the involuntary culling decreased, the voluntary level to maximize production increased.

Table 3.4. Simulated level of total culling to maximize objectives under varying

 

 

conditions.
Level of involuntm filling
15% 20% 25%

Maximum culling 42% 39% 36%
Culling to maximize milk production 33% 38% 36%1
Culling to maximize annual NPV per cow, low 42%1 39%‘ 36%‘
rearing costs2
Culling to maximize annual NPV per cow, medium 30% 34% 36%‘
rearing cost3
Culling to maximize annual NPV per cow, high 23% 27% 30%
rearing cost4

 

1. Maximum allowable.

2. Ratio of heifer cost to cull value of 0.94.

3. Ratio of heifer cost to cull value of 1.26.

4. Ratio of heifer cost to cull value of 1.57.

The authors conclude that the intensity of voluntary culling should at most be 3 to 8% in
addition to the involuntary replacement rate when the market price for a replacement
exceeds 1.5 times their cull value. Under the high rearing cost, the maximum culling rate
lowered NPV by about 10% from its maximum.

Allaire (1981) subsequently updated the model to study the effect of genetic

improvement. Six policies were modeled:

62

 

Replacements enter with no change in genetic merit for yield.

(1) Involuntary culling only at a rate of 20%.
(2) Voluntary culling above the 20% base rate of involuntary culling.

Replacements enter at genetic merit for yield reflecting expected response from selection.
(3) Voluntary culling, replacements from random sample of dams, no change in
sire merit.
(4) Voluntary culling, replacements fi'om best of potential dams, no change in sire
merit.
(5) Voluntary culling, replacements from best of potential dams, sire merit
increases 0.5% annually (5 28 kg milk).
(6) Voluntary culling, replacements from best of potential danrs, sire merit
increases 1.0% annually (-:- 56 kg milk).
All voluntary culling was for milk yield based on estimated real producing ability. Potential

replacements were selected based on their estimated breeding values using only dam
information, as for a given policy sire merit was constant. Replacement rates of 20 to 39%
were considered over a 20 year time horizon. The rearing cost of a replacement was 1.3
times her beef value. All replacements entered the milking herd at 2 years of age, all
culling occurred at the end of each year.

The 30 to 35% replacement rate which maximized milk production exceeded that
of the 25 to 27% rate which maximized NPV per cow. A 40% culling rate lowered NPV
by approximately 15 to 20% versus the rate to maximize NPV per cow. The greater the
genetic progress the higher the NPV, but “cow replacement rates necessary to maximize
NPV of the herd are insensitive to the influence of a continuously genetic trend in milk

between 0% to 1.0% per year”. The results appeared insensitive to the relative cost of

63

 

replacements. The author noted that in 1979, Ohio Dairy Herd Improvement herds
consisted of 30% first lactation cows.

The net marginal revenue model reviewed in Chapter 2, was used to compare
various strategies of selecting cows during the first lactation (Korver and Renkema, 1979).
Given a particular strategy, the culling decisions were optimized. It was assumed that
potentially 40% of the herd could annually be replaced by heifers. Four culling policies
were considered:(1) Ill-health or insufficient production are reasons for disposal. (2)
Removal only for ill-health. (3) Low producing cows are removed at the highest possible
rate during the first lactation. (4) Similar to 3 but no calves are raised from cows that
leave the herd due to low production. Policy one was considered to reflect typical Dutch
culling policy. The effect of the culling policies on future genetics of the herd was ignored.
The results are presented in Table 3.5.

Table 3.5. Results of Dutch simulation of four selection policies during first lactation.
Cullin Poli

 

1 2 3 4
Percent of Available Heifers Used 58% 51% 100% 85%
Ave. Culling rate 23% 20% 40% 34%
Ave. Productive Life (lact.) 4.3 4.9 3.5 3.9
Ave. Annual Earned Income per Cow (Dfl) 1363 1351 1246 1315
Percent Increase in Income Over Policy 3 9.4% 8.4% 0.0% 5.5%

 

While the authors correctly note that these results do not imply policy one is
optimal (since this is a simulation study), the disadvantage of heavy culling policies (3 and
4) is apparent, at least under the assumptions and conditions of the study. The higher milk
production of the older animals is insufficient to compensate for the higher cost of rearing

young stock and the unfavorable age distribution (more younger animals in the herd). Also

 

the critical production levels, relative to the average first lactation production for the herd,
below which first lactation cows should be culled were calculated. Using this information
and assuming a normal distribution of milk production with a 15% standard deviation of
phenotypic intra-herd milk yield, about 20% of the first lactation cows should be culled
based on production.

The authors suggested that instead of assuming replacement heifers have average
productive capacity, they could be ranked according to their estimated breeding value.
This ranking, as well as, lower rearing cost or higher salvage values will increase the
culling of first lactation cows.

Pearson and Freeman (1973) used a deterministic model to simulate the efi'ect of
four culling systems on the milk production of US commercial Holstein operations. The
four cow selection systems were: (1) only enough females were raised to replace
involuntary losses, estimated breeding value (EBV) was used to select those calves to
raise; (2) selection on EBV based only on the cow’s own record; (3) selection of the
youngest cows; (4) all voluntary selection during the first lactation based on EBV. Under
systems 2 to 4 all females were allowed to fi'eshen. Each herd was simulated for twenty
calvings with 13 month calving intervals. Profit functions were quite complete including
returns from milk, bull and cull heifer calves, cull dairy cows and those sold for dairy,
fixed costs per herd, interest costs for livestock, expenses associated with artificial
insemination, feed, rearing of heifers and an additional cost of milk for higher production
based on rolling herd average. While the study included the effects of different herd sizes

and levels of herd production, the rank of selection systems remained unchanged.

65

 

Table 3.6. Results of US simulation of four selection policies'.

 

 

 

Culling Policy
1 2 3 4
Genetic gain/period (kg) 56 57 53 56
Actual production gain/period (kg) 67 68 62 69
Ave. Profit with low rearing costs” $211 $229 $192 $218
Ave. profit with medium rearing costs“ $189 $188 $151 $218
Ave. profit withflh rearing costs?"5 $167 $147 $110 $136

 

1. Using intense sire selection (equivalent to PD>363 kg).

2. Salvage values held constant.

3. Estimated ratio of replacement cost to salvage value of cull is 1.3.

4. Estimated ratio of replacement cost to salvage value of cull is 1.5.

5. Estimated ratio of replacement cost to salvage value of cull is 1.9.

Selection can be characterized as: (1) high intensity low accuracy; (2) accuracy increases
with each lactation, but intensity decreases; (3) selection pressure is zero; (4) intensity and
accuracy of selection are high. Policies 1, 2, and 4 result in genetic improvement at nearly
the same rate. The higher production of 2 and 4 is due to the positive environmental efi'ect
resulting fi'om selecting on the cows’ own records. With sire selection intensity equal to
the 2% theoretical rate of genetic improvement, policy 1 had the slowest rate of genetic
and phenotypic improvement.

System 1 was more profitable than expected by its lower production due to the
smaller cost of replacements. The advantage of system 1 increased with the cost of
replacements as this overshadowed the increased returns associated with higher milk
production. The low and medium rearing costs appear inappropriate for application to the
present day. The effect of the sire selection system (2% genetic gain/year, sires with PD >
363 kg, or sires 91 kg 2 PD 2 -91 kg) on profits were greater than that of the cow

selection. The authors note that if calf selection is required, some selection should be

reserved to remove the mistakes made earlier and allow for an adverse sex ratio.

66

A simulation model estimated milk yield based on dams’ and sires’ breeding
values, the efl‘ects of health problems including reproductive disorders and mastitis were
included (Congleton and King, 1984). It is not clear why culling was based on cost per
unit of production during present lactation including the preceding dry period. This basis
for the culling decision would likely be biased against first lactation animals and increase
their culling. Pregnant cows could be culled at the end of lactation, while infertile cows
could be culled after 305 days of lactation. The time horizon was 30 years. Using this
questionable decision criterion the optimum culling rate was 20% to maximize discounted
retums. Increasing herd life fiom 3.4 lactations to 4.2 linearly increased discounted income
by $518.96 per lactation.

Congleton (1988a) conducted a simulation on the effect of culling on predicted
production or income on herd profitability and genetic trend. Future production was
estimated using previous performance and repeatability of complete lactation production.
Culling on profitability was a two stage affair. First the herd was screened with Kuiper’s
short-term MaxAMR to identify potential cull candidates (estimate returns from cows
through the next ten months of the next lactation). These animals were then ranked on
long-term income using cow attrition data. The order of the use of the two screens seems
intuitively backwards.

One of these two criteria were used to select a cow for culling when a heifer
fieshened (20% as many bred heifers as milking cows were targeted for cow replacement
resulting in a constant herd life regardless of culling method, heifers were selected on

breeding values). Since this would entail culling many cows afier breeding, the third

67

1'
.11“

IE".

.r-O
Jun- L
1 't.‘

.luk '11

323:
U

‘ r

12::

..'
a."

in.

H

culling option was not to breed the lowest producing cows and remove them when
maximum income had been produced for the lactation. Herds were simulated with varying
intensities of nonbreeding on production and then culled on either production or income.
A 20 year time horizon was simulated.

Culling on projected income increased herd income by 4.3 to 4.8%, and decreased
the genetic gain to 65.3 to 86.2% of that observed with cow selection on production.
(Breeding only the highest producing cows increased herd income by 4.3% to 5.7%).

Even the simulation model of Kuipers (1980) which, as discussed in Chapter 2,
underestimated the value of younger animals by using an inappropriately short time
horizon, found selling 30% of the heifer calves, rather than raising them all, gave higher
returns to management of about $50/cow/year for all selection methods. This simulation
model considered culling of cows based on their genetic index.

The results of these simulation studies agree with those of the dynamic
programming models. The cost of replacements is such that maximizing genetic progress
or milk production through increased culling is not economical. The realization that the
use of all replacements does not maximize profit is comforting fiom an economic
perspective. These models are simply portraying the firndamental, at least under certainty,
decision rule of profit maximization which appeared trivial in the previous chapter since it
was so obvious. Narnely continue to use more of an input (heifers) until the cost of doing
so exceeds the return (i.e. use an input until the marginal input cost exceeds the marginal

value product).

68

 

Marginal value product is the product of the physical marginal product and the
additional gross revenue generated by the employment of the extra unit of input. In a
competitive market the cost of inputs and outputs is constant, so the marginal physical
product essentially drives the profit maximizing decision. These studies then suggest that
as more and more heifers are used their returns decrease, this is simply a statement of the
well accepted theory of diminishing marginal productivity of inputs.

By focusing on using all potential replacements the industry is reflecting its output
bias. To maximize output, inputs will be used as long as they contribute anything to output
(i.e. as long as the marginal physical product is greater than zero). Obviously this decision
rule will result in more use of the input than the profit maximizing level of input. This is
supported by the studies which found that the replacement rate which maximizes milk
production exceeds that which maximized economic returns. The two levels of input will
be equal only if the price of the input is zero. The traditional biological focus of herd
management then has implicitly assumed zero cost of inputs.

The optimal culling rates are sensitive to the cost of the replacements relative to
the salvage value of the cull. However, if replacement rates are excessive it appears the
greatest gain to be made is through lowering involuntary culling.

Prior to considering the actual culling rates in Michigan and whether all
replacements are being used, a more careful assessment of the definition of culling rate is
required. Up until now culling rate has been loosely defined. Following this, the

partitioning of culling into voluntary and involuntary will be critically considered.

69

Many of the studies mentioned previously focused on average productive herd life
as a measure of culling rate. Productive herd life is the average length of time an animal
spends in the herd starting with the first calving. Average productive life can also be
considered the expectation of life (Rendel and Robertson, 1950). They state “if one-
quarter of the population is replaced each year, then obviously the expectation of life is 4
years.”

The definition of culling rate (C) is not explicitly defined in the optimization papers.
Rather the studies report the culling rate (i.e. 25%) and the optimal average herd life (i.e.
47.8 months). The average herd life (H) is actually the average productive life of the
animals in the herd. This is calculated as the summation of the proportion of the herd in

each respective lactation, assuming the distribution of ages is in a steady state.

H = in}, (3. 33)

1-1

where n, = the length of lactation number i in months,

P,- = proportion of herd in lactation 1'.
Pi is equal to the marginal culling rate for lactation 1' (Ci) multiplied by those animals that
have not been previously culled.

Equation (3.33) can then be restated as

H=1C,+2C2(1—C,)+3C3(1—C1)(1—C2)+ .......... +nC,,fi(1—C,) (3.34)

181

70

 

To calculate the average culling rate based on the average herd life, the marginal

propensity of culling in each lactation is assumed equal.

5

H = 1C + 2C(1- C) + 3C(1— C)2 + 4C(1- C)3 + .......... + nC(1- C)"'1 (3. 35)
H = inC(1- c)"" (3. 36)
m1

Equation (3.35) reveals the sum to be an infinite linear geometric series. Using the

notation ofRobison and Barry (1996, pg 95) R = 0, and a= r = C/(l-C) so (1+r) =
(1.0".

H(1- c) = 1C(1— C) + 2C(1- C)2 + 3C(1- C)3+ .......... + nC(1- C)" (3.37)
Subtracting (3.37) fiom (3.35),

H-H(1-C)=C+C(1—C)+C(1-C)2 +C(1-C)3+ .......... + C(1-C)”‘1

HC=C[1+(1—C)+(1-C)2+(1—C)3+ .......... +(1—C)"“]—nc(1-C)"

=-1—f—C[(1-C)+(1—C)2+(1-C)3+ .......... +(1—C)"]-nc(1-C)"
Dividing by C, and substituting b for (I-C),

H = 1[b+62 +b3+ ........ +b"]—nb"
b

71

 

 

= g-US,[——1;b.n] - nb"

where Lbb-=1-(1-C):1-1+C:_

C
1-C 1-C 1-C

 

As n :3 co, US,(1-b/b,n) =5 b/1— b.
Since C > 0. (1 — C) < 1. as n a co, n(1— C)" :5 0. Therefore as the potential number of

lacations approaches infinity, the value of the series converges to

_ 1 .1-C_l
1—c C C

 

Ifthe average culling rate is 25%, the average herd life should then equal the inverse of
the culling rate which is 4.0 years or 48 months which is that predicted. The 47.8 in the
article differs slightly because not all culling is done at the end of a year and the marginal
culling rates per lactation are not all equal. The age structure of the herd is a function of
the marginal culling rate each lactation or year, so herds with identical culling rates can
vary in average age (Sattler and Dentine, 1989). This then establishes the definition of
culling rate to be the number of animals culled in a year divided by the average herd size.
The average culling rate among Michigan herds on official DHIA test was 37% in
1996 (Michigan Dairy Herd Improvement Association, 1996). However, few Michigan
DHIA herds will have this high a number reported on their production records. For

Michigan DHIA calculates the culling rate as the number culled divided by all the animals

72

that were present in the herd at anytime through the year (Ferris, 1987), this is commonly

referred to as the herd turnover rate ( T 0).

 

S S
- ——
20—1 VCI'SUSC—I

where S is the number of animals sold, and I is the average herd inventory.

1 _I+S or1.70 _1_
T0 5 T0 3

 

Therefore, the culling rate, C, equals 1 T0

80 a herd turnover of 30% is equivalent to a culling rate of 43%. It is likely that
Rogers et al. (1988a) in stating the average US culling rate was 31%, and thus not
significantly difi‘erent than their determined optimal culling rate, assumed this was directly
comparable to their definition of culling rate (i.e. C). Whereas, this 31% truely represented
the herd turnover, with a resulting C of 45% which is considerably higher than the optimal
culling rate as determined by the dynamic program.

The proportion of first lactation animals in a herd is not an appropriate indicator of
herd culling. For consider that while at any one time only a single first calf heifer may be in
the herd, one could actually have culled 52 in a year if they were, on average, kept in the
herd for only 1 week. It does, however, represent the minimum culling rate.

Evidence from Ontario (Ontario DHI Corporation, 1997) suggests the distribution
of culling rates across herds is symmetrical. So it is quite possible that over halfthe

Michigan dairy herds have culling rates significantly in excess of the suggested optimum.

73

Only a high death loss or sales of heifer calves would prevent the average producer from
possessing an excessive heifer inventory. The presence of excess heifers is supported by a
survey of heifer management practices of producers comprising 78% of the US dairy cow
population and calf rearing p0pulation (Heinrichs, Wells et al., 1994). This survey found
that only 25.8% of herds sold any heifers prior to parturition.

If producers are interested in selecting heifers to lower their culling rate, it must be
recognized that this culling rate is dynamic. In making the transition to freshening fewer
heifers, the culling rate among the heifer population will decrease as the number of heifer
calves produced by first calf heifers drops.

Assuming herd size is constant and no heifer culling in practiced, 00, is the number
of ofi‘spring (as a percent of herd size) in period 0 due to calve in the next year, and Co is
the excessive nonoptimal herd culling rate in period 0. In period one a level of heifer

culling, C ,, , is required to achieve the desired adult herd culling rate C. in period 1 (i.e., C 1

=C‘)
C' = 00(1- CM)

C.
C :1-
” 0

 

1
0

As a result of heifer culling prior to their calving, there will be a smaller subsequent heifer

population, 0,.

 

01 = 0, -[C° 5C] (3. 38)

The denominator accounts for the sex ratio.

74

Because after the first period the culling rate in the adult herd equals the optimal

rate, if the optimal herd culling rate remains constant, so too will the number of ofl‘spring

 

 

 

 

 

 

andtheheifercullingrate.
02:01-[C1'C]=0,-[C ’C J=0,, .-.0,=02=03= .......... =0,,
2 2
C‘ C‘ C‘ C' .
Ch2=1-01=1-02=1-03= .......... =1"0n, "Ch2=Ch3=Ch4: .......... =Ch"

Heifer culling settles into a lower steady state rate after the initial period of higher culling,

noting that in this context a period is the length of time for a heifer to freshen.
IfCo = 43%,C' = 30% and 0,, =5

3 =1-%=0.40=40%

C =1—
”1 a 0.5

 

0,=00_[£q;_0]=0,5_(952%23_0)=044=44%

C’ 0.3
=1——= 0.32 = 32%
01 0.44

 

Chz =1“

Algebraic manipulation reveals the relationship between the initial heifer culling

rate and the subsequent steady state level.

 

 

(3. 39)

(1"Ch1) =1_[Co 4‘]

(1’92) 20°
As discussed previously the culling models assumed a base level of involuntary
culling for the pragmatic reason of not having to model the underlying biological

processes. Consideration of Table 3.1 is disturbing because it suggests that many

75

producers have been culling animals for infertility, generally accepted as an involuntary
cull, when the optimal choice would have been to continue to inseminate. In analyzing
culling records of Canadian dairy cows, Dohoo and Dijkhuizen (1993) determined that
reproductive culling was occurring too early, and that it was economically justifiable to
persist in breeding dairy cows much later into lactation than is often considered feasible.
The recommendations of the optimization and simulation models were to lower culling
rate preferably by decreasing the rate of involuntary culls. It would appear that this is
possible, simply by reconsidering the definition of voluntary and involuntary culling.

Webster (1994) defines involuntary as “not done of one’s own fiee will, not done
by choice” or “not consciously controlled, automatic”, an example given for the latter is
sneezing. I propose that in the use of the term involuntary with regards to culling, the first
definition reflects the accepted meaning of the adjective. However, the second definition
actually portrays involuntary culling in the US dairy industry.

It is apparent that with the exception of death and absolute sterility there is no such
thing as an involuntary cull. Culling is solely an economic decision. To recognize this
consider the following question: given a four year old cow rated excellent in
conformation, with a 30,000 lb ME would you cull her for reason X? Where reason X is
any reason for which an animal may be culled. If she has mastitis or infertility, commonly
cited reasons for involuntary culling, and yet you would not sell her then by definition it
can not be an involuntary cull.

The recognition that culling is firndamentally an economic issue has been proposed

(Fetrow, 1987). However, this distinction purely for the purpose of semantics serves little

76

purpose. The distinction is important if the involuntary/voluntary culling paradigm is
resulting in “not consciously controlled” culling decisions. Stressing the economic nature
of culling decisions may be a method to encourage the dairy industry to critically evaluate
their culling decisions.

The presence of producers who purchase the culls of others at beef price and then
milk these cows is empirical evidence that many in the dairy industry are not correctly
valuing their culls. This incorrect valuation could be due to their failure to critically
evaluate their assumptions and beliefs concerning the optimal culling rate and involuntary
culls.

I am troubled by my suggestion that the market is operating ineficiently by having
too high a culling rate. I base this irrationality on the following arguments: as the past few
chapters have highlighted, the optimal culling rate is highly complex and not intuitively
obvious, furthermore its relation to profit is even less clear. The behavior of producers is
consistent with bounded rationality, where knowledge of the optimal cull lies outside the
bounds. Milk output is easier to measure and maximization of milk production is often
used as a proxy for profit maximization. The suggestion that the higher than optimal
industry culling rates are the result of attempting to maximize milk production is
supported by some of the culling studies. Then we need to resolve why we never realized
that at these higher culling rates the marginal value product of a heifer was less than her
marginal cost. I believe this is due to overestimation of the value of genetic improvement
in the heifer and her potential future offspring. As the models show, the rate of genetic

improvement has little effect on the optimal culling rate and vice versa. Finally, the focus

77

by veterinarians and animal scientists on herd health and heifer rearing has resulted in
increased survivability of heifers and thereby provided the potential fire] to fire the
excessive culling rates due to increased heifer pressure.

In summary, the evidence is overwhelming that the optimal culling rates for
Michigan dairy producers are 20 to 30%, assuming culls are sold for salvage value. This
assumption is appropriate for most commercial herds. Despite varying models with
varying assumptions and conditions, as well as, different solution techniques these results
are very robust. Perhaps the greatest criticism of these models, in terms of truely
optimizing the culling rate, is in the widespread use of the industry involuntary culling rate
as a baseline parameter. Despite the uniformity of the studies’ results the industry has been
slow to adopt the recommendation of lower culling rates. In order to lower the culling rate
some potential replacements must be culled. The models described assumed random
selection or arbitrary selection of heifers based on some genetic estimate of potential for
milk production. Selected results of the culling optimization and simulation studies are
presented in Tables 3.7 and 3.8, respectively. Chapter 4, will explore the issue of heifer

selection.

78

BIBLIOGRAPHY

Allaire, F. R 1981. Economic consequences of replacing cows with genetically improved
heifers. J. Dairy Sci. 64:1985-1995.

Allaire, F. IL, and E. P. Cunningham. 1980. Culling on low milk yield and its economic
consequences for the dairy herd. Livest. Prod. Sci. 7 2349-359.

Bellman, R 1957. Chapter 1 (page 40) in Dynamic Programming. Princeton University
Press, Princeton, New Jersey.

Ben-Ari, Y., I. Amir, and S. Sharar. 1983. Operational replacement decision models for
dairy herds. J. Dairy Sci. 66: 1747-1759.

Burt, O. R 1965. Optimal replacement under risk. J. Farm Econ. 47:324-346.

Congleton, W. R 1984. Dynamic model for combined simulation of dairy management
strategies. J. Dairy Sci. 67:644-660.

Congleton, W. R. 1988a. Dairy cow culling decisions. 2. Profitability and genetic trend in
herds with culling on production versus income. J. Dairy Sci. 71 : 1905-1915.

Congleton, W. R 1988b. Dairy cow culling decisions. 3. Risk of culling on predicted
income (An application of Bayes criterion) J. Dairy Sci. 71: 1916-1925.

Congleton, W. R., and L. W. King. 1984. Profitability of dairy cow herd life. J. Dairy Sci.
67:661-674.

Congleton, W. R., and L. W. King. 1985. Culling cows on predicted fisture profitability
with a variable planning horizon. J. Dairy Sci. 68:2970-2983.

79

DeLorenzo, M. A, T. H. Spreen, G. R. Bryan, D. K. Beede, and J. A. M. Van Arendonk.
1993. Optimizing model: Insemination, replacement, seasonal production, and cash flow.
J. Dairy Sci. 75:885-896. '

Dijkhuizen, A A, and J. Stelwagen. 1988. An economic comparison of four insemination
and culling policies in dairy herds, by method of stochastic simulation. Livest. Prod. Sci.
18:239-253.

Dohoo, I. R, and A. A Dijkhuizen. 1993. Techniques involved in making dairy cow
culling decisions. Compend Contin Educ Pract Vet 15:515-520.

Ferris, T. 1987. Michigan DHI records manual. Guide to record use, herd evaluation and
explanation of report forms. Cooperative Extension Service, Michigan State University.

Fetrow, J. 1987. Culling dairy cows. Proc 20th Annu Meet AABP2102-107.

Frasier, W. M., and G. H. Pfeiffer. 1994. Optimal replacement and management policies
for beef cows. Amer. J. Agr. Econ. 76:847-858.

Galton, D. M., R. Everett, M. Van Amburgh, and D. Dauman. 1996. Economics of
extended calving intervals with use of bST. Proceedings 1996 NE Dairy Producers
Conference.

Heinrichs, A. J., S. J. Wells, H. S. Hurd, G. W. Hill, and D. A. Dargatz. 1994. The
national dairy heifer evaluation project: A profile of heifer management practices in the
United States. J. Dairy Sci. 77:1548-1555.

Houben, E. H. P., R B. M. Huime, A. A. Dijkhuizen, and A R. Kristensen. 1994.
Optimal replacement of mastitic cows determined by a hierarchic markov process. J. Dairy
Sci. 77:2975-2993.

Howard, R A 1960. Dynamic Programming and Markov Processes. MIT Press,
Cambridge, Massachusetts.

80

Kennedy, J. O. 1986. Chapter 7 Livestock Management, pg 194 in Dynamic
Programming: Applications to Agriculture and Natural Resources. Elsevier, London.

Killen, L., and B. Kearney. 1978. Optimal dairy cow replacement policy. Ir. J. Agric.
Econ. Rur. Sociol. 7:33-40.

Korver, S., and J. A Renkema. 1979. Economic evaluation of replacement rates in dairy
herds 1]. Selection of cows during the first lactation. Livest. Prod. Sci. 6:29-37.

Kristensen, A. R 1987. Optimal replacement and ranking of dairy cows determined by a
hierarchic process. Livest. Prod. Sci. 16:131-144.

Kuipers, A 1980. Development and comparison of selection criteria for cows and bulls
with a dairy herd simulation model. Ph.D. Thesis, Univ. Wisconsin, Madison.

McCullough, D. A, and M. DeLorenzo. 1996. Effects of price and management level on
optimal replacement and insemination decisions. J. Dairy Sci. 79:242-253.

Michigan Dairy Herd Improvement Association. 1996. Annual summary of production
records.

Neufeldt, V. (ed). 1994. Webster’s New World Dictionary of American English, 3rd
College Edition. Macmillan, New York, New York.

Ontario DHI Corporation. 1997. Provincial management centre benchmarks - 1996.

Pearson, R. E., and A. E. Freeman. 1973. Effect of female culling and age distribution of
the dairy herd on profitability. J. Dairy Sci. 56: 1459-1471.

Rendel, J. M., and A. Robertson. 1950. Some aspects of longevity in dairy cows. Empire
Journ. of Exper. Agric. 18:49-56.

Robison, L. J., and P. J. Barry. 1996. Present Value Models and Investment Analysis. lst
ed. The Academic Page, Northport, Alabama. Ch 1, 2, 5, 8, 11, 21.

81

Rogers, G. W., J. A. M. Van Arendonk, and B.T. McDaniel. 1988a. Influence of
production and prices on optimum culling rates and annualized net revenues. J. Dairy Sci.
71 :3453-3462.

Rogers, G. W., J. A M. Van Arendonk, and B.T. McDaniel. 1988b. Influence of
involuntary culling on optimum culling rates and annualized net revenues. J. Dairy Sci.
71 :3462-3469.

Sattler, C. G., and M. R. Dentline. 1989. Trends in herd age structure and the relationship
with management characteristics in Wisconsin Holstein herds. J. Dairy Sci. 72: 1027-1034.

Stewart, H. M., E. B. Burnside, and W. C. Pfeifi‘er. 1978. Optimal culling strategies for
dairy cows of different breeds. J. Dairy Sci. 61 : 1605-1615.

Stewart, H. M., E. B. Burnside, J. W. Mlton, and W. C. Pfeifl‘er. 1977. A dynamic
programming approach to culling decisions in commercial dairy herds. J. Dairy Sci.
60:602-617.

Tronstad, R., and R. Gum. 1994. Cow culling decisions adapted for management with
CART. Amer. J. Agr. Econ. 76:237-249.

Van Arendonk, J. A. M. 1985. A model to estimate the performance, revenues and costs
of dairy cows under different production and price situations. Agric. Syst. 16:157-189.

Van Arendonk, J. A. M. 1985a. Studies on the replacement policies in dairy cattle. II.
Optimum policy and influence of changes in production and prices. Livest. Prod. Sci.
13:101-121.

Van Arendonk, J. A. M. 1986. Studies on the replacement policies in dairy cattle. IV.
Influence of seasonal variation in performance and prices. Livest. Prod. Sci. 14:15-28.

Van Arendonk, J. A. M. 1987. Contribution of variables to the optimum policy for
insemination and replacement of dairy cows. J. Anim. Breed. Genet. 104:3 5-43.

82

Van Arendonk, J. A M. 1988. Management guides for insemination and replacement
decisions. J. Dairy Sci. 71:1050-1057.

Van Arendonk, J. A M. and A. A. Dijkhuizen. 1985. Studies on the replacement policies
in dairy cattle. III. Influence of variation in reproduction and production. Livest. Prod. Sci.
13:333-349.

83

 

 

 

84

 

   
  

 

 

 

 

 

 

 

 

 

 

 

2. .2. 88%.. .88.. 8... £8... .8
2. .2. H.2... =5 8.9... 8.... 22.8... - 2 .8... .88... 2 2% .2: 8m 8...
2. .. .o 22> .22 co 2 .88.... .2... 2% .22 28.82,.
2. .2. ”8.... 2...... .3 .o. .3 a 2. 8.28... 2 828...... 8.5 22.2. 28.8....
a. .- .o. .- H8.... 8...... .83" :2. z 22...... .8 - 2 8.8.8.... .882 8 .22 8>
.8... 2.8... L o 88880.... use: .8 .8...
a... 22.8... - 2 .8. .88... 2 2% 2...). 28.82..
2.- .8. 8.5. 88:8... 8.88.... 2 88.2.. .8. 2% 2...: 88.22.
N... .2. ”88.28.. A. .o .82. 2.8... i. - .8 2 88.8.... 8% 88...... e .288...
S- .... .28.... 8...: .2." .5... .- 52. 2.8...- N. 88.8. .8 8282 2 .32 8>
.8... 5:2: in o cacao-9:8 mo mega-
2. .2. .2... =5 8.9... 88 22.8... - 2 .8. .88... 2 2% .22 .3 .25 a
.8 .2. 22> .22 88.8.... 2 88.8. .2... 2% .22 2.89.2
.2. .2. 82... .22 G... .. .2. 58... i - ta 2 8.8.2.... .85 .m: 8 28.8....
.2- 88-8.... 8...: 2.2 .a 8.. 2.8...- N. 8258:8382 2 .82 8> .88...
.2." 8... 58... an 2 .8... £82
5:383... 8%. 25:9: - 5 20:92.8 he 9:8; .8 En— ._-
.823 88.8.... 2 .8. .88... 2 2% .22 .m... 88.38
no. .2- 85. 8.28080 58... 5.. - an 2 8.88.... ems... .228... e
.2..- ”8.... 8...: .83 :2.- .82 2.8...- N. 8.8822882 m .82 88.3.:
8.889 8822.. .552... 38 92.5 .88... 8.88:
Na... M225 3.32.20 :5 882.288. 8.825 2.5 8%... 2.88 8.8.
380888822... 288 8.2. 8 ”22¢ «a: 8.8» .§..._<
2.2.3. 192388... 8.8.588. 88.888222282833233 .8888. 28m

8.2... 8.8.58 8...... .o .88. 8.8.8 .2 2...;

3320:. ..\.n._ 9 2.2. .3 302.83 8 an? .3 2:8... 8: 8322.. v.2. 38.8. 8 n. =3 3.5.—9:. .8. 0:0

 

 

 

 

 

 

 

 

 

 

 

.28.
.3— .28.".—
.m< .H .h—
a .8 .. 2...... 8.8.2
8.... 3.. .- 8 8.8.8.... .382 8 .82 8 8:...
8.8.8.
.88... 2... 8.... 2% 2......
.. 38... .8... .8. 8.5 ..
.2 8.2. 8.8.... .28... 8.8.8... 8.: 8... 8
8.: ea. .88... 8.. 8.8822 .0 o. .22 8.2.80
2.28.5 8......
88.82 88...... 8.8.8.9.... 38 825 88... 8.28....
H.8. 88.5 .8. M8...... =5 8 8.8.88... 8......8> .8... 8...... 888 .582
85...... .o 822.8... a... .........o 8...: .o 88.8 .88 8 8... .88...

 

 

8.88. .2. 2.3

85

 

 

 

 

 

 

 

 

 

86

 

 

 

 

 

 

 

9.: 3:3 $8 2: 8
3.... “:5...“ $3 .23 $3 2 m. 32:95 E2.— an 0:05
£69 £8.58...» 630 8%
.2 “a 9... £3 38 .2. Sfiaueq a? 2:532 8:58 a 88sec a. .m: .3 E5 ..
$3 2 8 as =3 .53 $8 2.: 95.3 has? :2 go as was: 09.3.52» 8 .52 232
.385 0.55:8: ca «89:. 8:88.
3?: as ”5:8 38.8 £825 $3. 52.. 2.085% 883% .3 ES ..
2822.. §u§_ an 2 3 Eat a: 22. 332... 538.. mo .538 .m.: 39
.§§8_ 3. see; .553 he 2o ”5:8 8 was ”5:5 2. .32 339.5
$8 2 S 2.2. 238
3328v v5 .o\cm 8 v .3 2:35 F.2— eomaouu... x09: 005.com .m.D ._um baa ._.
Suave.“ 3%? 0885 3.8.83 .5 was: so cocoa—om hobo:— oa .32 83350
$~ $m 8: 2.: .23 EE— 3322.:
$2 $3 32 £335? a €2.52 55 “5:8 Egg...
Spa 0.2 mafia... veg—9:3 mo om: - .ooqaccofi
$3 $2 8. .5 3 5:85 .8 2 =5 . 3:838
j ”3% .22» .55 “maze . .3 .85
2am >23:on mac cm 5 3225 “—0982:—
“oonascoton £3.35 Ev on biota—.02 28:0: 533.05 a.
9:838. 2a 83%.: Boss .35 8.9: 3:5 :3 s 58:. gm - am. 2 ans. 5:3 2 .32 .3353
.82 .65:on $9. a .8 8:532 a8. 5:9: 8:90
3: :_ 3335 o\oe a mam—3 .28 succumua $3 a .8 2m 8% >222: . .«o 35:: H o— 5. .mom bin .n
$3. 85:95.. .2. 983.8... «.03— :ouuavEQ 5 «83a c0388! 32 .855? .m.D 3:03qu
05 5 38 >5 3 =0.—.83. mo 5:08 i. boas 38 .«o 5:08 5L a. 5.5.300 3.3—m fi
age: .8 a... as: .3?.. :38 go a 3.? - an 52. 585. .3 .82. ms» 2 .82 $8.30:
80$
333. B. saga 38 «£3 €85 3082
=5 a Sausage. :85; P58 :58.
5%: «Saga 83...; 25m 25 as; .33

 

 

.863» coma—55mm

3% £39 383 .3 as:

 

E9:— =.
83.5.: $3 a Ba .8659:— ouocow 3:8? by, J:

EEL

 

 

 

 

 

 

 

 

 

 

 

 

 

3.: .\.n a. 3.38. >m 8 .58.»... .2... 83...... _ .3 .35 ..
magi... 5... 3...... .93 2.? 332.. 8.2...» 5 8.32.. .6 an E. 9.3.8 5.5... 2 .m: 5.58...
828 mecca—um E... .830. 55.32.. baa mafia—no .6; o. .25 maize on- 05.5.5809 NN $.3— fi swan.—
58... a» .3 .5
{ovfi 28 .026— 5? Eu...— uouaflui .a .9082 b.“
accosta Bo. nouns—com...

.83 88.858 8 $65: .8 .9682- 283 v5 9...» 30:82.— : ._0m .35
8.5.8. a... ... 3% 2.8... N . its... .0385

$3. cocoa—v95 3o. .8 mafia c.5838 - .3385 $2.55 9:28 5:2: N. van—.0: «ES—cox
3 ”flag ”5:8 8.3.8. L- 08.55.25 «3. a .39..

:8 .x... W :2
$2 $3 $8 .3 do...
$2 $2 $2 $42»; 38.8....—
EE: .5... a. .5... 5... oh 5.5.9.... .5... so. 8 3:5 2......— Sficafiu
U8.... 9.....6 3.. cages; 2 .8... a 2.22
82...
93.3. a... 8:83 38 «5.3 385 30.32

:5 a 8.8.58... :85. .858 .252

3...»: .6 use: 8.....E> 25m 0...: dB; .853.

.333 .3 as:

87

Chapter 4

HEIFER SELECTION

In order to lower the replacement rate, not all heifers can enter the milking herd.
The models reviewed in Chapter 3 generally assumed random selection of heifers or
selection based on genetic parameters. The failure of producers to accept random selection
may be one reason for the industry’s failure to reduce the culling rate. The discovery of a
method to predicting culling or production of heifers in the first lactation may encourage
producers to select heifers and thereby reduce their culling rate. This chapter will review
the goal of heifer selection, namely profitability, and the parameters available to estimate
animal profit. Methods of selection for profitability will then be reviewed.

The goal of heifer selection is to maximize profitability. Given a choice among
heifers, those with the highest profit are desirable. Unfortunately it is not a simple matter
to assess which animals are most profitable. Absolute profit is not available for any
animals, rather the dairy industry uses profit functions to estimate the profit of a given
animal so the importance of various traits in the explanation of animal profit can be
determined. These profit fiinctions vary in method of estimation and completeness
(Pearson and Miller, 1981). As will be discussed in Chapter 5, these equations are also
used to weight the genetic value of traits used in multiple trait selection indices.

Tigges et al. (1984) found relative net income (RNI), calculated using information
available from dairy herd improvement associations, to be highly correlated with total

profit (r = 0.97) and profit per day of herd life (r = 0.92) calculated using more extensive

88

economic information such as returns from protein yield and cost of mastitis and breeding.
In comparing varying measures of profitability Cassell et al. (1993) found such high
correlations between relative net income and discounted relative net income, the latter was
of little utility.

Aside fiom the issue of the completeness of the profit fimction, the measure of
profitability is a contentious issue. As discussed in Chapter 2, Van Arendonk (1991)
noticed that use of lifetime profit implied no replacement while profit per day implicitly
assumed identical replacement. He noted that each of these ignore the opportunity cost of
replacement and therefore neither is an appropriate measure of animal profitability. Van
Arendonk showed these erroneous assumptions have drastic ramifications on the value
estimates of the two traits, production and herd life (Weigel et al., 1995a), most important
in the determination of profit. He showed that using total lifetime profit as the dependent
variable, the regression coefficient for first lactation milk production was correct, while
using profit per day of productive life as the dependent variable correctly estimated the
herd life coefficient. But in each case the value of herd life relative to production was
greatly overestimated. Weigel et al. (1995b) further refined this point by estimating an
opportunity cost for each lactation, removing the assumption of a constant opportunity
cost.

Considerable confusion exists in the literature over results comparing measures of
profit that don’t include opportunity cost and those that do. Weigel et al. (1995a)
determined the genetic correlation between RN] and relative net income corrected for

opportunity cost (RNIOC) to be 0.97. The phenotypic correlation between RN] and

89

RNIOC has been reported as 0.95 (Cassell et al., 1993), and 0.92 (Weigel et al., 1995a).
Some (Norman et al., 1996) have incorrectly interpreted these results as evidence for
ignoring the opportunity cost adjustment. This despite Cassell et al.’s (1993) conclusion
that remlts of herd management or selection practices using RNI corrected for
Opportunity cost could difi'er from those using RNI. This is not surprising given that the
variance of the RNI has been reported as 2.5 (Cassell et al., 1993), and 3.5 (Weigel et al.,
1995a) times that of the corrected RNI. Correcting the profit increased its heritability to
0.17 from 0.12 (Weigel et al., 1995a). Since the validity of the literature using profit
equations not corrected for opportunity cost is questionable, and studies using the
adjusted equations are available, the results of the former will not be considered.

Another source of confusion exists in the literature concerning Van Arendonk’s
paper (1991). He showed the regression coefiicient between profit not including
opportunity cost, and herd life was overestimated. Since this result was in the context of
multiple regression, including milk production as an independent variable, this implies that
the partial correlation between profit not including opportunity cost and herd life, given
production, was overestimated. It does not necessarily imply that the simple correlation
between profit not including opportunity cost and herd life was overestimated as has been
reported (Cassell et al., 1993).

Given, that lifetime profitability is not apparent until the end of an animal’s life, it is
not a useful method of selection prior to the end of that life. The literature has tended to
focus on indicators of lifetime profitability that are apparent early in the productive life of

milk cows. No work has considered indicators apparent prior to productive life.

90

The phenotypic correlations between first lactation actual milk and total lifetime
milk are 0.54 and 0.88 for RNIOC, and 0.52 and 0.96 for RNI (Cassell, Smith and
Pearson, 1993). First lactation milk production was found to have a 0.86 genetic
correlation, and a 0.57 phenotypic correlation with lifetime production among Canadian
Holsteins (Jairath et al., 1995). Weigel et al. (1995a) determined the genetic correlation
between the RNI corrected for opportunity costs and first lactation ME milk production to
be 0.80, and similar to the previous study, a phenotypic correlation of 0.51. Some physical
traits such as dairy form and udder depth had reasonably high correlations with RNIOC
(Weigel et al., 1995a). Age at first calving has a small negative correlation with RNIOC
(Cassell, Smith and Pearson, 1993). Both Weigel et al. (1995a) and Cassell et al. (1993)
found herd life had a correlation of approximately 0.70 with RNIOC.

Weigel et al. (1995a) considered the reliability of indirect prediction of sires’
RNIOC using type, first lactation yield, and proxies for herd life. Table 4.1 is a partial
reproduction of a table in the paper.

Table 4.1. Approximate reliabilities of RNIOC from combination of mature equivalent

yield of milk and fat during first lactation (yield), 14 linear type traits and final score
(type), and total months in milk to 84 months of age (MIM), and MIM predicted from 36

months of age (PMIM).
Approximate reliability of RNIOC

 

 

Variables Maximum 80 EPN‘
Type .48 .32
Yield .67 .59
Yield and type .75 .64
Type and PMIM .70 .36
Yield and PMIM .7 5 .60
Yield, type and PMIM .80 .65

91

Table 4.1 (cont’d).
Yield and MIM .90 .70
Yield, type and NIIM .91 .72
1. Efl’ective progeny number, represents the accuracy of a first-crop evaluation of a well-sampled AI bull.

 

Type characteristics provide little additional information in the prediction of
RNIOC, and are unsuitable as a method of heifer selection as only animals that have
fieshened are evaluated.

Herd life, or a proxy, is an important variable in the determination of profit.
However, it is unsuitable as a method of heifer selection. Its heritability is low, and while
the USDA is producing productive life genetic evaluations for cows and bulls, heifer
evaluations of productive life are not produced.

To restate, ideally heifer selection would be based on lifetime profit corrected for
the opportunity cost of postponed replacement. While RNIOC can not be measured and is
not recorded, first lactation milk production is widely available and moderately correlated
to RNIOC. This then suggests that selection of the highest producing first lactation heifers
will likely result in selection of the most profitable animals. We then turn to prediction of
first lactation milk production.

Compared to herd life, the yield traits have higher heritability. And studies have
been completed on the ability to predict future production. These studies fall into general
types of predictors - genetic and physiological. As few studies have considered the ability
of genetic estimates fi'om the USDA’s current Animal Model to predict subsequent heifer
performance, performance of estimates fiom the earlier Modified Contemporary

Comparison will also be considered.

92

Dams’ genetic index for milk production from the Modified Contemporary
Comparison was found to have a 0.18 correlation with daughter first lactation ME
production deviated from herdmates. This result was slightly less than the 0.20 expected
(McGilliard and Freeman, 1976). Linear regression of the measure of daughter first
lactation production on dam index was 1.15 which differed from the expected unity.

Powell et al. (1977) determined the average correlation between actual daughter
performance for milk production, deviated from contemporaries, and that predicted fiom
sire pedigree index using the Modified Contemporary Comparison was 0.40 for Holsteins,
slightly below the expected 0.45. The regression coefficient between the two was near the
expected 1.0 suggesting selection based on sire predicted differences should be successfirl.

Powell and Norman (1988) found an average correlation of 0. 19 between the
daughter’s deviation of milk yield from contemporaries and the dams’ genetic index for
milk production from the Modified Contemporary Comparison, which was lower than
expected. The correlation increased as the repeatability of the dams’ indices increased, as
the number of daughter records rose, and with a higher level of herd milk production.
Selection was expected to be effective given a regression coefiicient of approximately one
between daughters’ deviation and the dams’ index.

A problem with each of the preceding studies is that a considerable period of time
was used to collect the data, so parent indices were calculated years after a daughter’s
production. In heifer selection, it is the estimate of the parents’ indices at the time of

selection that is relevant.

93

Since 1989, when the USDA began using the Animal Model for estimating genetic
merit, only two domestic empirical studies have considered the ability of parents’ PTAs to
predict a heifer’s subsequent performance. Estimated genetic indices are produced for all
dairy animals with identified parents who are on a production test program. The indices,
which are updated every six months, and are generally provided at no extra cost to those
enrolled in a production test program.

Other empirical studies (Ferris and Wiggans 1990, 1991; Samuelson and Pearson,
1995) have considered the ability of parents’ PTAs to estimate the performance of
artificial insemination sires, through the performance of the sires’ daughters. Since these
sires tend to have genetically elite parents, with generally high reliabilities, the results of
these studies are not directly applicable to daughter performance.

Based on 76,871 daughters, VanRaden et al. (1989) found the regression
coeficients of daughter performance on EBVs “were near the (theoretically) expected
value of 1.0” with correlations that ranged from 0.20 to 0.49. Performance of daughter a
was measured as yield deviation (YD,) which according to Wiggans and VanRaden (1989)
is defined as

YD. = y, - M. - PE, - (siren-herd“)
such that y represents standardized milk yield; M, PE and sire ~herd are the effects of
management group, permanent environment and herd-sire interaction, respectively for a.
Management group includes effects for herd, year and season of calving, parity and
registration status, as well as their interaction. This equation for YD can be compared with

the USDA animal model of milk yield standardized for season and age at first calving,

94

y, = G, + M, + PE, + (sire, herd.) + U, + e,
where G is EBV, U is unknown parent group, and e is the error for a.

In heifer selection the producer is interested in predicting y,. While the use of YD
as a dependent variable serves to decrease the error variance, the results are not directly
applicable to heifer selection. For a producer is paid based on is unlikely to know the value
of M., PE, (sire, -herd,) and therefore, can not derive I’D, from y,. The prediction of y,,
not 1D,, is of primary importance to the farmer because this is the basis on which the
producer is paid.

On a small sample of 37 Holstein heifers, Mao et al. (1991) measured the serum [3-
lactoglobulin from 16 to 1 week prepartum. They found consistent positive correlations (r
s 0.3 to 0.6) between the serum protein over this 16 week period and first lactation,
second lactation or the sum of the two, ME milk yield. In a subsequent paper (Mao and
Bremel, 1995), the 0.034 coefficient of determination for prediction of first lactation 305d
milk yield (i.e., y,) of 26 Holstein heifers using only parent average of genetic merit for
milk production was improved to 0.211 with the inclusion of the serum B-lactoglobulin
level at the 26th week of pregnancy. Stating the residual (u,) fi'om the Mao and Bremel’s
regression in terms of effects accounted for in the VanRaden study, the lower multiple
coefiicient of determination of the former is expected.

it, = M, + PE, + (sire, -herd,) + U, + e, - herd,

The correlations between 305d milk yield and the log of B-lactoglobulin, and

parent averages for milk, protein and fat were, respectively, 0.456, 0.185, 0.149, and -

0.003. The authors note “The PA (parent average) is normally used as a selection criterion

95

for heifer replacements.” Similar suggestions have been made to sell heifer calves from
inferior parents (Jones and Rogers, 1995; Franck, 1994) however no specific
recommendations as to what method or when to conduct selection is presented.

While physiological measurements present potential methods for heifer selection
they are not routinely collected on heifers, and B-lactoglobulin is not produced until late in
the rearing period when the animal is pregnant.

Weight gain from 350 to 462 days of age had a genetic and phenotypic
correlations of 0.33 and 0.67 with first lactation 308 day milk yield (Lin, McAllister et al.,
1985)

The ability of at least the genetic indices for milk to predict production, which is a
major component of animal profitability, suggests their potential utility in heifer selection.
However, two issues are not resolved. Firstly, there is no proof that the heifers’ milk index
which equally weights the PTA milk of each parent is optimal. Furthermore, given the
potential 5 genetic indices for each parent in each period it is not clear how each of these
should be used. Lastly, the genetic indices are simply estimates which are updated over
time. It is not clear when in a heifer’s life selection should occur. Specifically, which of the
10 pieces of genetic information, available every six months during rearing, should be used
and how they should be used in heifer selection.

There are also other benefits from heifer selection besides increasing herd
profitability. Aside fi'om short-term improvement in cash flow, heifer selection will also

reduce the nutrient loading on the land, this issue will become increasingly important as

96

environmental restrictions become more prevalent and strict (Schmit and Knoblauch,
1995).

The importance of opportunity cost has resurfaced in this chapter. Likely, due to
the industry’s ignoring the work on the optimal culling rate, and therefore the need for
heifer selection, no work has been completed on heifer selection for maximum profit.
From the literature it can be determined that production is important to animal
profitability. Genetic indices for milk have been shown to hold some promise for the
selection of higher producing animals. The current availability and ease of access make
genetic indices a potential method of heifer selection. While the use of genetic indices for
heifer selection has been suggested, details and empirical results to support these

statements are lacking. Chapter 5 will consider the estimation of genetic values of animals.

97

BIBLIOGRAPHY

Cassell, B. G., B. B. Smith, and R. E. Pearson. 1993. Influence of herd-life opportunity
and characteristics of cows and herds on different net income functions. J. Dairy Sci.
76:1182-1190.

Franck, R 1994. Cashing in on your cull rate. Dairy Herd Management 31:57-58.

Ferris, T. A, and G. R “Wiggans. 1990. Accuracy of animal model parent evaluations in
predicting PTA for AI sampled bulls. J. Dairy Sci. 73(Suppl. 1): 141 .(Abstr.)

Ferris, T. A, and G. R VVrggans. 1991. Accuracy of animal model parent evaluations in
predicting Daughter Yield Deviations for AI sampled bulls. J. Dairy Sci. 74(Suppl.
1):265.(Abstr.)

Jairath, L. K., J. F. Hayes, and R. I. Cue. 1995. Correlations between first lactation and
lifetime performance traits of Canadian Holsteins. J. Dairy Sci. 78:438-448.

Jones, L. R, and G. W. Rogers. 1995. Management strategies: Culling. Proc 27th Am
Assoc Bovine Pract 93-96.

Lin, C. Y., A. J. McAllister, and A. J. Lee. 1985. Multitrait estimation of relationships of
first-lactation yields to body weight changes in Holstein heifers. J. Dairy Sci. 6812954-
2963.

Mao, F. C., R. D. Bremel, and M. R. Dentine. 1991. Serum concentrations of the milk

proteins a-lactalbumin and B—lactoglobulin in pregnancy and lactation: Correlations with
milk and fat yields in dairy cattle. J. Dairy Sci. 74:2952-2958.

Mao, F. C., and R. D. Bremel. 1995. Prediction of milk yields from serum B-lactoglobulin
concentrations in pregnant heifers. J. Dairy Sci. 78:291-295.

McGilliard, M. L., and A. B. Freeman. 1976. Predicting daughter milk yield production
from dam index. J. Dairy Sci. 59:1140-1146.

98

Norman, H. D., R L. Powell, J. R. Wright, and R. E. Pearson. 1996. Phenotypic
relationship of yield and type scores from first lactation with herd life and profitability. J.
Dairy Sci. 79:689-701.

Pearson, R E., and R H. Miller. 1981. Economic definition of total performance,
breeding goals, and breeding values for dairy cattle. J. Dairy Sci. 64:857-869.

Powell, R L., and H. D. Norman. 1988. Accuracy of cow indexes according to
repeatability, evaluation, herd yield, and registry statue. J. Dairy Sci. 71 :2232-2240.

Powell, R. L., H. D. Norman, and F. N. Dickinson. 1977. Relationships between bulls’
pedigree indexes and daughter performance in the modified contemporary comparison. J.
Dairy Sci. 60:961-966.

Samuelson, D. J ., and R E. Pearson. 1995. Accuracy of predicting genetic merit from

pedigree information for bulls entering stud sampling programs. J. Dairy Sci. 7822057-
2066.

Schmit, T. M., and W. A. Knoblauch. 1995. The impact of nutrient loading restrictions on
dairy farm profitability. J. Dairy Sci. 78: 1267-1281.

Tigges, R J ., R E. Pearson, and W. E. Vinson. 1984. Use of dairy herd improvement
variables to predict lifetime profitability. J. Dairy Sci. 67: 180-184.

Van Arendonk, J. A. M. 1991. Use of profit equations to determine relative economic
value of dairy herd life and production fiom field data. J. Dairy Sci. 74: 1 101-1 107.

VanRaden, P. M., C. A. Ernst, G. R. Wiggans, and H. D. Norman. 1989. Ability of animal
model and Modified Contemporary Comparison evaluations of parents to predict firture
progeny performance. J. Dairy Sci. 72(Suppl. 1):72.(Abstr.)

Weigel, D. J ., B. G. Cassel, I. Hoeschele, and R. E. Pearson. 1995a. Multiple trait
prediction of transmitting abilities for herd life and estimation of economic weights using
relative net income adjusted for opportunity cost. J. Dairy Sci. 78:639-647.

99

Weigel, D. J., B. G. Cassel, and R E. Pearson. 1995b. Adjustment of a net income
function for opportunity cost of postponed replacement on a lactation basis. J. Dairy Sci.
78:648-654.

Wiggans, G. R, and P. M. VanRaden. 1989. Developments in the USDA animal model
system for genetic evaluation of yield traits. J. Dairy Sci. 72(Suppl. 1):71.(Abstr.)

100

Chapter 5

SELECTION INDEX THEORY

From Chapter 4 it is apparent that genetic evaluations currently present the only
potentially viable methods on which to base selection of heifers prior to calving. A
familiarity with the methods and assumptions made in estimating an animal’s genetic value
is then warranted. This chapter will start with a brief review of genetic efi‘ects and the
assessment of genetic variance and covariance. The goals, methodology, and assumptions
of the Selection Index (81) model for a single trait are examined. The effects of the various
assumptions on the model are explored. Multiple trait selection is then reviewed, followed
by miscellaneous issues which firrther cast doubt on the optimality of SI. By the
conclusion of this chapter it will be apparent that, due to the assumptions made, SI
evaluation of an animal’s value is not optimal, even more critical to heifer selection, it is
not clear that heifers will even be correctly ranked. Based on this inability to correctly rank
heifers for selection, combined with the effect of Mendelian sampling which greatly
increases the prediction error of the index, the value of genetic indices in heifer selection is
questionable.

The approach to genetic evaluation and S1 in this chapter is heavily based on
Selection Index and Introduction to Mixed Model Methods by Van Vleck (1993),
supplemented with relevant information from the literature. Given the context of this
chapter is understanding the estimation of a heifer’s value based on her parents’

information, examples will appropriately focus on such estimations.

101

The theoretical composition of genotype, as well as, the determination of genetic
covariance between individuals is fundamental to understanding selection theory and its
assumptions. So each of these issues will be addressed prior to reviewing selection theory.

The genotype of an animal can be broken down into linear genetic efl’ects and
epistatic or nonlinear genetic interaction efi‘ects. An additive genetic effect is the aggregate
efl‘ect of allelic substitution at a particular locus. The sum of all additive genetic effects
over all loci is the additive genetic value (0,) of the animal, also known as the estimated
breeding value (EBV).

Let A; = 5, A2 = 2 where A; is the 1’ll allele of gene A. If only additive efl‘ects exist,
the phenotypes resulting fi'om each allelic combination are:

AIAI = 5 + 5 = 10
A1A2 = 5 + 2 = 7
AzAz = 2 + 2 = 4

A dominance genetic effect is the average effect of allelic substitution at a
particular locus less the additive genetic effects. The sum of all dominance genetic effects
over all loci is the dominance genetic value (Go) of the animal.

Suppose, similar to the above example, B1 = 5, B2 = 2 but a dominance effect of
+1 occurs then

B1B, = 10
B1B; = 7 + 1 = 8
B2B; = 4

An additive by additive genetic effect is the average effect of substituting a pair of

102

alleles at different loci less the additive genetics effects. The sum of all specific efi‘ects of
non-allelic gene pairs is the additive by additive genetic value (GM) of the animal.
Expanding the first example, let A. = 5, A2 = 2, C; = 1, C2 = 0, and an additive by additive

efi‘ect of +3 occurs when A; and C1 are in common.

 

 

 

 

AIA. ArAz AzAz
C1C| 12+6=18 9+3=12 6
C1C2 11+3=14 8+3=11 5
C2C2 10 7 4

 

 

 

 

 

 

An additive by dominance genetic effect is the average efi‘ect of substituting an
allele at one locus and an allelic pair at another locus. It is calculated as the difference
from the additive, dominance, and additive by additive genetic effects. The sum of all such
effects is the additive by dominance genetic value (GAD) of the animal.

As in the second example, let B1 = 5, B2 = 2, C1 = 1, C2 = 0, with a dominance efl‘ect of
+1 on the B locus, and an additive by dominance effect of -4 when the genotype BlB2C1 is

present.

 

B 181 Ble B232

 

 

 

CIC1 12 10-4=6 6
C1C2 1 1 94:5 5
C2C2 10 8 4

 

 

 

 

 

 

Further higher order epistatic effects can be defined. Given the number of genes

103

and alleles in any mammal’s genome, the number of potential effects quickly approaches
infinity.
The total genetic value of an animal can then be represented by

G=GA+GD+GM+GDD+GAD+GAM+GMD+GADD+HH (5.1)

Selection theory also relies on the genetic variance or covariance. With the
reasonable assumption of independence among the different type of genetic efi'ects, the

genetic variance can be represented by

2 _ 2 _ 2 2 2 2 2 2 2
HG 1-00 “00‘ +000 +00“ +00” +00“ +00” +0.64“) “1"” (5. 2)

where E is the expectation operator, and the genetic value is defined to have zero mean.

Now consider the genetic covariance between individuals 1' and j.

_ _102 012 202 022112
E[G,-Gj] — 06,0, — a”. d” 00.. +a,j (1,1000 «ray. d, 00.“ +a,j d”. 0600a,]. d, 060
(5. 3)

3 0 2 2 1 2 1 2 2
+0” d0 06‘“ +00 d0 OGMD +0011”- OGADD +‘"

where a, = the fraction of additive gene effects in common between individuals 1' and j,
and d.-,- = the fraction of dominance effects in common by descent.
These coefi'rcients are calculated using simple probability theory. Equation (5.4) is the

model of genetic covariance between a parent (P) and an offspring (0).

1 ' 1 ° 1 2 1 ° 1 ‘
0,0,, = (5] 0003,, {3) 0'02,” {5) 0°03,“ {5) 020:,” {5) 0'03,” +

1 3 l 2 l I
(a) °°"3~« {5) ““3- ”“(51 0’03- +°"

Given that the offspring receives half of its genes from each parent there is a (0.5)“

104

probability that the two share n non-allelic genes in common. Notice the fiaction of
dominance effects in common by descent is zero between this pair. In fact this null
coeficient for a biallelic locus is used for all but firll siblings for whom it is 0.25. Given
that at least one parent has a heterozygous genotype and exhibits dominance efi‘ects, the
probability of an ofi'spring having the same genotype at this locus is 0.50 not zero,
regardless of the other parent’s genotype.

The unexpected null coemcient is the result of the model used to determine the
relationship between individuals. These coefficients consider only genes in common which
source from a common ancestor. This is certainly a reasonable assumption for additive
genes which are simply linear interactions. The null coefficient is appropriate when using
the identical by descent criteria, since heterozygosity is required for dominance efi‘ects and
one allele of each locus is transmitted fiom parent to offspring there is no opportunity for
the dominance efi‘ect to be identical by descent. Considering the purpose of Equation (5.3)
is to quantify the similarity of the genetic effects of the two individuals, regardless of the
source of that similarity, ignoring the identical in state genes, those not fi'om a common
ancestor, misrepresents the true composition of genetic covariance and hence the interest
in DNA sequencing.

It is also clear that as the number of non-allelic genes and/or allelic pairs increases
the coefficients decrease in magnitude. This is the frequently cited reason for routinely
ignoring any higher order epistatic effects. However, in determining whether a right hand
side element of Equation (5.3) is so small as to be ignored, the variance of the genetic

effect itself should also be considered.

105

The result of this assumption of no interaction effects of order two or higher, and

the assumption of identical by descent is to reduce Equation (5.4) to

1
0006,. = 5031,. (5. 5)

This simplification is used for all but full sib genetic covariances for whom a dominance
efl‘ect may be included.

Reliable estimation of variance/covariance components for genetic parameters is
one of the major tasks for any livestock improvement scheme. Maximum likelihood is a
method of estimation that empirically results in asymptotically unbiased estimates in the
face of selection (Rothschild, Henderson, and Quaas, 1978; Schaeffer, and Song, 1978). A
major drawback of this method is the variance estimates of a mixed model are biased since
the fixed effects are treated as known, resulting in overestimation of the degrees of
fi’eedom. According to Meyer (1983) and Harville (1972) restricted maximum likelihood
(REML) overcomes this problem by maximizing only that part of the likelihood that is
independent of the fixed effects. Empirically REML also appears to remove any selection
bias from the estimates (Meyer, 1983). In the genetics literature it is common to ignore the
potentially complex distributional properties of the estimator. This may be due to the work
by Harville (1972) where it was suggested that the REML estimates derived under
normality may be reasonable estimators even when the form of the distribution of the
estimate is unspecified.

The REML covariance estimate is then set equal to the product of the additive

genetic variance and the appropriate a, The result is a potentially exaggerated additive

106

genetic variance for all relationships, since all the elements of the right hand side of
Equation (5.3) are allocated to either the additive genetic variance or the residual term.

Tempelman and Burnside (1990) showed that this exaggeration can be sizable.
Their study revealed the identical in state dominance variation for Canadian Holsteins’ first
lactation milk and fat were, respectively, 15 and 75 percent of the additive genetic
variance. Despite similar standard errors for each of the estimates, the former was not
statistically significant due to a high coefficient of variation (i.e., standard error divided by
the mean).

A second Canadian paper (McAllister, Lee, Batra, et al., 1994) indirectly
investigated the presence of dominance effects between dairy breeds. In acceptance with
previous studies, this experiment showed a significant heterotic effect in Holstein x
Aryshire F; females. Heterosis is defined as the realized phenotype less the expected
phenotype calculated as the average additive effect of the two breeds. The necessary
conditions for a heterotic or crossbreeding effect are the presence of dominance efi‘ects
and difi’erent gene fiequency between the two breeds.

This paper is of interest for two reasons. Firstly, while heterotic effects were
generally accepted to occur, conventional wisdom among animal breeders held that the
dominance efi‘ects of crossbreeding were insufficient to compensate for the much lower
production of the Aryshire parent. As a result it was accepted that, despite heterosis, the
F; would have lower production than the higher producing Holstein parent. This study
found no significant difference between the Holstein x Aryshire F; female and the purebred

Holstein line for lifetime milk production. This suggests the presence of a much stronger

107

dominance efi'ect than generally accepted by the industry. This paper also gave evidence of
how heterosis may vary with the period over which a given trait is considered, thereby
bringing into question the ability to extrapolate the aforementioned first lactation results of
Tempelman and Burnside to production in general. For example the heterosis effect was
3.7% for first lactation yield and 16.5% for lifetime milk production. Lifetime heterosis
efl'ects were 20.0, 17.2, 17.9 and 20.6% for lifetime fat yield, protein yield, milk value and
annualized discounted returns.

Clearly it is extremely difficult if not impossible to estimate dominance efi‘ects
remlting fi'om identical in state genes. Estimation of other variance effects would require
supplemental equations to solve for the extra parameters. It is only recently that inroads
have been made in the inclusion of dominance effects in genetic evaluations (Misztal,
1997). Furthermore, the number of equations required to estimate higher order genetic
variance components is overwhelming. This model, as all others is a tractable
simplification of reality, and should not be judged on its realism. As with any model it
should be judged on its performance, this may include theoretical results or the ability of
the model to mirror reality. It is in this assessment of the model that its assumptions
become critical. For failing to recognize the assumptions, and their effects on the results,
leads to the model being accepted as reality, and therefore beyond reproach.

Given these theoretical models, the fundamental challenge in selection is to choose
those individuals with the greatest genetic merit for the trait(s) of choice to be used as

parents of the next generation (110.), i.e.,

maxAG=ug.-uq (5.6)

108

where Ila is the average genetics of the total population.

This challenge is heightened by the fact that rarely can the genetic effects be
directly observed, and this is especially common in economically important production
traits. Rather the breeder must attempt to deduce the genetic value of the animal when

only phenotype (P) is apparent.

Pa. 511+G, +PE, +7120 (5. 7)

Where P, = the phenotype of individual i in period j
p = the population mean
G; = the total genetic value of r' deviated from the population mean
PE; = the permanent environmental effect on P, deviated from the population
mean
7E,- = the temporary environmental effect of period j on P.- deviated fi'om the
population mean.

E,- the total environmental effect on P,- is equal to the sum of W.,-and PE,
Heritability in the narrow (h’) and the broad (hf) sense can now be defined, the former is

the most common definition,

 

O
h’ = a"; (5.8)
P
2 0%
h, z? (5.9)
P

where a}. is the phenotypic variance in the population, and is easily estimated.

109

Notice that overestimation of the additive genetic variance results in overestimation of
narrow heritability.

Armed with an understanding of the various genetic effects and an understanding
of the assumptions made in estimating genetic covariance we can proceed to review
selection theory. Two models have been used for animal selection: Selection Index (SI)
and the Animal Model, the latter is the current method of genetic analysis used by the
USDA. Hazel et al. (1994) and Philipsson et al. (1994) provide reviews of the past,
present and future of SI. SI employs all the same assumptions as the Animal Model. For
81 all phenotypic and genetic variances and heritabilities, genetic and phenotypic
covariances for using multiple traits are assumed known (V andepitte, and Hazel, 1977).
The former also requires all fixed factors be known. Fixed factors include all those
nongenetic factors, excluding temporary and permanent environmental effects, which
afl'ect phenotype. Fixed factors, represented by p in Equation (5.7), include effects due to
year and age of the animal. Despite this one difference in the practical application of the
models, all the theoretical assumptions and results of each model are identical. Since these
assumptions are more explicit in the SI, it will be the method reviewed here.

The goals of the SI are:

1) Minimize the prediction error variance or E[T-l]2 where T is the true value and

I is the index. This is equivalent to maximizing m, the correlation between the true

value and its prediction.

2) Maximize the probability of correctly ranking animals.

3) Maximize the true value of the selected group.

110

In the development of selection theory the main emphasis, rightly so, was on goals 2 and
3. However, the solution to even a simple formulation of this problem was mathematically
intractable without the assumption that the P., the phenotype of individual i, and T must
have a bivariate normal distribution (Williams, 1962). Under these conditions, goals two
and three can be maximized through maximizing r77. Clearly selection will result in a
truncated distribution if the r17 > 0, and as a result the normality assumption will be
violated to some degree, and goals 2 and 3 may not be realized. Potential violation of this
normality assumption is not a trivial problem, fi'om the perspective of individual animal
breeders it is the correct parametric ranking of animals which is of importance, and will
permit maximum genetic progress. Under certain conditions the SI indices are unbiased
when the population consists of individuals who were themselves selected (Gomnet,
1983).

Geneticists have considered the effect of ignoring the changing population
parameters as selection occurs. This change is frequently ignored, and the selection
coemcients are not updated as the population changes. Under conditions of an
infinitesimal genetic model and an infinite population size, the effect of updating the
selection coefficients was found to be negligible with less than a 1.5% loss of aggregate
genetic progress per generation (Villanueva and Kennedy, 1993).

The only data available to predict an animal’s true genetic value is the phenotype
of relatives, and perhaps its own phenotype. By definition, relatives are expected to have
more genes in common with the individual than the population at large. The general linear

selection index model for individual a (1,) can then be expressed as dependent on the

111

phenotypes of a number of animals (X 1. X 2, XN ), where a is not necessarily a member

ofN.
Ia=b1X1+ b2X2+ . . . +bNXN (5. 10)

WhereX;=Pi'fl.

All that remains then is to determine the values of the bs.

an
max r = (5. 11)
1511:; n Jafiaf

A logarithmic transformation results in a linear objective function.

1 1
{“123} 108021 ) = 108(011) ’ 3103(0% ) " 3198(012 ) (5- 12)
1 -1
where BUY] = or, = bran. + b207,“ +- - -+ bNO'rxN (5. l3)

2 _ 2 2 2 2

(5. l4)
‘1'”“1‘ ”szaxsz +"'+ b13032“
This unconstrained maximization problem is solved by taking the first derivative of the

objective function with respect to each b, recognizing that the variance of T is independent
of the current bs. (Notice, however, that a; is a firnction of the index, assuming the index

is at all usefiil (rn>0) and has been used for selection, the variance of T is then dependent

on previous bs.)

2
6108017) = 0W1 _b‘axr +b20X1X2 +.+bNaX1XN

2 =0
($1 an a,

 

 

112

2
610g(rn) _ 01x, _ bro'xpr2 +b20'x, +"'*’bN‘7x,xN

2
‘92 an 01

 

 

= o (5. 15)

 

2
6log(r,,)_ 01x, _ b10331,” +b2‘7.1',.r,., *""'bN0xN

2 :0
&N on 01

 

Redefining this system of simultaneous equations in standard linear notation, with the

primed symbol designating a transpose, reveals
Pb = Ice (5. 16)

where
P = variance - covariance matrix of the X ,. = E(XX')

X=[X, X2 XNI
b‘lbr b2 bNI

2

k = thescalargl—
01']

c = covariance vector of T and X ,-

Since the scalar It does not affect the relative magnitude of b, it can arbitrarily be set equal

to unity. Equation (5.10) can then be restated in linear notation as
'i = I = b'X (s. 17)
with
b = We (5. 18)

Since the b vector is a fiinction of P and c it is important to consider how each of these
matrices are estimated. Once an understanding of the assumptions made in estimating b

are gained, we will turn to actual use of an index. First derivation of e, the covariance

113

between the true value and the phenotypes, is reviewed. This covariance is a function of
the desired true value which is being selected for. The most commonly used true value in
the dairy cattle industry is estimated breeding value (EBV) or the additive genetic efi‘ect of

individual a, then the first element of c is
on. =00“, (5.19)
=E[GA.X,]
=E[GAOGI]+E[GAUE,]
=E[GA,GA, +0400. +GA,GAA. +04,qu +GA,GDD, +GA.GAAA;

+GAaGMDl +041qu +"°]+E[GAnE-l]

07x. = 0.03;, (s. 20)

where an, is defined analogously to that used in Equation (5.3).

In arriving at the final expression of Equation (5.20), the phenotype of individual 1
has been decomposed into a genetic and environmental effect. Using the results of
Equations (5.2) and (5.3), the covariance of the true breeding value and 1’s genetic efi‘ect
is estimated to be represented by Equation (5.20). Since the strong assumption of zero
expected additive genetic-environmental effect covariance was made, the genetic
covariance equals the covariance between the EBV and X ;. Clearly this later assumption
abstracts fi'om reality. It is not hard to imagine additive genetic-environmental interaction.
Intuitively, consider a genetic trait such as nervousness or shyness, individuals possessing

such a trait will perform better with a quiet gentle manager (environment) than not. The

114

independence between the genetics and environment is then refined and furthermore, a
role of management is to capitalize on this potential interaction. Preferential treatment,
which will be discussed more later in the chapter, can be considered selective exploitation
of a environment-genetic interaction. While a model to account for this genetic-
environment covariance has been developed by Schaeffer and Henderson (1983), it is not
employed by the USDA

Even ignoring the assumption of zero additive genetic-environmental covariance,
Equation (5 .20) does not accurately represent the covariance between the additive genetic
value of a and the genotype of individual 1 for the desired trait. Recall that earlier in the
discussion of estimation of genetic covariance it was shown that the additive genetic
covariance was overestimated because, it in effect, included some genetic covariance other
than the additive portion. Tempelman and Burnside (1991) showed that omitting the
dominance portion from an animal model resulted in a 4% overestimation of the additive
variance for milk. If the goal of selection is to maximize progress in the additive genetic
effect of a trait, the net effect on c is ambiguous, since ignoring the genetic-environment
covariance underestimates c, while the additive genetic covariance is slightly
overestimated.

The diagonal elements of P are phenotypic variances while the remaining elements
are covariances. Each type of element may be calculated using individual records or the
average of multiple records. The estimation of each will be assessed in turn, starting with
estimations using single records.

Since the phenotypic variance of an individuals single record is the expectation of

115

the population variance, the variance elements of P based on single records per individual
are readily available from the population being studied and therefore no extra estimation is
required for SI. However, this assumes homogenous variance among all lactations. Van
der Werf et al. showed that ignoring variance heterogeneity resulted in sire evaluations for
milk based on progeny records being 157 kg lower than that expected based the sires’
EBVs.

The nondiagonal elements of P estimated fi'om the data are again unbiased, and are
represented by Equation (5.21). However, as can be seen fiom (5.22), in SI the
assumptions made in the estimation of the non-diagonal elements of P are similar to those
used in determining c. Specifically, SI assumes that the only source of phenotypic
covariance is additive genetic variance, and as a result the additive genetic variance is
overestimated.

am, = 1=.[(Gl + 5,)(G, + E,)]

(5. 21)
= ElG,G, + 6.1:, + (3,15, + 5,15,]

=a120é‘ (5.22)

Equation (5.22) then does not accurately reflect the additive genetic covariance.
Also, many strong assumptions are made concerning the environmental interactions. It; as
argued previously, the environment of an animal may interact with its genotype, given that
the genotypes of the animals are related, it would not be surprising to find that the
genotype of one animal covaries with the environment of the other. The assumption of no

covariance between the environmental effects of the two animals may also be strong.

116

These assumptions will result in the underestimation of these non-diagonal elements of P.

The elements of P may themselves be an average if multiple records are available.
The source of these multiple records include an individual with more than a single record,
or records from a homogenous group of relatives (i.e., 20 daughters all with comparable
single records). Taking the first example of multiple records on an individual animal, the
phenotypic variance of this individual is then actually the squared standard error (i.e. the
variance of a mean), which will be less than the variance of a single record. The initial
assumptions made in estimating the standard error are: the variance of each record is the

same, and the covariance between each pair of records is identical.

2
_ noJr +n(n-1)ax‘x‘. _ oi. +(n-l)a'x‘x‘.

 

 

a}! n, n (5. 23)
The covariance between records of a single animal, X.-, can be represented as
a”; = 1~:[(G, + PE, + 715,, )(G, + PE, + 773,. )]
(5. 24)

_ 2 2
- 0c; + 0P5 + 01150225,, +0073, + ‘7va +0012, +OPETE0 + 20'6”
Making the strong assumption of zero covariance among all terms leads to

2 2 2
axx;=aG+OPE-=-0A (5.25)

I

Where A, the animal effect, is the sum of all permanent effects on the animal (i.e., G +

PE). While Equation (5.23) can then accurately be denoted as

2 _ 2
a}! = ”X +0; 00, (s. 26)

117

geneticists commonly employ the concept of repeatability (R) in this calculation where

 

2 2 2
RE 200 :O'PE 2 :9? (5.27)
By ignoring the interaction effects in Equation (5.24), Equation (5.23) is then
underestimated through the use of
2 02. + (n —1)Ra2
a - = ’1 X (5. 28)

 

X, n

When X.- is the average of a single group of equally related relatives then Equation

(5.24) becomes

on = Elm.- + PE. + 75,)(0, + m. + 7E7 )1

(5.29)
= 00,0, + 0mm + “751175;; + ”OWE: 1" 061717; + 09517!!!

‘1' Gama, + 06,175,, + 0' 135,115,,
The strong assumption that all terms, other than the first, are zero is generally made. Note
this implies no environmental covariance even though animals may be in the same herd
(i.e., environment). The inclusion of herd in the model would account for this
environmental covariance. Given that genetic-environmental interactions occur for a given
animal, that the environments of two animals aren’t independent, and that genetic
covariance exists it logically follows that the genetic-environmental interactions in
Equation (5.29) are not zero either. The assumption of importance only of additive genetic

effects is invoked to arrive at

a —. = 0-03;" (5. 30)

118

It is then apparent that P , is an unbiased estimate of P. However, in using these
estimates of phenotypic variance, SI makes simplifying assumptions. Given the bias in E ,
the estimate of c, and that many of the elements of P are not equivalent to that required

for SI theory, 5 is not equal to the optimal b, however the degree of inoptimality is

indeterminant. Progress in selection will then be less than potentially possible, but more
importantly, it is likely that the level of bias in the relative values of the I; s (the elements
of b) will vary across individuals so the paramount rankings of individuals may be
incorrect and less reliable than suggested by 81 theory.

Having assessed the RHS and LHS of Equation (5.16), b for EBV can now be
derived. Consider estimating the EBV of a heifer (H) for milk production given the single
milk production record of her dam (D) and the effect of the sire. Since the sire cannot
produce milk, his effect is represented by his relatives. For this simple example, we
consider that the sire has a daughter who has completed a record. This daughter is a half
sib (S) to H, as they are assumed to have different dams. This example can be extended to
consider multiple records on multiple relatives of each of H’s parents. The index can then
be represented as

1,, = [11,, b31231 (5.31)

From Equation (5.16)

Pb

II
a

where

119

2
0 0x
0.502
c = G;
02500,,

The oE-diagonal elements of P are zero since the dam and half sib, are assumed to be
unrelated (i.e. aDS= 0). From Equation (5.20), the elements of c are expressions of the
additive genetic variance weighted by one half for D as the heifer receives halfof her
genes fi'om each parent. The weight for S is one quarter, reflecting that she and H have

difl'erent dams.

b = P"c
=[af o I 0.50;, H 0.512] (5.32)
o a}? 0.2503,, 0.25};2

Upon reviewing the definition of heritability in Equation (5.8), it is not surprising that the
bs will always be firnctions of the heritability of the trait (in this case, milk production)
since P is the phenotypic variance-covariance matrix, while c contains the additive genetic
variance. Equation (5.33) represents the more general index coefficients for a parent, as
derived by Van Vleck, where n is the number of records of the parent, and R is
repeatability.

0th2

bl)... = _—1 + (n _1)R (5. 33)

The relative size of b ; is proportional to the strength of the relationship between
the individual being indexed and individual 1. Ceteris paribus, the size of b; will also

increase as the number of 1 ’5 records increase. In reality, each parent would likely have

120

more than a single record so the bs would vary among parents. In general, sires would be
expected to have more records and therefore contribute more information to their
daughters’ EBV.

The estimation of the average additive genetic value that will be transmitted to
progeny is widely used in the dairy industry. Every six months the USDA calculates the
equivalent of bDXD, and bsXs and labels them predicted transmitting abilities (PTAs). In
other words, PTAs are one halfthe estimated additive value of an individual. So to
determine the milk EBV of a heifer all that is required is to sum the PTA milk of the dam
and sire. PT As are routinely published for pounds of milk, fat, and protein, as well as,
dollar value of milk and fat yield in cheese production.

The EBV of an animal is an estimate of the mean additive genetic value. We are,
therefore, not only interested in the variance of that estimate, but as well, its accuracy, or
the degree of correlation between the estimate and the true value.

Since X is a random variable, I is also a random variable with a variance. Recalling
the linear notation for I used in Equation (5.17) and the definitions of P and c fi'om

Equation (5.16),

a} = E[b'XX'b] = b'E[XX')1 = b'Pb (5. 34)

a, = E[Tb'X] = b'E[TX] = b'c (5. 35)

As a result the variance of the index and the covariance between the trait and index can be

shown to be equal.

a} = b'Pb = c'P"PP"c = c'P"c = b'c = a, (5. 36)

121

0'1 b'c b'c

r17 = —-—I— : —— = —
2 2 2
Jena, Jb'ca} 0r

However, since b at b,&,2 at a}, and 6,, at an and therefore, FT, t r”. The variance of I

(5. 37)

and the covariance of IT are also firnctions of h), the relationship weighting and the
additive variance, while accuracy is a function of h’. For a given trait, heritability is
constant, as is the additive genetic variation. Accuracy is then a function of the strength of
the relationship between the 1 and a, with increased records helping to increase accuracy.
As a pedagogical device in deriving the accuracy of the heifer’s EBV index, it will be
assumed that sires have records. Then the accuracy of the heifer’s EBV index based on her

parents’ records is

 

 

 

[ 0.5nDh2 + 0.5nsh2 05022;,
b'c 1+(nD -I)RD 1+(ns -1)RS : 0.50é“

a, 05,,
(5. 38)

 

 

 

1 21th 113112
=;/Z[1+(n,_1)n, ’ 1+(n,_1)1e,]
where n, and R, are respectively the number of records and repeatability of parent j. The
previously determined values of the b and e vectors were inserted into Equation (5.37),
and since the true value being estimated is additive genetic value, Esz J: E161]: aé‘ .
Notice that Equation (5.38) can be explicitly stated as a firnction of the accuracy of

the parents’ own estimated additive genetic value. The index to predict parent j’s additive

genetic value using only its own records is simply twice its PTA or b,. And the equivalent

122

of Equation (5.37) to determine the accuracy of j’s PTA is

 

 

 

 

 

, A A 06.1 2
r7!) = 23 = ”1"!” I = "I” - (s. 39)
a} 0% 1+(nj —11Rj

where r}, is the accuracy of parent j’s own additive genetic value.
The coeficient on e is one since, in the absence of inbreeding, this is the relationship of an
animal to itself.

Equation (5.38) is then equivalent to

1 2 2
r, = J?” + r}; ) (5. 4o)

 

As will be shown, while sires don’t have phenotypic records, their squared accuracies are
widely reported and available.

The accuracy of a progeny’s record is considerably less than that of her parents’.
Even if the parents’ values are known with certainty, the EBV of progeny would explain
only 50% of the variation in the animals’ true additive genetic value. This perhaps
surprising finding is the result of Mendelian sampling. Even if the genetic effects of the
parents are known with absolute accuracy and certainty, a progeny is the summed effects
of its entire genome. In the production of gametes a distribution of genotypes is produced,
with a mean equal to the respective parents’ genotype.

Now consider the prediction of this daughter’s first record. Returning to Equation
(5.18), since the previous relatives, in this example the dam and half sib, are still the only

‘ individuals upon which the index is based the phenotypic variance-covariance matrix P and

123

its inverse will be the same as that used in the estimation of EBV. e represents the
covariance between each relative and the trait of interest, namely the heifer’s phenotype.

axx
c=[ ””] (5.41)

axaxu

Making the standard selection theory assumptions of:
-zero expected covariance between genetic efl‘ects;
-only the expected genetic variance due to the additive portion is non zero;
-no expected genetic-environmental covariance;
-no environmental covariance despite the fact that heifer and dam may be in the
same herd;
-if either parent has multiple records no covariance between animal efi‘ects and TE,
and zero TE variance;
the first element of c in Equation (5.41) can be expanded as
axnxh = EKG,” + PE, + 7E, XGAD + PED + IE,)]

(5. 42)

= 0.503%
The elements of I"l and c, and therefore also b, are all the same for estimation of a heifer’s
EBV or next record. This is the result of the assumptions made as well as the inability to
predict the random variable of environmental effects. Since the indices are identical so are
their variances.

Now consider the variances of the true values. The variance of EBV“ is simmy

- the additive genetic variance, while for the first milk record it is the phenotypic variance.

124

The ratio of these two variances is then the heritability of milk production. The USDA
uses a heritability of milk production for all lactations of 0.25 (personal communication,

George Wiggans, Animal Improvement Programs Laboratory, Agricultural Research

 

 

Center, USDA).
0.5}1203;
’mvrd... 0.3 “G: J}?
r _ 2 2 - 2 .-
BV‘. 0.5h 06 0x
CG:

(5. 43)

fawn,“ = ‘1 hiilk rBV = V 0.25))”, = 0.50fBV“

So while the accuracy of estimating an offspring’s additive genetic value was low,
the accuracy of estimating a heifer’s first record is only half as large.

Had the assumptions made in calculating P and c held true, I would have been an
unbiased estimate of T, but the index would still have prediction error. The variance in the

true value given a particular index, or the prediction error is
”414,317; (5.44)

The USDA calculates reliability of PTAs which is defined as the squared
correlation of an animals predicted transmitting ability and its true transmitting ability
variance (Van Raden and Wiggans, 1991). Reliability is then equivalent to our previous

definition of accuracy (see Equation (5.38)) squared. The confidence interval for an index

 

is then \[1 - Reliability * a "' Z , combining this with Equation (5.40) the confidence

. interval can be expressed as a function of parent reliabilities

125

 

 

 

J 4 - Rehabtht YD._m - Relrabrlrt >11; 4 a 4 z , where o' is the population standard deviation

4
of the index, and Z is the appropriate standard normal probability for the specified
confidence interval width.

Table 5.1 presents for EBV milk and first lactation production indices the 95%
confidence intervals of the indices based on various combinations of parents’
repeatabilities of milk PTA These combinations of reliabilities are what can be expected in
the population. The mean of all indices is 20,000 lb.

Table 5.1. Estimated breeding value, and first lactation production 95% confidence
intervals for the range of parent reliabilities of milk production genetic value.
1 :

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Milk Production (lb) 20,000 1 ;
I > 1
Repeatability 95% Confidence htervals
Breeding Value Milk 1st Lactation Production
Sire Darn Within Herd l , Between Herds
Variance‘: 8,880,400 Variance”: 12,882,870] éVariance‘: 18,404,100
0.99 0.6 15,466 24,534 14,539 25,461 13,473 26,527
0.8 0.6 15,291 24,709 14,328 25,672 1 13,221 26,779
0.6 0.6 15,113 24,887 14,114 25,886 1 f 12,965 27,035
0.4 0.6 14,942 25,058 13,908 26,092 12,718 27,282
0.99 0.5 15,373 24,627 14,427 25,573 ' 13,339 26,661
0.8 1 0.5 15,201 24,799 14,220 25,780 _ 13,092 26,908
0.6 0.5 15,027 24,973 14,010 25,990 1 12,841 1 27,159
0.4 0.5 14,858 25,142 13,807 26,193 1 12,598 1 27,402
0.99 0.4 15,282 24,718 14,317 25,683 13,208 26,792
0.8 0.4 15,113 24,887 14,114 25,886 12,965 i 27,035
0. 6 0. 4 14, 942 25, 058 13, 908 26,092 12, 718 i 27, 282
0. 4 0. 4 14, 776 25, 224 13,708 26,292 12,479 1 27,521

 

 

1) From INTERBULL Centre Routine Genetic Evaluation, August 1996. Table 2. Estimated sire standard
deviation and genetic correlations between countries considered 1n the INTERBULL Centre evaluation of
August 1996. The sire standard deviation multiplied by four results in the genetic standard deviation
(personal communication, George Wiggans, Animal Improvement Programs Laboratory, Agricultural
Research Center, USDA).

2) Within herd variance assumed to be 70% of between herd variance.

3) Jairath et al. (1995) phenotypic variation first lact production 1950 kg with a mean of5155.

or Weigel et al. phenotypic variance first lactation milk ME 1686 kg mean 8500.

The phenotypic variance of milk production in the population was taken from the

126

literature. Weigel et al.’s (1995) estimate of the variance of first lactation ME milk was
13,758,165 lb’. Notice that the genetic variance is only 48% of the phenotypic variance,
while a heritability of 0.25 implies that it should be 25%. The confidence intervals reported
may then be too narrow.

Even the most accurate confidence intervals are quite wide. With the range of the
expected or mean EBV milk being :t 1,000 lb in the population there will be considerable
overlap of the prediction intervals for all animals in the population. While the intervals do
narrow as the reliabilities of the parents increase, given the still considerable width of the
interval for the progeny of even the most reliable parents, it is not clear whether this
increase is practically significant.

Despite SI or the equivalent animal model evaluations being theoretically superior
to previous methods of predicting genetic merit, this did not stop Samuelson and Pearson
(1995) from recently investigating this very issue. They explored whether the daughter
yield deviation and EBV of young A] bulls were more accurately predicted through the
use of the parents’ PTAs (i.e. EBV as defined earlier in the chapter), use of sire PTA, sire
PTA and dam PTA or a pedigree index based on the PTA of the young sires’ sire and
maternal grandsire. EBV was found to explain the most variation in both the young sires’
PTA based on daughter information, as well as, his daughters’ actual yield deviations.
Although, the regression coeflicients were less than the expected 0.5, with sire coefiicients
generally slightly greater than dam coefficients, and “accuracy of prediction appeared to be
declining erratically with time”. They also investigated the change in the EBV over 3

periods prior to the realization of daughter information. Over this time, the R2 between

127

EBV and young sires’ PTA and daughter yield deviation increased, with the increase being
due to improving accuracy of the dams’ PTAs.

The genetic improvement per year under normality and other assumptions is
AG/ yr = (”fwd/L (5. 45)

Where D is a measure of selection intensity and L is the generation interval.

The rate of genetic gain has consistently been less than theoretically possible and
expected using SI (Van Tassel], and Van Vleck, 1991). While numerous reasons for this
shortfall have suggested, incorrect assumptions in the genetic models has not been
included. Given the assumption violations, resulting in unoptimal indices, it is not
surprising that fiequently actual genetic progress lags that which is expected from the
model. However, the method of calculating the gain must also be questioned. From
Equation (5.37), the estimated m is not equal to the true correlation, and this is assuming

correct estimation of of. The inverted variance of the true value will likely be unbiased if

the trait is observable, while if unobservable, such as additive genetic effect, this estimate
may be biased as well. Inaccurate estimation of population parameters can lead to high
overprediction of expected gains and loss in the efficiency of the index (V illanueva, and
Kennedy, 1993; Hill, and Myer, 1984). The appropriateness of Equation (5.45) as a
measure of genetic improvement is then unclear.

Until now only selection for a single trait using that single trait has been
considered. Selection using multiple traits will now be briefly reviewed. Selection for

multiple traits presents an analogous problem to that encountered previously, but now we

128

are concerned not only with estimating the weights for different individuals, but also, how
to weight the different traits in the index. Again Van Vleck’s development of the solution
is followed.
Let g = vector of genetic values of m economically important traits for animal a,

v = vector of economic values for the traits in g,

G = E(gg') = E(Xg') = the genetic variance-covariance matrix for m traits,

Consider the derivation of m single trait index equations analogous to Equation (5.17),
'1‘J = I J = bjx (5. 46)

If T1 is additive genetic values, g ,- can replace c in Equation (5.18), where

E[X(genetic value of trait j)] = g j.

Equation (5.18) can then be restated as
bJ = P“gJ (5. 47)
wherej = 1, ..., 111.

Notice that b; = (P"g,) = g’,(P") = g3(P')" = ggP'l , recalling since it is a symmetric
matrix P' = P.

The overall index (I) is the sum of the I, weighted by v,

 

 

11 bi ( Sip-i 1
I b' r -I
I=v f =v ,1 sz' 311: X=v'GP"X=B'X (5. 48)
1.. b2. 18:3“;
. Van Vleck shows that

129

 

a} = v'Gv
r _ ’B'Gv
Tl V'Gv
r = v'g (5. 49)

The true value is now expressed in economic units, while only relative economic value of
the traits is required for selection. Applying Equation (5.45), if progress (AT) is to be

measured in dollars, absolute economic values are required.
AT = rnDoT = Do, (5. 50)

The response of a trait j, whether included in the index or not, to selection can be

measured, by regressing G, on I.

 

 

a 0
AG, = “foo, = “”19 (5.51)
01 a,
where
a = G.,BX+---+BX
0,) E( j 1 1 N N) (5.52)

= 310616. + 820-616) '1" ' .BNOGIGN

As portrayed in Equation (5.53), selection can still be practiced for a single trait by

setting the economic value of all other traits equal to zero.

ifv, =0Vi¢j,thenlet v}. =1
(5. 53)
06,1 = 1100p, +bj'2acp, +'”b,NUGJG,,

The use of other traits will result in more progress being made than if selection was

130

conducted using only the trait of interest. This incremental progress is a firnction of the
heritability of the other traits as well as the genetic correlation between the trait of interest

and the other traits

Intuitively, in order to make progress in selection, variance is required in the trait
of interest, such that individuals can be difi‘erentiated and selection can occur. Ifother
traits are heritable and genetically correlated with the trait of interest, they can aid in the
assessment of an individual’s genetic value of the trait of interest and therefore aid
selection. As shown in Table 5.2, first lactation dairy production traits of interest tend to
be highly correlated.

Table 5.2. Heritabilities (on diagonal), phenotypic (above diagonal) and genetic (below
diagonal) correlations among first lactation yield traits (Jairath et al., 1995).

 

 

Trait Milk Fat Protein
Milk .25 .95 .98
Fat .89 .23 .97
Protein .96 .92 .22

 

 

 

To make economic progress through SI, the economic weights must be equi-
proportional to their net marginal value per unit per trait. To measure selection progress in
actual dollar units the absolute values are required. Generally the national per unit price of
a trait (i.e., Slpound of milk) is used as the respective weight. Prices resulting from
perfectly competitive markets meet the above requirements for measurement of selection
progress. For in such markets, price equals the marginal cost of production, with the latter

1 including an allowance for “normal profit”.

131

The criteria for a perfectly competitive market are:

1) complete markets;

2) more is preferred to less;

3) numerous consumers and producers;

4) no increasing returns to scale;

5) no externalities;

6) perfect information;

The market for raw milk meets the first two criteria, the last four are questionable.
Firstly, there is concern in the dairy industry that the dairy processors, who purchase the
milk from dairy producers, are few in number and possess potential market power.
Recently Kraft Foods Inc. has come under investigation for manipulation of the exchange
price of cheddar cheese. This price forms the basis of that received by farmers for their
product.

A topic of much debate among the dairy industry has been whether increasing
returns to scale exist. Increasing returns to scale implies that marginal costs are
decreasing. The literature generally does not reject the null hypothesis that the industry
possesses constant returns to scale. Continued nation wide expansion trends among
producers appear to support the concept of increasing returns to scale.

Extemalities are prevalent in agriculture in general. Extemalities may be defined as
production by a firm affecting the preferences of a consumer or vice versa. It is important
to recognize that this is a direct effect, not one transmitted through the price mechanism.

Examples of exterrialities in dairy production are neighbor displeasure with the smell of

132

manure, and animal welfare concerns.

Lastly, for this market to be perfectly competitive all market participants must be
omniscient. Consumers must be fully aware of the value of dairy products and all others,
and producers must have perfect knowledge about their own operations as well as that of
others. Given that a significant number of producers are not even aware of their cost of
production per hundred weight of milk, much less their marginal cost of production for
milk fat, milk protein, and milk less the components, the perfect information criteria can be
refuted. Clearly then the dairy market is not perfectly competitive. What is not so clear is
whether the violations are significant enough to suggest the withdrawal of prices as the
economic weights, even to the point of not using any weights under the auspices that more
harm may be ensuing under this inappropriate plan than without.

A simulation revealed that a i 50% error in any single economic weight in a seven
trait swine selection index caused less than a one percent loss in relative efficiency of the
index (Vandepitte, and Hazel, 1977). The study showed that underestimation of economic
weights generally caused greater errors than overestimation, and the effect were nonlinear,
and could be sizable with greater than 50% error. As the authors point out the results are
specific to the P and G matrices and economic weights used in the example and any
generalization must be approached cautiously. It is not clear how well these results
extrapolate to stableness of an index such as PTAS with fewer traits, different covariances,
and difi‘ering relative magnitudes of the economic weights.

Lin and James also found that errors in relative economic weights result in loss of

efficiency with selection index, but the predicted losses are not highly sensitive to this type

133

of error (Villanueva, and Kennedy, 1993).

While SI is based on linear estimations, Dekkers, Van Erp, and Schukken, (1996)
showed that the economic weight for savings in penalty costs by reducing somatic cell
count in Ontario were not linear. The authors recognized that penalty costs are only a part
of the savings associated with improved SCC, however, there is no mention of the
importance of considering the cost of attaining lowered somatic cell counts. The authors’
recommendation of a target SCC appears quite arbitrary. Similarly, the use of the average
marginal value of SCC reduction across herds is suggested as the proper economic weight
for use in a population wide breeding program. It is not clear what objective firnction this
suggestion is intended to maximize or the basis on which this decision was made. Given
the availability of the marginal economic value of lowered SCC for a given herd SCC
level, and the advent of personal computer it is surprising that the creation of individually
customized indexes based on farm customized economic weights was not proposed.

Sampling error variance in estimating economic weights has been noted to also
increase the estimated genetic gain, overestimate the attainable genetic gain, as well as, the
realized gain (Vandepitte, and Hazel, 1977).

Other pragmatic problems also arise with the use of price as the economic weights.
Many traits of interest do not have readily available prices, for example somatic cell count
and type traits. The weights for these traits appear to be arbitrarily chosen. Clearly the net
marginal returns per trait is a function of individual operations. This is not however
reflected in the price. Furthermore, national price estimates are used despite many regions

of the country increasingly having separate milk pricing formulas, especially with the

134

advent of component pricing. Hazel (1943), a pioneer in SI development, recognized that
the proper economic values may vary by region. The relationship of national prices to the
true net economic values appears ambiguous. He also suggested that since the results of
current selection are not apparent until the future that the economic values should
represent the anticipated future values.

A number of technical difficulties are also encountered by the USDA in doing the
calculations. The USDA Animal Model includes a sire x herd interaction term. Recent
work (Dimov, Albuquerque, Keown, et al., 1995) suggests that the estimate used in the
model may overestimate the true effect of this interaction term by a magnitude of five to
ten. Overestirnation of this effect may lead to underestimation of genetic merit of superior
sire and dams, and therefore incorrect animal ranking. Differences in the heritability of
milk yield (0.21 versus 0.28) were also recorded among different regions of the country.

Preferential treatment of animals also appears to be a problem in the US.
Preferential treatment can include any management practice that increases production and

is applied to one or several cows but not contemporaries (1900,;- at 0, for some

individuals). Assuming preferential treatment among dams with high cow indexes, Powell
and Norman (1988) found the milk yield deviations of daughters of these dams were
higher than expected. This underestimation of daughter performance was thought to have
occurred due to preferential treatment of this group of daughters. There is also evidence
that such effects occur in the animal model estimation of cow PTAs (Kuhn, Boettcher, and
Freeman, 1994). The bias of PTAs resulting fiom preferential treatment can be sizable,

with 0.06 to 0.39 of the increased production resulting from the special treatment being

135

transferred to her estimate of PTA. Preferential treatment is also considered to occur in
genetic evaluations of cows for selection of their sons as sires (Murphy, Everett, and Van
Vleclg 1982). Such work has resulted in the suggestion to consider only the first record of
a sire’s dam be used in predicting a bull’s evaluation.

In estimating the EBV of animals, an adjustment must be made to remove the
variation in yield records due to systematic environmental effects. “Genetic trend in milk
yield was sensitive to the choice of adjustment procedure”(Khan, and Shook, 1996). The
current method of adjustment by USDA biases the estimates of poor sires downwards and
good sires upward, although it is not clear whether the rankings of cows or bulls are
changed.

Up until this point only nuclear genetics have been considered. While the embryo
receives half of its nuclear genetic material from each the sperm and ovum, the latter
contributes all the cytoplasm. This maternal cytoplasm which contains mitochondrial DNA
can contribute to the offspring’s genome. While the effects of cytoplasmic inheritance on
dairy production traits is disputed in the literature, failing to account for cytoplasmic
variance, if present, causes overestimation of additive genetic variance and
underestimation of permanent environmental variance, as a result heritability is
overestimated (Boettcher, Kuhn, and Freeman, 1996). This also causes slight errors in the
accuracy of selection of cows. The USDA Animal Model does not currently include
cytoplasmic effects.

The purely tactical application of SI can also present some problems. There are a

number of commonly used indices in which may include such factors as production, type

136

traits, and possibly genetic evaluations of somatic cell counts and herd life. Even
overlooking the caveats from chapter 2 on the effect of ignoring opportunity costs on the
relative value of herd life and production, it is not clear how the value of herdlife can be
assessed when, based on the information on optimal culling rates in Chapter 3, it would
appear that in general herdlifes are less than optimal. There are still other pragmatic
problems with the determination of which of many indices to use. Despite similar
correlations between Canadian indices, the reranking of animals was found to be
substantial (Dekkers, Kolstad, Jairath et al., 1996).

It has been noted that the average gain expected from use of an index is not
appropriate for comparison of indexes since sampling is only occurring once. Rather the
variance and expected gain should be considered when comparing indexes (Williams,
1962). Expected gain is an average of realized gain over all samples and is unaffected by
the sample-to-sample index variance. The probability of correctly ranking individuals is a
function of the error variance of the index. So while two indices can vary in expected
genetic gain there may be only negligible differences in the probability of correctly making
of individuals, and therefore little real difference between indices. In terms of heifer
selection there is little to indicate what index would best rank animals on future profit.

The dynamics of selection have not been totally ignored by geneticists. Weigel,
Cassel, and Pearson (1995) considered what proportion of each decile of potential Al
bulls, ranked by EBVS at 1 and 3 years of age, subsequently proved to be above the 85th
percentile of their sampling group. The analysis was simplistic and unsophisticated often

relying on inaccurate charts. The final recommendations of percentages of bulls to cull at

137

each age are arbitrary. Conclusions were made as to the economic impact on the Al
centers, this despite the fact that neither the costs of raising the bulls, nor the costs of the
incorrect classifications were considered. All incorrect classifications were considered
equal despite the great range in returns generated by the bulls.

Fortunately, more sophisticated and appropriate techniques have been used to
assess the dynamics of animal selection. Interest has stemmed fiom the potential economic
savings of sequentially selecting animals as traits of interest are revealed over time, rather
than the traditional selection index approach of waiting until all traits of interest can be
measured before selection occurs. Given the multivariate normally distributed traits of
interest and their economic weights(v), P, G, the number of stages over which selection is
to occur, and the total proportion to be selected numerical integration can be used to
optimize an objective function such as aggregate breeding value. The information on those
individuals which haven’t been culled in previous periods is accumulated and used in
selection at subsequent periods. Conventional independent culling, multistage index
selection, multistage selection, and selection index updating are equivalent terms used to
denote this technique which will solve for the for the optimal truncation points at each
selection period.

An orthogonal transformation procedure of the traits has recently been developed
(Xu and Muir, 1992). Since subsequent distributions are no longer a function of previous
selection, numerical integration is not needed, with the concomitant saving in CPU time.
However, the required computer time with numerical integration was not inhibitive. The

relative efficiency of this transformation compared to numerical integration in maximizing

138

aggregate breeding value for various scenarios was reported at 87% to 300% (Xu and
Muir, 1991). Given that the transformation places a restriction on the solution space, it is
not clear how it can result in increased efficiency. The cited references do not support the
explanation provided in the paper.

Given the cost of measuring the various traits, the objective firnction of multistage
index selection can include indicators of profit. For a particular example it was shown that
when the objective function included the cost of the increase in aggregate breeding value,
the results of multistage index selection varied greatly from the results where the focus
was purely on increasing aggregate breeding value without regard to cost (Xu and Muir,
1992). In particular, both the aggregate breeding value and costs were lower when cost
was included in the firnction to be maximized. However, it should be noted that the profit
function used in the study was incorrect.

There is no record of these techniques having been used for selection. It would
seem that an approach that lies between the lax method of analysis used by Weigel et al.
and the restrictiveness of multistage index selection would be appropriate. Note that the
latter maintains the hypotheses of multivariate normal distributions and uses estimates of
P, G, and v, which as discussed earlier in the chapter will not result in the optimal indices
in each period, and the level of bias across individuals may well vary. Also multistage
index selection requires the intertemporal phenotypic and genetic correlations among the
traits used for selection. These correlations are currently unknown and genetic theory is
silent on potential values.

In summary the methodology of selection index for single and multiple traits has

139

been considered. Even assuming optimal weights for the indices, the proportion of
explained variance of an animal’s EBV, much less first record, is very low due to
Mendelian sampling. Furthermore, based on all the assumptions made in deriving the
variances, covariances, and economic values, the index weights clearly are not optimal.
But even more worrisome is the fact that, the level of inoptimality may vary across
individuals resulting in inconsistent rankings. SI will also fail to correctly rank individuals
if the genetic and phenotypic values of the trait of interest are not normally distributed.
Given these difficulties, it is not clear whether any genetic indices will be of greater
value in heifer selection than their cost. Furthermore, should they be of some value, it is
unknown which indices should be used and at what time periods in a heifer’s prepartum

life they should be applied to permit heifer selection to maximize producer profit.

140

BIBLIOGRAPHY

Boettcher, P. J ., M. T. Kuhn, and A. E. Freeman. 1996. Impacts of cytoplasmic
inheritance on genetic evaluation. J. Dairy Sci. 79:663-675.

Dekkers, J. C. M., B. Kolstad, L. K. Jairath, and Z. Liu. 1996. 1996 Dairy research report,
University of Guelph, Publ. No. 0396: 15-18.

Dekkers, J. C. M., T. Van Erp, and Y. H. Schukken. 1996. Economic benefits of reducing
somatic cell count under the milk quality program of Ontario. J. Dairy Sci. 79:396-401.

Dirnov, G., L. G. Albuquerque, J. F. Keown, L. D. Van Vleck, and H. D. Norman. 1995.
Variance of interaction effects of sire and herd for yield traits of Holsteins in California,
New York, and Pennsylvania with an animal model. J. Dairy Sci. 78:939-946.

Goflinet, B. 1983. Selection on selected records. Genet. Sel. Evol. 15:91-98.

Hazel, L. N., G. E. Dickerson, and A. E. Freeman. 1994. Symposium: selection index
theory; the selection index-then, now and for the firture. J. Dairy Sci. 77:3236—3251.

Hazel, L. N. 1943. The genetic basis for constructing selection indexes. Genetics. 28:476-
490.

Harville, D. A 1972. Maximum likelihood approach to variance component estimation
and to related problems. I. Am. Stat. Assoc. 72:320-338.

Hill, W. G., and K. Myer. 1984. Effects of errors in parameter estimates on efiiciency of
restricted genetic selection of indices. 1n: Hinkelmann, K (ed). Experimental Design,
Statistical Models, and Genetic Statistics. Essays in honor of O. Kempthome. Marcel
Dekker, New York, pp 345-367.

Jairath, L. K., J. F. Hayes, and R. 1. Cue. 1995. Correlations between first lactation and
lifetime performance traits of Canadian Holsteins. J. Dairy Sci. 78:438-448.

141

Khan, M. S., and G. E. Shook. 1996. Effects of age on milk yield: Time trends and
method of adjustment. J. Dairy Sci. 79: 1057-1064.

Kuhn, M. T., P. J. Boettcher, and A. E. Freeman. 1994. Potential biases in predicted
transmitting abilities of females from preferential treatment. J. Dairy Sci. 77:2428-2437.

McAllister, A J., A J. Lee, T. R. Batra, C. Y. Lin, G. L. Roy, J. A Vesely, J. M. Wauthy,
and K. A Wmter. 1994. The influence of additive and nonadditive gene action on lifetime
yields and profitability of dairy cattle. J. Dairy Sci. 77:2400-2414.

Meyer, K. 1983. Maximum likelihood procedures for estimating genetic parameters for
later lactations of dairy cattle. J. Dairy Sci. 66: 1988-1997.

Misztal. I. 1997. Estimation of variance components with large-scale dominance models.
J. Dairy Sci. 80:965-974.

Murphy, P. A, R. W. Everett, and L. D. Van Vleck. 1982. Comparison of first lactations
and all lactations of dams to predict sons’ milk evaluations. J. Dairy Sci. 65: 1999-2005.

Philipsson, J., G. Bangos, and T. Amason. 1994. Present and future uses of selection
index methodology in dairy cattle. J. Dairy Sci. 77:3252-3261.

Powell, R L., and H. D. Norman. 1988. Accuracy of cow indexes according to
repeatability, evaluation, herd yield, and registry status. J. Dairy Sci. 71 :2232-2240.

Rothschild, M. F ., C. R. Henderson, and R. L. Quaas. 1978. Comparison of maximum
likelihood and Henderson’s method 1 estimates of variances and covariances for two traits
when selection is present. J. Dairy Sci. 61 (Suppl. 1):91. (Abstr)

Samuelson, D. J ., and R. E. Pearson. 1995. Accuracy of predicting genetic merit fi'om
pedigree information for bulls entering stud sampling programs. J. Dairy Sci. 78:2057-
2066.

142

Schaefi‘er, L. R., and C. R. Henderson. 1983. Best linear unbiased prediction when error
vector is correlated with other random vectors in the model. Genet. Sel. Evol. 15:395-
400.

Schaefl‘er, L. R, and H. Song. 1978. Selection bias and REML variance-covariance
component estimation. J. Dairy Sci. 61 (Suppl. 1):91. (Abstr.)

Tempelman, R. J ., and E. B. Burnside. 1990. Additive and nonadditive genetic variation
for production traits in Canadian Holsteins. J. Dairy Sci. 73:2206-2213.

Tempelman, R. J., and E. B. Burnside. 1991. Additive and dominance genetic variation for
production traits under an animal model. J. Anim. Breed. Genet. 108:330-342.

Van der Werf, J. H. J., T. H. E. Meuwissen, and G. De Jong. 1994. Effects of correction
for heterogeneity of variance on bias and accuracy of breeding value estimation for Dutch
dairy cattle. J. Dairy Sci. 77:3174-3184.

Van Raden, P. M., and G. R. Wiggans. 1991. Derivation, calculation, and use of national
animal model information. J. Dairy Sci. 74:2737-2746.

Van Tassel, C. P., and L. D. Van Vleck. 1991. Estimates of genetic selection difi’erentials
and generation intervals for four paths of selection. J. Dairy Sci. 74:1078-1086.

Van Vleck, D. L. 1993. Selection Index and Introduction to Mixed Model Methods. CRC
Press, Boca Raton

Vandepitte, W. M., and L. N. Hazel. 1977. The effect of errors in the economic weights
on the accuracy of selection indexes. Ann. Genet. Sel. Anim. 9:87-103.

Villanueva, B., and B. W. Kennedy. 1993. Index versus tandem selection after repeated
generations of selection. Theor. Appl. Genet. 85:706-712.

Weigel, D. J., B. G. Cassel, I. Hoeschele, and R. E. Pearson. 1995. Multiple trait
prediction of transmitting abilities for herd life and estimation of economic weights using
relative net income adjusted for opportunity cost. J. Dairy Sci. 78:639-647.

143

Weigel, D. J., B. G. Cassel, and R. E. Pearson. 1995 Relative genetic merit and
efl‘ectiveness of selection of young sires for artificial insemination. J. Dairy Sci. 78:2481-
2485.

Williams, J. S. 1962. The evaluation of a selection index. Biometrics 18:375-393.

Xu, 8., and W. M. Muir. 1992. Selection index updating. Theor. Appl. Genet. 83 :451-
458.

Xu, 8., and W. M. Muir. 1991. Multistage selection for genetic gain by orthogonal
transformation. Genetics 129:963-974.

144

Chapter 6

LITERATURE SUMMARY AND HYPOTHESES

Literature Summary

The literature review can be summarized by the following points:
1) Optimal culling rates in dairy production are a complex economic question. This issue is
best structured as an infinite capital asset replacement model with nonidentical
replacement. The question is best solved with stochastic dynamic programming.
2) Stochastic dynamic programming and simulation models suggest the optimal culling
rate is between 20 and 30%, assuming risk neutrality of producers, a constant herd size,
culls are sold for beef not dairy, and no strong seasonal effects on milk production or
reproduction. Higher culling rates are associated with increased milk production but not
increased profit. Optimal culling rates are not impacted by the assumed rate of genetic
improvement. Mth the cost of replacements being high relative to the cost of culls, as
occurs in North America, maximizing genetic progress or milk production through
maximum culling rates is not economically optimal.
3) The excessive culling rate in Michigan indicates producers have excess heifers, and a
means of heifer selection is required. To maximize profit, heifers should be selected for
lifetime profit corrected for the opportunity cost of postponed replacement. First lactation
milk is a proxy for lifetime profit corrected for the opportunity cost of postponed
replacement.

4) Genetic evaluations by the USDA are the only widely available means of selecting

145

heifers. Theoretically, animal model genetic evaluations should be useful in predicting first
lactation milk production, and therefore, lifetime profit corrected for the opportunity cost
of postponed replacement. However, there is little empirical support for the ability of
Holstein parent PTAs to predict subsequent daughter milk production.

5) Given that parent PTAs are intuitively useful in heifer selection, no consideration has
been given to a selection scheme for heifers. This scheme would need to address not only
which of the potential genetic evaluations (i.e., milk, fat, protein, reliabilities) of each
parent would be used, and how they would be used, but also, in which period(s) would
they be used, for the dynamic nature of the selection process needs to be accommodated.
New genetic estimates are produced every six months providing 4 complete sets of
estimates over the course a heifer being raised.

Based on the literature review the following hypotheses were generated:

Statement of Hypotheses

Hypothesis 1: Parents’ PTAs from the last rearing period will not predict first lactation
milk.

Hypothesis 11: Genetic indices based on parents’ PTAs from the first rearing period will
be of little value in estimating whether culling occurred prior to, or during, the first
lactation.

Hypothesis III: Prepartum heifer selection based on their parents’ PTAs and associated
reliabilities, of milk, fat, and protein fiom throughout the rearing process, will be

more profitable than random heifer selection.

146

Hypothesis 1, along with various estimation methods are evaluated in Chapter 7.
Chapter 8 considers hypothesis H, again evaluating various methods to estimate the
relationship between culling and parents’ PTA. And the value of genetic evaluations in

prepartum heifer selection is addressed in Chapter 9.

147

Chapter 7
PREDICTION AND RANKING OF FIRST LACTA'I‘ION MILK USING PARENT

PTA

ABSTRACT
The objectives of this study included determining the ability of parents’ PTA for milk, fat
and protein to predict subsequent first lactation milk production of daughters; the ability
of the predictions of first lactation milk production to correctly rank heifers based on
actual milk production; and finally whether, in estimating this relationship, the herd efi'ect
needed to be estimated as fixed efi‘ects. The data consisted of first lactation mature
equivalent milk production of 5,123 lvfichigan Holstein heifers, and the heifers’ parents’
animal model evaluations immediately prior to the heifers’ fi'eshenings. While statistical
comparison of the estimates resulting from treating herd as a fixed or random efi’ect
revealed they were different, the practical differences were not as clear. Regardless of
modeling technique, parents’ genetic evaluations explained less than 7% of the variation in
subsequent daughter first lactation milk production. A nonparametric comparison of the
within herd rankings of the predictions from the random and fixed effect models suggested
the rankings were substantially similar. While diffuse, the distribution of within herd rank
correlations between predicted mature equivalent milk production, based on parents’
genetic evaluations, and the actual mature equivalent milk production, was skewed to the
left and suggested the evaluations may be usefirl in prepartum heifer selection. The

omission of heifers which were culled prior to first lactation, did not appear to cause

148

selection bias in the sample data.
(Key words: random effect, fixed effect, prediction first lactation mature equivalent)
Abbreviation key: FE = fixed effect, MEM = mature equivalent milk, 01.8 = ordinary

least squares, RE = random effect.

INTRODUCTION

Given a constant herd size, no seasonal effects on reproduction and production,
and culled cows being sold for meat (not dairy) purposes, optimization (5, 18, 23, 28, 33)
and simulation (1, 2) studies were unanimous in their conclusions that a culling rate of
20% to 30% optimizes producer profit. While other simulation studies (4, 19, 20, 25),
which considered subjectively selected culling rates or strategies for comparison, indicated
that profit would be increased with culling rates below the average 37% culling rate1 in the
Michigan dairy industry in 1996 (24). Comparability of the actual culling rate and those in
the studies assumes no herd contraction or internal expansion. The results of these
multiple studies are compelling because, not only did the various models employ difl‘erent
assumptions and analytical techniques, but as well, the models operated under economic
conditions representative of a number of countries (i.e., the USA, England, Ireland,
Netherlands). In the models, the increased profit associated with use of the lowered culling
rates varied from 2% to 40%.

Comparison of only the mean profit by the studies implies risk neutrality, whereas

a majority of Michigan and Dutch dairy producers are risk adverse (17). The risk

 

' Culling rate, annually defined as, the number of animals leaving the herd due to sale or death divided by
the average herd size.

149

preferences of decision makers can have significant effects on their choices, so the optimal
rule assuming risk neutrality cannot necessarily be extrapolated to those who are risk
averse.

At least under risk neutrality, one direct ramification of Michigan’s excessive
culling rate, assuming herd sizes have at least been constant, is that Michigan dairy
producers do not need all potential replacement dairy heifers to enter the lactating herd,
necessitating a method of heifer selection.

This need for heifer selection is in direct contrast to the present situation in which
most producers rear and fieshen the majority of heifers produced on the farm (6, 14). The
simulation studies revealed that while this practice maximizes genetic progress and milk
production, the economic performance is lower due to failure to consider the opportunity
costs of higher culling rates.

While the culling models suggest the need for heifer selection, they lend little
insight into how selection should be conducted. Most optimization studies employed
random replacement selection. Some simulation studies arbitrarily selected replacements
based on their simulated EBV of milk. While this approach to selection has intuitive
appeal and indeed theoretical justification (15), empirical support for heifer selection on
this basis is lacking. If a heifer selection rule more profitable than those used in the
optimization and simulation models can be developed, producer profit may increase more
than 2% to 40%.

While use of genetics to select heifers is appealing, application of finance theory

may suggest otherwise. A standard result of capital asset replacement theory is that as the

150

expected profitability of the potential replacements rises relative to the existing assets, the
optimal length of the utilized assets’ lives decrease (27) which is equivalent to an increase
in the replacement rate. Therefore, if EBV were valuable in selecting more profitable
heifers, the simulation model which utilized this method of heifer selection should have
had higher optimal culling rates than those studies which used random heifer selection.
However, the study (1) which based replacement selection on EBV had a lower optimal
culling rate than all of the studies which randomly selected heifers (2, 5, 18, 28, 33),
except one (23). One possible explanation for the similarity between the most profitable
culling rates of the studies, despite the use of selection based on EBV milk versus random
replacement selection, is that the genetic information is not highly valuable in selecting
heifers for profitability. An alternative explanation is that differences among the models
may preclude the recognition of the value of EBV in heifer selection.

Given that heifer selection is required, the first step in determining a method of
selection is defining the objective. The objective of heifer selection is to choose the most
profitable animals. Van Arendonk (32) showed that lifetime profit corrected for the
opportunity cost of postponed replacement is the correct method for ranking animals in
terms of profitability. Unfortunately, corrected lifetime profit of animals is generally not
available and furthermore, can not be determined prior to the culling of the animals.
However, the literature has focused on indicators of lifetime profitability that are apparent
early in the productive life of milk cows. No studies have considered the relationship
between information available prior to productive life and any measure of subsequent

lifetime profit, much less lifetime profit corrected for the opportunity cost of postponed

151

replacement.

The phenotypic correlation between first lactation actual milk and lifetime profit
corrected for opportunity cost has been reported at 0.54 (3). Weigel et al. (3 5) determined
the phenotypic and genetic correlations between corrected lifetime profit and first lactation
mature equivalent milk production to be 0.51, and 0.80, respectively. For purposes of
heifer selection, first lactation milk production may then be a suitable proxy for corrected
lifetime profit.

Presently, alternative predictors of subsequent milk production, such as the
concentration of serum B-lactoglobulin 26 weeks prepartum (22) and genetic molecular
markers (7, 8), are not widely available. From Lush and Hazel (13) to the more recent
mixed model genetic prediction by Henderson (15), there is a strong theoretical basis for
using EBV in predicting most probable producing ability. EBV provide estimates of the
genetic merits for milk production which should on average manifest as phenotypic
performance. Prior to utilizing any management technique, including heifer selection,
managers reasonably require, if possible, some evidence of the technique’s expected
performance.

Since 1989, when the USDA began using the animal model for estimating genetic
merit, only two domestic empirical studies have, in a temporally correct sense, considered
the ability of parents’ PTA to predict a heifer’s subsequent milk production (22, 31).
These studies are temporally correct for the purpose of heifer selection in that the parents’
PTA were those available prior to the heifers’ production. Contrast this with the situation

in which the parents’ PTA were generated some years after the heifers calved and

152

therefore were based on information that was unavailable at the time of prepartum heifer
selection. Notice this information unavailable at the time of selection would likely include
the very production records of the daughters which the parents’ PTA were to predict.
Neither of the two temporally correct studies included EBV fat or EBV protein, along
with EBV milk, in the prediction of daughter performance.
Based on 76,871 daughters, VanRaden et al. (31) found the correlations of
average parent animal model evaluations with daughter milk and fat performance ranged
fiom 0.20 to 0.49. Daughter performance was measured as yield deviation of standardized
milk yield.
In heifer selection the producer is interested in predicting standardized milk yield.
While the use of yield deviation as a dependent variable serves to decrease the error
variance, the results are not directly applicable to heifer selection for a producer is unlikely
to know the value of management group, permanent environment and herd-sire interaction
efl'ects for each heifer prior to parturition, and therefore, could not derive predicted milk
yield fi'om predicted yield deviation. Given that producers are paid based on milk yield,
their interest would lie in prediction of milk yield, perhaps standardized for age and season
0f calving. Even if the expectation of yield deviation was identical to the expectation of
milk yield, the accuracy of the milk yield predictions would affect producers’ acceptance
Of this method of heifer selection.
The second study (22), based on a small group of 26 heifers in a single herd, found

that the EBV milk had negligible value (R2 = 0.034) in prediction of first lactation 305d

milk yield.

153

\Vrth heifer rearing, and the associated potential costs and revenues, occurring
over the span of many months and new genetic information provided by USDA quarterly,
heifer selection utilizing genetic evaluations is potentially a dynamic issue. As suggested by
Quinton and Smith (26), there has been little empirical checking of the efficacy of genetic
evaluations. They found that the rank correlations with progeny phenotype were 36% and
27% higher for parental BLUP genetic evaluation than for parental phenotypes of fat
depth and days to 100 kg, respectively. Prior to developing a dynamic heifer selection rule
to maximize profit based on EBV, the empirical ability of EBV to predict milk production
should be ascertained.

A producer, prior to making an informed decision about implementing a heifer
selection scheme based on parents’ PTA, would prefer to see empirical evidence of the
scheme’s performance in the real world. This preference is only reasonable especially since
some selection has already occurred in terms of which sires and dams to breed. Managers
are faced with limited time to complete a potentially nearly infinite number of management

138k!» With some understanding of the rule’s potential performance, the producer could
eVillllate whether the performance was sufficient to warrant foregoing some other
m‘I-“agement task in order to employ the selection scheme.

Given the high genetic correlations between milk and fat, and between milk and
PTOtein, inclusion of parents’ PTA fat and PTA protein, along with PTA milk, would
increase the ability to predict first lactation milk production of daughters. The
disadvantage of selecting for the real producing ability of a single trait (i.e., milk) based on

mUItiple traits (i.e., milk, fat, and protein) is the theoretical weights for the index are

154

no- '3 9
indifi h'

s

at»

..E

WW

.Vw

 

unique for each combination of the number and source of records that contributed to each
parents’ PTA. Specifically, the nondiagonal elements of the variance-covariance matrix of
the multiple traits are firnctions of the number and source of records that contribute to the
records of the traits. Furthermore, given that the weights for any type of record (i.e., milk,
fat, protein) would rarely be the same for sire and dam, there is no theoretical basis to
equally weight the same type of record of each parent and base multitrait analysis of MEM
0n heifers’ EBV.

This inability to determine the theoretical weights is not critical in an empirical
analysis where the focus is on the ability of information, available to producers, to predict
heifer milk production. Furthermore, this recognition that there is not a single optimal
Vector of theoretical weights for an analysis, while explicit, is not unprecedented in
Selection Index or mixed models. Relaxing assumptions in genetic evaluations can easily
result in multiple optimal values for any single theoretical weight. For example, the bias in
breeding values resulting from assuming homogeneity of variances at a minimum dictate
the need for herd level corrections, if not animal level corrections in the weights (Van der
we“). The assessment of whether to base milk production prediction on a single trait
”ms multiple traits reduces to whether the theoretical ability to derive a single vector of
Optimal weights is preferred over the additional information provided by the additional
tl’tlits.

In selection schemes the actual prediction of performance is not as important as the
‘1’th to correctly rank the animals in relative terms. Parents’ PTA would be useful in

heifer, Selection if models using this information resulted in within herd heifer rankings

155

identical to those based on actual first lactation milk production.

Finally, the pragmatic issue of appropriate estimation technique must be
considered. The apparent paradox of estimating herd effect, which intuitively can be
considered a random variable, as a fixed effect (FE) is explained by Henderson (16). He
stated that, should the other independent variables be correlated with the herd efi’ect, the
bias in estimation can be eliminated through estimating the herd efl’ects as fixed, though at
the price of larger variance of the estimates.

It can be inferred that Henderson recognized that estimates resulting fi'om treating
the unobserved effect, or categorical variable, as random or fixed were not necessarily
difl‘erent (21). Specifically, as herd size increases, or the variance of the herd efl‘ect
increases relative to the variance of the error term, the estimates from random effects
(RE) and fixed effects estimation converge, even if the herd effect is correlated with the
independent variables. A priori, it is difficult to assess whether RE will converge to FE. So
SiVen a likely correlation between genetics of a herd and the herd effect, and small herd
sizes, Henderson reasonably opted for the more conservative FE estimation. A few years
3381' Henderson’s publication, Hausman (12) developed a statistical test of the equivalence
OfRE and FE estimates, permitting statistical evaluation of whether RE is consistent, or if
FE is required.

The objectives of this study were to determine: 1) the ability of parents’ PTA for
millg fat and protein to predict subsequent first lactation milk production of daughters; 2)
the ability of the predictions of first lactation milk production to correctly rank heifers

based on actual milk production; and 3) whether, in estimating the relationship in objective

156

one, the herd effect needed to be estimated as fixed effects.

MATERIALS AND METHODS

Data Collection
A total of 10,854 Holstein heifer records were collected from Michigan DHI. Only
Holstein heifers that met all the following criteria were included: 1) as of June 30, 1995,
identified as prepartum with at least one parent’s PTA for milk, fat, or protein recorded;
2) born between July 1, 1992 and December 31, 1992; 3) a calving date between July 1,
1994 and December 31, 1994; 4) first lactation mature equivalent milk (MEM)
production of greater than zero.
Genetic data were the heifers’ parent PTA for milk, fat and protein from the July
1994 sire and dam animal model evaluations calculated by the Animal Improvement
Programs Laboratory, USDA (Beltsville, MD). Heifer information including birthdate and
“Wing date were collected from Michigan DHI animal identification records. The
Pdeuction data, which consisted of MEM production, were collected from Michigan DHI
COW production records. MEM, standardizes production for age and season of at calving.
After deleting heifers that were culled or missing the genetic information of either
par eat, records of 5,123 heifers were available for estimation of the empirical selection
index weights. In effect then this data sample consisted of, for each of the 716 herds, a
001km of heifers which were born within a six month window, and then, two years later,

calved in a six month window. The number of heifers in a cohort for a given herd varied in

157

size fi'om 1 to 165 heifers with median and mean sizes of 4 and 7.2, respectively.

Estimation of Empirical Model

The unbalanced general population model to empirically estimate the relationship
between parents’ PTA and standardized (i.e., deviated from the mean and divided by the
standard deviation) first lactation MEM was:

Yb. : Ill-B + Ch '1' Uh.
where

y;.. = standardized first lactation MEM of animal a in herd h;

x... = l x 6 row vector. The six elements of x... are the standardized milk, fat
and protein PTA of individual a’s parents, fiom the July 1994 animal
model evaluations calculated by the Animal Improvement Programs
Laboratory, USDA;

B = vector of empirical regression coefficients;

0;. = unobserved random variable representing the efi‘ect of herd h;

u... = the idiosyncratic error term.

Three estimation techniques were considered to determine the nature of the
unobserved herd effect. The first method of analysis was ordinary least squares (OLS)
pooled over all observations. In the OLS regression, MEM was regressed on parents’
PTA ignoring the herd effect. This technique results in the herd effect being included with
the idiosyncratic error term, to create a composite error term. Standard OLS regression

mechanics indicates the estimates of the empirical selection weights will be consistent if,

158

for each heifer, neither component of the composite error term is correlated with the PTA
of the heifers’ parents. A variance estimator robust to correlation among the residuals was
required to account for the presence of the unobserved herd effect in the composite error
term.

Making the assumption that the herd effects all originated from the same
distribution, and therefore had a common variance, generalized least squares was used to
estimate a RE model. The estimates will be asymptotically consistent and more eficient
than OLS if the unobserved herd effect and the idiosyncratic error term for any heifer in a
herd were uncorrelated with the parents’ PTA of all heifers in the same herd.

Finally, the herd effect was estimated as a FE. Fixed Effects estimation or within
estimation in the econometrics literature (10), can be viewed as conducting OLS
estimation on transformed data. Because the value of the herd effect was not of interest,
the data were transformed to eliminate the unobserved effect. The transformation, also
known as herdmate deviation or within herd deviation, involved herd demeaning each
observed variable in the regression (e. g., y;,., 1...). For example, the value of sire PTA milk
averaged over all heifers in a given herd was subtracted from the sire PTA milk of each
heifer in the herd. Because the mean of a constant is itself, and by definition the herd efi‘ect
is constant for all heifers in the herd, this transformation removed the unobserved herd
efi'ect. The resulting FE estimates were identical to those from a regression where a
dummy variable was inserted for each herd. These estimates will be consistent assuming
the idiosyncratic error of each heifer in a herd is uncorrelated with the parents’ PTA of all

heifers in the same herd.

159

The RE estimator lies within the continuum spanned by OLS and FE (21). The RE
estimates approach the OLS estimates as the ratio of the unobserved effect variance to the
idiosyncratic error variance approaches zero. In contrast, the RE estimates approach those
of the FE as the ratio of the variances approaches infinity, or as the number of animals in

each herd gets large.

Statistical Comparison of Empirical Regression Coefficients

Hausman (12) developed a general specification test for the null hypothesis that an
asymptotically eflicient estimator must have zero asymptotic covariance with its difi‘erence
fiom a consistent but asymptotically inefficient estimator. The Hausman test is a Wald
based test. The Hausman test can be applied in the comparison of the RE estimates with
either the FE or OLS estimates. Maintained hypotheses of the Hausman test are that the
idiosyncratic errors are homoskedastic, are not correlated with themselves, and are not
correlated with the independent variables.

Ifthe unobserved effect and the idiosyncratic error are uncorrelated with the
independent variables, the OLS and RE estimates should be asymptotically consistent,
with RE being efficient, and the Hausman test would not be expected to reject the null
hypothesis. Should either of these assumptions be false, each set of estimators would be
inconsistent with generally different probability limits, and the Hausman test should have
some power to detect this.

The more traditional use of the Hausman test is in comparison of the RE and FE

estimates. If herd is uncorrelated with PTA, both sets of estimates would be

160

asymptotically consistent, and RE estimators efficient, again assuming the independent
variables are uncorrelated with the idiosyncratic error. However, should herd be correlated
with PTA, the FE estimates would still be consistent while the RE estimates would not,
and the Hausman statistic should reject the null hypothesis.

To assess the practical significance of difi‘erent empirical weights in prediction of
milk production, a nonparametric approach was used. For with the focus of using these
weights to select animals, it is the resulting within herd rankings of animals, not the
prediction error, that is paramount. Spearman’s within herd rank correlations, compared
the predicted first lactation MEM based on the RE and FE estimators to each other, as

well as, to the rankings based on actual MEM.

Statistical Evaluation for Potential Selection Bias

Of the total 10,854 Holstein heifer records collected from Michigan DHI, it was
determined that 1,546 heifers with complete parent PTA fi'om the USDA July 1994 animal
model evaluations were culled prior to first lactation. A heifer was identified as culled
based on the following criteria: 1) as of June 30, 1995, identified as prepartum with at
least one parent’s PTA for milk, fat, or protein recorded; 2) born between July 1, 1992
and December 31, 1992; 3) if culled, the culling date was prior to July 1, 1994; 4) if
culled, a culling code was recorded; 5) as of January 4, 1996 no first lactation MEM
record.

Culled animals were then born within the same time window as the 5,123 heifers

which had a first lactation. The culled heifers were either identified as culled prior to first

161

lactation, or had not initiated their first lactation by at least three years of age. Heifer
information including birthdate, calving date, culling date and culling code were collected
from Michigan DHI animal identification records. The MEM values were collected from
Michigan DHI cow production records.

“0th 23% of the heifers with parent PTA being culled prior to first lactation, the
potential for selection bias existed. Selection or sample bias exists if the presence of MEM
is based on the idiosyncratic error (i.e., the portion of MEM not explained by parent
PTA). For example, selection bias could exist if producers assessed heifers’ milk
production, then kept and identified to Michigan DHI for record evaluation only the
higher producing heifers. In the presence of selection bias, the model results are applicable
only to the sample, rather than to the population in general.

To be representative of the population in the presence of selection bias, the model
based on the selected sample has another term, denoted the inverse Mills ratio (36). The
inverse Mills ratio reflects that only the selected animals have been included in the sample.
The addition of this term, in effect, transforms the population model into a tobit model or
censored model (Greene) which, in the canonical form, explicitly recognizes that the
dependent variable is observed only if it exceeds some value (e.g., y;,. > 0).

Following the approach of Wooldridge (36), the sample was examined for
evidence of selection bias using a standard two step estimation procedure. In the first step
the inverse Mills ratio was estimated using a pooled probit, accounting for the herd efl‘ect
as a FE. In this first step, the data sample consisted of 6,669 animals, the 1,546 culled and

the, 5,123 not culled. Specifically, the dependent variable for the probit model was whether

162

or not the heifer was culled prior to first lactation. Technically, the inverse Mills ratio is
the linear prediction, in the Z metric, from this estimated probit selection model evaluated
at the standard normal probability density function divided by the linear prediction
evaluated at the cumulative normal distribution function.

The second step of the procedure used only the data of the 5,123 heifers which had
a first lactation. In the second step the linear FE model, including the inverse Mills ratio
term, was re-fitted. Statistical significance of the inverse Mills ratio in the second step,
suggests selection bias is present, otherwise the sample can be considered to be
representative of the general population.

In the presence of selection bias with an unobserved effect, the estimates of the
second stage model would not be consistent for the population parameters. Therefore, this
procedure is appropriate only for testing, not correcting, sample selection bias. The FE
analysis was chosen over the RE in each of the two steps due to consistency of the
estimates in the presence of arbitrary correlation between the unobserved effect and either
selection or the independent variables.

All models and tests were conducted with Stata Statistical Software (30), except

for the Spearman’s rank correlations which were computed by SAS (29).

RESULTS AND DISCUSSION
Means, standard deviations, minima and maxima for parent genetic information,
including reliabilities, and first lactation MEM are in Table 7.1.

Regression coefficients, and standard errors for each of the three regressions of

163

standardized parent PTA of milk, fat and protein on standardized MEM are in Table 7.2.

The addition of PTA fat and protein of each parent as independent variables
increased the predictive ability for first lactation milk. Though the inclusion of these
variables likely resulted in multicollinearity, best linear unbiased estimates were still
produced by regression analysis (9). While the variance of the estimates would have been
smaller if correlation among the independent variables was not present, the point estimates
and their variance estimators remain unbiased. The large standard errors of the estimates
did, however, cause larger P values.

In comparing the estimates from the OLS and RE models, the relative change in
the PTA fat of sire coefficient was considerably larger than for the other independent
variables, but the estimate was still not significantly different than zero. As expected, the
standard errors of the RE estimates were less than the respective OLS estimates. In the
RE analysis the coeflicient on PTA fat of dam was statistically significant at P s 0.05.

The FE estimates were similar to those from RE. The standard errors of the FE
model were much closer to those from the RE model, as opposed to the considerably
higher OLS standard errors. The level of significance among the FE estimates was similar
to those of RE except PTA fat and protein of dam were now significant at the 0.01 level.

As indicated by the R2 listed in Table 7.2, parents’ PTA explained 6.86%, 6.80%,
and 6.69% of the within herd variation in first lactation MEM using the OLS, RE, and FE
models respectively. The low coeflicient of determination of all models was consistent
with sampling error in genetic evaluations and the large environmental variance relative to

genetic variance in first lactation milk production. Estimating an OLS model which

164

included the unobserved herd effect as 8 FE by including a dummy variable for each herd
resulted in an R2 of 0.4692. This indicates that parents’ PTA and herd explain 46.92% of
the variation in first lactation MEM. In other words, within the herds parents’ PTA explain
less than 7% of the variation. Then conditional on parent PTA, herd explains roughly 40%
of the variation in first lactation MEM among heifers fi'om difl‘erent herds.

The Hausman test of the OLS and RE estimates was rejected at the 0.0154 level,
rejecting the null hypothesis that the two vectors of estimates varied only due to sampling
error. The presence of a unobserved effect which is correlated with the independent
variables, would cause inconsistency among the OLS estimates and the RE estimates and
would be compatible with the rejection, implying that the OLS and RE estimates varied
significantly from each other.

The Hausman test of the RE and FE estimates was rejected at less than the 0.0001
level. Thus statistically rejecting the null hypothesis that the asymptotically eficient RE
estimates had a zero asymptotic covariance with its difference from the asymptotically
ineficient FE estimates. This would be compatible with the herd effect being correlated
with the independent variables, causing inconsistency among the RE estimates. Either of
the Hausman test rejections could also be due to the idiosyncratic error being correlated
with PTA which would result in the OLS, RE and FE estimates being inconsistent.

Statistically, the results of each of the above tests suggest herd is correlated with
parents’ PTA, necessitating 8 FE model. However, it is not as clear that the RE and FE
estimates difl'er in practically important ways, rather the rejection may be a vestige of the

large sample size. The closeness of the estimates is surprising given the median and

165

average number of animals in each cohort was only four and 7.2, respectively. Also, the
ratio of the variance of the herd efl‘ect to the variance of the idiosyncratic error was only
2.2 and 1.2 for the RE and FE models, respectively.

To investigate the practical effect of these different weights on heifer selection, the
within herd rankings of the heifers based on their predicted MEM by the RE and FE
models were compared using Spearman’s rank correlation. Based on the similarity of the
RE and FE estimates, it was hypothesized that, within herd, heifers would be ranked the
same, resulting in a rank correlation of one.

Figures 7.1 through 7.3 present histograms of the rank correlations. The
distributions are bounded by plus and minus unity, the minimum and maximum potential
values of the Spearman’s rank correlation. A rank correlation could not be computed for
herds of a single heifer and thus 139 herds were excluded from the analysis for this reason.
A herd which consisted of a pair of identical twins was also excluded as the tie in the
rankings precluded calculating the correlation. The within herd rank correlations between
the predicted MEM based on the RE and FE models are presented in Figure 7.1. The four
herds with a correlation of -1 each consisted of two heifers, and as a result, the possible
values of rank correlation were limited to -1 or +1. All of the rank correlations less than
0.70 were associated with herd sizes of four or less. Based on the within herd rank
correlations, the RE and FE estimates appear much more similar in a pragmatic sense than
indicated by the statistical Hausman test.

Figures 7.2 and 7.3 present the within herd rank correlations between actual MEM

and its estimates based on RE or FE, respectively. The distributions of rank correlations

166

were difi‘use in each case. Each of the distributions were skewed to the left, with
approximately 65% of the herds having a positive rank correlation between the actual
MEM rankings and the predicted MEM rankings. This suggests that, while the amount of
MEM phenotypic variation explained by genetic evaluations is low, the evaluations may,
nevertheless, be useful in heifer selection. The utility of genetic estimates may be increased
by evaluating the estimates as they are updated over the course of heifer rearing.

Finally, no evidence of selection bias was found as the level of significance of the
inverse Mills ratio in the FE equation of the second stage of the test for sample selection
bias was 0.440. This suggests these results are applicable to those heifers which were

culled prior to first lactation.

CONCLUSIONS

Estimation of the empirical selection weights for each parent PTA of milk, fat, and
protein using a RE model resulted in more efficient estimates than OLS as expected. While
the RE and FE empirical weights appeared quite similar numerically, this was strongly
rejected statistically. However, an investigation of the comparability of within herd heifer
rankings based on predictions from the RE and FE models supported the similarity of the
estimates with over 96% of the herds having a Spearman’s rank correlation between the
rankings of predicted MEM from the models of greater than 0.90. Given the similar
performance of the two models, consideration should be given to modeling the herd effect
as a random efi‘ect due to the corresponding increased efficiency of the estimates.

It is important for producers to temper any expectations of heifer selection for

167

MEM based on genetic evaluations with the realization that parents’ PTA explain only a
small amount of the variation in MEM. However, the within herd rank correlations
between predicted MEM, based on genetic evaluations, and the actual MEM suggest the
evaluations may be useful in prepartum heifer selection.

The omission of heifers which were culled prior to first lactation, did not appear to
cause selection bias in the sample data. As a result, these results were applicable to all

heifers enrolled in Michigan DHI.

ACKNOWLEDGMENTS
The authors thank Al Thelen, Terry Speer and Lyle Smith at Michigan DHI, as
well as, Leigh Walton and H. D. Norman at the Animal Improvement Laboratory, USDA
for assistance with data collection. I. M. Wooldridge’s econometric comments were highly
appreciated. The financial support of the primary author by the Department of Agricultural
Economics through the Animal Industry Initiative funds provided by the Agricultural
Experiment Station at Michigan State University while this work was conducted is

gratefully acknowledged.

168

REFERENCES

1 Allaire, F. R 1981. Economic consequences of replacing cows with genetically
improved heifers. J. Dairy Sci. 64: 1985-1995.

2 Allaire, F. R, and E. P. Cunningham. 1980. Culling on low milk yield and its economic
consequences for the dairy herd. Livest. Prod. Sci. 7:349-359.

3 Cassell, B. G., B. B. Smith, and R. E. Pearson. 1993. Influence of herd-life opportunity
and characteristics of cows and herds on different net income filnctions. J. Dairy Sci.
76:1182-1190.

4 Congleton, W. R. 1988. Dairy cow culling decisions. 2. Profitability and genetic trend in
herds with culling on production versus income. J. Dairy Sci. 71 : 1905-1915.

5 Congleton, W. R., and L. W. King. 1985. Culling cows on predicted future profitability
with a variable planning horizon. J. Dairy Sci. 68:2970-2983.

6 Connor, L. J., L. G. Hamnr, S. B. Nott, D. Darling, W. Bickert, R. Mellenberger, H. A
Tucker, 0. B. Hesterman, J. A. Partridge, and J. H. Kirk. 1989. Summary of the 1987
Michigan State University dairy farm survey. Research Report 498, Michigan Ag. Exper.
Station, East Lansing.

7 Cowan, C. M., M. R. Dentine, R. L. Ax, and L. A. Schuler. 1990. Structural variation
around prolactin gene linked to quantitative traits in an elite Holstein sire family. Theor.
Appl. Genet. 79:577-582.

8 Georges, M., D. Nielsen, M. Mackinnon, A. Mishra, R. Okimoto, A. T. Pasquino, L. S.
Sargeant, A. Sorensen, M. R. Steele, X. Zhao, J. E. Womack, and I. Hoeschele. 1995.
Mapping quantitative trait loci controlling milk production in dairy cattle by exploiting
progeny testing. Genetics 139:907-920.

9 Greene, w. H. 1993. 9.2 Multicollinearity. Page 266 in Econometric Analysis, 2'“l ed.,
Macmillan Publishing Company, New York.

169

10 Greene, w. H. 1993. 16.4. Longitudinal Data. Page 464 in Econometric Analysis, 2"
ed., Macmillan Publishing Company, New York.

11 Greene, W. H. 1993. 22.3.2. The Censored Regression Model - Tobit Analysis. Page
694 in Econometric Analysis, 2"l ed., Macmillan Publishing Company, New York.

12 Hausman, J. A 1978. Specification tests in econometrics. Econometrica 46: 1251-
1270. .

13 Hazel, L. N., and J. L. Lush. 1942. The efficiency of three methods of selection. Jour.
Hered. 33:393.

14 Heinrichs, A J., S. J. Wells, H. S. Hurd, G. W. Hill, and D. A. Dargatz. 1994. The
national dairy herd evaluation project: A profile of heifer management practices in the
United States. J. Dairy Sci. 77:1548.

15 Henderson, C. R. 1963. Selection index and expected genetic advance. Page 141 in
Statistical Genetics and Plant Breeding ; a symposium and workshop. W. D. Hanson and
H. F. Robinson, ed. Natl. Acad. Sci-NRC, Washington, DC.

16 Henderson, C. R. 1973. Sire evaluation and genetic trend. Page 10 in Proc. Anim.
Breeding Genet. Symp in Honor of Dr. J. L. Lush, Am. Soc. Anim. Sci., Am. Dairy Sci.
Assoc, Charnpaign, IL.

17 Huime, R., A. Dijkhuizen, R. King, and S. Harsh. 1993. Goal, critical success factors,
and information systems needs on swine and dairy farms - Identified by workshops.
Zeitschrifi ffir agrainformatik 3:61-65.

18 Killen, L., and B. Kearney. 1978. Optimal dairy cow replacement policy. Ir. J. Agric.
Econ. Rur. Sociol. 7:33-40.

19 Korver, S., and J. A. Renkema. 1979. Economic evaluation of replacement rates in
dairy herds II. Selection of cows during the first lactation. Livest. Prod. Sci. 6129-3 7.

170

20 Kuipers, A 1980. Development and comparison of selection criteria for cows and bulls
with a dairy herd simulation model. Ph.D. Thesis, Univ. Wisconsin, Madison.

21 Maddala, G. S. 1971. The use of variance components models in pooling cross section
and time series data. Econometrica 39:341-358.

22 Mao, F. C., and R. D. Bremel. 1995. Prediction of milk yields from serum [3-
lactoglobulin concentrations in pregnant heifers. J. Dairy Sci. 78:291-295.

23 McCullough, D. A, and M. DeLorenzo. 1996. Effects of price and management level
on optimal replacement and insemination decisions. J. Dairy Sci. 79:242-253.

24 Michigan Dairy Herd Improvement Association. 1996. Michigan dairy production herd
evaluation report.

25 Pearson, R. E., and A. B. Freeman. 1973. Effect of female culling and age distribution
of the dairy herd on profitability. J. Dairy Sci. 56: 1459-1471.

26 Quinton, V. M., and C. Smith. 1997. An empirical check on best linear unbiased
prediction genetic evaluation using pig field recording data. Can. J. Anim. Sci. 77 :21 1-
216.

27 Robison, L. J., and P. J. Barry. 1996. Chapter 21, Replacement Principles in Present
Value Models and Investment Analysis. 1st ed. The Academic Page, Northport, Alabama.

28 Rogers, G. W., J AM. van Arendonk, and B. T. McDaniel. 1988. Influence of
production and prices on optimum culling rates and annualized net revenue. J. Dairy Sci.
71:3453-3462.

29 8A5db User’s Guide: Statistics, 6th ed. 1989. SAS Inst., Inc., Cary, NC.

30 StataCorp. 1997. Statado Statistical Software: Release 5.0. Stata Press, College Station,
TX.

171

31 VanRaden, P. M., C. A Ernst, G. R. \Vlggans, and H. D. Norman. 1989. Ability of
animal model and Modified Contemporary Comparison evaluations of parents to predict
future progeny performance. J . Dairy Sci. 72(Suppl. 1):72.(Abstr.)

32 Van Arendonk, J. A M. 1991. Use of profit equations to determine relative economic
value of dairy herd life and production from field data. J. Dairy Sci. 74:1101-1107.

33 Van Arendonk, J. A M. and A. A Dijkhuizen. 1985. Studies on the replacement
policies in dairy cattle. III. Influence of variation in reproduction and production. Livest.
Prod. Sci. 13:333-349.

34 Van der Werf, J. H. J., T. H. E. Meuwissen, and G. De Jong. 1994. Efi'ects of
correction for heterogeneity of variance on bias and accuracy of breeding value estimation
for Dutch dairy cattle. J. Dairy Sci. 77:3174-3184.

35 Weigel, D. J., B. G. Cassel, I. Hoeschele, and R. E. Pearson. 1995. Multiple trait
prediction of transmitting abilities for herd life and estimation of economic weights using
relative net income adjusted for opportunity cost. J. Dairy Sci. 78:639-647.

36 Wooldridge, J. M. 1995. Selection corrections for panel data models under conditional
mean independence assumptions. J. Econometrics. 68:115-132.

172

Table 7.1. Parent PTA of milk, fat and protein, reliabilities for milk-fat, and first lactation
mature equivalent milk production for 5,123 animals.

 

 

Variable‘ 7 s1) Minima Maxima
PTAMilkofSire, kg 916.5 261.1 -947.1 1518.2
PTA Fat of Sire, kg 28.4 9.8 -39.9 57.2
PTA Protein of Sire, kg 24.7 8.1 -29.9 42.6
PTA Milk of Darn, kg 409.0 298.6 -819.7 1579.9
PTA Fat of Dam, kg 12.2 11.4 -32.2 64.9
PTA Protein of Dam, kg 11.4 8.7 -25.4 47.6
REL of Sire Milk-Fat, % 93.7 10.8 17 99
REL of Dam Milk-Fat, % 49.3 6.0 14 85
MEM, kg 11084.7 2451.6 1362.6 18875.2

 

lMature equivalent milk production (MEM), reliability (REL).

173

.eoeesoe cheese 2.. 3 82% use see a... sea Benson r

._oo.o w Kan...
Aodmk:
.modwks

$3.85 ea noes—2: 523 .39: a Eat m_ Nd vouch—ox -
«Ste 2: mecca scum—2.50 can 3638333830; c. 3.53— n
.mcozmtomno =a 68 e262. moans—cm «meo— baEEO a

 

 

 

occoe .88e £85 .3.
38¢ :38? 28° .38.? need .28.? 58 co sees <2
m :3 :83? :85 .536. 88° 98.0. See E <2
2 8e :63? N. Se ...ememo 23o ...mmono cameo é: <2
88¢ 88° 83° 885 £86 $85 25 co sac... <2
88... 38.? 8:3 2 So- 285 88.? an co .2 <2
5:3 t... .86 3:3 ...286 285 ...mn. ..o an ..o ca: <2

mm .2. mm a ME A.
aeoem seem sooem Eocene NSo acne;

 

mm 23 356580 commemom

 

den; 33.. 5.. cocoa .35 as .35.: Lo <2 ache .8855: .8 8.8.383 as.

«533:? 832: .5588. FE noses—83 me 8282»?— ooh: 2.. mo :08 .8 Echo cheese 23 556803 comes—mom .2. 035.

174

Table 7.3. Frequency distributions of within herd Spearman’s rank correlations comparing
predictions and actual first lactation milk production for 576 herds.
Variables Compared with Spearman’s

 

 

Rank Correlations
Midpoint of RE‘, F52 RE, Actual3 FE, Actual
category
-095 2 44 45
-O.85 0 4 4
.075 0 4 3
-O.65 0 1 3
-055 0 5 5
.045 0 28 26
.035 0 12 7
.025 0 18 17
.015 0 27 28
-005 0 23 25
0.05 o 41 45
0.15 0 39 43
0.25 0 47 41
0.35 1 42 44
0.45 0 43 39
0.55 4 48 48
0.65 0 25 27
0.75 3 20 20
0.85 12 23 26
0.95 554 82 80

 

‘ Predicted MEM from the RE model.
2 Predicted MEM from the FE model.
3 Actual MEM.

175

3.9. 05 .0. 22.235
Ems. mu. new Ems. mm 5923 2.2.22.8 xcm. m.:m=..mmaw 4.... 939..

5.5.2.8 xcw. m.cm:..moaw
mad mod mud mod and mvd mad mmd 9.0 mod mod. 9.6. mud- mad- mvd- and- mod- mud- mad- mad-

» h h b p L b P — p n
— d u q 1 J a d J u a

N o m o _. o o o o o o o o o o o o N

h P h in P 1— p
u q u a u a -

 

    

NF

t cow

team

rcan

48v

1 can

 

0mm
- can

176

spiau io JaquinN

.622. 2.... .6. 22.6.8...
Ems. mm and sms. .32. 5652 6832.8 6.5. 6.88.88 .2 6.3..

8.8.2.8 xcu. 985.800
00.0 00.0 05.0 00.0 00.0 020 00.0 0N0 0'0 00.0 00.0- 0w.0- 0N0- 00.0- 0v.0- 00.0- 00.0- 05.0- 00.0- 00.0-

 

-0..

-0N

-00

-0v

 

spJau io JequinN

177

.86.. es .6. 22.6.8...
EmE m". 08 EmE .884. c853 80.5.2.8 v.8. Maw—E800 .0.» «50...

8.5.2.8 xcm. 988.800
00.0 00.0 0E0 00.0 00.0 0v.0 00.0 0N.0 0...0 00.0 00.0- 0'0- 0N.0- 00.0- 020- 00.0- 00.0- 0N0- 000- 00.0-

 

-0

-0..

-0N

r0n

r0v

-00

.00

-0~.

 

-00

spiau io JeqmnN

178

Chapter 8
PREDICTION OF CULLIN G PRIOR TO SECOND LACTATION BASED ON

PARENTS’ PREDICTED TRANSMITTING ABILITIES

Abstract

The objective of this study was to determine the ability of parents’ PTA for milk, fat and
protein to predict subsequent culling of heifers during rearing or first lactation. The data
consisted of the culling outcome of 5,619 Michigan Holstein heifers during rearing or first
lactation, and the heifers’ parents’ animal model evaluations fi'om the period in which the
heifer was born. To ensure asymptotic unbiasedness of the estimates the unobserved herd
efi‘ect was modeled as a fixed effect. Despite evidence that first lactation milk production
did influence producers’ culling decisions, the PTA of milk, fat, and protein for each
heifers’ parents did not predict subsequent culling during rearing or first lactation. This
result was not surprising given that first lactation milk production explains less than 20%
of the variation in culling and from Chapter 7 is was apparent that parents’ genetic
estimates explained less than 7% of the variation in first lactation milk production. This
poor predictive ability was consistent regardless of the distributional specification. Weak
evidence for the necessity of modeling the herd efi‘ect as fixed was present. Furthermore,
the inability to correctly predict culling reason broken down into voluntary and involuntary
reasons, coupled with the counterintuitive effects of parents’ PTA on the probability of
being milled for each of the reasons raised questions as to the existence and utility of

classifying culling reasons according to the voluntary nature of the culling decision. Also,

179

it can be concluded that Michigan producers are not using parents’ genetics in heifer

culling decisions.

Keywords: dairy culling, prediction, fixed effect

1. Introduction

Given a constant herd size, no seasonal effects on reproduction and production,
and culled cows being sold for meat (not dairy) purposes, optimization (McCullough and
DeLorenzo, 1996; Rogers et al., 1988; Congleton and King, 1985; Van Arendonk and
Dijkhuizen, 1985; Killen and Kearney, 1978) and simulation (Allaire, 1981; Allaire and
Cunningham, 1980) studies were unanimous in their conclusions that a culling rate of 20%
to 30% optimizes producer profit. While other simulation studies (Congleton, 1988;
Kuipers, 1980; Korver and Renkema, 1979; Pearson and Freeman, 1973), which
considered subjectively selected culling rates or strategies for comparison, indicated that
profit would be increased with culling rates below the average 37% culling rate1 in the
Michigan dairy industry in 1996 (Michigan Dairy Herd Improvement Association, 1996).
Comparability of the actual and optimal culling rate assumes no herd contraction or
internal expansion. The results of these multiple studies are compelling because, not only
did the various models employ different assumptions and analytical techniques, but as well,
the models operated under economic conditions representative of a number of countries

(i.e., the USA, England, Ireland, Netherlands). In the models, the increased profit

 

' Culling rate, annually defined as, the number of animals leaving the herd due to sale or death divided by
the average herd size.

180

associated with use of the optimal culling rates varied from 2% to 40%.

Comparison of only the mean profit by the studies implies risk neutrality, whereas
a majority of Michigan and Dutch dairy producers are risk averse (Huime et al., 1993).
The risk preferences of decision makers can have significant effects on their choices, so
the optimal rule assuming risk neutrality cannot necessarily be extrapolated to those who
are risk averse.

At least under risk neutrality, one direct ramification of Michigan’s excessive
culling rate is that Michigan dairy producers do not need all potential replacement dairy
heifers to enter the lactating herd, necessitating a method of heifer selection.

This need for heifer selection is in direct contrast to the present situation in which
most producers rear and freshen the majority of heifers produced on the farm (Connor et
al., 1989; Heinrichs et al., 1994). Animals are then selected largely on the basis of the level
of first lactation and expected future milk production. The simulation studies revealed that
while this practice maximizes genetic progress and milk production, the economic
performance is lower due to failure to consider the opportunity costs of higher culling
rates.

The ultimate objective of heifer selection is to choose the most profitable animals.
Van Arendonk (1991) showed that lifetime profit corrected for the opportunity cost of
postponed replacement is the correct method for ranking animals in terms of profitability.
Unfortunately, corrected lifetime profit of animals is generally not available and
furthermore, can not be determined prior to the culling of the animals. However, the

literature has focused on indicators of lifetime profitability that are apparent early in the

181

productive life of milk cows. No work has considered indicators apparent prior to
productive life.

The phenotypic correlation between first lactation actual milk and lifetime profit
corrected for opportunity cost has been reported at 0.54 (Cassell et al., 1993). Weigel et
al. (1995) determined the phenotypic and genetic correlations between corrected lifetime
profit and first lactation mature equivalent milk production to be 0.51, and 0.80,
respectively. For purposes of heifer selection, first lactation milk production may then be a
suitable proxy for corrected lifetime profit. Then by choosing animals based on first
lactation milk production producers have selected the correct animals.

However, this strategy of milking all heifers and heavily culling the herd to select
the highest producing animals is costly. Given the large price differential between a heifer
just prior to entering milk production and her value as a cull cow, the increased profit
fi'om even randomly selling some heifers prior to milking, and thereby lowering the culling
rate, more than compensates for the decreased ability to identify and select the highest
producing animals. In addition, with fewer heifers in the milking herd the proportion of
older, more profitable cows in the herd rises.

Currently the information provided to Michigan producers upon which to base
culling predictions is limited to the heifers’ estimated breeding values (EBV) of milk, fat
and protein production. The EBV are the average of the parents’ predicted transmitting
abilities (PTA). Because EBV milk provide estimates of the genetic merits for milk
production which should on average manifest as phenotypic performance, EBV milk may

have a role in heifer selection. While the culling models suggest the need for heifer

182

selection, they lend little insight into how selection should be conducted. The optimization
studies employed random replacement selection, and many simulation studies selected
replacements based on their simulated EBV milk. While this latter approach to selection
has intuitive appeal and indeed theoretical justification (Henderson, 1963), empirical
support for heifer selection on this basis is lacking. Working with swine, Quinton and
Smith (1997) found the within herd rank correlation with progeny performance was higher
for parents’ genetic estimates than for parents’ phenotypes. However they warn the
benefits of genetic estimates over phenotype in selection may be less in practice than in
theory.

Due to the positive correlation between EBV fat and protein with EBV milk,
genetic selection index theory, as well as statistical theory, suggest prediction of milk
production will be improved by incorporating information on the components. Because
selection is based on milk production, it is possible that the EBV or, alternatively, parents’
PTA of milk, fat and protein can explain subsequent culling in first lactation.

The objective of this study was to determine the ability of parents’ PTA of milk,

fat and protein to explain culling prior to, or during first lactation.

2. Materials and methods

2.1 Data collection

A total of 15,116 Michigan Holstein heifer records were collected from Michigan

DHI. Only Holstein heifers that met the following criteria were included in the data set to

183

evaluate the ability of parents’ genetic estimates to predict culling: l) as of January 1,
1995, identified as prepartum; 2) born between July 1, 1992 and December 31, 1992; 3) at
least one parent’s PTA for milk, fat, or protein recorded; 4) either a calving date between
July 1, 1994 and June 30, 1995, or if no calving date, a culling date prior to July 1, 1994;
5) if culled, a culling code was recorded.

The 15,116 heifer records were also used to generate a second data set to
evaluate the ability of first lactation mature equivalent milk production (MEM) to predict
culling in first lactation. To be included in this second data set, records were required to
meet criteria 1, 2, 3 and 5 from above as well, the heifer must have calved between July 1,
1994 and June 30, 1995.

Genetic data were the heifers’ parent PTA of milk, fat and protein fi'om the
January 1993 sire and dam animal model evaluations calculated by the Animal
Improvement Programs Laboratory, USDA (Beltsville, MD).

Culled heifers included those that died or were culled prior to second lactation.
Therefore, heifers considered culled included those animals that were sold or died during
first lactation, as well as, those heifers without a reported calving date.

After deleting heifers that were missing the genetic information of either parent,
records of 5,619 heifers were available for statistical analysis of the relationship between
parents’ genetic estimates and culling. These heifers were present in 781 herds. In efi‘ect
then, this data sample consisted of, for each herd, a cohort of heifers which were born
within a six month window, and then either calved in a six month window or were culled

in the two years that separated the windows. The number of heifers in a cohort for a given

184

herd varied in size from 1 to 157 heifers with median and mean sizes of 4 and 7.2,
respectively.

The second data set of MEM and culling outcome in the first lactation contained
12,128 heifers in 1,100 herds. Again, each herd was represented by a single cohort of
heifers which were born and calved within the prescribed time frames. The cohorts of

heifersvaried from 1 to 244 heifers, withamediansizeof 7and mean ofll.0.

2.2 Estimation technique

2.2.1 Population Model
Consider the unbalanced general population panel data or cluster probability model
P[Culled... =1 I pm, Herd-.1 = own-.8 + Herdt.) (1)
a = 1, ..., A1,; )2 == , ..., H.
where
P[Culled 1. =1 IPTA., Herd..] is the response probability of being culled conditional on
PTA and the unobserved herd efi‘ect. Culledt. is equal to one if animal a in herd h
was culled during the rearing period or first lactation, and zero otherwise;
G is a binary response fiinction with values in the open unit interval;
PTA. is a 1 x 6 row vector. The six elements of PTA... contain the first milk, fat and
protein PTA of individual a’s parents, generated following a’s birth;

and Herd. is an unobserved random variable representing the effect of herd h.

185

Given the need to empirically estimate the relationship between culling and parent
PTA, the pragmatic issue of appropriate estimation technique must be considered. This
consideration should include the distribution of the response probability as well as the
nature of the unobserved herd effect.

As is often the case, it was unclear, a priori, whether a normal or logistic
cumulative distribution function (cdf) best represented the probability of culled conditional
on the independent variables. The normal cdf would imply a probit model is appropriate,
while a logistic cdf would lead to a logit model. Pragrnatically, the probability density
fimctions of these two distributions differ only in the thickness of the tails and are
generally considered to be sufficiently similar that either is appropriate. Given the
ambiguity over the a priori appropriate distributional assumption, probit and logit models
were each considered.

However, decisions with regard to modeling the herd effect are potentially not so
innocuous. The herd effect, more generally termed an unobserved effect, cluster effect, or
repeated measure indicator, denotes that this effect captures similarity among some
observations in the data set. A priori, the magnitude of this effect is unobservable or
unquantifiable. With regard to the herd effect, the issue to be resolved is whether it should
be modeled as a random effect (RE) or a fixed effect (FE), with the former being used if
' the unobserved effect is assumed to have a distribution and the latter if the effect has no
distribution. But this classification of an unobserved effect based on its theoretical
distributional assumption ignores the pragmatically more important issue of whether the

unobserved effect is uncorrelated with the independent variables. Notice that, by

186

definition, if an unobserved effect is thought to be fixed (i.e., not have a distribution) it can
not be correlated with anything, including the independent variables.

Modeling unobserved effects in linear probability models is relatively straight
forward. Ifthe effect is considered random, it can be accounted for through generalized
least squares estimation, or least squares with a variance estimator robust to serial
correlation among the error term. But notice that, if the RE is correlated with any of the
independent variables, the resulting coeflicients are biased (Greene, 1993), due to an
omitted variable, namely the RE. If the unobserved effect is considered to be
distributionless, or of more pragmatic importance, to be correlated with the independent
variables, it can be modeled in a variety ways as a FE resulting in unbiased regression
coeficients. Finally, the assessment of the preferred modeling technique has been aided
through the test developed by Hausman (1978) to determine if FE estimation is required.

Unfortunately, for maximum likelihood estimation (MLE) of models with a
dichotomous dependent variable, no such general test is available to determine whether an
unobserved effect should be modeled as a RE or FE. While tests comparing the parameter
estimates fi'om RE and FE specifications may be available for specific models, the
modeler’s judgement must often guide the estimation.

Modeling an unobserved effect as a RE in a nonlinear probability model assumes
the unobserved effect has a distribution and is independent of the explanatory variables
(Hsiao, 1986, pp. 164-167). The FE approach does not reqmre the latter assumption. RE
estimation of nonlinear probability models generally involves integrating the unobserved

. efi‘ect out of the response function. Intuitively, if the unobserved effect is a firnction of the

187

independent variables, performing such an integration must also affect the independent
variables. Given that it is unclear what the independent variables would then represent, it is
even less clear how to interpret the resulting regression coemcients.

In considering the nature of the herd effect, it must be recognized that this efi‘ect
includes a cohort effect in addition to a herd management efi'ect, which in turn selects the
herd’s genetics. It is therefore plausible that the herd effect is correlated with the
independent variables. This suggests a FE estimation technique is appropriate, for the
results from a RE model would be asymptotically biased.

While the unobserved effect can technically be modeled as a RE or FE, modeling it
with a dummy variable should generally be avoided. This assumes that software that can
add the potentially large number of dummy variables is available. To model an unobserved
efi‘ect, herd effect, cluster effect or repeated measures indicator as a dummy variable
results in all parameter estimates, including those associated with the independent
variables, being asymptotically biased. This result is not dependent on the unobserved
efl‘ect being correlated with the explanatory variables.

In a limited set of Monte Carlo experiments, Heckrnan (1981) showed that
inclusion of a dummy variable to represent an unobserved effect in a probit specification
resulted in asymptotically biased estimates for the coefficients of the independent
variables. While the bias varied with the value of the coefficient, with 8 observations
contributing to each dummy variable and as long as no lagged values of the dummy
variables were included in the model, the bias was never more than ten percent. The bias

was towards zero, and increased as the variation of the unobserved effect increased. Of

188

course to permit predictions an estimate of the coefficient of the dummy variables is
required. This may be problematic for out of sample predictions.

Abrevaya (1997) generalizing the work of Hsiao (1986, pp. 159-161) and
Chamberlain (1984, pp. 1275-1276) showed that, for the case of two observations
contributing to each dummy variable, the maximum likelihood logit estimates of the
coeficients associated with the independent variables are asymptotically twice their true
value. While a closed form proof for the case of greater than two observations
contributing to an unobserved efi‘ect has not been discovered, there is evidence that the
bias may be minimal for sufficiently sized clusters. Monte Carlo simulation revealed that
when 20 observations contributed to each dummy variable the estimates were virtually
unbiased compared to those from FE logit (Hsiao, 1986, pp. 228).

In general, the use of dummy variables does not necessarily result in asymptotic
bias of the estimators. For example, inclusion of a gender dummy variable in a nonlinear
model will not adversely affect the properties of the estimators, as the gender coefficient is
based on the entire sample. If MLE is being used, this implies the sample size is
sufiiciently large for asymptotic analysis to be appropriate.

The bias originates with the fact that there are a limited number of observations
associated with any given unobserved effect. In a panel data set, the asymptotic properties
of the estimates are derivable only if the number of observations per unobserved efi‘ect are
held fixed, while the number of unobserved effects gets large. As a result, any estimation
of the coefficient associated with a dummy variable for an unobserved effect is dubious

. given that maximum likelihood results in large sample estimators. While statistically the

189

number of observations per unobserved effect can be large with the number of efl‘ects
fixed, a data structure traditionally dealt with in time series analysis, other issues arise such
as the stationary properties of the variables. Taking the example where the unobserved
effect is a herd efi‘ect, letting herd size tend to infinity is intuitively objectionable. It is
much more natural to assume the number of animals per herd is fixed and the number of
herds is large, than the reverse.

All analyses discussed will be asymptotic, making the reasonable assumptions that
herd sizes are fixed, while the number of herds is large, and the herd effect is potentially

correlated with the independent variables.

2.2.2 Probit model

While the FE probit can be conducted using firll maximum likelihood which
requires specifying the joint distribution of (Culledm, ..., CulledM), a common alternative
is that proposed by Chamberlain (1984). In general, he suggested assuming the
relationship between the unobserved effect and independent variables was such that the
expectation of Herd.I conditional on PTA.i was linear. A parsimonious form of this
assumption is

Hard. = w + m. it + d, (2)

where it is further assumed d.., the error term, is distributed Normal(0, 0'3 ).

PTA- represents, for herd h, the herd average values of each of the six elements of the

PTA]. VGCIOI'.

190

If it is assumed that G, the binary response function fiom Equation (1), is the
normal cdf the assumed conditional expectation of the unobserved effect (2) can be

substituted into the population model (1) resulting in
P[Culled-=1lprA-.rierd-i=G(PTA-.0-+ tunfixtx-i (3)
where p.- = 0/(1+af,)°-’, w = w/(1+a§)°-’, and it.- = 210+ 63 )°-’.

With this approach, only the probability of an individual animal being culled, and
not the joint probability distribution of all the animals in a herd being culled, is assumed to
be correctly specified. For this partial ML firnction to be correctly specified, P[Culled 1..
=1] conditional on the genetics of all the animals in the herd must be uncorrelated with all
firnctions of the independent variables for all other animals in the same herd, a reasonable
assumption. If the unobserved effect truly is a RE then M = 0. This can be tested with a
likelihood ratio test.

The normality assumption of the probit specification is met since the addition of
the two normally distributed variables ( i.e., the error terms from the population model and
(1..) results in a normally distributed composite variable (i.e., error term). Equation (3) can
then be estimated by running probit on all the observations pooled together. However, as a
result of using pooled probit, the residuals will necessarily be conditionally correlated
within herd due to the presence of d.., necessitating a robust variance estimator.

The presence of (1.. also causes attenuation bias in the resulting coefficients. As
shown above, the 134 are biased away from B towards zero. In considering the
interpretation of B and 34. the attenuation biased estimates are likely preferable. B, if they

' were possible to estimate, would reflect the effect of changing PTA... evaluated at the

191

mean herd effect which is assumed to be zero. As few, if any of the herds, actually have
the mean herd efi‘ect, the representativeness of such a result is unclear. In contrast, B.
represents the efl’ect of a change in PTA... averaged across the distribution of the herd
efi‘ect in the population, a more appropriate evaluation for a random heifer.

To further evaluate performance of the probit model, alternative firnctional forms
were considered. First, quadratic terms of each element of PTA... were added. A
likelihood ratio test was used to test the hypothesis that the coefficients of the quadratic
terms are jointly zero. Second, the assumption that the binary response firnction, G, in the
population model of Equation (1) was symmetrical was relaxed through specifying it to be
a log-log link firnction. Specifically the negative natural logarithm of one minus the
probability of being culled is equal to the exponential of PTA..B. The herd average values
were excluded as this approach to modeling the herd effect is specific to G being the
normal distribution. By ignoring the herd effect, the estimates that resulted fiom this link

function may have been biased.

2.2.3 Logit model

If it is assumed that the idiosyncratic errors have a logistic distribution, a FE logit
approach has the advantage of not imposing any restrictions about the nature of the
arbitrary relationship between Herd. and PTA... It does however, require that within a
herd the outcomes are independent. In the present analysis this amounts to assuming that
after controlling for genetics, the culling of each heifer is independent of the culling of all

. other heifers in a given herd. Violations of this assumption would include a producer

192

predetermining the number of members of the cohort to be culled, or a common
environmental shock among some, but not all, heifers increasing their probability of being
culled. Of course, the errors are also assumed to follow a logistic distribution.

Beginning with the joint distribution of (Culled..., Culledu) which is then
A.
conditioned on the genetics of all animals in a herd, the herd effect, and z Culledh , the
.21

FE logit model can be derived. The resulting estimates of B are asymptotically unbiased.
The outcomes, for a given herd size, are the different possible combinations of culled
heifers that result in the total number of heifers sold. Those animals in herds in which all or
none of the animals in the sample were culled do not contribute information to the
estimation since their likelihood firnction is one. As Chamberlain (1984) shows, due to the
logistic function, for each combination of herd size and number of animals culled the result
is a multinomial logistic distribution where the independent variables are differences of the
right hand side variables of equation (1). As a result, the unobserved herd efi‘ect cancels,
leaving the joint distribution of the outcomes conditional on the data and number of
successful outcomes. This approach is also termed conditional logistic, presumably due to
the conditioning on the herd outcomes. See Appendix A for a fixed efi‘ect logit
(conditional logistic) proof for a cluster of 3 with one success.

The FE logit estimates are for the observations evaluated at the level of the
average herd. The average herd effect where, for a given PTA..., the estimate is averaged
across the distribution of Herd. in the population, requires a distributional assumption on

the unobserved herd effect.

193

In summary, the FE probit has the disadvantage of requiring strong assumptions
about the relationship between the unobserved effect and the independent variables, it
does however result in the average treatment effects. In contrast, the FE logit
specification, while not requiring any assumption about the relationship between the Herd.
and PTA.., or the distribution of the herd efi‘ect, does demand conditional independence
of the outcomes. The FE logit discards all data for clusters with invariant outcomes, and
the coeficients are evaluated at the average unobserved efi'ect. Finally, to be useful in
prediction the FE logit requires a priori knowledge of the herd size and number to be

culled.

2.2.4 Multinomial logit and ordered probit models

The ability to correctly classify culls conditional on PTA... and the fact that the
animals had been culled was evaluated with a multinomial logit model and an ordered FE
probit model. These models are extensions of their respective logit or FE probit models
with a dichotomous dependent variables generalized to encompass multiple positive
outcomes (Greene, 1993) which in the case of ordered models follow a logical ordering.
For these models the dependent variable was recoded into culling classifications fiom most
to least valuable animal where O, 1, 2, 3 corresponded to not culled, culled during first
lactation for reasons other than low production, culled during first lactation for low
production, and culled prior to first lactation, respectively. A value of 1 then represented
what has traditionally been considered an involuntary cull, and a 2 corresponded to a

_ voluntary cull.

194

Theoretically, the ordered model, by utilizing the information contained in the
ordering of the outcomes, should result in performance superior to the multinomial
specification. Among the ordered models, the probit was selected over the logit because
the probit permitted modeling of the fixed herd effect. Failure to account for a herd efi‘ect
which was correlated with the independent variables would result in biased estimators.
Assuming the herd efl‘ect existed, even if it is not correlated with PTA, it is unclear how a
the logistic distributional assumption could be true for a general model such as Equation
(1) extended to categorical responses. For there are no known distributions for the
population model error term and the herd effect which when summed will result in a
logistic distribution. Instead an approximate two step analysis of random effects ordered
logit estimation with an extra distributional assumption has been developed (Stiratelli et
al., 1984). In the first stage the within cluster variation is modeled assuming a logistic
distribution, followed by modeling of the assumed normally distributed random efi‘ect in

the second stage.

2.2.5 Prediction of first lactation culling based on first lactation MEM
The assumption that producers based selection on milk production can be
represented by the following model
P[CulledLact ..- =1 IMEM.., Herd..] = G(pMEM ., + Herd.) (4)

where

195

P[CulledLact .. =1 1PTA.., Herd..] is the response probability of being culled conditional on
MEM and the unobserved herd efi’ect. CulledLact .. is one if animal a in herd h
was culled during first lactation;

MEM..- is the first lactation mature equivalent milk production for animal a in herd h;

and Herd. is an unobserved random variable representing the effect of herd h.

Using an approach similar to that discussed earlier in the estimation of equation

(1), G in equation (4) was assumed to be either a normal or logistic distribution and the

unobserved herd effect was modeled as a fixed effect.

2.2.6 Model evaluation

Prior to considering culling predictions, evaluation of the marginal efl‘ects of parent
PTA on the probability of being culled serves as another measure of model performance.
To evaluate the efiem of a change in one element of PTA... on the probability of being
culled requires taking the derivative of the probability with respect to the element of
interest. This resulting expression will be a firnction of all the elements of PTA... As is
typical when dealing with continuous variables, the function was evaluated at the mean of
each of these elements, including the mean heifer cohort size for the probit estimates.

As the primary focus of these analyses were to ascertain the ability of parent PTA
to predict subsequent daughter culling during rearing or the first lactation, the predictive
value of each model was of interest. For each model the predicted probability of being
culled was generated. However, for the FE logit the predicted probabilities were limited to

. those animals in cohorts where a single animal was culled. This limitation is due to the fact

196

that the calculation of predicted probabilities, which is a firnction of the all the potential
combinations of culling outcomes for all animals in the cohort that results in the total
number of animals culled, is difficult for cases where more than a single animal was culled
in each cohort. Specifically, as is apparent from simple extension of Appendix A to larger
cohort sizes, the predicted probabilities are functions of difi‘erent independent variables for
each combination of cohort outcome. The 95% confidence intervals of the FE probit
predicted probabilities of being culled were also calculated and evaluated.

A threshold of 0.5 was used in assigning the predicted probabilities to predicted
outcomes. The predicted outcomes of each model were then compared with the actual

culling outcome.

3. Results and Discussion

Means, standard deviations, minima and maxima for parent genetic information are
in Table 8.1. Notice that the mean PTA for fat and protein is less than 30 kg for sires and
under 12 kg for dams. Of the 5,619 animals, 9.0% were culled prior to first lactation and
15.4% were culled during first lactation for a total culling rate prior to second lactation of
24.4%.

Table 8.2 reports the results of the regression analyses with dichotomous
outcomes of culled on parent PTA. The Chi-square near the bottom of the table reveals
each model explains a statistically significant amount of the total variation. However, the
pseudo R2 values suggest the models explain little of the variation in culling prior to

second lactation. This result of model statistical significance, in the face of poor

197

explanatory nature of the models, is consistent with large sample size driving the statistical
significance. As explained earlier, one drawback of the logit specification over the probit
in a FE fiamework is the potential loss of observations. In the present analysis, 967
observations could not be used in the logit analysis as these animals belonged to cohorts
which were invariant with respect to culling prior to second lactation.

The likelihood ratio test of A... = 0 resulted in a Chi-square statistic of 10.64, with 6
degrees of fieedom or a level of significance of 10.0%. While this provides only weak
evidence that the herd effect nwded to be modeled as 8 FE, this conservative approach is
appropriate given the alternative of potentially biased estimates fiom the RE specification.

The absolute size of the respective model coefiicients reported in Table 8.2 are not
directly comparable due to scale differences, attenuation bias in the probit estimates, and
evaluation of the herd effect at different values. The coefficients of the FE logit model
indicate the direction of the change in the probability of being culled given an increase in
the independent variable. Care must be taken in making the same assessment in the FE
probit models due to the presence of the herd mean variables. To assess the efi’ect of an
independent variable in the FE probit model on the response probability, its indirect
influence through the herd mean of the respective variable must also be considered. Sire
protein, dam milk and darn protein pose potential problems because their signs difl‘er from
their respective herd mean coefficients. The sire protein coefficient dominates the herd
mean effect, so the direction of this effect will remain negative. And since the efi‘ect
mediated through the herd mean effect will be divided by the size of the heifer cohort, for

‘ the average cohort size of 7.2 and for all but the smallest cohorts, the signs on dam milk

198

and dam protein will not change. The signs of the coefficients between the FE probit and
FE logit models are then identical, suggesting the qualitative results are not sensitive to the
differing assumptions made between the models. Ignoring the effects of the herd mean
PTA, the coefiicients’ levels of statistical significance were also similar between the two
models, thereby supporting the robustness of the results. The high degree of correlation
between PTA for milk, fat and protein likely contributed to the lack of statistical
significance of many of the variables. In the presence of multicollinearity the coeficients
remain unbiased (Greene, 1993, pp. 266) although the standard errors are increased over
the situation where multicollinearity is not present.

Three of the PTA, namely sire fat, dam fat and darn protein, each possess a
positive sign, suggesting increases in any of these, keeping all other independent variables
the same, resulted in an increased probability of being culled prior to second lactation.
This selection against fat, while not significantly different fiom zero at a P value of 5%, is
consistent with component milk pricing. However, component milk pricing which places
higher relative value on protein should have resulted in sire and dam protein PTA having a
protective efi'ect. The weak statistical significance of the positive sign of dam PTA protein
is then unexpected. Rational expectations, or the ability of Michigan producers to survey
the operating environment and surmise component pricing would be forthcoming, could
explain why selection may have been consistent with such a pricing scheme although the
scheme was not in effect at the time of selection.

To evaluate the practical significance of a 100 kg change in a parent’s PTA on the

probability of being culled, the derivative of the respective model was taken with respect

199

to the PTA of interest. This in effect rescales the logit and probit coeficients from their
logistic and Z scales, respectively. Table 8.3 represents the mean efi‘ect and 95%
confidence intervals for a 100 kg change in a parent’s PTA on the probability of being
culled prior to second lactation.

As expected, based on the discussion of the signs of the coefficients in Table 8.1,
the signs of the derivatives are identical between the two models. The size of the
derivatives, for a given PTA, are the same order of magnitude for the FE probit and FE
logit models. The attenuation bias evident in Equation (3) caused the FE probit estimates
averaged over the herd effect distribution to be smaller in absolute size than estimates
evaluated at the average value of the herd efi‘ect, as were the FE logit estimates. With the
exception of sire milk, the FE probit estimates were smaller in absolute size than the
respective FE logit derivatives. Given the difference in the evaluation of the herd efi'ect,
along with the other variant assumptions including the distributional assumption, and the
relationship between the herd effect and the independent variables, the effect of a change
in a given PTA on the probability of being culled is strikingly similar between the two
models. This similarity suggests the results of the models are robust to the assumptions
made.

Increasing the milk PTA of sire and dam by 100 kg decreased the probability of
being culled by less than one percent and less than three percent respectively with narrow
95% confidence intervals. While changing the fat and protein PTA of either parent appears
to have a large efl‘ect on the probability of being culled, reconsideration of Table 8.1

reveals that with respective means of less than 50 kg, the effect of changes in PTA on

200

probability of being culled is minimal. The evaluation of the derivative for what amounts
to a very large change in the independent variable explains why the upper bound of the
95% confidence interval of the FE logit derivative with respect to dam protein extends
beyond unity. The confidence intervals for the derivatives with respect to fat and protein
of each parent were quite wide.

Care should be taken in the ceteris paribus interpretation of the efi‘ect of PTA on
the probability of being culled. Due to the high correlation between the PTA of a given
parent, single PTA do not generally change in isolation. When evaluating the relative
probability of two heifers being culled, the joint effect of changes in all of a parents’ PTA
should be considered. However, the summing of positive and negative effects tends to
cancel the effects leaving little change in probability of being culled with difi‘erent parents.
This finding is consistent with the low pseudo R2 of the models. The multicollinearity also
widened the confidence intervals listed in Table 8.3.

Table 8.4 represents the predictive ability of the FE probit model. The model’s
performance is poor. The probit model did not predict the culling of any animals much less
the 1,370 animals that were actually culled. The poor predictive ability of the FE probit
model is consistent with the low pseudo R2 reported in Table 8.2, and the small efi‘ect of
reasonabley sized changes in the PTA on probability of being culled. Model performance
is just as poor when the confidence intervals of the probabilities of being culled were
considered. The 95% confidence intervals of only five animals included 0.5. Of these five

animals, three were actually culled.

201

Use of the traditional 0.5 threshold in classifying predicted probabilities into
predicted outcomes does afl‘ect the predictive ability of the models. The low pseudo R2
suggests the models are explaining little of the variation in culling beyond the mean. Using
a threshold of 0.5 when the mean level of culling of was 24.4% predisposed the models to
low specificity, and thus predictive value. This is simply an application of the well known
result that the prevalence afl‘ects the predictive ability of a model.

To firrther evaluate the poor performance of the model, alternative firnctional
forms of the model were considered. First, the addition of quadratic terms of each element
of PTA... did not alter the coefiicient signs of the original variables, nor did the predicted
culling outcome change for any animal. The hypothesis that the coefficients of the
quadratic terms were jointly zero was not rejected at a probability of 0.38 by a likelihood
ratio test with a chi squared statistic of 6.43 and 6 degrees of freedom. Second, G, the
binary response firnction, in the population model of Equation (1) was specified as the
asymmetrical log-log link function. Again, the coefficient signs of PTA..., and the culling
predictions were identical to the FE probit model.

Given the inability of the FE probit model to predict culling prior to second
lactation, the predictive ability of parent PTA conditional on more information was
explored. First predictions from the FE logit model were considered. Unlike the FE probit
model which made predictions conditional on PTA... and the herd average of PTA, the FE
logit predicts conditional on PTA... and the number of animals in the herd that were culled.

Due to the conditioning on the latter the FE logit is not useful in a priori prediction of

202

Table 8.5 presents the culling predictions for cohorts in which a single animal of
the heifer cohort was culled prior to second lactation. The difficulty in calculating
predicted probabilities where more than one animal is culled ji’mher problem with using
the FE logit model for out of sample predictions of which animals would subsequently be
culled. Of the 174 cohorts that culled a single animal the FE logit model correctly
predicted the number of animals culled in 35 cohorts. However, for 13 of those 35 cohorts
the incorrect animal was predicted to have been culled. The poor predictive abilities of the
models are again consistent with the low pseudo R2 reported in Table 8.2, the small efi'ect
of reasonable changes in the PTA on probability of being culled, and the FE probit
predictions.

It was considered that the poor predictive ability of the model may be due to the
attempt to predict voluntary and involuntary culling. Specifically, since the producer
presumably has little choice with regards to involuntary culls, or alternatively, the culling is
not due to an animal’s genetics but rather a random environmental shocls parent’s PTA
would be expected to predict voluntary but not involuntary culling. This ability to
correctly classify culls conditional on PTA” and the fact that the animals had been culled
was evaluated with a multinomial logit model and an ordered probit model. Pragrnatically
the multinomial logit model was rejected for the nonsensical results that the probability of
being culled for low production increased with sire PTA milk, and that increasing dam
PTA milk increased the probability of a heifer being culled prior to first lactation. Ifthe

herd efl‘ect was correlated with PTA the estimates from the multinomial logit model would

. be asymptotically biased.

203

Theoretically, the ordered FE probit by accounting for the herd efl‘ect and utilizing
the information contained in the ordering of the outcomes, should result in performance
superior to the multinomial specification. The results of the ordered FE probit model are
presented in Table 8.6.

The coeflicients fi'om ordered models are very difficult to interpret (Greene, 1993).
While in general, even the signs of the coefi'rcient can not be interpreted, in the present
case the coemcient signs do represent the direction a change in the respective independent
will have on the probability of each type of cull. The signs of all the coefi'rcients except for
that associated with sire PTA milk are identical to those fi'om the FE probit. As the
standard errors have not been corrected for correlation among the residuals, the true P
values will be larger than those reported. Table 8.7 represents the mean efl‘ect and 95%
confidence intervals of a 100 kg change in PTA on the probability of being culled for each
culling classification evaluated at the mean cohort size. Except for the effect of dam PTA
protein, the efl‘ects fi'om the ordered FE probit model were of the same sign and similar
magnitude as those efl‘ects from the FE probit and logit models. The reason for the
negative effect of dam protein is that negative herd mean dam protein effect overwhelms
the positive individual dam protein. The standard errors of the derivatives are generally
large relative to the mean effects, multicollinearity would increase the standard errors
while leaving the mean effects asymptotically unbiased.

Table 8.8 represents the predicted culling classification fi'om the ordered FE probit
model versus the actual culling classification for only those animals which were actually

. culled. Similar to the dichotomous FE probit model predictions, the predicted outcomes

204

performed poorly and fell overwhelmingly into the most prevalent classification which for
the ordered FE probit model was culled for other reasons.

The results in Table 8.7 suggest that the direction of the effect of changing a given
parent’s PTA is consistent for each of the three culling classifications. In general the
magnitude of that effect is lowest for production culls and highest for heifers culled prior
to lactation, although it is doubtful that the differences among the classifications are
practically significant. The relatively large standard errors indicate the difl‘erences among
the classifications are not statistically different either. While it is plausible for genetics to
have the greatest efl‘ect on heifer culling, one would have expected genetics to have
greater effects on culling for production than other culling reasons. The results in Table
8.7 indicate that genetics play a greater role in culling of first lactation animals for what
has traditionally been considered involuntary reasons and that genetics play a lesser role in
culling for low production which has generally considered a voluntary culling decision.
This result is dificult to accept. A more plausible conclusion is that, at a minimum, the
' classification of culls into voluntary and involuntary for lactating animals is inappropriate
and rather these results are a function of modeling nonexistent classifications. The failure
to correctly predict the culling classification supports this argument. Of course, the
alternative explanation for the unexpected results with regard to Table 8.7 is classification
error. But again this explanation by suggesting producers have such difficulty classifying
animals into voluntary and involuntary classes further supports the lack of utility of this

approach.

205

Upon reconsideration, the failure to find results consistent with the concept of
voluntary and involuntary culling is not surprising. Every culling decision involves a
voluntary choice as supported by Grohn et al. (1998) whose “research indicates dairy
farmers consider many factors, including diseases, milk yield, conception status, parity,
and stage of lactation, when deciding whether and when to cull a cow.” The dairy industry
must resolve this intuitively obvious statement by Grohn et al. with another equally
generally accepted view that “Dairy cows may be culled for either involuntary reasons
(i.e., death, acute disease, infertility) or voluntary reasons (i.e., low yield)” which is
included in their introduction. Sudden death and sterility, but not infertility, are the only
afllictions of a milk cow which result in a producer having no choice in the culling
decision. This is not to suggest animal removal reasons have no place in herd management,
but rather the concept of the voluntary nature of the removals has little place, particularly
in culling and replacement strategies.

Moreover, the issue of voluntary versus involuntary culling is not merely a matter
of schematics, but rather may have practical importance in improving producers’
decisions. The concept of involuntary and voluntary culls may be the greatest impediment
to producers realizing increased profit through the strategy of lower culling rates. For if
producers and their advisors continue to believe they have no choice in many of their
culling decisions, their opportunity to change these decisions and their culling rates is
severely inhibited.

Table 8.9 contains the results of regressing culling on MEM using the FE probit

. and logit models. As expected in each model the coefficient associated with MEM is

206

negative indicating higher production has a protective effect on the probability of being
culled. The probability of being culled drops by 7.3% and 0.1% for every extra 1,000 kg.
of MEM for the FE probit and FE logit models, respectively. In the FE probit model, first
lactation milk production is explaining almost 15% of the variation in the probability of
first lactation culling, and when conditioned on the number of heifers in the cohort that
were culled during first lactation, MEM explained over 20% of the variation in milling in
the FE logit model.

While first lactation production did not explain a large portion of the variation in
first lactation milk production it’s predictive ability, as presented in Tables 8.10 and 8.11,
was fair and improved over the predictive ability of parents’ genetics. Of those animals the
model predicted would be culled 71.5% actually were, while 83.0% of the animals
predicted to be kept were correct. The primary shortcoming of the FE probit model’s
predictive performance was predicting only 656 (5.4%) of the heifers would be culled.
While 469 of these predictions were correct, the predictions underestimated the 20%
culling that actually occurred in the population. Among the 238 cohorts with a single
animal culled, the FE logit model correctly predicted a single animal was culled in 106 of
these heifer cohorts. And for 73.6% of these 106 cohorts the FE logit model correctly
predicted which animal was actually culled.

So despite a mean probability of being culled of 20%, using MEM the models still
had reasonable predictive performance. This suggests the poor predictive ability of the
models based on parents’ genetics was not due to the low mean culling outcome of 0.24,

. but rather the poor model performance was due to the inability of parents’ genetic

207

estimates to explain the variation in culling.

Inability of parents’ genetic estimates to predict culling, despite milk production
afl‘ecting producers culling decisions can be explained based on Chapter 7 results. While
milk production plays a role in producers’ culling decisions, since fi'om Chapter 7 it was
revealed that parents’ genetic estimates explained less than 7% of the variation in first
lactation milk production, it is not surprising that genetics explaining virtually none of the
culling decisions.

4. Conclusion

Despite evidence in the present data set that first lactation milk production plays a
role in producer culling decisions, parents’ PTA of milk, fat, and protein available
following the daughters’ births did not predict subsequent culling during rearing or first
lactation. Even when conditioning on information that would not be available for culling
predictions, parents’ PTA were still unable to predict culling. Given that first lactation
milk production explains less than 20% of the variation in culling, the poor predictive
ability of parents’ PTAs for subsequent culling are consistent with the Chapter 7 results
that parents’ genetic estimates explain less than 7% of the within herd variation in first
lactation milk production. Furthermore, the failure to predict whether a culled animal was
culled for voluntary or involuntary reasons, along with the unexpected effects that parents’
genetics had on the probability of being culled for each of these classifications suggest
voluntary and involuntary culls do not exist, and may be a barrier to management.

Moreover, the consistency of the result across the unordered FE probit and logit models,

208

as well as the ordered FE probit model suggests these results are not a fimction of the
distributional and other assumptions made in each of the models. These results also

suggest that Michigan producers are not using parent genetics in heifer culling decisions.

Acknowledgments

The authors thank Al Thelen, Terry Speer and Lyle Smith at Michigan DHI, as
well as, Leigh Walton and H. D. Norman at the Animal Improvement Laboratory, USDA
for assistance with data collection. Jeff Wooldridge’s econometric comments were greatly
appreciated. The financial support of the primary author by the Department of Agricultural
Economics through the Animal Industry Initiative firnds provided by the Agricultural
Experiment Station at Michigan State University while this work was conducted is

gratefully acknowledged.

209

References

Abrevaya, J. 1997 . The equivalence of two estimators of the fixed-effects logit model.
Econonrics Letters, 55:41-43.

Allaire, F. R 1981. Economic consequences of replacing cows with genetically improved
heifers. J. Dairy Sci. 64:1985-1995.

Allaire, F. R., and E. P. Cunningham. 1980. Culling on low milk yield and its economic
consequences for the dairy herd. Livest. Prod. Sci. 7:349-3 59.

Cassell, B. G., B. B. Smith, and R. E. Pearson. 1993. Influence of herd-life opportunity
and characteristics of cows and herds on different net income firnctions. J. Dairy Sci.
76:1182-1190.

Chamberlain, G. 1984. Ch. 22 Panel data. In: Z. Griliches and M. D. Intriligator (Editors),
Handbook of Econometrics, Volume II. Elsevier, North-Holland, pp. 1247-1318.

Congleton, W. R. 1988. Dairy cow culling decisions. 2. Profitability and genetic trend in
herds with culling on production versus income. J. Dairy Sci. 71 : 1905-1915.

Congleton, W. R, and L. W. King. 1985. Culling cows on predicted firture profitability
with a variable planning horizon. J. Dairy Sci. 68:2970-2983.

Connor, L. J., L. G. Hamm, S. B. Nott, D. Darling, W. Bickert, R. Mellenberger, H. A
Tucker, 0. B. Hesterman, J. A. Partridge, and J. H. Kirk. 1989. Summary of the 1987
Michigan State University dairy farm survey. Research Report 498, Michigan Ag. Exper.
Station, East Lansing.

Greene, w. H. 1993. Econometric Analysis, 2"“ ed., Macmillan Publishing Company, New
York.

Grohn, Y. T., S. W. Eicker, V. Ducrocq and J. A. Hertl. 1998. Effect of diseases on the
culling of Holstein dairy cows in New York state. J. Dairy Sci. 81:966-978.

210

Hausman, J. A. 1978. Specification tests in econometrics. Econometrica 46:1251-1270.

Heckman, 1981. Ch. 4 The incidental parameters problem and the problem of initial
conditions in estimating a discrete time-discrete data stochastic process. In: C. F. Manski
and D. McFadden (Editors), Structural Analysis of Discrete Data with Econometric
Applications, the MIT Press, Cambridge, Massachusetts, pp. 179-195.

Heinrichs, A J., S. J. Wells, H. S. Hurd, G. W. Hill, and D. A. Dargatz. 1994. The
national dairy herd evaluation project: A profile of heifer management practices in the
United States. J. Dairy Sci. 77:1548.

Henderson, C. R. 1963. Selection index and expected genetic advance. Page 141 in
Statistical Genetics and Plant Breeding ; a symposium and workshop. W. D. Hanson and
H. F. Robinson, ed. Natl. Acad. Sci-NRC, Washington, DC.

Hsiao, C. 1986. Analysis of Panel Data. Cambridge University Press, Cambridge.

Huime, R., A. Dijkhuizen, R. King, and S. Harsh. 1993. Goal, critical success factors, and
information systems needs on swine and dairy farms — Identified by workshops. Zeitschrift
ffir agrainformatik 3:61-65.

Killen, L., and B. Kearney. 1978. Optimal dairy cow replacement policy. Ir. J. Agric.
Econ. Rur. Sociol. 7:33-40.

Korver, S., and J. A. Renkema. 1979. Economic evaluation of replacement rates in dairy
herds II. Selection of cows during the first lactation. Livest. Prod. Sci. 6129-37.

Kuipers, A. 1980. Development and comparison of selection criteria for cows and bulls
with a dairy herd simulation model. PhD. Thesis, Univ. Wisconsin, Madison.

McCullough, D. A., and M. DeLorenzo. 1996. Effects of price and management level on
optimal replacement and insemination decisions. J. Dairy Sci. 79:242-253.

211

Michigan Dairy Herd Improvement Association. 1996. Michigan dairy production herd
evaluation report.

Pearson, R E., and A. E. Freeman. 1973. Effect of female culling and age distribution of
the dairy herd on profitability. J. Dairy Sci. 56:1459-1471.

Quinton, V. M., and C. Smith. 1997. An empirical check on best linear unbiased
prediction genetic evaluation using pig field recording data. Can. J. Anim. Sci. 77 :21 1-
216.

Rogers, G. W., J. A M. van Arendonk, and B. T. McDaniel. 1988. Influence of
production and prices on optimum culling rates and annualized net revenue. J. Dairy Sci.
71 :3453-3462.

Stiratelli, R, N. Laird, and J. H. Ware. 1984. Random-effects models for serial
observations with binary response. Biometrics 40:961-971.

Van Arendonk, J. A. M. 1991. Use of profit equations to determine relative economic
value of dairy herd life and production from field data. J. Dairy Sci. 74:1101-1107.

Van Arendonk, J. A. M. and A. A. Dijkhuizen. 1985. Studies on the replacement policies
in dairy cattle. III. Influence of variation in reproduction and production. Livest. Prod. Sci.
13:333-349.

Weigel, D. J ., B. G. Cassel, I. Hoeschele, and R. E. Pearson. 1995. Multiple trait
prediction of transmitting abilities for herd life and estimation of economic weights using
relative net income adjusted for opportunity cost. J. Dairy Sci. 78:639-647.

212

Table 8.1.

 

 

 

Parent PTA of milk, fat and protein for 5,619 animals.

Variable’ i so Minima Maxima
PTA Milk of Sire (100 kg) 8.43 2.38 -9.53 14.02
PTA Fat of Sire (100 kg) 0.26 0.09 -0.40 0.53
PTA Protein of Sire (100 kg) 0.23 0.07 -0.30 0.43
PTA Milk ofDam (100 kg) 3.66 2.71 -8.54 14.24
PTA Fat ofDam (100 kg) 0.11 0.10 -0.35 0.57
PTA Protein of Dam (100 kg) 0.10 0.08 -0.29 0.41
Table 8.2.

Coeficients and their standard errors from the FE probit and logit models of regressing
culling on parent’s PTA.

 

 

 

FE Probit FE Logit
Variable b SE‘ 0 SE
Sire Milk -0.001 0.012 -0.005 0.023
Sire Fat 0.221 0.270 0.427 0.515
Sire Protein -0.799' 0.461 -1.478 0.905
Dam Milk -0.056" 0.018 -0.106“ 0.034
Dam Fat 0.690" 0.298 1.332“ 0.634
Dam Protein 1.248' 0.684 2.327' 1.258
Herd Mean Sire Milk -0.026 0.031
Herd Mean Sire Fat 0.471 0.732
Herd Mean Sire Protein 0.535 1.328
Herd Mean Dam Milk 0.118“ 0.057
Herd Mean Dam Fat 0.696 0.970
Herd Mean Dam Protein -4.376' 2.313
x2 28.6" 12 df 18.6" 6 df
Pseudo R2 0.0045 0.0048
n 5,619 4,652

 

lRobust to correlation among the residuals.
'P S 0.10; ‘P S 0.05; "'P S 0.01.

213

Table 8.3.
Mean effect of a 100 kg change in PTA on probability of being culled, and associated 95%

confidence intervals (CI)

 

 

 

 

 

 

 

 

 

FE Probit FE Logit
Variable Mean 95% CI M 95% I
Sire Milk -0.002 (-0.009, 0.005) -0.001 (-0.012, 0.010)
Sire Fat 0.090 (-0.064, 0.244) 0.105 (-0.146, 0.355)
Sire Protein -0.227 (-0.493, 0.039) -0.363 (-0.794, 0.069)
Darn Milk -0.012 (-0.023, -0.001) -0.026 (-0.042, -0.010)
Dam Fat 0.246 (0.063, 0.429) 0.327 (0.021, 0.633)
Dam Protein 0.200 (-0.217, 0.617) 0.571 (-0.034, 1.181)
Table 8.4.
Predictive ability of FE probit model.
Actual
Predicted Culled Kept Total
Culled O 0 0
Kept 1370 4249 5619
Total 1370 4249 5619
Table 8.5.

Predictive ability of FE logit model for the 174 heifer cohorts with a single animal culled.
Predicted Number of Number of Herds Percentage of

 

Animals Culled per Herd Herds
0 137 78.7
1 35 20.1
2 2 1.1
Total 174 99.9

 

214

Table 8.6.

Coefficients and their standard errors from the ordered FE probit model of regressing
culling classification orgarent’s PTA.

 

 

Variable b‘ SE
Sire Milk 0.001 0.012
Sire Fat 0.237 0.274
Sire Protein -0.750 0.484
Darn Milk -0.040" 0.018
Darn Fat 0.724" 0.341
Dam Protein 0.554 0.681
Herd Mean Sire Milk -0.025 0.024
Herd Mean Sire Fat 0.672 0.558
Herd Mean Sire Protein 0.322 0.972
Herd Mean Dam Milk 0.111" 0.042
Herd Mean Dam Fat 0.462 0.786
Herd Mean Dam Protein -3.998* 1.686
p. 0.673 0.111
.12 1.140 0.112
p; 1.325 0.112
x2 244* 12 df
Pseudo R2 0.0027

n 5,619

 

‘P s 0.05; "P s 0.01. As the standard errors have not been corrected for correlation

among the residuals the reported P values understate the true values.

Table 8.7.

Mean effect and standard errors (SE) of a 100 kg change in PTA on the probability of

beingculled for each culling classification.

 

 

Culling Classification

 

 

 

Variable Otherl Production2 Heifer3
Mean .8; Mans 38 Maria .813

Sire Milk -0.0003 0.0012 -0.0001 0.0005 -0.0004 0.0018
Sire Fat 0.0367 0.0271 0.0132 0.0104 0.0469 0.0409
Sire Protein -0.0783 0.0478 -0.0283 0.0182 -0. 1000 0.0715
Dam Milk -0.0027 0.0018 -0.0010 0.0007 -0.0035 0.0027
Dam Fat 0.0874 0.0342 0.0316 0.0131 0.1116 0.0528
Dam Protein -0.0001 0.0935 -0.0001 0.0263 -0.0002 0.1005

 

I Culled during first lactation for reasons other than low production.

2 Culled during first lactation for low production.
3 Culled prior to first lactation.

215

Table 8.8.
Predictive ability of ordered FE probit model for those animals which were actually culled.

 

 

 

Actual
Predicted Otherl Productiori2 Heifer3 Total
Other1 678 187 502 1367
Production2 0 0 0 0
Heifer’ 0 2 1 3
Total 678 189 503 1370

 

’ Culled during first lactation for reasons other than low production.
2 Culled during first lactation for low production.
3 Culled prior to first lactation.

Table 8.9.
Coeficients and their standard errors from the FE probit and logit models of regressing
culling on MEM.

 

 

 

 

 

 

 

 

FE Probit FE Logit
Variable b SE‘ b SE
MEM -0.032“* 0.001 -0.060"* 0.002
Herd Mean MEM 0023"“ 0.002
x2 1023.8""‘ 2 df 1798.46"* 1 df
Pseudo 112 0.148 0.209
11 12,128 10,760
lRobust to correlation among the residuals.
"*P s 0.001.
Table 8.10.
Predictive ability of FE probit model of regressing culling on MEM.
Actual
Predicted Culled Kept Total
Culled 469 1 87 656
Kept 1,952 9,520 1 1,472
Total 2,421 9,707 12,128

 

216

Table 8.11.
Culling predictive ability of FE logit model using MEM for the 238 herds with a single
animal culled.

Predicted Number of Number of Herds Percentage of

 

 

Animals Culled per Herd Herds
0 132 55.5

1 106 44.5
Total 238 100.0

 

217

Chapter 9
COMPLEX SOLUTION TO THE VALUE OF GENETIC INFORMATION IN

PREPARTUM HOLSTEIN HEIFER SELECTION

Introduction

Genetics is a source of information that aids livestock producers in animal breeding
and selection decisions. Like other information sources, the value of genetic information is
dificult to discern. Previous studies on the value of genetic information have addressed
the issue of how genetic information should be valued, but did not address the value of the
genetic information per se. Hedonic pricing has been used to marginally value genetic
traits of sires to form a pricing mechanism for semen (Schroeder et al. 1992; Richards and
Jeffrey, 1996). At a more aggregate level, consideration has been given to assessing the
proper economic weights of various genetic traits to maximize producer income (Harris
and Freeman, 1993). But again the value of this genetic information was not directly
addressed.

The value of information (Antonovitz and Roe, 1987; Nerrnuth, 1982) is a natural
fi’amework to assess the value of genetics in producer decision making. In this fiamework
the information is represented by some arbitrarily accurate signal of an underlying random
variable or event. Based on the signal, the information user makes a decision. The value of
the information is represented by the increase in the decision maker’s welfare by using the
signal versus not using the signal in the decision. This framework suggests that the value

of the information is dependent on the signal and the decision to be made based on the

218

signal.

Genetic evaluations are signals of an animal’s true genetic ability. Furthermore
these signals are dynamic — as more information is collected on the animals and their
relatives more informative signals are produced. The value of the genetic information is
specific to the context of the decision, here we focus on the value of genetic information in
selection of prepartum Holstein heifers.

Given a constant herd size, no seasonal effects on reproduction and production,
and culled cows being sold for meat (not dairy) purposes, optimization (McCullough and
DeLorenzo, 1996; Rogers et al., 1988; Congleton and King, 1985; Van Arendonk and
Dijkhuizen, 1985; Killen and Kearney, 1978) and simulation (Allaire, 1981; Allaire and
' Cunningham, 1980) studies were unanimous in their conclusions that a culling rate of 20%
to 30% optimizes producer profit. While other simulation studies (Congleton, 1988;
Kuipers, 1980; Korver and Renkema, 1979; Pearson and Freeman, 1973), which
considered subjectively selected culling rates or strategies for comparison, indicated that
profit would be increased with culling rates below the average 37% culling rate1 in the
Michigan dairy industry in 1996 (Michigan Dairy Herd Improvement Association, 1996).

The results of these multiple studies are compelling because, not only did the
various models employ different assumptions and analytical techniques, but as well, the
models operated under economic conditions representative of a number of countries (i.e.,
the USA, England, Ireland, Netherlands). In the models, the increased profit associated

with use of the optimal culling rates varied from 2% to 40%.

 

‘ Culling rate, annually defined as, the number of animals leaving the herd due to sale or death divided by
the average herd size.

219

Comparison of only the mean profit by the studies implies risk neutrality, whereas
a majority of Michigan and Dutch dairy producers are risk adverse (Huime et al., 1993).
The risk preferences of decision makers can have significant effects on their choices, so
the optimal rule assuming risk neutrality cannot necessarily be extrapolated to those who
are risk averse.

At least under risk neutrality, one direct ramification of Michigan’s excessive
culling rate is that Michigan dairy producers do not need all potential replacement dairy
heifers to enter the lactating herd, necessitating a method of heifer selection.

This need for heifer selection is in direct contrast to the present situation in which
most producers rear and fieshen nearly all of heifers produced on the farm (Connor et al.,
1989; Heinrichs et al., 1994). Animals are then selected largely on the basis of the level of
first lactation and expected future milk production. The simulation studies revealed that
while this practice maximizes genetic progress and milk production, the economic
performance is lower due to failure to consider the opportunity costs of higher culling
rates.

The culling models can lend little insight into heifer selection. For while the
lactating herd, which was the primary focus of these studies, was modeled with
sophistication, the heifer modeling was simplistic. None of the culling studies dealt with
the issue of when in the rearing period was the most profitable time to make the selection
decision. Generally the models assumed that replacement heifers were available when

required. This was operationalized in the studies through either the sale of all animals at

220

birth and purchase of required replacements, or raising of all young stock with subsequent
sale of the excess animals.

Regardless of the method of heifer procurement, most optimization studies
assumed naive expectations with respect to future heifer profitability, and as a result
employed random replacement selection. Many simulation studies arbitrarily selected
replacements based on their simulated estimated breeding value (EBV) of milk. The EBV
milk, which represents the animal’s estimated genetic ability to produce milk, is the
average of the parents’ potential transmitting abilities (PTAs) for milk. PTAs, which are
estimated by the USDA, represent the genetic ability the parents are expected to transmit
to their ofi‘spring.

While selection based on genetics has intuitive appeal and indeed theoretical
justification (Henderson, 1963), empirical support for heifer selection on this basis is
lacking.

The objective of this study was to determine the value of parents’ PT As of milk,
fat and protein and their associated reliabilities in the prepartum selection of Holstein
heifers, and assess whether this value was sufficient to prompt producers to select heifers
on this basis. In order to accomplish the above objective it was necessary to derive an

optimal heifer decision rule.

Theoretical Framework for the Value of Genetic Information in Prepartum Heifer

Selection

The ultimate objective of heifer selection is to choose the most profitable animals.

221

Van Arendonk showed that lifetime profit corrected for the opportunity cost of postponed
replacement is the correct method for making animals in terms of profitability. This
measure of an animal’s lifetime profit explicitly recognizes that, in retaining the animal in
the herd, the average potential replacement’s profit was foregone.

With perfect information, heifer selection would be a two step procedure for
producers. Given that they desire to select N number of heifers, the heifers would be
ranked by lifetime profit corrected for opportunity cost and the top N selected. The
remaining heifers would then be culled. The second decision is when in the rearing period
these remaining heifers should be sold. The profit maximizing producer would sell the
animals in the period in which the gross sale price less the cumulative average variable

rearing cost is the greatest. This can be mathematically represented as

(1)

maxl'lf’fi“ = ;[[t—zs,]n, +233“, {(02%).}, a=1,...,A,,;t =1,...,r;

r 1 1:1
subject to

5;: s‘, 6 (0,1)
ZS, = n,

0

Where

II. is the profit of herd h from heifer selection

{1 if animal a is not selected (i.e. is culled) in period t
s _

0 otherwise

a, is the lifetime profit of animal a corrected for opportunity cost

222

r. is the revenue from the sale of a heifer in period t
c. is the average variable rearing cost of a heifer in period I
m. is the number of heifers, in herd h, to be sold
A. is, for herd h, the total number of heifers which form a cohort for selection purposes
Tis the total number ofperiods in the rearing process.

The simplicity of the objective filnction (1) is apparent when animals to be culled
and those not culled are separately considered. For that portion of the herd culled in
period 1’ the profit firnction simplifies to

1'1?“ =Z[r, -:c,], a=1,...,A,f""‘"; t=1,...,T.

[=1

While for the portion of the heifer herd that enters the lactating herd the profit firnction is

T
HW=Z[7ra-Zc,:l, a=1,...,A,f""“"; t=1,...,T.
n (:1

Unfortunately, corrected lifetime profit of animals is generally not available and
furthermore, can not be determined with certainty prior to the culling of the animals.
However, the literature has focused on indicators of lifetime profitability that are apparent
early in the productive life of milk cows. No work has considered signals apparent prior to
productive life.

The correlation between first lactation actual milk and lifetime profit corrected for
opportunity cost has been reported at 0.54 (Cassell, Smith and Pearson, 1993). While
Weigel et al. determined the correlation between corrected lifetime profit and first
lactation mature equivalent milk production, which is actual milk production standardized

for efl‘ect of age and season of calving, to be 0.51. For purposes of heifer selection, first

223

lactation milk production may then be a suitable proxy for corrected lifetime profit. Then
by choosing animals based on first lactation milk production producers have selected the
correct animals.

However, this strategy of milking all heifers and heavily culling the herd to select
the highest producing animals is costly. Given the large price difi‘erential between a heifer
just prior to entering milk production and her value as a cull cow, the increased profit
from even randomly selling some heifers prior to milking, and thereby lowering the culling
rate, more than compensates for the decreased ability to identify and select the highest
producing animals. In addition, with fewer heifers in the milking herd the proportion of
older, more profitable cows in the herd rises.

Currently the information provided to Michigan producers upon which to base
culling predictions is limited to the heifers’ EBV of milk, fat and protein production. The
accuracies of these signals, termed reliabilities, are also estimated by the USDA Reliability
is the squared correlation of an animal’s estimated genetic value and its true genetic value.
Thus reliabilities are the measures of the genetic signal’s infonnativeness. Because EBV
milk provide signals of the genetic merits for milk production, which should on average
manifest as actual performance, EBV milk or alternatively parents’ PTA milk, may have a
role in heifer selection.

Working with swine, Quinton and Smith found the within herd rank correlation
with progeny performance was higher for parents’ genetic estimates than for parents’
phenotypes. However they warn the benefits of genetic estimates over phenotype in

selection may be less in practice than in theory. From Chapter 7 it was apparent heifers

224

parents’ PT As explained less than 7% of the within herd variation in subsequent first
lactation milk production, and Pearson rank correlations between the linear model
predictions of first lactation milk production and actual production cluster around the
value of one-third.

This result is not surprising for, due to Mendelian sampling and environmental
variance, the reliability of a prepartum heifer’s genetic estimated producing ability is only
one-sixteenth of the parents’ summed PTA reliabilities. At a maximum, a heifer’s parents’
PTAs will have reliabilities of 0.40 and 0.10 in predicting the daughter’s genetic ability to
produce milk, and actual subsequent milk production, respectively. So genetics can be
considered a distorted or incomplete signal of lifetime profit corrected for the opportunity
cost of postponed replacement.

Due to the positive correlation between PTA fat and protein with PTA milk,
genetic selection index theory, as well as statistical theory, suggest the signal will be
improved by incorporating information on these three PTAs of the parents. Moreover, by
indicating the degree of distortion in the PTA signals, the reliabilities may themselves be
valuable in heifer selection. Following the definition of Nerrnuth, the signal incorporating
information on the parents’ three PTAs and associated reliabilities must be more
informative than a signal based solely on parents’ PTAs of milk, given that the latter is a
subset of the former. Furthermore, regardless of the source of information for the signal,
the signal will improve in infonnativeness over the course of the rearing period of the
heifers as the genetic estimates are periodically re-estimated as more information on the

heifer’s relatives are collected.

225

The application of capital asset replacement theory presents a caution, in addition
to that provided by Quinton and Smith, in the utility of genetics in heifer selection. The
study (Allaire, 1981) which based replacement selection on EBV had a lower optimal
arlling rate than all of the studies which randomly selected heifers (Allaire, and
Cunningham, 1980; Congleton, and King, 1985; Killen, and Kearney, 1978; Rogers, van
Arendonk, and McDaniel, 1988; Rogers, van Arendonk, McDaniel, 1988), except one
(McCullough, and DeLorenzo, 1996). Standard capital asset replacement theory indicates
if the profitability of the replacements selected based on genetics was higher than those
selected randomly, the optimal length of the more profitable heifers’ lives would decrease
(Robison and Barry) which is equivalent to an increase in the culling rate of the simulation
studies relative to those of the optimization studies. Then one possible explanation for the
similarity between the most profitable culling rates of the optimization and simulation
studies, is that the genetic information is not valuable in heifer selection.

Uncertainty in predicting a heifer’s subsequent profitability can be incorporated
into the initial objective function by noting that lifetime profit corrected for opportunity
cost is a firnction of first lactation milk production (L) which in turn is a function of the
heifer’s estimated genetic information (0..) which should increase in infonnativeness over
time.

(2)
nlaxII, = Z[(1‘ZS.)”.(L.(Ga-»+ str, -Z[1- H sq},], a =1,...,A,,;r =1,...,T
' a r r r 1

j:

The profit associated with random selection can be represented as

226

iii-Hr“ =;[[l-;S-]E[fl.(la.(0.))l+Zar- —Z[1-§:sj]£,}, a =1,...,A,;r =1,...,T

1-1

The lifetime profits corrected for opportunity costs is the expectation (E) of the heifer
cohort, and the only selection decision is with respect to which period to sell the excess ,
heifers.

The ex ante value of genetics in heifer selection can then be assessed by IT. -

1'11“”. Ifgenetics are of no value in heifer selection then 1'1.I = 1'17”“. The ex post value

of genetic information in heifer selection is represented by I'lf'fi“ -— 1'12"“.

Theoretical Decision Rule for use of Genetic Information in Prepartum Holstein Heifer
Selection

To assess the value of genetic information in prepartum Holstein heifer selection, a
decision rule transforming the information into action by the decision maker is required.
Given genetic theory’s silence on how to optimally select heifers based on dynamic genetic
information, a decision rule was developed.

Let the optimal decision rule for prepartum Holstein heifer selection be s,.(I, p)
subject to

05p.sl

T
S=Hp-

1:1

Where

227

I is a partitioned matrix of dimension .4. by T, composed of elements g..w., where g. is a l
x B vector, containing the B pieces of genetic information for animal 0 available at
time t, w. is a B x 1 vector of weights for the B pieces of genetic information in
time period. g..w. then is a composite signal of animal a’s genetic information at
time t.

p is a 1 x T vector whose elements, p., are the proportions of animals to be selected in
period t.

Sis the overall proportion of animals to be selected.

The decision rule begins with the first period (i=1) in which the heifers are ranked
by 9w. and the top p, -A;. are selected, and the remaining (I-p,) 24;. are sold. The number
of animals to be selected, p, oAh, is rounded to the nearest integer. The remaining animals
are then re-ranked in the second period (1 =2) by the second period composite signals, with
p;- $, A.) selected and the remaining sold. This process continues to the T "' period, by
which time only the desired overall percentage of heifers remain. These animals will then
realize their first lactation and associated estimated profit. Heifer selection can then be
considered in a dynamic value of information issue, weighing the value of the more

informative signal versus the cost of its attainment.

Deriving and Testing the Optimal Decision Rule - Empirical Methods
Mth the herd, as opposed to the individual animal, being the unit of interest, and
without knowledge of either the temporal relationship among the genetic information or

between the information and lifetime profit, the dynamic decision rule of how and when to

228

select heifers is not amenable to standard optimization techniques such as dynamic
programming. Rather Modified Box-Complex, a derivative free non-linear search
technique was employed. This method of sequential searching has been found useful in
globally optimizing nonlinear multivariable objective functions subject to constraints
(Harris, 1981).

Based on the description by Harris, Kuester and Mize’s procedure for Box’s
original Complex algorithm was modified to include Nelder-Mead’s flexible polyhedral
procedures.

In exploring the effect of the input parameters on the objective value it was noted
Modified Box-Complex was not consistently converging to a global optimum. Rather,
over a set of 10 runs with a given set of parameters and 100 randomly generated vertices
or vectors of search variables it was noted the final objective values varied by as much as
4.4%. The random creation of 1,000 initial vertices, of which only the 100 with the highest
objective values were subsequently used in the algorithm improved the consistency of the
results. Using this technique, for the same set of input parameters, the range of the
objective values decreased 25% despite increasing the number of runs to 20. The search
variables were limited to three decimal places for the 10 runs and then two decimal places
for the runs of 20.

The data set consisted of Michigan Holstein heifers that were born between July 1,
1992 and December 31, 1992 and first calved between July 1, 1994 and December 31,
1994. Genetic information collected consisted of the heifers’ parents’ PTA for milk, fat

and protein along with the milk-fat reliability and protein reliability from the 1993 to 1994

229

semi-annual sire and dam animal model evaluations calculated by the Animal Improvement
Programs Laboratory, USDA (Beltsville, MD), the exception being the dam genetic
estimates fi'om July 1993 which were unavailable. Heifer birth dates, calving dates and first
lactation mature equivalent milk production were collected fi'om Michigan DHI.

In efi‘ect then the data sample consisted of, for each herd, a cohort of heifers that
were born within a six month window of time and calved within a six month window of
time, with the two windows being separated by two years. Herds whose heifer cohort
contained less than 10 heifers were deleted, leaving a sample consisting of 1,982 heifers in
115 herds. The lifetime profit corrected for opportunity cost, associated with a given level
of mature equivalent first lactation milk production, was deterrninistically estimated for
each heifer based on information provided by Weigel et al. Generation of this profit
measure likely misrepresented the true distribution of heifer profits in two ways. The use
of a linear relationship between lifetime profit corrected for opportunity cost and first
lactation milk production assumed a constant marginal profit for milk, and as a result over
estimated the profit of high producing animals. Secondly, Weigel et al. generated the
relationship based on nonoptimal culling practices, which corresponded to lower heifer
profits. The resulting distribution of heifer profits would be narrower than the true
distribution due to compression of the upper tail of the distribution.

Table 9.1 contains descriptive statistics for g“, g“, L., and Ira from the data
sample. The milk PTAs are one to two orders of magnitude greater than the other genetic
information. Relative to their mean, the reliabilities have less variation than the PTAs. The

respective means of the genetic information rose consistently over the four periods. Within

230

herd, It. generally had a range of under $2,500.

Thirty-eight search variables then impacted the objective value: the ten weights for
the genetic information in periods one, three and four, along with the five weights in
period two, and the proportion of heifers to be selected in each of periods one through
three.

The number of animals selected in the fourth and final period was equal to S, so
the number sold in the final period was p, ‘pfpg‘Ah - S‘AI. if the result was positive. To
increase eficiency of the algorithm, the implicit constraint on the number selected, was
incorporated in the objective function as an artificial activity. Specifically, during the last
of the four periods, if S‘A. > p, ‘p2*p3‘A,. heifers could be purchased for $1,000 and a $0
lifetime profit corrected for opportunity cost. To prevent difficulties with finding corner
point solutions, p1. p2 and p3, were permitted to be greater than one, however the number
of animals available in periods prior to the fourth was constrained to be less than or equal
to the number in the previous period.

The input parameters required for the heifer selection problem included S and the
n. To make the transition from Michigan’s 1996 average culling rate of 37% to a 30%
culling rate suggests 80% of heifers need to be selected. In their calculation of 1994 7a.,
Weigel et al. included the cost of rearing, so only the net revenue from the sale of heifers
in each period was required for the empirical objective firnction. To generate the net
revenues associated with heifer sale in a given period, it was assumed that fixed costs,
such as building and machinery overhead, were sunk. Based on dairy heifer budgets for

1993 by Karszes, the total cost to raise a heifer in each period were calculated. The net

231

return per period from sale of a heifer was then the 15% sunk portion of total cost to raise
the animal in the respective period.

Given that the USDA genetic estimates are overwritten when re-estimated, an
extended time We would be required to collect multiple heifer cohorts for each herd in
order to generate an optimal selection rule for each herd. Rather a cross-sectional
approach to generate a single decision rule, which maximizes the summed profit of the
individual herds was opted for. The advantage of this method, besides the decreased data
requirements, is greater ease of decision rule implementation. However, the herd profits
generated with the cross-section approach cannot be greater than those of the time-series

approach. The new objective firnction associated with the cross-section approach can then

be represented as

(4) maxZQh, h =1,...,H.
S. h

Where Q). = IIh/Ah.

The use of omega, the average herd profit per heifer, imposes equal weighting of
each herd, regardless of its size, in the objective function.

Sample size calculations determined the requisite number of herds for one-tailed
analysis of paired t tests to detect a $20 difference in profit per heifer between random
selection and that associated with some selection rule which required information. Based
on the calculations, to detect a $20 difference the 115 herds were randomly divided in two
groups following stratification on herd size. For the average herd in Michigan, a $20
difi'erence would result in roughly a $1,000 annual increase in herd profit which we felt

would be suficient to interest producers in collecting the necessary information and using

232

the rule. The 58 herds used by Complex to derive the optimal decision rule, denoted as the
rule deriving sample, were comprised of 1,034 animals and ranged in size fiom ten to one-
hundred and thirty-two heifers, average of 17.8 and median of 14. The remaining 948
hdfers in S7 herds of ten to seventy-three heifers, average of 16.3 and median of 14,
termed the rule testing sample, was used to test the performance of the heifer selection
rules.

To evaluate the effect of the input parameters S and r, on rule performance the
three difi'erent scenarios listed in Table 9.2 were considered. In scenario 2, the efl‘ect of
the percentage of heifers selected on the optimal rule was evaluated through specifying a
70% selection percentage. This latter selection rate represents the lower selection
percentage required by producers with minimal heifer death loss or desiring a lower cow
culling rate than 30%. While in scenario 3 the effect of the r, were evaluated. Specifically,
later sale of heifers incurred increasing losses and the absolute difference between r, and
r;, and r; and r, was less relative to the other scenarios. Modified Box-Complex was used
to develop an optimal heifer selection rule for each scenario using the rule deriving sample
of 58 herds. Complex was used to generate twenty solutions for each scenario. The
optimal rule for each scenario, was the one of the twenty runs with the highest value of the
objective fimction.

Using a simulation program and the rule testing sample of 57 herds, the Complex
selection rule was compared with random selection and another selection rule under each
scenario. The simulation program simply calculated the average profit per heifer for each

herd given S, n, W., p and the animal information. Members of the dairy industry felt

233

random selection was naive, and suggested a rule based on static genetic Selection Index
theory which consisted of zero weights on all the genetic information except the parents’
PTA milk which were equally weighted. This was the same rule used in many of the
culling simulation studies. The signal based on PTA milk was less informative than the
composite signal consisting of PTA milk, fat, protein, and associated reliabilities. For both
the random and theoretical genetic rules, all selection was assumed to occur in the period
during which the net revenue from heifer sales was the greatest, which was period four for
scenarios 1 and 2, and period one for scenario 3. It was assumed that heifers were sold in
the same periods that the decision that they would not be selected to remain in the heifer
herd was made. In comparing Complex’s rules with the theoretical genetic selection rule
paired t tests were again employed, however, a zero difference was tested using a two-

tailed distribution.

Results

Unless otherwise stated, all results presented are evaluating the performance of the
rules on the 57 herd rule testing data which was not used in deriving Complex’s optimal
rules. The artificial activity of purchasing heifers did not enter the solution of any of the
rules generated by Complex indicating the constraint on the number selected was met.

The rules with the highest objective value for each scenario developed by Complex
using the rule deriving sample, along with the rule from selection index theory, are listed in

Table 9.3.

234

In conducting the twenty runs of each scenario occasionally rules, but never the
rule with the highest objective value, would generate the same value of the objective
fimction but with difi‘erent p and w.. Instances were found where a given herd’s profit
varied between the two rules, suggesting selection rules that make different selection
decisions for a considerable portion of the herds can result in the same value of (4).

More interestingly, in scenario 1 a case also existed where the profit for each herd
was identical between the rules suggesting more than a single vector of search variables
existed which resulted in the same animal selection decisions, w. for these rules are
reported in Table 9.4 (each rule conducted all selection in the fourth period). A rule
created by the average of the weights resulted in the same profit as the original rules for
each herd but one which experienced a relative loss of $58. This suggested the search
space between these two local optima was a shallow valley. The great similarity of the
values of the elements of w. which correspond to each parents’ milk PTA, and to a lesser
extent sire protein reliability and dam milk-fat reliability suggest these weights are driving
the similar results.

Averaging the weights from the twenty scenario 1 runs, which varied in objective
values fi'om $9,926 to $9,653, and applying the rule to the rule deriving sample resulted in
a value of an objective function of $9, 191 which was above random selection’s objective
value of $8,008. This suggests the valley between the local optima could be quite deep,
extending almost half way down to the floor of random selection. This low value of the

average also suggests why Complex encountered difficulty optimizing this objective

23S

 

function, for this average, denoted as the centroid in Complex, is used extensively as a
reference point for movement of the vertices.

Optimal selection of each herd would have resulted in an objective fimction of
$15,519, however focusing on the final period data for even a single herd it was
sometimes unclear how to derive a mle to optimally rank the animals. So this figure only
represents the absolute maximum any optimal rule could ever achieve, and it is highly
unlikely that any cross-section rule would approach this level of performance. Using the
genetic information, Complex accounted for approximately 25% of the range in profit
between perfect selection and random selection. This performance is consistent with the
low positive rank correlations between predictions of first lactation production and actual
production (presented in Chapter 7), and therefore lifetime profit corrected for the
opportunity cost of postponed replacement. For a given herd in the rule deriving sample,
Complex accounted for between -63% and 100% of the range in profit between perfect
selection and random selection.

For scenario 1, the optimal Complex rule selected all heifers in the final period.
Figure 9.1 presents a comparison of the performances of Complex’s optimal rule derived
under scenario 1 conditions and the theoretical genetic selection rule with random
selection. Specifically, the difference in average heifer profit for each herd for each rule
less the profit fi'om random selection is presented.

Random selection was not dominated in the first nor second degree by either of the
rules which utilized the information. While for the majority of herds random selection was

inferior to at least one of the other two selection rules, random selection provided the

236

“":;v' v . :0.

a. ‘,.IVI"I.

 

-.. !' HIM»... .4. ',,,.. ..
H.- .lllbl‘an_c-I_

, .
PM"

"is

highest profit of all three rules for ten of the 57 herds. Including ties, Complex’s rule had
the highest objective value for 39 herds under scenario 1, while for 33 herds the highest
average profit per heifer was associated with the theoretical genetic selection rule.

Table 9.5 presents the means and sample variances of the increase in herd average
profit per heifer for each of the two rules utilizing genetic information over random
selection for the three scenarios. Using a one-tailed t-Test the hypothesis that Complex’s
rule was not $20 more profitable than random selection was rejected at P = 0.045 (t
statistic = 1.73). The mean profit difference of $27.39 between selection based on genetic
theory and random selection was not significantly greater than $20 (P = 0.12, t statistic =
1.20).

Figure 9.2 presents the increase in average profit per herd through use of
Complex’s rule versus the rule based on genetic Selection Index theory. For the majority
of herds the difi‘erence in profit is minimal, although a few herds would greatly benefit
from use of one rule over the other. The less than $4 difference in mean profit of the rules
was not significantly difi‘erent statistically (P = 0.28 two-tailed, 1 statistic = 1.09) nor
pragmatically.

For scenario 2, Complex was used to generate a new optimal selection rule with
the 58 herds in the rule deriving sample. Again, this rule conducted all animal selection in
the fourth period.

First this rule was evaluated against the Complex rule generated under scenario 1
by specifying in the simulation of the scenario 1 rule on the 57 herd rule testing sample

that 30% of the heifers would need to be sold in the final period rather than 20%. While a

237

 

few herds in the rule testing sample would greatly profit over the use of one rule over the
other, the average loss per heifer by using the Complex rule derived under scenario 1
when applied to scenario 2 was $2.46 per heifer as opposed to using the optimal rule for
scenario 2. (Under scenario 2 with the rule deriving data sample the optimal rule derived
under scenario 2 was $4.76 more profitable than the optimal rule derived under scenario
1.) This small difi‘erence was not statistically different than zero (P = 0.45 two-tailed, 1
statistic = 0.76) and was not felt to be sufficient to warrant the industry employing another

rule for the difi‘erent proportion to be selected.

 

The Complex optimal rule derived under scenario 1 was then tested against the
theoretical genetic and random selection rules under the conditions of scenario 2. The
comparison of the performances of Complex’s optimal rule and the theoretical genetic
selection rule with random selection under scenario 1 was substantially similar to that
under scenario 1 presented in Figure 9.1. The distribution was slightly wider and flatter
than that in Figure 9.1, with random selection being the most profitable method of heifer
selection for 15 herds, 5 of which random selection was the most profitable method of
selection under scenarios one and two. Neither Complex’s rule nor the theoretical genetic
selection rule exhibited first or second degree stochastic dominance over random
selection. Including ties, Complex’s rule and the selection rule based on genetic theory
represented the maximum profit for 32 and 33 herds, respectively.

The $32.07 increase in mean profit associated with Complex’s rule was not

statistically $20 greater than random selection (P = 0.06, r statistic = 1.54). However, the

238

mean herd $35.21 increase in average profit per heifer as a result of selection based on
genetic theory was statistically greater than $20 (P = 0.03, r statistic = 1.96).

Again, while slightly flatter and wider, the histogram comparing the average profit
per herd through use of Complex’s rule versus the rule based on genetic theory was very
similar to that of Figure 9.2. Similar to the results under scenario 1, the $3.14 difi‘erence in
profits associated with the two rules which make use of the information were not
statistically (P = 0.45 two-tailed, 1 statistic = -0.76) nor practically different.

However, this result that the profit resulting from selection based on genetic theory
was greater than the profit of even the rule derived by Complex under scenario 2 to
maximize the objective firnction was unexpected. This result suggested the existence of a
rule within Complex’s search space superior to the rules Complex found. Applying the
static theoretical genetic selection rule to the 58 herds which Complex used to derive its
selection rules resulted in a herd average profit per heifer of $166 versus $177 for
Complex’s best rule under scenario 2. Applying Complex’s optimal rule fi'om scenario 1 to
the conditions of scenario 2 and the rule deriving data sample resulted in simulated profits
of $1 72. As such, the suggestion that Complex did not find a global maximum can not be
supported. Rather this unexpected result is due to sampling error among the choice
variables, such that the apparently optimal w. and p fi'om the rule deriving sample were
not optimal for the rule testing sample.

The presence of sampling error then indicates the existence of sampling variation
between the rule deriving and rule testing samples. However, it is unclear whether

sampling variation is necessarily the sole or primary source of the sampling error. The

239

 

 

creation of the two samples through random assignment of herds following stratification
on herd size, and the reasonably large size of the data samples would decrease the
probability of large sampling variation. It is possible that the level of sampling variation
was not great, but the variation was magnified by over-fitting Complex to the rule deriving
data resulting in the difi‘erent optimal choice vectors for the two samples.

However it is not clear that over-fitting of the model would be expected to
account for these results either. Over-fitting could be due to exploiting individual herd
efi‘ects in the rule deriving sample or by over-parameterization across the entire rule
deriving sample. With 57 or 58 randomly assigned herds in each sample and equal
weighting of each herd in the objective function, any individual, or group of idiosyncratic
herd efl'ects would be expected to be countered by other herd effects in the sample and as
a result it is not clear that herd effects could have been exploited in rule development. In
terms of model parameterization, of the 38 search variables, only thirteen parameters, the
ten of w. and the three of p, played any role in the scenario 1 and 2 results. Furthermore,
within w. the ten variables are not independent. While sire PTAs were expected to be
uncorrelated with dam PTAs, within each parent the PTA for sire, fat and protein were
highly correlated, and the reliabilities were also highly correlated, and as discussed
previously, the reliabilities also had limited variability. Given the modest number of
parameters, the high degree of correlation between many of them, and data sample sizes of
nearly 1,000, it was not clear to what degree the model was over-parameterized. So it was
unclear to what degree over-fitting of the optimal rule to either individual herd efi‘ects or

the entire rule deriving data sample was possible.

240

 

By way of analogy, assume that interest lies in forecasting using either a simple
rule or complicated rule, each of which are optimized through linear regression. Further
assume that the simple rule is the globally optimal rule, and is within the parameter space
of the complicated rule. It is very possible that the more complicated rule derived using
the rule deriving sample data could exploit the idiosyncrasies contained in that data sample
to produce a model with a higher R2. Statistically, the more complicated rule cannot result
in a lower 18 (although the adjusted R2 may in fact be lower).

However, when applied to the out of sample rule testing data the forecast ability of
the more complicated rule will be worse as the idiosyncrasies of the rule deriving data
sample will not be present in the out of sample data. In general, increases in the number of
parameters in the complicated rule’s parameter space increases the probability of over
fitting the rule to the in sample data. Increasing the size of the samples would lower the
probability of getting samples which differ in important ways, and would also minimize the
efl’ect of any one observation or cluster of observations on the deriving of the optimal
weights for the complicated rule.

Under scenario 3 with the increasingly negative returns to heifer rearing, only one
of the twenty rules generated by Complex completed all the necessary heifer selling in the
first period, and this rule had the third lowest value of the objective firnction. The
remaining rules generally selected 83 to 93% of the heifers from the rule deriving sample
herds in the first period, the remaining majority of the selection occurred in the second
period in 11 of the rules and in the third period in three rules. Commonly, a few herds sold

animals in the fourth period. As the latter tended to occur primarily in the larger herds, the

241

efi’ect on the objective firnction would be small as the loss in profit was averaged over the
entire herd before entering the objective function. The optimal rule selected 0.9, 0.91, 1.0
proportion of the heifers in periods one to three respectively, with a few herds selling
heifers in the fourth period. Of the three scenarios, this one had the greatest range in
objective values at 5.1%.

The variability of the periods in which selection occurred under the various r, may
reveal the factors that impacted the simple appearing trade-ofi‘s made in heifer selection.
“frthin a herd, the range in It, was generally between $1,000 to $2,500, so the profit fi'om
making a better choice in terms of which animals were selected could potentially more
than compensate for conducting some selection in periods which were nonoptimal in terms
of maximizing the herd profit from n. Under scenarios 1 and 2, with the improved
information in the fourth period and the $44 cost to total herd profit for each animal
selected in the third period all selection was conducted in the fourth period. However,
under scenario 3, Complex was routinely willing to pay $20 and occasionally $50 to attain
the improved genetic information and/or simply spread selection out over multiple periods
to improve those selected. It is not clear which effect dominated.

Similar to the approach taken with scenario 2, the Complex rule fi'om scenario 1
was adapted to operate in scenario 3 by forcing all selection to occur in the first period
based on w... The resulting rule, applied to the rule testing data, was $9.72 more profitable
per heifer for the average herd than using the Complex rule developed under scenario 3

conditions. In contrast, simulating the profit from use of this adapted rule on the rule

242

deriving sample data resulted in a loss of $1 1.80 as compared to the scenario 3 developed
Complex rule. This again suggested the increased profit was the result of sampling error.

The modified rule fi'om scenario 1 was compared against selection based on
genetic theory and random selection. The distributions for scenario 3 corresponding to
Figure 9.1 were again substantially similar. The selection rules based on the information
were not first or second order stochastic dominant to random selection. The majority of
herds profited from at least one selection rule based on the information versus random
selection. Interestingly, of the 15 herds for whom random selection was the most
profitable selection method under this scenario, for 5 of these herds random selection was
the superior rule regardless of scenario. Under scenario 3, the rule generated by Complex
accounted for the highest profit per heifer in 35 herds and while selection based on genetic
theory resulted in the highest profit for 34 herds.

Complex’s rule’s $32.04 was statistically greater than $20 more profitable per
heifer than random selection (P = 0.04, t statistic = 1.76), while the $25.09 difference
between the theoretical genetic selection rule and random selection was not (P = 0.21, r
statistic = 0.82).

The $6.95 difference in mean average profit per heifer between the two selection
rules which used the information was not statistically different than zero (P = 0.12 two-
tailed, t statistic = 1.58), and similar to previous scenarios a few herds would profit greatly

from use of one rule over the other.

243

Summary and Conclusions

Modified Box-Complex had some difficulty in optimizing the heifer selection rule,
which is not surprising given the 38 potentially relevant search variables and the
discontinuous nature of the objective function. Even more troubling was the sampling
error which was sumciently large that the optimal decision rule derived under one sample
was clearly non-optimal when applied to the other sample. It was not clear whether the
sampling error was due solely to sampling variation, or whether the variation was
magnified through over-fitting of the model to the rule deriving data sample. The sampling
error not only makes interpretation of the profits resulting fiom nrle use difficult, but also
does not bode well for herd individualized rules. For even under the USDA’s current
quarterly schedule of genetic data estimation, a decade of prospective data collection
could easily be necessary to amass sufficient information for developing and testing
individual herd rules.

The suggestion that for some herds random heifer selection would consistently be
more profitable than selection based on the genetic information warrants reconsideration.
While a priori it was plausible that genetics may have been of little value in heifer
selection, it is dificult to understand how, for a given herd, genetics could have negative
value resulting in random selection of heifers consistently providing higher profits than
selection based on genetics. It is possible that the single cohorts of heifers which
represented these five herds experienced a random event which led to poor performance of
the genetically superior heifers. In this situation, it is not surprising that selection based on

_ genetics would consistently have resulted in lower profits for this cohort of heifers

244

regardless of scenario. This suggests some of the differences between herds in terms of
profit per heifer was likely due to systematic herd effects and some was due to random
noise.

While the relative profit distributions evaluated were for a cross-section of herds,
with each herd consisting of a single cohort of heifers, the profit distribution of a given
herd for a number of cohorts would be of primary interest to dairy producers. The random
noise experienced across the herds would also be present across cohorts of a given herd.
However, the systematic herd effects can be decomposed into individual cohort efi‘ects
and a herd management effect constant across cohorts. Then for a given herd while the
herd management efi‘ect would afi‘ect the mean of the distribution of profits fi'om repeated
rule use on numerous cohorts it would not contribute to the width of the profit
distribution resulting in a narrower distribution than displayed in Figures 9.1 and 9.2.

As mentioned in the introduction, the selection of replacements which are more
profitable than the current defenders dictates the replacement rate should rise
concomitantly. However, it is difficult to determine to what degree the optimal culling rate
in the cow herd and therefore the selection rate in the heifer herd would change. Following
the initial period of heifer selection the rate of heifer selection would enter a steady state.
It is unclear what this steady state rate would be because it depends on the relative
fecundity of the cows, which are no longer being culled, compared to the heifers which are
now being sold. These are issues best addressed through linking the output of a heifer

selection model into a stochastic dynamic programming model of the cow herd.

24S

With the similar performance of the two mles, the rule suggested by genetic theory
of selecting on EBV of milk is preferred over Complex’s rule which is more complex and
requires information not currently reported to producers. It can be concluded that while
the composite signal based on parents’ PTAs of milk, fat and protein and associated
reliabilities was more informative than the signal based solely on parents’ PTA of milk, the
values of the two signals were pragrnatically equivalent. This similar performance of the
two rules which utilized the information suggests the value of the genetics information
presides primarily in the parents milk PTA, while the other PTAs and reliabilities have
relatively little value in prepartum Holstein heifer selection.

While it is likely that the solids PTAs are not valuable in heifer selection due to the
high correlation with PTA milk, the limited value may also have sourced from limited
variability in the data, this is likely more relevant for the reliabilities. Given that Complex’s
performance did not suffer when the decimal places for the weights was limited fi'om three
to two it is unlikely that the apparent unimportance of the genetic information other than
PTA milk was due to scale problems whereby the one to two larger order of magnitude of
PTA milk relative to the other genetic information simply overwhelmed the signal.

Complex’s optimal selection rule from scenario I appeared robust to adaptation to
the various other scenarios. For any of the three scenarios there appeared to be little
systematic difference between the distribution of profits resulting from the rules derived by
Complex and that suggested by genetic Selection Index theory. The robustness of these
results suggests genetic information is valuable in heifer selection, and the ex ante value is

estimated at greater than $20 per heifer realized over the course of the heifers’ lives. The

246

herd profit associated with a $20 per heifer increase in profit would make non-random
heifer selection a potentially worthwhile endeavor for Michigan producers.

While profitable, selection based on EBV of milk is likely capturing only a portion
of the profit potentially available through heifer selection. Recalling that lifetime profit
corrected for the opportunity cost of postponed replacement was a deterministic function
of first lactation milk production, this low capture ratio is then consistent with the results
presented in Chapter 7 revealing a diffuse distribution of within herd rank correlations
between parent PTA based predictions of first lactation milk production and actual
production. To capture this elusive profit the dairy industry needs to continue to explore
potential information sources other than EBV on which to base methods of heifer

selection.

Acknowledgments

The authors thank Al Thelen, Terry Speer and Lyle Smith at Michigan DHI, as
well as, Leigh Walton and H. D. Norman at the Animal Improvement Laboratory, USDA
for assistance with data collection. Jason Shelagowski’s assistance in FORTRAN
programming is acknowledged. The financial support of the primary author by the
Department of Agricultural Economics through the Animal Industry Initiative funds
provided by the Agricultural Experiment Station at Michigan State University while this

work was conducted is gratefully acknowledged.

247

References

Allaire, F. R “Economic consequences of replacing cows with genetically improved
heifers.” J. Dairy Sci. 64(1981):l985-1995.

Allaire, F. R, and E. P. Cunningham. “Culling on low milk yield and its economic
consequences for the dairy herd.” Livest. Prod Sci. 7:(1980):349-3 59.

Antonovitz, F ., and T. Roe. ”Chapter 9: Economic Efiiciency and Market Information” in
Economic Efiiciency in Agricultural and Food Marketing, ed. R L. Kilmer and W. J.
Arrnbruster, Ameszlowa State University Press, 1987.

Box, M. J. “A new method of constrained optimization and a comparison with other
methods.” Comput. J. 8(1965):42-52.

Cassell, B. G., B. B. Smith, and R. E. Pearson. “Influence of herd-life opportunity and
characteristics of cows and herds on different net income firnctions.” J. Dairy Sci.
76(1993):l 182-1190.

Congleton, W. R. “Dairy cow culling decisions. 2. Profitability and genetic trend in herds
with culling on production versus income.” J. Dairy Sci. 71(1988): 1905-1915.

Congleton, W. R, and L. W. King. “Culling cows on predicted future profitability with a
variable planning horizon.” J. Dairy Sci. 68(1985):2970-2983.

Connor, L. J ., L. G. Hamm, S. B. Nott, D. Darling, W. Bickert, R Mellenberger, H. A
Tucker, 0. B. Hesterman, J. A. Partridge, and J. H. Kirk. “Summary of the 1987 Michigan
State University dairy farm survey.” Research Report 498, Michigan Ag. Exper. Station,
East Lansing, 1989.

Harris, B. L., and A. E. Freeman. “Economic weights for milk yield traits and herd life
_ under various economic conditions and production quotas.” J. Dairy Sci. 76(1993):868-
879.

248

Harris, T. R “Appendix A: The Modified Box-Complex.” Analysis of Irrigation
Scheduling for Grain Sorghum in the Oklahoma Panhandle. Oklahoma State University
PhD. Dissertation, 1981.

Heinrichs, A. J., S. J. Wells, H. S. Hurd, G. W. Hill, and D. A Dargatz. “The national
dairy herd evaluation project: A profile of heifer management practices in the United
States.” J. Dairy Sci. 77(1994):1548.

Henderson, C. R “Selection index and expected genetic advance.” Page 141 in Statistical
Genetics and Plant Breeding ; a symposium and workshop. W. D. Hanson and H. F.
Robinson, ed. Natl. Acad. Sci-NRC, Washington, DC, 1963.

Huime, R, A. Dijkhuizen, R. King, and S. Harsh. “Goal, critical success factors, and
information systems needs on swine and dairy farms — Identified by workshops.”
Zeitschriftfiir agrainformatik 3(1993):61-65.

Karszes, J. “Dairy replacement programs: costs and analysis Western New York, 1993.”
1994 Heifer Management Symposium Mimeograph Series Cornell Cooperative Extension,

Killen, L., and B. Kearney. “Optimal dairy cow replacement policy.” Ir. J. Agric. Econ.
Rur. Sociol. 7(1978):33-40.

Korver, S., and J. A. Renkema. “Economic evaluation of replacement rates in dairy herds
II. Selection of cows during the first lactation.” Livest. Prod Sci. 6(1979):29-37.

Kuester, IL, and 1H. Mize. Optimization Techniques with Fortan. pp 368-385. New
York: McGRaw-Hill Book Co., 1973.

Kuipers, A. Development and Comparison of Selection Criteria for Cows and Bulls with
a Dairy Herd Simulation Model. Univ. Wisconsin, Madison Ph.D. Thesis, 1980.

McCullough, D. A., and M. DeLorenzo. “Effects of price and management level on
optimal replacement and insemination decisions.” J. Dairy Sci. 79(1996):242-253.

249

Michigan Dairy Herd Improvement Association. Michigan Dairy Production Herd
Evaluation Report, 1996.

Nelder, 1A, and R. Mead. “A simplex method for firnction minimization.” Comput. J.
7(1964):308-313.

Nerrnuth, M. “Information Structures in Economics” in Lecture Notes in Economics turd
Mathematical Systems. Monograph 196, ed. M. Beckman and H. P. Kunzi, pp 12-44.
Berlin:Springer-Verlag, 1982.

Pearson, R. E., and A. E. Freeman. “Effect of female culling and age distribution of the
dairy herd on profitability.” J. Dairy Sci. 56(1973): 1459-1471.

Quinton, V. M., and C. Smith. “An empirical check on best linear unbiased prediction
genetic evaluation using pig field recording data.” Can. J. Anim. Sci. 77(1997):211-216.

Richards, T. J ., and S. R. Jeffrey. “Establishing indices of genetic merit using hedonic
pricing: AN application to dairy bulls in Alberta.” Can. J. Ag. Econ. 44(1996) 251-264.

Robison, L. J., and P. J. Barry. “Chapter 21, Replacement Principles” in Present Value
Models and Investment Analysis. lst ed. The Academic Page, Northport, Alabama, 1996.

Rogers, G. W., J.A.M. van Arendonk, and B. T. McDaniel. “Influence of production and
prices on optimum culling rates and annualized net revenue.” J. Dairy Sci. 71(1988):3453-
3462.

Schroeder, T. C., J. A. Espinosa, and B. K. Goodwin. “The value of genetic traits in
purebred dairy bull services.” Rev. Ag. Econ. 14(1992):215-226.

Van Arendonk, J. A. M. “Use of profit equations to determine relative economic value of
dairy herd life and production from field data.” J. Dairy Sci. 74(1991):1101-1107.

Van Arendonk, J. A. M. and A. A. Dijkhuizen. “Studies on the replacement policies in

250

dairy cattle. III. Influence of variation in reproduction and production.” Livest. Prod Sci.
13(1985):333-349.

Weigel, D. J ., B. G. Cassel, I. Hoeschele, and R. E. Pearson. “Multiple trait prediction of
transmitting abilities for herd life and estimation of economic weights using relative net
income adjusted for opportunity cost.” J. Dairy Sci. 78(1995):639-647.

251

Table 9.1. Genetic information for the first and fourth periods, along with first lactation
milk production and lifetime profit corrected for the opportunity cost of postponed

 

 

 

252

replacement.
Variable‘ i so Minima Maxima
an
PTA Milk of Sire, lb 1,894.2 470.1 -1086 3093
PTA Fat of Sire, lb 59.3 19.6 -28 110
PTA Protein of Sire, lb 51.0 14.2 -19 94
REL of Sire Milk-Fat, % 88.8 8.6 24 99
REL of Sire Protein, % 88.2 8.9 24 99
PTA Milk of Darn, lb 874.4 574.7 -1269 2575
PTA Fat ofDam, lb 25.6 21.5 -44 109
PTA Protein of Danr, lb 23.8 16.5 -39 82
REL of Dam Milk-Fat, % 47.9 6.2 12 74
REL of Dam Protein, % 47.7 6.2 12 74
8-4
PTA Milk of Sire, lb 2,084.1 515.0 -1073 3347
PTA Fat of Sire, lb 64.0 21.0 -28 116
PTA Protein of Sire, lb 55.1 16.8 -19 94
REL of Sire Milk-Fat, % 95.3 7.1 28 99
REL of Sire Protein, % 95.1 7.3 28 99
PTA Milk of Dam, lb 928.2 647.6 -1240 2912
PTA Fat of Dam, lb 27.1 24.6 -57 117
PTA Protein of Dam, lb 25.4 18.8 -35 103
REL of Dam Milk-Fat, % 51.4 5.23 26 85
REL of Dam Protein, % 51.3 5.2 24 85
La, lb 25,9956 5,293.1 6,746.0 41,612.0
_lrt_.,, $ 142.0 602.9 -2,050.6 1,920.7
Reliability (REL).

Table 9.2. Scenarios used to develop and test heifer selection rules.

 

 

 

 

 

 

Scenario
Parameter’ 1 2 3
S 0.8 0.7 0.8
n $41 $41 410
r: $75 $75 430
r, $114 $114 -$60
r4 $158 $158 4140

 

’ S - Overall proportion of heifers to be selected in each herd.
r, - Net revenue fiom sale of heifer in period 1.
r2 - Net revenue from sale of heifer in period 2.
r3 - Net revenue from sale of heifer in period 3.
r4 - Net revenue from sale of heifer in period 4.

253

nauseoumummm.

dunnliabillypruah
'Notappicabla.

Variable
Murmur

X(1)'
Xlzl'
N3)“

Xm-
X(5)'
Xm-
Xm-
Xm-
X19)-

XUO)‘
Xflll'
X02)'
X03”
X04)"
X(15)'
XUB) "
X(17)-
X63)"
X09”

X00)'
X(2‘|)'
M'
W)'
X00"
X(25)'
m5)-
X(27)-
WP
X(29)'

”30)"
XGW'
WP
X(33l'
WP
Xl35)'I
N33)"
X(37)'
X03”
X09)"

N40) '
X011)"
N42) '
N43) '
N44) '
X(45) I
X015) '
X(47) '
N43) '
N49) '

Rules comm Rub based on
Scamrio 1 Scanarb 2 Scenario 3 Selection 11111911
1.10 124 0.90 1.00’
120 122 091 1.00’
1.05 1.15 1.14 1.00’
0.40 4 .15 0.15 0.00
0.13 0.97 0.19 0.00
0.55 025 0.15 0.00
0.19 1.07 025 0.00
0.17 0.15 0.04 0.00
40000.00 40000.00 40000.00 0.00
0.25 0.54 0.01 0.50
0.55 0.55 0.75 0.00
0.42 0.55 0.19 0.00
0.15 025 021 0.00
0.05 029 0.59 0.00
0.02 0.51 0.21 0.50
0.02 0.42 0.52 0.00
0.52 0.73 0.59 0.00
0.15 0.51 0.23 0.00
0.75 0.03 0.43 0.00
0.45 0.15 0.35 0.50
0.15 0.71 0.12 0.00
0.47 0.95 025 0.00
0.24 0.55 0.52 0.00
0.29 0.75 0.13 0.00
0.50 0.47 0.70 0.00
0.47 0.09 0.15 0.00
0.55 025 0.05 0.00
0.75 0.73 0.15 0.00
0.52 0.75 0.07 0.00
0.45 0.75 0.17 0.50
0.07 0.59 0.34 0.00
0.15 0.54 0.43 0.00
0.33 0.45 0.55 0.00
0.15 0.40 0.55 0.00
0.00 0.45 0.12 0.50
0.54 0.59 0.01 0.00
0.05 0.92 0.09 0.00
0.09 0.95 0.05 0.00
0.74 0.09 0.05 0.00
0.07 0.30 0.45 0.50
0.70 0.99 0.15 0.00
0.55 0.59 0.95 0.00
0.19 0.55 0.15 0.00
0.31 0.95 0.05 0.00
0.11 0.33 029 0.50
0.11 0.70 0.17 0.00
0.11 0.53 0.14 0.00
0.20 0.75 0.15 0.00
0.25 0.42 027 0.00

’mmmwmmanmwmnpmpammuw
bythadaalredculllngratahthacmharu.
’Fuflnnflabaaadeahcfionlndaxmlladecficncccrnhfinpubdwhichgum
MMratmntotnflermrhgfuunmdiaphyedmmhmmm

mmummmmamr 5:2

254

Table 9.4. Fourth period weights, and their average, for two scenario 1
rules with identical selection decisions for the 58 rule deriving herds.

Variable
w. Number Rule A Rule 8 Average
sire milk X(40) = 0.69 0.70 0.70
sire fat X(41) = 0.50 0.43 0.47
sire protein X(42) = 0.52 0.43 0.48
sire reliability milk-fat X(43) = 0.01 -0.06 -0.03
sire reliability protein X(44) = -0.24 -0.26 -0.25
dam milk X(45) = 0.29 0.30 0.30
dam fat X(46) = -0.05 -0.03 -0.04
dam protein X(47) = 0.41 0.23 0.32
dam reliability milk-fat X(48) = 0.13 0.11 0.12
dam reliability protein X(49) = -0.14 0.25 0.06

255

Table 9.5. Means and variances of the increase in herd average heifer profit of the
selection rules utilizing genetic infomation over random selection for the three scenarios.

 

 

 

 

 

 

 

Mean and Variance of Profit/Heifer
Scenario Profit from Heifer Sale % fi'om Rule vs Random Selection
Selected Rule
Period Complex Genetic Difi‘erence
1 2 3 4 Theory
1 $45 $71 $114 $158 80% 331* $27 $4
2443 2159
2 $45 $71 $114 $158 70% $32 $35‘ -$3
3497 3413
3 -$10 -$30 -$60 -$l40 80% $32‘ $25 $7
2660 2184

 

‘Significantly different (P s 0.05) than $20, using one-tailed t-Test.

256

.020; no .6. v 25:00» .095 5.60.0» E85. 2 0220.0. 00.2 5.80.00
oz0c0m 30.20505 new x0388 Ea: 0:0; .00 E05 09803.. 4.0 059”.

0:0; .8 ESQ 09205 c. 00:05:.0
com 09 8F 03 our cor om cm 04 cm 0 cm. on. om.

TIA-+31!

-r-

5.80.00 Seven.

30. 0.2 >605 0.20:3 do ESQ a
5:00.00

E85. $0. 0.2 08.9.50 .0 Each. I

 

 

2S7

spreu 10 requrnN

 

 

.020: R0 5.. P 0.20500 .09.: 00.2 5.50.00 0.00200 .8550... 05
58.950 503.00 52.0.. .00 .55 00205. c. 00:05.50 .0 50520.: N0 050....

0.2 5.60.00 0:050 00.20505
5.; 00.0.3000 :55 000. 0.2 0.0.950 0503 5.80.00 50.0.. E9: 50.0: .00 .55 0020}.

00 cm ow ON 0 ON. av. co. co.

 

 

spreu ro requrnN

258

Chapter 10

CONCLUSIONS

In terms of dairy production, the literature revealed that assuming risk neutrality, a
constant herd size, culls are sold for beef not dairy purposes, and no strong seasonal
efi‘ects on milk production or reproduction, the optimal culling rate is likely between 20
and 30%. Prior to addressing the research conducted for this dissertation, it is worthwhile
considering why Michigan apparently has a higher than optimal culling rate, for until
producers recognize they may have an excessive culling rate no consideration will be given
to heifer selection. While a violation of any of the four assumptions could support the
Michigan DHI average culling rate of 37%, these four assumptions are not as restrictive as
perhaps they initially appear.

While the objective functions of the stochastic dynamic programs and simulations
implicitly assumed risk neutrality, the decision to lower the culling rate by selling
prepartum heifers also has advantageous risk characteristics- early positive cash flow and
highly reversible in the short and long term. Turning to the second assumption, while for
herds which are contracting in size, a higher culling rate could be accepted as the herd is
liquidated, the majority of Michigan dairy herds which are not maintaining a constant herd
size are likely expanding. The high average culling rate is more difficult to explain in these
herds where, due to expansion, the facility is under capacity. As a result, the zero
opportunity cost of postponed replacement should result in lower culling rates which

would be expected to rise as livestock capacity is reached.

259

In terms of the third assumption, while some producers certainly can sell their
culled first lactation heifers as dairy instead of beef, the logic of this is unclear. Assume
that, despite having been culled (i.e., is inferior to the majority of non-culled animals in the
seller’s herd), the culled animal has higher milk production than the average animal in a
potential buyer’s herd.

It is possible that the better performance of the culled animal relative to the buyer’s
herd is due to genetics. “0th the heritability of milk being between 0.25 to 0.30, genetics
account for less than one-third of the variation in milk production. However, fiom Chapter
7 it was evident that only 7% of the within herd variation in first lactation production was
explained by estimates of parents’ genetics. This poor performance of the parents’
predicted transmitting ability (PTA) is due to the fact that the PTA are only estimates of
the parents’, and therefore their daughters’, additive genetic values of milk production.

Once she has produced milk, and thereby contributes to her own PTA, the
accuracy of the culled animal’s estimated genetic ability to milk will likely improve to
explaining over 10% of the variation in milk production. So while it is possible that the
cull may be genetically superior, given the large estimation error in the cull’s estimated
genetic value it is difficult to tell how much of her performance is due to genetics and how
much is due to environment.

The amount of variation explained by genetic estimates is small compared to the
40% of variation in first lactation milk production that herd, conditional on parent PTA,

explained in the Chapter 7 analysis. So unless a large part of this herd effect is permanent

260

 

 

(i.e., is a permanent environmental efl‘ect) it is highly likely that the culled animal would
rank similarly in the buyer’s herd as in the seller’s herd, namely at the bottom.

Furthermore, initially it would seem that having been identified as a cull, the
animal’s value would be less than one not so identified. However, given the altered
probability distribution of the animal’s potential returns following calving, as no
uncertainty exists as to whether the animal will survive calving and come to milk, it is
possible the animal may be worth the same or more.

Turning to the last assumption of no strong seasonal effects, it is accepted that
compared to areas such as Florida and Texas, the level of heat stress encountered by
Michigan cattle can be effectively managed so it does not have prolonged efl‘ects on
reproduction or milk production.

With this literature on optimal culling rates being applicable to much of the North
American dairy industry, it is certainly disconcerting that this wealth of information,
concentrated in the Journal of Dairy Science and Livestock Production Science, could
have been overlooked by the academics of the dairy industry. My only explanation for this
is agricultural economists tend not to subscribe to such applied journals, while the
biologically based veterinarians and animal scientists are not versatile in the techniques
used in these studies. The other potential impediment to embracement and dissemination
of these studies is that the results may not have coincided with some of the scientists’
assumptions and beliefs that maximizing the inflow of heifers, which represent the best

genetics and thus milk production potential of the herd, would maximize profit.

261

Of course, as discussed in Chapter 3, the other reason the industry may not have
recognized the nonoptimality of its culling rates is simply failure to recognize that the
culling rates in the models and those reported by many dairy herd improvement
associations are defined and calculated differently.

The suggestion, based on this information of optimal culling rates, for producers
and their advisors to consider lowering their culling rates is generally not embraced, at
least not initially. My explanation for this resistance is that culling and culling rates are not
largely an active decision. Rather rules of thumb have been developed to guide under what
conditions animals should be culled. These culls are then usually designated as involuntary
and account for roughly a 20% culling rate. According to another rule of thumb the
producer then culls an extra 10 to 15% for voluntary purposes. Some of the culling studies
(see Chapter 3, especially Table 3.1) suggest these rules of thumb for involuntary culling
are wrong, resulting in the culling of animals that should not have been. More
firndamentally, as discussed, the category of involuntary culls is so small as to be
negligible. Rather producers must focus on culling the least profitable animals and not
worry about classifying the culled animals into categories of voluntary and involuntary
culls. Results fi'om Chapter 8 support the finding that there is little difl‘erence in the
voluntary nature of culls classified as voluntary or involuntary. As long as producers and
their advisors are convinced that they have no choice in the culling decisions of the
majority of their cows they will be ill prepared to critically evaluate their culling decisions.
Certainly, it is worthwhile for producers to monitor why cows are being culled but for the

purposes of monitoring management and not to aid in culling decision making.

262

A counter argument to the above explanation is that producers and their advisors
are making active decisions concerning culling, and based on the poor milk production of
the current assets and/or the low probability of the current animals’ reproductive success
the producer is making the optimal decision to maximize the inflow of heifers. While this
argument is logical in the short-term, the fact that this actively assessed culling rate always
exactly equals the number of available heifers leaves this argument ringing hollow.
Furthermore, if heifers were sold or purchased this was likely the result of an active
decision with regard to individual heifers and not reflective of an active decision at the
herd level.

It is quite possible, that heifer pressure is driving the high culling rate. It may be
that the culling rules of thumb were developed in response to the need to cull suficient
cows to make room for the first lactation animals. A prescription to lower the culling rate
then simply starts with determining a target culling rate. Keep a sufficient number of
heifers to meet this target rate and then rank the cows in terms of the order that they
would be culled. Starting at the bottom of the list with the cows the producer would most
prefer to cull, cows are designated to be culled until the proper number to meet the target
rate has been reached. The optimization and simulation studies on culling rates suggest 5%
of the first calf heifers should be culled for low production.

We then arrive at the issue of how to select those heifers. Certainly random
selection is one option. The derivation of a heifer selection method which can select the
better heifers would result in more than the 5 to 10% increase in profit for the average

Michigan producer through use of a 20 to 30% culling rate and random heifer selection.

263

First, I define better as higher lifetime profit corrected for the opportunity cost of
postponed replacement. Of the production traits apparent early in life, first lactation milk
production has the highest correlation with this profit number. Based on genetic Selection
Indextheory, heifers’ parents’ PTA should be unbiased predictors of daughters’
subsequent first lactation milk production. Therefore, the parents’ PTA should also be
usefirl in predicting heifers’ lifetime profit corrected for the opportunity cost of postponed
replacement.

However, there is little in the genetics literature to suggest how well parents’ PTA
actually predict daughters’ subsequent milk production. Based on a sample of 5,123
Michigan heifers, parents’ PTA available just prior to the heifers’ calvings explained less
than 7% of the within herd variation in first lactation mature equivalent milk production.
Despite geneticists consideration of herd as a fixed effect, my Chapter 7 analysis revealed
that it can be treated as random resulting in greater efliciency of the estimates.

To be useful in heifer selection it is the rankings of heifers based on predicted first
lactation production that is of importance, not how much of the variation is explained.
That PTA may be useful in heifer selection was suggested by the results that the within
herd rank correlations between predicted and actual first lactation mature equivalent were
positive for 65% of the herds, and clustered in the area of 0.3. The omission of heifers
which were culled prior to first lactation, did not appear to cause selection bias in the
sample data suggesting that these results are not limited only to the sample used in the

statistical analyses.

264

Given some evidence that parents’ PTA may be useful in predicting milk
production and therefore lifetime profit corrected for the opportunity cost of postponed
replacement, I next turned to the evaluation of an optimal heifer selection rule.

First, in Chapter 8 I considered whether parents’ PTA from the period in which a
cohort of 5,619 heifers were born could predict subsequent heifer cullings during rearing
or first lactation. The logic behind such an approach is that producers are currently likely
making optimal decisions with regard to which heifers to cull and which to keep in the
herd. However, by waiting to freshen the heifers to make this decision they are in essence
paying too high a price to make this decision. Rather the culling literature reviewed in
Chapter 3 suggests making the heifer selection decisions prior to calving would be more
profitable even after taking into account the cost of decreased ability to pick the most
profitable heifers. Then if genetics could be used to identify the heifers which subsequently
would be culled, an optimal heifer selection rule would largely be accomplished.

Regardless of the model specification, including the distributional assumption for
the probability of being culled, PTA of milk, fat, and protein for each heifer’s parents did
not predict subsequent culling during rearing or first lactation. This despite evidence that
first lactation milk production did influence producers’ culling decisions. This inability of
genetic estimates to predict culling outcomes surprisingly also indicated producers were
not using genetics in their culling decisions. The culling reasons were classified into the
broad categories of supposed voluntary and involuntary, and then the ability of genetics to
predict culling in each of these classifications, as well as, the effect of a change in PTA on

the propensity of a heifer being culled were evaluated. Based on these analyses it was

265

concluded that there was no difference in the voluntary nature of the culling decisions in
the two classifications, and thus no evidence of involuntary or voluntary culls.

Given the failure of regression techniques to predict which heifers would be culled.
1 turned to Modified Box-Complex to solve the issue of how to select heifers to maximize
profit and assess the expected increase in profit versus random selection. The objective of
the heifer selection rule was to maximize herd profit, given the proportion of heifers to be
selected. Herd profit then consisted of the returns fiom those heifers that were sold and
the lifetime profit corrected for the opportunity cost of postponed replacement of those
that were selected. Due to data limitations, herd profit was summed over a number of
herds, resulting in a single rule for all farms. Herd profit was on a per heifer basis so each
herd, regardless of herd size, had equal weighting in the objective function.

While Complex was used to generate an optimal rule for each of three scenarios, it
was found the performance of the rules varied little when adapted to a particular scenario,
suggesting robustness of the rule. As a result, the performance of a single Complex rule
was tested against random selection and selection based on estimate breeding value (EBV)
of milk under three scenarios. EBV milk is the average of the parents’ PTA of milk.
Generally selection was assumed to occur at the most profitable time in terms of returns
from those sold.

Regardless of scenario, Complex’s rule and the EBV of milk selection rule
performed very comparably, suggesting parents’ PTA fat, protein and the associated

reliabilities provide little value in heifer selection. Given the decreased information

266

 

requirements for heifer selection on EBV of milk versus Complex’s rule, EBV of milk is
the recommended method of heifer selection.

Generally, for every heifer in the cohort, selection based on EBV of milk was $20
more profitable than random selection. While the performance of the rule would vary by
cohort, on average a producer with a herd of 100 cows would benefit by roughly $1,000
per annum. This increase in profit, due solely to heifer selection, would be realizediover
the course of those selected heifers’ lives. Other profit would also accrue if, in addition to
the heifer selection, movement was made to a more optimal culling rate among the cow
herd.

The derivation and testing of the heifer selection rules on different data sets also
provided some insight into the presence and potential causes of sampling error. The 115
herds were randomly assigned, following stratification on herd size, to one of the two data
sets resulting in nearly 1,000 animals in each data set. Despite the size of these randomly
created data sets, the optimal rule derived under one sample was non-optimal when
applied to the other data sample. This presence of sampling error indicated the existence
of sampling variation between the rule deriving and rule testing data samples. It was not
clear whether the sampling error was due solely to sampling variation, or whether the
variation was magnified through over-fitting of the model to the data sample used to
derive the selection rule.

The sampling error not only makes interpretation of the profits resulting fiom rule
use difficult, but also does not bode well for herd individualized rules. For even under the

.USDA’s current quarterly schedule of genetic data estimation, a decade of prospective

267

data collection could easily be necessary to amass sufficient data for developing and
testing individual herd rules. The possibility of the sampling error being due to samng
variation serves as a caution to anyone basing conclusions on data analysis of single
medium size or small data sets, especially in the absence of a rigorous theoretical basis to
aid assessment of the results.

If parents’ PTA and reliabilities were the only information available to select
heifers, Complex revealed that EBV milk represented a nearly optimal method of using
this genetic information in heifer selection. While heifer selection based on EBV milk is
certainly worthwhile, evidence was presented that this method of selection is capturing
only a fraction of the profit potentially available though heifer selection. Given that the
lifetime profit corrected for the opportunity cost of postponed replacement was
deterministically estimated fi'om first lactation production, this then suggests EBV are not
optimally ranking the heifers in terms of milk production which is consistent with the
results presented in Chapter 7. More accurate methods of identifying a heifer’s lifetime
profit corrected for the opportunity cost of postponed replacement, or alternatively her
subsequent milk production would enable producers to profit more fiom heifer selection.
Serum B-lactoglobulin, discussed in Chapters 4 and 7, and genetic molecular markers,
mentioned in Chapters 5 and 7, each have potential as more accurate information on which
to base selection. Whereas the molecular markers could likely be identified at birth or
earlier, the serum protein has the interesting dynamic component of not being available

until mid-pregnancy.

268

The present work does not address the issue of the optimal heifer selection rate,
rather it is taken as given. This rate is impacted by two factors: the desired optimal culling
rate in the cow herd, and the second is the number of heifers available. As the profitability
of the heifers rise, due to more accurate selection, the culling rate in the cow herd should
also rise, but it is not clear to what degree. The answer to the optimal cow culling rate is
then best answered through integrating a heifer selection model into the dynamic
optimization models discussed in Chapter 3.

The percentage of heifers selected will also be a function of the number available.
Clearly, once selection has occurred in a cohort of heifers, those heifers who were sold
prior to calving will not have contributed a calf to the farm so the number of heifers will
drop. But this will be compensated, to some degree, by calvings of those cows which
otherwise would have been culled to make room for the heifers. This issue would most
easily be addressed through a simulation model of a cow herd which could track animal
numbers.

Selection based on EBV milk also has future potential applications when sexed
semen becomes commercially available. By largely removing the uncertainty of which
pregnancy will result in a heifer, the selection decision turns to selecting the parents of the
heifer.

Aside fiom the issue of heifer selection, a number of interesting aberrations of
animal valuation have become apparent. As discussed earlier, this has included producers
being able to sell cull first calf heifers at the same price as springing heifers. Another

example is the ability of producers to buy cull cows and then proceed to milk these cows

269

and then eventually re-sell these cows as culls. While the latter example can be explained
by producers exploiting the dairy industry’s nonoptimal culling strategy it is nonetheless
interesting that the phenomenon persists. These observations indicate the need for firrther
work on the valuation of cattle. Real options, a method which unlike traditional net
present value analysis, explicitly accounts for the ability to sell the asset throughout its life
may be a fi'uitfirl extension of the current cattle valuation work.

In summary, it is my hope that this research has provided two firndamental pieces
of information to the Michigan dairy industry. Firstly, there is an optimal culling rate of
20% to 30%, and I expect the upper half of this range would be an appropriate starting
target for many herds. Where culling rate is defined as the number of animals sold in a year
divided by the average herd inventory. This range of optimal culling rates assumes a
constant herd size, so for a herd undergoing expansion an even lower rate is dictated.
Studies suggest moving from an non-optimal to optimal culling rate will likely improve
profit by 2% - 40%. My feeling is that for most Michigan herds 5% to 10% is the more
likely return associated with moving from, say a 40% culling rate to a 30% culling rate.

Secondly, as a result of the lower culling rate, producers will have excessive
heifers. This dissertation provides empirical support for those suggesting selection based
on EBV milk. The prescription for heifer selection is then to, based on a heifer enterprise
budget, determine the raising period in which the maximum return from selling heifers is
achieved. Then sell the heifers with the lowest EBV of milk. I expect this to return $20
more per heifer in the group than picking them randomly. This return will be generated

over the course of the lifetimes of the animals in the cohort.

270

 

If the history of the culling literature is any indication, these results will need an
active extension efi‘ort to disseminate the information to producers and their advisors. This
dissertation can serve as the basis for an intuitive understanding of the important issues of
cow culling and heifer selection. But I would suggest an appr0priate place to begin any
extension effort directed to dairy producers, their advisors, or extension educators would
be with the following two questions:

“What is your current culling rate? ”

16W? 9’

271

 

 

 

APPENDICES

272

APPENDIX A

FIXED EFFECT LOGIT (CONDITIONAL LOGISTIC) PROOF FOR A CLUSTER OF
3 WITH ONE SUCCESS

273

The log-likelihood function for clusters of 3 (i.e., t= 1, T. and T = 3),

conditioning on fly“, for i = 1, N (where N is assumed to approach infinity) is

 

N
L(fi)=er(/3)

where r, = ln[exp(fiéx,y,) 26x fiixldt)}

dd; '81

K, = {a = (d,,d,,d,)|d, e {0.1} and id, =gy,}

{=1

 

 

 

 

Preliminaries:
let P(y = 1|X) =A(XB) = 13:32”
a) if A, B, and C are independent then P(ABC) = P(A)-P(B) - P(C)
b) P(ABCID) = Fag?)
_ _ exptxm _1+exp(xm— exptxm _ 1
°) Mm” '1 [1+exprxml’ 1+exp<xm ’1+exp(xrs)
d) E;- = eH
6

3
Assume one success in the cluster of three (i.e., 2 y, = 1), as a result there are
[:1

three possible alternative outcomes. Notice that if the value of the dependent variables are
invariant within a cluster, the value of yr is completely determined. As a result, these

observations are dropped from the analysis as they contribute no information to the log-

274

 

likelihood firnction. In general the number of potential outcomes (combinations) can be

calculated from

 

In step one, of the following proof (a) and (b) are used. An assumption of fixed
effect logit is that $1.70. Yrs are independent conditional on Xi. Cr. and Ly“, as a result, (a) is
assumed to be true. In step two the probabilities are reported in the logistic distribution.

Notice that in step two the denominator is the sum of the numerators since, by definition,

3
z y, = 1. In step three the exponential formulation of the logit model is explicitly stated,

1:1
and (c) is applied. (Steps two and three have been omitted for the second and third joint
conditional probabilities.) Step four results fiom arithmetic simplification of the expression
in step three, notice the denominators of each fraction are the same and, as a result,
cancel. The final step involves dividing both the numerator and denominator by the
independent variables of one of the observations raised to the power of c. This is
equivalent to dividing by one. In this case, those of the third observation (i.e., X13 + Cl) have
arbitrarily been chosen. As a result of the logistic function, namely the property noted in
(d), the unobserved effect falls out, leaving differences of the observed independent
variables. Notice that, within cluster, the independent variables are being difl'erenced
between observations. As a result, explanatory variables which do not vary among

observations within a cluster do not contribute information to the assessment of B for that

275

 

 

 

variable (i.e., observations that are concordant with respect to a epidemiological risk

factor are dropped fi'om the calculation of B for that risk factor).

276

... - .2... .4..- 2......-
..... - 5......
Q0 + Q0598 . Q0 + $598 . Q0 + @2398
Q0 + $1598 1
A 0..kwo.1° 1 .Ar+ A 0..RF._.O11 if... A 0:03 3." if" ...AWMW...0..H
A.0..HFI 1 0.29% . A.0..xr u «SYN . Adah? n ear IF"

 

...Iu acne-F " N~A.° " t

=2 1 «with. =50 1 exits. In

 

AA». 1 59.98
Q0 + @0598. A 0 + nexv98. Q0 + $598"

 

.5 + 0.598

r... + 05.095 + L. r... + 0.595 + L. 7.5 + 0:598 + .—

Q0 2. $2398 + Q0 + 9.598 + Q0 + 9.398
AA .0 + Q0598 + L. —Q0 + 9.598 + L . AA .0 + @5098 + %
Q0 + 9598

Q0 + an.av<. AQO + neav<1 L. AQO + azav<1 L+ _Q 0 + nexv<1 1.L Q0 + n«.xv<. _Q 0 + nuxv<1 L+ _+A.0 + neav<1 L. AA. 0 + canv<1 L. Q0 + new?"

 

 

AA .0 + ne¢<1 L. AQO + n«.xv<1L.Q0 + acav<

...,...T e. 4...... e. 44...... e. . .. .. _ . .
A0393 0 2...? .623? 0 «car .A.0..Rr u 5.0% u w" AW .0 a H 0A. O u «..Afifl 0;.

277

 

 

.... - 5.5.... - 5...... .. -
_.

..e + .2395 . ..e + 0.595 . ..e + 0.595 1
A...» + aflkvgxm l
A.0..xr.c.o u it + A.0..k_o.w.o u ...Ql + A.0..aF.o.w n ..O&

A.,-czar u 2.9% .A.0..x_o u «if . A.0..x_o n 2.9%

 

 

wan

Full QAN:U:R N healc " fluidc N :3.
n

278

APPENDIX B

MODIFIED BOX-COMPLEX FORTRAN CODE FOR THE HEIFER SELECTION
PROBLEM

279

 

 

 

00 0:0
00 0:0
00 0cm
0 0 .22....000000.00
0 4 xx 0 an
a 000 .00. xx 0 0.0:: 00
o 0 xv
0 4 00 0 as
. 000 .0.. as 0 0.0:: 00
0 + .. 0 ..
0 0 an
a n .0.. .. 0 0..:3 00
o 0 ya
0 0 as
0 0 ..
>0000 000.00 0:0 000s .

.0:¢.000:0..00.0000.00.:::000z.0:0.0000.ux.e.00< 4
.0..x.350.300.0fi.<.0£00..0_.x!0_.0.....=50.: :8

«or I No>o.
map I Pu>00
New I 0:0.
won I 0>0.
0 I an

. . I .ncw
00— I a
00 I 8

ow I x

0.: I 00.00

a u . ew‘u‘

..08.-000<00 38:28.35. 4
..06.»...806..5.0..0.:.30:339-088.. .00.:02 .08

c000< mCLOuUOLIfi.
.20 0.030220
33!... .32: ..0. .9860 .038 .20... .00..
8:2. 8.283
00 .2. .: ..8....0£.88.S.08.e0 0.608...
880:. .8833... .096. .026. .26. ...6. 0.60:...
2.0.0 ..8....0:.3..:08..0 .32: 50.33.. 0.60:...
2... .33.. .5. .0000 .273... .0500. 0.6.. .0. 0.68....
an. 083:. .59-.0.. .0000 .0.-.00 .500. .0 .x ... 09.0000...-

..s.0.50-..8 8 an- 80. 300:... .4

.0350 ...8:3......... 8.650 .50 8:6... 6.9.... «..0 .82. .0 .8. .
..<.n .000.0: 00 0000000000 00.0.00: 0:0 00 00.0003000 000 0.000 .
65:-tea... 5;... 0.0.3.5 330 15:23 6.9.2.5.. 0.5.3.0 .

2.0 c. 5.0.50 0.9.0 .50 8:26:00 8:02.... 0e 0:52.. .
.0.-:38

55.. 8.0.8.. 0.: 2. 5950000.. 8.8.: it... 80 .80-80 .
00 .n0000 0000000 :00: 00:00::000 00.0-0.0.000 00.: 0:0 0000000 .
000 00.0sou.x0¢ e. 000: 00.:0.00> 00000000 0:0 00 00.0.0.000 000 .
.0 00 0 0: ..0: 0x 0:0 .

.6... 6 0. 8 e6 88.0 .0208 ~ ..0... 8.. z .88... .868 n .

8 z... a. 20 .8250... 0. e: a. 2.0 .e 8.0.020 0.20:8 02.. .
.000000000

000 000.000> 000m >..0.0¢0000 0:0 00 0:00 :0: 00:.I00000 0:0 .
.0508 05.08.. 6.0.80.3. 8.0.8.. 2.9.30 05 .8. >2... .

x 0:0 00 00000000000 000 00:.0> 0>.000—00 000:u.: 0:0 :0.) 000.000> .
0 0:0 0e 0.. 0:. .80—06> .5 .605: 2000.6. .5525 000:0 32. .
.x 000 .000.00 ..0000 .0000000 0: :00 000.000) .0c.0.00 0000 000: .
.000000000. 0 0>0: 0000.0: 0:000: >0< ..0: .

000.

00. :0 0000.... 000000000 00: 00:0 .000...000.000 00500000 00 0000 .
>0.::000000 0:0 000 000000000 000.0: 0:0 000 0.0000 000000.. 0:00 .
0000000. 000 .00000: 0 00 0.00 .000 000.000 00.0000 0>.0000000 .

0:0 00 :000 0. 0000000 00: 0:00 0¢.n¢.~¢.—¢ 00 00.00000 0 0. .
0.0000 000: .0000: ..0 00>0 000300 00:0 000 000: 0:0 0. 0000.0: .

00 000:0: 0:0 0: 000.>.0 0.0000 000: 0:0 0. 0:.0> 0>.000.00 0:0 .
.00...00..00 0.00000 .>0...00..00 000.0.00 .0000000 .

.000 .0..s <00 .500 >0 00:0..00 00.0 "0000 0.00000 0:0 00 00000 .
.80 8.60 80. o... 050 3.0.0.. 2.0 0... e0... 8 8... .

.00.000 00. 0:0 0. 0000 0.00000 0:0 000 00:0.03 0:0 000 00: 00 00x .
0.. 00 00.00.00 00.00 000000 00.300:00 :0 0:00:00 >0.00.00.0 .
0c0>000 00 m~.. 00:0 000. 00 000.0000000 000 00.00.00> 000.00 0:0 .
.0000.0: 0. ..00 000:0 >00 000 0: 0:0: 00 0:0 00 000020 .

0:0 000 00.000 :00 0:0 c. 0000.0: 00.00: 00 >0.>.000 .0.0.0.000 0:0 .
00.000

00 00000.0 :00: 00: 0000.00 00.0005 00 00.0000000 0.0..00. 0:0 .
.n000.00 0:0 .

00000.00 0: 00 0000.0: 00 0000000000 0:0 00 00.0000000 0:0 0000.00 .
.6038 080600... 2.0 0. 8.60 5.. 2.0 e. 038.8 .605: 0.: .

.n 00 0 000.000 0. 00000.00 0.00.00 00 0000000000 0:0 000 nx 00 .x .
.000000.00 0000000000 000 000000 000 0000.0: 0:0 :0.:: :0 000:. :0 .
000000

00 000:0.0: 0: 0.00:0 00.000 :000 c. 00:. 0.00000 0:0 :0: 0:0 .

00000.00 0: 0.00:0 0000.0: oc.>.00 00 00.00 000.000 0 0:0 0. 00:3 .

00.000000 00 00.0000 00.000.00 000.0: .0I.000 0:0 003.00 0000 0.:0 .
.

00.0-00 3000000
999999999999999999999999999999999999999999999999999999999999999999999 ~

280

.98....5000....20.0=0.8..0.08.00.....2...E.00.6...... .
.~.6..0..0....00.8.III0.00I0.0:0.<..I0.0.0.5.080..00 00

00 000
0. 00m
00 000
.00...0.x I .00....0.000¢0x
. + as I as
. ~n .0.. 00 0 0..:0 00
0 I as
00:. a a .0.. ...0..00¢ 0 0.
— o .. I .—
. 0:0 .0.. .. 0 0..:0 00
0 I .—
>0000 .x 0:0 .000 >0000 x 0:0 00 00.0) .
0>.000.00 000:0.: 0:0 00.0 000.000> 000 0:0 000 x 0:0 >000 .
00 000
00 000
0. 000

0 + .00..000¢ I «000.000.
00:0 . 0000.000. .00. ...0..00. . 0.
w + as I an
A 0:0 .0.. as v 0..:0 00
.. I an
. 0 .. I ..
a 0:0 .0.. .. v 0..:0 00
0 I _.
000.0000 .0 0. 00.0 >00 00000 .

00 00m
00 00m
0. 00m
- 0 a..0.ucos I 5....3000
00:» A «..0.0 .00. 0000.0 0 0.
0 o as I as
a 0:¢ .0.. as 0 0..:3 00
a I an
— I A__v.xcos
0 0 .. I ..
.000 .0.. ..0 0..:3 00
o I _.
.xCIc 0. 000.0000 00000 000 .0 0008 .

0.. 0...
.0.8.8....50§. .
..0..0....00<§0.80.....08.I0....0.............:200 ...0

. + . I .
a 0:0 .0.. . 0 0..:0 00
0 I .
00000

...00..I00.o.....0.~. .

. .IEu. .xm.0- 0 .IOuOD. 0xm0N.mm. .I-fi.<. .XNvugnK

00.6.. .0.-.8 .0060 .20.. 3.53....
.0....0...x~.0. +
35800..£0.01}...£0.01}.....0.~...I......~.0E000

0. .00.... .0 .0 .. 2.500....

..00600I000...0..§...3 .0 0.500800 00.089.580.800

8.9.3....

00 000
0 I .....0000
0 + .. I ..

. 0000 .0.. .. 0 0..:0 00
0 I _.

>0000 .0000 0:0 000~

00 000
o I ....0
0 + .. I ..

a 00. .0.. .. 0 0..:0 00
o I .—

00000 0 0:0 000~

00 000
0 I ...0.0
0 + .. I .—

. 0000 .0.. .. 0 0..:0 00
0 I ..

>0000 .0 0:0 000~

00 000
00 000
o I ....000x
. + an I as
. 00. .0.. as 0 0..:0 00
0 I an
. + .. I ..
. ~m .0.. .. 0 0..:0 00
0 I 00
o I ..
>0000 x 0:0 0000

or

281

acum

. .n:o.o u.cu so. oc_ououosa toga.c.m no; luggagn. .cuc..a
. . .Cuc—La
. . ..u:?.&

8 Eu
.o.m.... . ...~....x..x~.\.uus.o.
.2255. .2. 8338....
z.. u as on

8 .5
.8...:8..\2...~>o....:8.. . 235....
p o __ u .—
.2 .5. .. .32.. 8
o u :

8 .5
. .xn.o..n .. . uo:.o> woo.om .ac....x~.\..as.o.
....n.~>o..uuo.om .....«.~>u...uo.om.....n.~>u...uo.um .
.....~.~>o...uo.om.......~>o...uo.om .a..nn.o...a
. + .. u ..
. nausea: .... .. . o..;: on
o a ..

.m...u.q.u..x..n.....n.¢..x..n. +
3.8....56.31:....x..e.2..&.8.om.5.3.232

3... .3... .3... .2... .98.»... 5.83....

. n... a .35: 8:83.36... a as and... a 32: 1
£6... a .3. ..s... . Sioux—<6... . 83 1

£6... . 2...... ..o... .. «3...... 8...8......=.:.l.§.
.. .5. .32: ..2. .15.. .83 .21... .8... 5.83....

:83... x .2....x~.:.l.o..
312.27...
.2... u 8:83 2.. .o 3.... .2......~.§:.E£
3.6.: 5.3.3....
rind-:20» 5.30.5. +
.xo.......v38u§ .2. 83.83. .o .35.. of..x~<3.lco.
.5... .8683...
on.-.- 28.72:.

_: new
on 38
co... . .58.. .8. 58.. . :

new Angus» ...a. +

op

up

or

m—
-

ow
on

282

...~ .. 2......
3.23.1.3.
2...... 3.5.8...
. 2...... a .35. ...c...

on new
on new
a u ...av.x
p + a u a
A coon .pa. 3 v c..:: on
o a a
p + . a .
. ~m .».. . . o..g: on
o a a
a u .
>o... .x as. o.o~

on new
a I n-vu
— + . I —
v o..:3 on
o u _

a e .h..

o u .oo.nuo»
. a .5...
o u u.
6...... 5.... c. 3......- 8..3 .

3.3.8 3.8.2.. .0 8......8 .38.... 8.8... 8:33.89... .
$3.-Ea.

.8... $5.88.
3.2.2.... .o .85.. 5...... a .3... ...c...

8. u .3...

z a...
8 .2.
.. a...
. ..So. as... ...c...
82o... ...c...
~n . .8.
.83....581 .8..§8.....8.

n. .~w<..-I.o. —
.858. 25.8....
. ..oeuc .oguoca >.. ...xo .0: coat 0.... .c.c..a
a o...xw ..02. .0..:: on

.. new
. .83. 8... ...c...
«sac»... ...c...

~n .. .8.
.8.¢..u.<.8. . 32.5.8. . .28. .
3.8.1.52... ..<..a£a....8uuu<.§. ...w... .~nu.......
...8

cos» a nun—um u *—

.....x.u...x..nifi...u.....o...&c.
«scene». \\ o..mc. a «sac»...
..nu.o.nuouc.. a magenta»

Awp<vuus.o. n—.

.558. 8.6.3... .

..0... aunt oz. .0» «In: u... a .o.c~. .c.:..a .
.xua.s/.o. u o..wc.
can» a . .Om. an . w.

p + an u 2x
«0 o u lactgo:
coop u 05¢

2.... .23....
nun-nu .ou.oo.
. o.og»..ua.a.uo.om ..aococ .asn.< .oou
«taco... o~¢.o.uo.agu
«scenes. ~Pc.o.oa.asu
o...c. oa.o.ua.nzu
c.un< ma.o.ua.oau
use .5 .. .:.oo.. .00.. .c.l. ..g. .uoom c..ooo.c.
8...... :2... .55.... i... .8... .5. .33... 2.68....
.2688... ...,-8.. ...o. 9...... J. in... .. ... ... «.183...

:.oao.um:»<pm .a<~wxwaawa.ummwou<.ollc0..uuma—u. NMHh_z:v .

ammo
cosh . uuu.xw v .—

................I..c.....3...2.3...
all... t 3:... u 8.2....

.
35.32... .......8.om.§.....o...3...8.........8<....
..x..il.....3.-i.<..c.h..u....l.............2.... 8.5.3...

283

cosh a «$30: .wz. «vac: v v—

.. an.
:8... u .2...
~ . a
c... . . .... a . ..
.o........
...o: ......c..o...=o..u...
. u a
. u .
$3.0: $0 5085:: Gay venue
on a...
8 a...

...o......¢..=oc....c.c< . ......x
no.p-v+.~..:oco¢. . usocou
. ac. .. co.3.on on ..35 oc.x guao.g. ¢.x
..aoc.¢ .uoom.xoz<¢ ...u
g.oo.. . a...
...o....o....c........:..... ...u
. + a a a
. a .... a . ...g: on
n a a
. + . u .
. 0;. .... . . ...g: o.
o u .
woouuuo> -_C .09 Owux Cu «.8 Ounuucou

2. a...
8 9..

cc
on
o~

AAnasmv.coxcamk.+nougavdloaxam.caaa._v.xv.ao¢~vvuc_c<

. ......x
...o......¢..:o:....c.c. . ......x
38:... .3328... :8
..oo.. . u...
...oo........c........:..... ...u
. + a a a
A n .... a v ...g: on
o a a
. + . a .
. as. .... . . ...g: o.
o u a
. a .
. ac. o no.2... .. ..a. n.x .. ..x
.x...o> asp nag. .oguo Lo. n.x o. p.x o.-.ocoo

...... . .n....x
...... . .~....x

...... u ......x
..a..> .....c. .

.m...n§.o.
>3¢ “ow.¢vnoo¢
. .uo_.on gu.:o. oz. c. ..s.cc cu >35 0. anon 0:. ...cm. ...c..a

.0....l.o.
u... ..........
. o.na...> co...uo.a - .o.Cm. ...c...

on. u A...
... a .n..
m. . .~.¢
.. u ...a
.m....§.o.
...u ..........
. c. . ...c.. ...c...
......¢.o.
.n.. .m........
. mm s ...cm. ...c...
.m....s.o.
.~.. .~........
. ~¢ . ...cw. ...c...
.m....s.o.
.... ..........
. we . .oucm. ...c...
o. I laugh
........s.o.
...g. .o....u..¢
. ...g. . ...c.. ...c...
m.~ . ...
...sm...l.o.
... .....u..¢
. ... . ...c.. ...c...
a. . n.....m
...o......o.
a...... .........
. unannouson uuudom Laucw. .cucpga
N I in
.n........
I... $5....
. in a .35. ...c...
. . ....
.u......o.
.... .....u..¢
. .... . ...c.. ...c...

op

mp

P—-—-

N—

—.

OP.

0...-

284

 

..c..._ 0.. on
aux. Ac .ou. ..u.. .—
..Z... .0. .2. 2 too. 3!. a... .3 5-..“.

..azm
. a ..u.
o . emu»:
....aumm u ..Vu oo—
..c.... oo. o:
zwz. .o .... auwma. .—
.8332...
.2. .o. .80. 0.3 3...... 2.. 0. so... too. 2.. .09.... 5-..".
v0.3... 8...... co 3.... too. .085: 3a.... a... .0... 2...“.

............... ----.----.--.--..co...c.... .......> ...o. .0 ac...-.-u
.....u. ...o
.....n.....~..~o....o..~...... ...o

...a ........m .<..
.2.. ...u. .. ~......z.

-....-.......-...-..-.-..-...-co...c...o ..n....> ...o. .o ....m....-u

...--.................-..-...-..........o.. cos... a...gm .. ac....--u

.-.-.-.-....-.-..----.--.---------..._uo.n cilou no.2.“ .o ...um.---.u

-......-....................a.c.:..¢ ac. a..... ...n....> .0 ac.......

8.. o. 9.... acts... .35.. 8a.... a 52...... u
v... a... ..3. o. .215... 2... on: u o: 8.. u
.a... .....a ..a u o. .3..> a... oc.....m . om... u
3.2.: u

2:22.. ....

omwmz ecuwuupz_

----.--...-...............a.c.:... ac. a..... ...a....> .. ....m.....u

9.. o. a... I... ...i... 8!... I325 - 839... o. 0.3.8... 0
8°85. "octaokflfi u

azalz<¢.omwusvaoz<u wz_»:O¢n:m
iC.
:CCCCC‘...’C..COCC.C.C.:.£§C§CCC£C.CC:CCC.:CCCOGCC.C.U

ac.
53......

. . .CuC—LA

. . .cu¢~sa
. 6...... . .55.... ..o.o 0.2... ...c...

. . .Cuc—us

. . .cacmsm

u:o_ uc_so¢
_ a lacuna: oo—
on cuou
m. new
pu.o: n wvco:
p o _ n _

2535

 

 

o a .—
>o..a o>uu o.o~ .

I039... .8393: .032. 12.8 .

can...
out...

. + E... u .8...
. 50.2. .5. E... . 3...: 8

o a E...
o a .9...
8 u...
8 a...
on a...
o u .ss.._......uu.om
. + 5. .. a.
.8... .5. v... . 3...: 8
o u g.
. + a. u as
. 8. .5. 2. . 3...: 8 a...
. + : a .. 2.5....
o a 2.
. m .5. : . 3...: 8 .o.....3.5<..+.8.. o
a a 5. ..5.53.28.....~.....<...+.8....5.55.... .. 52......
o u 2. a... . 8. . .3. a .3.
o u _. co. \ ...u u .3..
>9..- ..uo_om o... 9.3 . a...» + 3.: u .3.
80' O gm» . “nun I anvx
=3... mize-.28 89 \ 3.3 n 5...
53¢ 9.3.3.26 .3... + .3. u an:
3920. .2... a... c 8. - 2.52 a .3.
~u..= co. \ .~.u u .x..
:22. ..5. ..5. .. .N ..58 .58... .2. .: 3.38.... 9... + a... .. .8.
.33.}... 4326.55.33 ..3...o......§. $.83... 82 . 5... - .5.. u .5.
8........oh.8...3...o...5.5.8...36653538 8... x ..x. u a...
$130.... .33 c n u A...
.98. ... ..N .5. .2... .56.... £3 .8... .3"... 3.33... .8. . n a .8.
.8."...u..8o...:.8.o.....8...:...~..8o.:.......... 3.38.... .5. . .3. . n u .3.
.-.-....-.-...-....-.-..-.--..-.-......-.--...--...--..--..--. .~.. . ...x . n . ...a
. .85.. 8...... 0... 0.3338 3:“.
.
3.8.8....go832. + .3...
.c.3<.§..8.......8.-.......=.........:u5. 858.8. . u ...u.

--..---.--..--.---.....--...-----..-.-.--............-..-.---. 9.... . ........ . .... o..

22865

. .. .... as .o...: on
c a as
8 ac.
.as.n-u. . .as.....-.-.
>I... .I.Ia o.c. I.ga u..ocoo an. .
. + a. a s.
. . .... sq .o..ca on
o a a.
.538. . 2:53:38.
....< ...o.. co..a.ua. aao. .
c... . .a... .a.. ~aco. . ..
.. ac.
.aco= . ~a.o=
cog. . . .a.. .. . ..

p + __ I .—

.o....x..o....q.x..o....cco.
.~ ...e..u.. .....ccu.. ..acoz .co.uac..o..~n.acu.

. .aco. .mx. .ac. . o..c3 on
o I .—
a.o; o.oc.o I .0. a... o.oo c. anon .

an ac.
on ac.
o . ....qa..a.oc
. + as I as

. on. .... as . o_.c: co
. o ._ I _.
o I as

. oo ..4. ._ . a..c: on
o I as
o I _.

>I... .I.aa 0c. o.o~ _

on new

F 0 __ I .—
00 new
a I a...aav—o>nm
— 9 as I aa

a on. .p.. aa v o..:: oo
o I as

a we .pa. __ a od—sa on
o I as
o I __

cc

>I... ..>Iu Ia. o.o~ .

on ac.
c u .......o.c.oo.
. + _. I .—
.om. .... _. . o..c: on
o I .—
>c..o ...o.. co..I.uI. cc. o.o~ .

8 ac.
o . ....cuo.
. + .. a ..

. .c .... .. . g...) on
o u ..

o... a.oa 0c. .0. >0..u >.I.onuu. 0c. .30 o.o~ .

an ac.
an ac.
c I .ss....xcoa

. + as I as
. o. .... as . o..c: on
a I as
. o __ I .—

. on. ..4. .. . o..c3 on
o I .—

>I..I scan 0:. o.o~ .

287

on ac.
on ac.
a I .sa._...xoac.
. o as I as
. o. .... as . o..ca an
o I as
. + .. I .—
. on. .... __ . o..c: on
o I .—
>I..I xoac. o.o~ .

on new
on new
p + aa I as
c I aaa.__vwo>aa
a «a .h.. as v o..c3 oo
o I as
w + __ I ~—
A on— .ha. __ v O._;3 on
a I aa

a: .5. .. .22.. on
o I ..
85...... .020... c. 3.» >5 .3... .

on a...
2 as.
on a...
m. a...

I. ac.
. I ......xco. I ......scc.
coc. . .......Ioac. .Iu. ...aa..xoac. . I.
c... . I .a.. .a.-..aa..u>am . I.
. I aa I aa
. a .I.. as . o..c: ea
c I aa
I I .......cI¢
cog. . . .o.. .o........o>um . I.
I I .. I ..
.x .I.. ... I..c: on
o I ..
..0—Ion 33.6.... o... c. ago. I. 9.3.0.. .23. .

an ac.
I. ac.
on ac.
I I as I sa
...o-aa......IuacI.aa....x
. I 3.2.2.8... I 3.2.53...
:3 .0: I ..I..ou c. I. 3...... at: .3 .
3.2.8 0.2.3.. o 3 £39 >9... I33 .023. .
a 3+... .3. as . 0...! on
I. I as
can. A p .0.. naoI.¢.....pu>Im . I.
I I .. I ..
c. .5. .8 0:5 on
o I Z
xoac. 333.8 .58.... I. .

9.....a. I a a. on

.acI=.I.....uo.o. . .acoz.n.....uo.o. I .acoc.......uo.om
.ac.cIc c.c=oI ac. c. .
v3.5.3
I.Ic.cI Io ceccac Ic. I. I»..Iaoc I. .acox.......oo.oa .
can...» I .acox.e.....uo.on
638.: an 3 .085... co .

2.8.2.3 .81 all c act... c. 3.8.: .805: .
I. ac.
.acoz.~.....uo.om I .aco=.n.....ua.oa
coc. ..acoz.~.....uo.I. .... .a.I=.n.....uo.om. I.
..uoc...oogI I
...aco=.~....uoo.o...oo¢I..n....x..II¢...c.cI
I .aco=.n.....uo.om
I. ac.
.acox.......uo.oa I .aco=.~.....oo.om
coc. ..acoz.......uo.om .... .a.o=.~....IIo.o.. I.
..uo....ao.. I
...a.I=......uuo.om..II¢I..~....x..Io¢...c.cI
I .aco=.~.....uo.om
I. ac.
acne. I .acos......uuo.om
coc. ..caou .... .acox.......oo.om. I.
..uo....Iu¢\...c:ou..ao.I.......x..oo¢...c.co
I .a.o=.......uo.om
n o. p 30...... c. 0300...... .005: 333.0,... .

on ac.
I I .a.....o>a.
II .. I Z
a «58 .S. 2 .03.... on
o I ..
.58 o. G. .8.» o a... c. I an an £2... .0)... 3:235 .

...c:ou..ao¢ I c.uo.o...c.c< I c.uo.om

.550 u x
Z I £509 3..
9.3.0.. Io Ion-Sc I .53 .

Nn 2.30.. 8—

on a... on

I. ac.
.c. 0.0.
I . .. I ..
N. 883.8.
03m
2. ac.
o I .aa.cco.
. I I. I a.

288

ac.
cIauoc

o I In...
Inca. I .....

2. ac.
..... I Inca. I IchI

I. ac.
...c:ou...o. \....acox.e.....uo.oa..II.. I

..acI=.n.....oo.oa..II¢.I.Ia...II.
I ........II....c.c< I ..I..

03.
...cscu..Io. I....a.o=.e....Iuo.om..ao¢- I
..acI=.n....Iuo.o...-I¢.I..I.¢...o.
I I.......Io¢..uc.c< I I....
cocI I .a.I=.I.....uo.om ..o. .aco=.n.....oo.o. . I.

.Icsm I
I ..a.I=.n....Iuo.om..aco=.~....uoo.o..I.n.. I

I ..a.I=.~....uoo.om-.aco=.I....Ioo.oa.II~.¢ I
I ..acos.I.....uo.om.=.I.I.¢. I .....

.a.o; c. .os.cn Ion u.Io.n an»; ooa.o>a I. o:.u> osquo—ao .

3 ac.
I.....Ioa.uI.I8......II>Im I I5. I I5...
I I .. I ..
I x .I.. .. . I..g: on
o I ..

03c.ucou or

an ac.
.— ac.
.. ac.
— I Aa.__v—o>om
cos» a “v.0:INI_v.uuo.om I... an._~vxcu¢ . .—

cosh a w .0.. aopIa.__vwo>am u .—
p + —_ I .—
n: .p.. ..v u_w:3 on
c I .—

.onuvuc. I ~
«a.oc.x.c..uuo.om cog. Iva-0.0 .cI. g..x I.0I.I: ..0. .

2. ac.
8 ac.
I. ac.
I. ac.
I. ac.
I I ...aa..cI¢ I ...aa.xcae
cocI I ...aa..cax .a.. .......caa . I.
cocI I I .a.. .oI...aI.II>Im . I.
cog. . I .a.. .oI-.....Io>Im . I.
I I aa I as
. z .I.. Ia . o...: co
.. I aa
I I .. I ..

289

I. ac.
I~>I... I goIg.

II. .oI.. I
. I coI.I.ouI acI I:.I> cquucaI uuogng aox..x~.\\\vuulco.

.. .I~>I... .aI.nn.I...a
cog. I ga... .I.. IN».... . I.

I. ac.
I~>o..I I ga...

IoII. I

I .00....) .IcIuIIO as. Io o:.a> cquucnI .IosoIz..x~.\\\..aI.o.

I~>I... ..I.nn.o.I.:
cog. I I .a.. .. . I.

IncIIco.
In: I ~>0—
o~I .cII .oII IIco... . I~>o.... I.
..~ I cu. o~I on
I I ~>o.
8.: 8.85. .88... g.... .58 ac: .

Iac..co.

.u— I P>0u

co .aoI .ooI IIca... . II>I.... I.
..~ I :0. ooI ac

I I I ‘08
I I I 9:00
I I I Ncou
I I I wa
I I I $08
I I I)».
2.3.2... I I..n.......n..IcoI
I..I I a ..a.. .a. I-.I.I...a
..co..uc:I Ig. Io II:.I>..I~.\..IIco.
.I~.I.I...:
NI .9. ..I ..c..a.. I.
3.: 8.35. .83. 5... .52. ac: .

o I I.
I I .cao.
85.8.
Ia.uI.Ia....c:caco=

..ac..cI8I§I.8.I....8.I......x..........5. :8

O

or

ON,
c—p

cc—

9»

..I I . a. on
u I I.

IacI.cau
I. ac.
cIIN cuoo
cog. I .c..... .a.. cIIoI . I.

.uoca.cIIu<.I..I..oa.ux.oao....I...x...z.z..uogu ..I.

.. I I.
.. I .
..N I .. no on

IIcII..mcou .IcIIuI IucIoa xo.ncuu .uogu I

oacIucou
I I oaou
oI.o~.o~ Iz-z.I.
o I oaox
— I .—
.IIII og. can. .ocuo Ioqu.o> x ..I 0.0.
.cuIIIQ nucIIIIIcou .IuI.nII II xuocu

a I ~>0—
«or I ~u>o—
now I pu>o_

Now I o>o_
—op I .>0_

upon. I... ..0.

II. o—c.o.uo.agu

a... ..co. :8. d... also-=5
cIoo< m..o.uo.ocu

.ac. nccoquIIgu

.oIVox coIIcasIo

c.uo.I. .I..oc .Ign.< .II.

go.g. ... .II .I II.III.c.

cacacoz ..a. .9... ..oquI...I.IoIII..uo.I. IIcouI.c.
I..I.. ..c.... ...m.: .I~..u ... IIcIIIIc.

3o... ..~..3I.x .32. it... .5.... ..I .I. ..I .1092.

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo .

Aauo:»I_Is

I».I..x....IQIIIIIQIIqu.Icn.<.xnluIIxIszvxIcou IcIuao.n:I

no

ow
o—

_
+

...uo.Im.c:cacI=..ac...:..uoc....I.I..c.IoI.I..cIcn..ux.=...~>o.

+

25K)

03..) :IIIuc3I II.....
Io: ..0... 0g. 0).; I.cI.ou so..3 ac... nouoo.I0I Io IIO.Iucou .

I... 38
III. I .
III.~u>I..I I III.I>I.II II.
I.. I II II. I. II.

III ..In ..In II~IIIIII.II>I.III I.

I. co..u3a.I ...3I0suo ...I0I... .0:.o..>:ou xuogu :0.» In... .
..0...
I. 03..) cOIIuc3I I 03..) coIIuc3I ..s In... ~I0.Iucoo I. .

.~uu..u:oo. I Ncou

IanI.I..u...:cacI= I
..ac..c.II<.I:..II.I.I.III.II.....x...x.I.Ic=I ..I.
I. ac.
oe~ o...
cIcI I .c..... .0.. cIIII I I.

IuIcI.c.II<.II.II.0a.ux.oaox...I...x...:.IIIIIcu ..Iu

~g)o. I .
IIIIII.I>I.Ix..II¢I I
IIIIIIIIIIIIIIIIIIIocI...IIIIcIc< I III.~u>o.Ix Inn
I.I I II In. an Inn
IIIIIIIIo .

Inc... ... I. 03..) cOIIuc3I ..030. ..I no. I
Ia... u0Iu0.I0I 0.0.3 ac... .3..) coIIuc3I ..0... I. Nuu.cucou I

o- ..0.
I>I. I .
III.I>...x I III.I)I..I I~n
I.I I II I.” o. o~n
I... 88
III. I .
III.I>I.II I III.I>I..I II.
I.I I II .In a. II.

c~n .c.» ..I. IIIII.II.II>I..II I.
03..) co..u:3I III-0.. In. ...3 .aouu..I0I Io co..:.ax0 .
.xouco)
0.. ...3 II)... .3..) :oIIuc3I ..030. 0:. cu.) IcIon ....a.« I

.co..c.nx.. I ax.

«Quad!» Iqugo: 0
..uc..c..u<.)3-.90...¢..o..0m.I...x.x.:.z.uc3. ...u
I. new
ov~ o...
Cbzu A IC—IOOI .am. c..o< v m—

IuocI.c...<.I...I.oc.ux.oaox...=.u.x.....IIIooco ...u

s— I ~
naaaaa.u)o..x...0¢...ua+~aaavuxvca..a
...IIIIc.c< I III.I)I.Ix II.
=.I I II II. on II.
«Sago
0>quuoa I Q. Inn 050‘. I933 smuufldvog v0 swixw -

on. ...n ...n IIc>o..I.II>I..I. I. can
can ..In ..In III)o.I.-I~)o.II. I.
I. 2.8.8.
0..)cosuo .I. ~IuIIIcou c0.. 03..) coIIucaI ..030. 0.. I. .
.um
acoaxo c... 03..) coIIoc3I ..0.... 0.. In. IcIOQ 30.00.I0I I. I

.co.u00.I0I. I won

IaIuo.o..xI.c:caco= I
Id2“.c—au‘usaueb‘ICQUUO—“mu—IKIxsxctczvgfi dd.”
I. ac.
ocw ..0.
cocI I .c..... .o.. c.... . I.

Iooca.c...<.I...I.oc.ux.oaoa...z...x...:.z..uogu ...u

I>I. I .
IIIII.I>I..I..IIIIIc..<
-IIIIIIIII...<Io.IIIIcIc< I III.c>u.Ix ..I
..I I II III on

III.x.ux.I)I..I.cIcou ..Iu ch
03..) co.Iuc3I ..033. guIz we... 0o..a0¢ I

I... .8. ...I III-8...}... I.
I I Icaou I Icao. III

amp 0.00
I I uc3ou omI
III .onI ..II IIIIIIIIIII..II.I~>I.III I.
IIIIIIIu cocooI.)cou xuogu I

25>]

I I Ioucza
anxQInuII I «IIN—IIIIInuIvaIvnvugcu O—
II.IIII .III.I>I..I .II .III.. III.I.II..:
I..c.on aouoIIIOI I. IIIIcIucoou..x~.\.I.I.o. v~
nquIVOu—IIJ OMN

In. ..- .on~ IIc.c... I. I-

IcIIo.II.II..:caco= I
..ac..c...<.§...8c......I8.0......x...£....I§I :8

I. ac.
oqw a...
cogI I .c..... .... cIIII I I.

III...c.I.<.II.II.II.II.IaoI...I...I.I...IIIIIII ..Iu
IIcIIcoI o-
9.. III.
.coIIIaaII. I an.

8.58. fl...
IaIuo.om.u..c:caco= I
..ac..cII.<.I:..Io.I.I.IIo.I......x...x.z.uc:. ..II
..I I . II» o.

I. new
o¢~ ouou
cog. I .c..u.. .au. c..u< I I.

03c.ucou «In
IIIII.c.I.<.II.II.II.II.oaoI...x...x...:...IIIcu ..Iu
I.I I . II. o.

2. ac.
8 ac.
I. ac.
I~IIIIII...III.IIIIIIII.~>I.Ix..IoI.IIc.cI I III.....
II..
III.~>I..x I III...Ix
cIcI I I .a.. IIII...Ix-III.~>I..x.IaI . I.
I I .. I ..
I I .II. .. . IIIc: o.
o I ..
I I II I II

a a .h.. as v o._g: on

o I ..
o I II .In
.3..) cOIIuc3I p.030. .
0;.
0).; o. .3:.Icou .ng1 .~ .0 I ...c... noun-IIcou Io cOIIuaoou .
o- a...
I)0. I .
III.I)I.Ix I III.I>I..x no.
2.. I II mom a. .0.
con ..0. .Ion III>I....III)0.... I. no.
II. .cIn .non IIIIII....II>I.... I. .0.
o- oIo.
I)». I .
III.II>I..x I III.I>I..x .0.
3.. I II Ion on com
..n ..0» ...m IIII>I..I-I~>I.... I.
..I0IIIo 00:09.0)cou 3.0:. ac. 03..) cOIIuc3I Im0guI. .
0;. In;
I0)0zo.g: ICIOQ IuquIIIcou Io 30.00.I0. ..0.0I I)0. 0....0. .
0II3I05Io
.I. co.II:aoc c... 02..) coIIIcaI IIIIo. III Inc I. I. .
..II0IIII 09:0..0)cou .00.. ac. .
I. cIII
I)0. 0...a0. 03..) coIIuc3I ..0co.z no; Ic.oa quogucou I. .
.puunsucouI I «too
IaIu0.om.....:cacI= I
..ac..c..I<.I:..IIII.I.III.II.....x.I...IIIc:I ...I
I. ac.
II. III.
cIcI I .c..... .a.. cIIII I I.
IIIII.c...<.II.II....ox.Iao....z...x.a.:.a..uocu ...u
«gun I —
IIIIII.I)I..x....I. I
IIIIcIIIIII.ox.IIIIIcI.o.IIIIc.c< I IIIIII)I.II Inn
I.I I II Inn on an.

Ivouogpvo .

iZSXZ

cc..ooI.o. III..II.IIIII I. as
II.II.Io IIII..II..III I.

IuIc..=.oIIIcou I...
:.II I I III on

I I z a a: co
oucIocuacoo ..0..nc. as» u.:.... uoocu .

II.III.III Ioao.I I.

I. ac.
oII ouoo
.cIaoo. n cIoo<
I I IcucIIa
I I IIucILa
..IIoo
Io .ucsnu .u:n.< acoIoIIIu IIIcuIII - on: «woo... .IIcII.
. aucI-II-cou
..0..axo ucI>IIIIao Ioc .. ausauuo .0.... .cuc...
cos» I can .Io. I I I.

I I I I I
«2.28 o.
I...ocIIc. . III: I II..Ix II
on.cn.oq III..Ix.III:I I. on
on one.
IIIIIIIIc. I III. I II..Ix o.
on.on.o~ IIII.-II..III I.
I.I I I c. on

Inc—aguncou .IuIIQIo IncIouo xuocu .

Iuoc..=.oIIIco. I...
o I I. II
o I I

I. .oao. .IoIIo.
II... I...
cIIuI accouooc-gu
8:“... 8.28....
u... .Ia cacooouc.
. ...III .I~.Io .I~I.IoIIx .. .x .I IIIIIIIc.
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
.
.
Iuogg.c..oI.I...I.Ia.ox.oao....I...x.a.x.II.oIgu IcIIaocaau

ac.

ccauox

II. .o. .o. Inca..-..I I.
I+u.uu.

. . .IIc.c.
I. ac.

.coIIuaaII. .IIc.c.
cocI I .coIIuaaoc. .o.. an. I I.
I. ac.

.I coIIIIIIcou. .IIcIII
8.: I III-IE3. 6.. I8. I I.
I. ac.

IN COIUOOuuCOUI IIuCILm
cog. I ..IuIcIcou. .a.. ~cou I I.
I. ac.

.co..caax.. .IucIII
cog. I .caIIcocxo. .o.. II. I I.
I. ac.

.coIIuo.IoI. .IIcIII
cogI I .caIIuo.Ioc. .a.. III I I.
I I I¢ucI5&

3.3.395: 8......3...I.~.IIII.3I

I. In~.IIoI.c:

IIx~.m.III.. I I..~.....qu.IIIIIc.oI

II.IIuI .IoIon .uII II~.III..I:

I.a.oI.cou cc. Io IIIIcIanou..x~.IIIIIcoI
I.~.IIII.I:

Iu.II. .I.I. ..I I-.IIoIII3

oe~
n-

n~
o~
m~

293

 

occacI I IoIIz
occacI- I IoIIo
uogacI I IoII:
ocuncpI I aopvu
cocacI I IIII:
occacI- I IIIIo
ooIch I IoII:
uIIIII. I IoIIo
occucI I InII:
uocacII u InIIu
ougacw I «III:
oagucwo I nerve
oocacI a InIIs
uagacwu I anpvo
ooIch I I~II=
uocaII. I I~IIo
uocnoI I IIIIz
uocacI. I IIIIu
avgacw I Aopvx
DOInI—I I nopvz
ooooooI. I IoIz
oooocoI I IoI:
ooooooI- I IoIo
cocoooI I Ion
ooooooI. I Iqu
ooooooI I III:
ooooooI. I III.
ooooocI I III:
ooooooI. I Iqu
ooooooI I ImI:
ooooooI. I ImIo
ooooooI I III:
ooocooI- I III.
IIuoIaIIIoacm~.IIuc.c< a InI:
IIuocaIIIoccmo.IIcIc< I InIo
IIuOIcIIIozcm~.IIIcIc< u INIz
IIUIIII.-oucmo.IucIc< a I~Iu
IIuocaI.-occm~.II.c.c< I III:
IIuochI-oacmo.IIcIcI I IIIu
IuInOIn cOIIooIoI .oIIoc IncIuno _

90.. .I~mI: .IunIo «CIooouc.

50055.2.uvuucoo IcIuaocnan

ac.
cI=III III
o..oI..o.I IIII I.

oncIIco. co.
2. ac.
I~IIIIIqu I IIII..IxI.IoIIIIc.cI I III..II
n.I I II on

II..x.ox.I>o..aIcIcou I...

I I on
. n I>oI on

125%4

2m
cgauoc

oosmI— I AOQv:
ooLme- I Aocvu
00.5: I as;
OOLAI—u I «Ovvo
UOLAIw I Anew:
OOLAIFI I ahcvu
UCLAI— I afifivz
oogacpu I AOQvo
uognIp I Anew:
Oduncpu I Amcva
UCLaIp I A'Qv:
UCLAIPI I «cdva
uouacp I “new:

uoumcpu I anevu
OILaIw I ach:
concur- I nchu
uouaow I arty:
UOLaI—u I Apcvu
oosacw I Acev:
ooumcr. I «ocvo
OOLaIw I Aonvs
«Oscar. I nonvu
OOLAIF I “any:
Quanta- I acnvu
uoLAIw I abnv:
905&IF. I ahnvu
Gustav I aonvx
cougar. I Aonvu
UOLAIP I Anny:
ovumIp. I Amnvu
uosacr I acnv:
OOLQIrI I Awnvu
concur I Anny:
ocuacp- I annvo
Dagmar I awnvx
uogacp. I a~nvu
uoLaIp I a—nvz
uogacwo I arnvw
8.5:. I aonv:
UOLAIwI I “One:
8009: I nay:
oooooop. I «ONvu
OOOOOOF I AQNV:
oooooow. I Ao~vu
oooooow I ath:
ooooocp. I athu
oooooop I AONVx
oooooop. I nowvu
Soccer I ava:
oooocop. I nvao
9055.? I Ava:
ocuacp. I aQNvu
OOLGCP I Aans
uogacp. I nnNuu
UOLQIP I aNNva
uoLaI—I I nNNvu
UOLQIP I A—Nv:
oosnop. I a—Nvu
9.5509 I AoNv:
uosnap. I aomvu

295

a amp .ha. aw v I._g3 on
p I __ I .—
o I as
A ov .pa. __ v o..g3 on
o I as
o I .—
>IIII you-u Isa ogo~ .

on new

— I __ I .—
09 new
0 I A__.sav—I>Im
p I as I qa

A on. .»4. as v I._g3 on
o I as

a ,e .p.. __ v I..g3 on
o I as
o I .—

>ILII po>II Isa ago~ .

on new
a I 2.3.3.98.
— I _. I .—
acm— .ha. __ v o._:: on
c I .—
>IILI u.»oIa co_quuI. as» ouo~ .

. 8 95
o I ~__vatop
p I __ I .—
a .c .h.. __ v o..g3 on
o I .—
o._v IuIv Isa gov >IIII >Logoniou as» «no oII~ .

on new
on new
a I ass.._.xcoa

— I an I as

A cc .»4. as . o..z: on
o I sq

p I __ I .—

A amp .p4. __ v I..sa on
o I .—
>ILII acne can oIIN .

3 2m
8 2..

o I .aa.~_..xouc_
. I as I an
A ow .p.. as v I_.;: on
o I as
p I __ I .—
5 amp .pa. __ v I..;3 on
o I .—
>IIII none. ogon _

on new
on new
p I sq I sq
o I nas.__ku>am
A «q .p.. as u o..c: on
o I as
p I ._ I ~—
A on? .p4. __ v o..:: on
o I .—
>IIII I>II oIIN .

50.noua co_uuo.on .oI_oc .Ie_uno .

a I ulna
a I Faun

— I v50: I ugox

. 553.. .5. Bo: . 3...... 2.
o I 20..
o I it.

=5 €38.28
c.Iu< nIgoqugIsu
98.8 .8.
N95:
.53.. .NS... .23 .I .~ J58 .58.»... .2. .: «2.33:.
2.3.9.: ..3...3.£:33 53695.2... {83...
8m:Sfohugé...13.2518.5.3.8:...38
I133...
.92 ... .2 .5. .2... .52.... S... .8... .33... {83...
.8.£.n:.3.£:8.8 .. .32: 5.32... ......... .2188...
on---IIIIII-..IoIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII--
.
.
3.8.8.§.gu....2m I
.c.3<.§..8._...¢.uuo.ou...u.x.u.z..:u§_ 82:83.

296

:3 no: I S30 5 I. rota III: ..3 .
039.8 0333 o 3 53.3 >I....I .33 I828 .
a 3+3 .m... 2. v 0:5 on
a I as
cos» A — .om. anQPIav.__vwc>am v I.
p I -_ I -
2. I»... :v 0:5 on
o I __
xouc— Iuodaudou «canons w- .

QPIQIIOF I a o— co

.u.o=.I...Ioo.oI . .u.o:.n..IIuI.Im I .ugux.n..IIII.Im
.312. 5.58 2: c. .
335......
38.5 II .35: I... I. 2,288 I. .Bo...m...IuI.om .
5933 I 3.523.330...»
.IIIII.II In on Ionssc co .
«5959.3 wool all I .823 5 $3033 .505: .
I. new
.uco=.~...Iuo.Im I .oaox.n...uoo.om
cosI ..uau=.~..IIuo.Im .Iu. .ocoz.n...uuo.om. I.
.IIIII..II¢\ I
...u.o=.~..IIII.III.IIII..I...x..II¢.IIc.cI
I .uaox.n...uuo.om
I. new
Iuao=.I..IIuo.II I .uao:.~..IIuo.om
cogI ..uaI=.I...IuI.II .Iu. .uII=.~..IIuo.omI I.
..uo.a...o¢\ I
...oao=.I..IIII.II..IIII..~...x..II¢.IIc.cI
I .u.o=.~...IuI.Im
I. new
Icaou I .uaoz.I..IIoo.Im
cogI .Icaou .Io. .uaI=.I..I.uo.IaI I.
..ooII..II¢I..IcaouI.II¢I..I..IxI.II¢IIIc.¢I I .III=.I..IIoo.om
n 3 _. £87.... 5 $393.. .895: 33:33 .

on Em
I I .o....II>Im
I I : I Z
I ~58 .5. 2 .025 on
o I 2
E80 3 .3 Ian. 0 05 c. I II In “>9...- —I>: 3:32.: .

..IcaouI.II¢ I anI.ImIIc.c< I cIoo.Im

acaou I z
: I ago
9.032. In .885: I ~58 .

Nn «In—30¢

8 I...

II new
For ouoo
I I __ I .—
~n ouonaxuau
oa_m
on new
a I 2.39.8
I I ss I ss
5 II .I.. ss Io._g: on
o u ss
on new
IaaInsoI I .ss....IIIIa
>IIII quou can. Iuov quocoo wood .

p I ss u ss
a a .5. ss 32.3 on
o I 2.

239.8 I 2.33238.
IIII< IIIoII coIIIIoI. III. .
cogI I III»: .am. ~III= I I.
I. new
Iago: I wucox
cogI I I .o». .. I I.

II:I:

.aI...xI.oI..II.x..o..III.oI
- 5....qu .33.»: .63.. .8.%c...o...~n.u8..

a .vco. Ina. .tcw v ad's) on
o I :
v.2. 39.: I ..0» 3: 33 c. .80. .

8 2m
8 new
a I ....IIIIIIIa
..I ss I ss

2:

oo—

on

cc

297

new
$3508

I I IIIII
I98. I I.II

on new
..II I InIII I InIoI

I. new
acaou

IIIIucI=.I..IIIIIII.IIII=.I..IIIIIIIII>:I I IIIII I I.II .
IIIcsouIIII. IIIIIIII=.I..IIIIIIIIIIII. I
IIu¢o=.n..IIIIIIIIIIIIIIIIauIIIII I II.III.II¢IIIc.c< I I.II
IIII
. Icaou
IIIIu.o=.I..IIuo.om-Iqu=.n..IIIIIIIIIIIII I I.I.I I ..I. I
IIIcaouIIIII IIIIIIII=.I..IIIIIIIIIIII. I
IIIII=.n..IIIIIIIIIIIIIIIIIIIIIIII
I II.III.IIIIII:.c< I ..I.
cIII I IuII=.I..IIuI.II .Io. IuII=.I..IIoI.om I I.

IInam I
I IIuII=.n..IIIIIII.IIII=.~..IIIIIIIIIIIII I
I IIuII=.~..IIoo.om.Iuao=.I..IIIIIIIIII~I¢ I
I IIuII=.I..IIIIIII.=IIIII¢I I ..I.
.qus :. .IE.:I Ion IIIoIa van; IoIII>I I. I:.I> o>.uuo.no I

8 I5
I..IIIIoIanIIIIoI...III>II I Ian» I Inam
I I .. I ..
I z .I.. .. I IIIII on
o I ..

Inc.ucou OI

oo nCm

I. new

I. new
I I In...III>Im

:2.» I IEI:.~.=uoo.Im .3. III—3353 I I.
:IsI I I .au. IoIIs....II>Im v I.

I I .. I ..

I: .I.. ..I o..:3 on

IIIIIIIc. I N
IvIostIquuIIII saga IIIIIII «III :I.: IIII.I: ..Im .

on new
on new
I. new
I. new
I. new
I I Ia.saIch¢ I IIIIIIxcoa
cog. I II.IIIIcI¢ .au. II...IacI¢ I I.
cogI I I .aw. IcI.a.ssIII>Im I I.
cos. I I .ow. IoI.a...IIo>Im I I.
I I a» I as
I a .II. as I I..g3 on
.. I as
I I .. u ..
I z .I.. .. I IIIca on
o I ..
Inc.chI IIIIIc c. II.I Ito onIu .

on new
I. new
on new
I. new

I. new
I I II...IchI I II...Ix:I¢
cos. I II...IIIIu:. .II. II.IIIIIIuc. I I.
cogI I I .am. III.I.IIIII>II I I.
I I II I as
I a .II. as I «..I: on
o I as
I I II...II:II
cogI I I .II. III.I...III>II I I.
I I .. I I.
I: .I.. ..I o..:: ea
c I ..
IIIIIQ Iao.>oIa on» c. vo>II I. IIII.ox scou I

8 new
I. new
8 new
I I II I as
I III-III...IIIIIIIIII..II
I I II...IIIIuc. I II...IIIIuc.

298

new
$5qu

8 new
Io..-IIIIIIIIIIIIIII.III.III.III-IIIIIII I IIIIII
II I II
8 2m
IIII...IxI.II¢ I IIIon I IIIIII
II.I I .. on
9.: I IIIqu
a.I I II on

IoIqu IIIIcIIII
II.....¢.II.I~I.IIIII.I>I..a IIIIIIII.
IIIIIIIIIIIIIIIIIIIIIII...IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII-
.
.
IIu.x.ux.I>I...IIIcou IIIIIoIaam

125%?

APPENDIX C

FORTRAN CODE TO SIMULATE HERD PROFITS GIVEN A HEIFER SELECTION
RULE

300

 

 

InsnucoxIo=o¢o=Iauoo.oa.>aa.¢.«9...»....Ix.a.:.a.ucau ..I9

8 Eu
OI.::
. I __ I .—
.8. .5. : .323 8
o I __
IIIII ...I cg“ oII~ .

3:8

an...Iu..xn.~...I:..xn.~_..Ix..x~.uI-Io.

a I: .a A...I.o~..:

...IIIIIIIII.Ian—III..xon Io IIsuouoIn no.nuou..xo..uaugo.
8.58::

on any
on tan
a I ._...vuuo.om
w. 2 I 2
«8.. .5. 2 v0.23 on
o I 2
p I _ I _
an .S. _ v3.5 on
o I 2
o I.
>9...- 323. a...» 9.3 .

.nuuo.oa.nacugox.Iau...x.u.x.:.u:nc_ ..Iu

Svuu
own:
09-:

06 2w
2 2w
. DCSOu on). ICuC—Ln
‘60.: Icuc-La
agw_Uh<h8—I .Bbbgngu

3
:—

II.o.5.-§=u ...<=x§wu.umawuu<I£_:.-ma:I «nub—83

8&8
:2.» A nun—Kw v :

283-55.2.23$353.3...
gluon \\ 3:5 I «.5.:

323.83 P
.85.: 2......»-
..o..lc .8528 >3 «2.8 an: .80 3:. £358
a 33.6 .8... 3:5 on

I. new
. Sn :3. Ida—3.5
Isuzu... .Iuc..a
.¢o¢¢m_Ip<~«o_..au——<xeoI.I:¢o. I
..aao.Ia:.<»a...<.pauaaum.Iaumuu«IIIIco..IIm..I.¢nIp.aa.
swan
cog» . III.xu . I.

.III_qu»m_xu.IIIcI..IIUJ_I.II.:u:_
«(9.93» xx 3:5 I 83:3:

awp<vu£0m N
.88.: 2.533.
..I: 2: .5859 as: slug 3:. a ..35. 82......
.zuc—E/uu. I 0::5—

a.§.ugh<pm I.ow:§m.§m +

..uuu..u.:oo.,ou.e/I9.Iud_u..J¢__aw=aum.quuuu< .mnIp_:=. ammo

ouu.xu .Iu.uo.

god: ONOLOuUILC—U

g» 39.33.35

0:3: OCLOuooulp-u
.oa_"..mup.uuo.am ..~m.x .auoo.om ..I.

asp: «9.000%:

.~.oo..ocucoz .Ian ..I".v¢ IIIIoIIc.
: Igog .u i I: 2183:.

.3: 59: 2: c. v.3: so: ..3 20:2. .o .803: 05 3 .
333v
~29... v.3... .5 cow-338 u...- Iohuooa u:- oE. 928.09 2: .
8.5-«coo 0:.— 33 2: n:- Ix 05 c. at... a. .538... .322. .
..0.— c.:._ :02. a 5.3 vow-.83- 23... 2: .333... I299... 2.: _
.

5.93 '58....

301

“macaw—uh<bmo.I .owhh<IBOu.I280u
I.oao.lm:h<hmI .4<.hzu:ouu.ummuou<.ailco..uuu..mI ~an.a:v
ammo
60:» a ouowxw v .—

..II_ImI.m.xu.IIIcI...I~4.I.og.:ac_
«Eleni.» xx «..Ic. I ounce...

.9338: .
Isuzu-o. ...I.oII¢
..osu: Losuoco >5» u¢.xo no: noon 0.... Iauc.ga
. ouo.xm .po:. .0..;3 on

$~ DCN
. BCJOm .03. qucqu
allcoduu Icucuga

~n I “no.
225328. . . 8:528. .539. I
..oao.Im:.<pm..4<..zm:aum.Iammuu<.osaco..IIu..II~nI».z:.
sumo

cosh A ovumxw v ».

.III.IuI.m_xu.IIIcI...Im4_..oI.:uc.
2.29! xx 3:... I I88...

.~.¢.ugugo. o.
Iguanas» .o..I.qu¢
..I..I cyan cg» Io. III: I... I Ioucu. .I.¢.II
.qumfiiuu. I Od—VC—

O I ISEULOS

auo.xu .ou.oo.
3m... $338 .58... .3.
lll60.—m o~150uufluagu
lien-o» 32.30935
«03.0) opcgouUOLlsu
3:... 9138.35
03.C> ”cucuuigazu
Nugoz ..ugox ..scuaoa .Ian ..I"... IIIIuouc_
a I. In I: I: «Igooouc.
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII.
.
.
.auuo.IaIsscquz.>:..¢.x.u.x.a.uan:_ Ic_u:oIn=I

new
noun

8.. 88

mason

. .naoLo m.:. so. u:.unouogn uoga.:.. no; Eagooga. IIuc.ga
. . Icucuua

. . Iauc-La

8 new
.o.m.I.. I ...~....x..x~.x.uuuco.
.svx .a .o~.nn.ou.ga
x.. I a on

8 new
. .xn...o..n.xn.o..xn.o_.xm.o_.x..uaucoI
....m.uuo.om ..._.I.uuo.om.....nvuuo_om.._..~.uuo.om I
.._...vuuo.um..~....ocuIIx.....I....._vocugo: .o..nn.ou..:
. I ._ I _.
.52... .5. : .32.. on
o I .—

. .muuo.om..x~..qauo.om..x~..nuoo.om..x~..~.oo.om.Ix~ I
...uuo_om..x~..uc:ou..x~.....o.......x~.. ocquz ..I-sac.
...__.ocucox .o..nn.ou_.:

.xIm... I ...c. I
.x..m_.. I Invu..x..m... I .~.¢..x..m... I ...u..x..u.ILoI
...g ..n.. ..Nvu ....¢ .nPInn.I..Ia

.x.m_.. I Ian..x..e.om.. I auuo.oa..x..u-Ico.
In. .n.uo.om .I..nn.ouII:

ow

or

on

n.

«F

302

III
canyon

on an»
35:58
I:.I> IIoIIucIIoIIInIIIIg
SfiIIIxNIIIIIIIEoI
IIII .II:.I> IIoIIucu.ao.InIIIII
I I I I I
I ~n .I.. . I I.I;: on
a I I

In ouonaxooo

8 I5
IIIIIIIIII
I:.I> Icm.InIIIII
a .x I waI 03.0) v U_'£3 co
I.II III.I.IIII III IIII ..II I. III. I

I I IcuCILa

I I Icucpgu
. .Izcucoz ..III IIIII. .IIcIII

I I Icuc.ga

I I I've-La

.IIII:

III. ucIIII
. I IIIIII:
on oIou
I. new
IIIII I NIIII
I I _ I I
cIII I NIIII .Ia. IIIII I I.
I. new
IIIII I NIIII
~ I I
chI I I .II. I I I.
IIIIIIIIII
IIIII IooIIIcI.oI.III.IIII¢
I I I
are... Io I093... 05 ~53
I I I

Am.va¢lL0u

In. IIIIIIIIII
. .uoIIon cuIaoI on» c. .IIIcI co Inn On uaoo as» Ioucu. IcucIIn

omI I III¢
III III I InII
an I INII
II I III¢
as ImIIII-IoI
III. IIIIIIIII.
co ..I. I IIIcI. IIIcIII
IIIIIIIIII
III. III.IIIIII
. mg I IIIcm. .IIcIII
IIIIIIIIII
I~II I~I.IIIIII
. ~¢ I IIIcu. IIIcIII
IIIIIIIIII
IIII III.IIIIII
on . II I IIucm. .IIcIII

o. I auoo.om
.v.oI.aIILou
IIII.II .I.IIIIII
. oooucougoa Iuo.om Ioucw. IIICIII

on new
9.9 I I..x
I I . I .
I ~m .II. . I I.I;: ea
c I .
co. IIIII x In» oIou

no new
9 I A..u
p I . I .

A I .ha. . v o.—‘3 oo
o I .

.uoIIoa :quI c. ..I-Ian oc.>:n I

II IIIIIIII .IIIIIIIII Io IIIIIIII IIIIIII IIIIIII IIcIIIIIcIIca .
on
8 8 IS

. I. III

. acne. II). IIIcIII
IIICI.II IIIcIIa

or mm I use.

P----

~I

pp

In

303

o I II
I on .I.. .. . I.Ig: oo
o I II
o I ..
>IIII III-o ecu ogon .

on new

I I .. I ..
09 tau
9 I ...Isavpo>am
I I II I II

I on— III. II I o..s: on
o I II

A II .I.. .. . o..:3 on
o I II
a I ..

>IIII II>II ecu oIon .

on new
a I a..vu.I0Iauoo.
p + .. I .—
.om— Iva. .. u o..:: 09
o I ..
IIIII u.IoIa :oIIIIoI. Isa OII~ I

on new
o I 239.2
I I .. I ..
I II .I.. .. . o..ga oo
o I ..
I..I oust ecu IoI >IIII >IIIon2II Isa use oIIN .

on new
on new
a I III....ch¢
p + as I as
I cw .I.. II I I.I;: on
o I II
I I .. I ..
I omI .I.. .. I o..g: on
o I ..
snug. scan Ina oIIN I

8 I5
8 I5
I... I 32:35.8.
I I II I II

A cc .I.. II I o..:a on
o I II
I I .. I ..
n on. .I.. .. I o..s: on
o I ..
IIII- xuuc. oguu .

on new
no new
a I IIII...Io>om
I I II I II
I II .I.. II . I.I:: on
o I II
I I .. I ..
a on- .I.. .. I o..g3 on
o I ..
>IIII o>II oIIN _

o I II
I. I :
5.8.... 828.8 IIIIII. ..I-:8 .

o I «23m

II...I..._
o I ..I;

p + . I .
a EJCULO: .h4. . v o..cn on

..I.
..I:

...5 ”13885
.SI.I.IIIII8.II. .8...I...II.IIIII2. .33.. .98...» .III
mugox

:22. :5... .I .N ..I-.8 6.8.3 J. .5. .II .: I183...

IIIIIIEII .3...I.RI.IIIIIII .8...I.I.II.II..:II I188...
3m:III.IohIII...88.omI.IIII>II «2.83...
56...... .I~.8II8I...I.. .NN $.- ISIIII IIIIIIIS
. .82: 9...... IIIIIIIS
III-IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII.
.
.
IIIEIIiEIIdIISIQIB£533...I.......I....IISI
oc.u:oIn:I

304

oIIocao 0.30909 0 In clanu In... III-o Ioo..ou I uczou I s
I IIIII .u.. II I I...a no I. I Icaou III
II I II 9.022. III .893: I ass I
cogI I I .am. IIoI.II...III>Im I I.
I I I. I .. NI IIIIII III
I: .I.. ..I III;: on
o I I.
Iona. III.:u.Iu IIII... I. . on new on
oI.oI.oI I . II on I. ac.
IoI oIoa
I - I. I ..
I..IIIIIIII . I..nIIuo.II I I..IIIuo.II ~n ouIaIqua
.uoIIIQ :I.:oI Isa c. I ouIm
vouugu.:n on new
I3...... II .II::: as. I. IIIIIuIc II Iu.ox.m..IIuo.Im . o I IIIIIIII
cIoo.om I I..IIIuI.Im I I II I II
688.3 I. 8 .35.. 8 . I I... .5. II I32: on
2:38.... «on! all .I not! ..I $38.3 ..oi... I o I II
I. new 8 new
I..~IIII.II I I..nIIuI.Im IIIIIIII I III...IIIIIa
cogI II..~IIII.II .Iu. I..nIIuI.ImI I. II... III-I oIc. IIII IIIocoo III.
I I II I II
IIooooo.oIII..~IIII.III.II¢IIImeI.II¢I.c.c< I I..nIIuI.II I a .I.. II II...) on
I. new a I II
2.8.8.2.. I 2.8.8.8 2392 I 2.3.3.88.
cogI II..IIIII.II .Iu. I..~IIII.III I. II..< IIIo.. IIIIIIII. III.
IIooooo.oIII..IIIII.III.II¢III~IxIIIIIIIcIc< I I..~IIII.II cogI I II.I= .ou. ~v.a= I I.
I. new I. new
.58 I 2.3.8.2.. 5.0.. I ~22.
cogI Iucaou .Iu. I..IIIII.III I. cog. I I .om. I. I I.
IIooooo.oIII::ouI.II¢IIIIIxIIIIIIIIIII I I..IIIII.II
n 3 I 38...... c. 8.8.8 .35.. 3:3... . I I I. I ..
IoI..IxI.oI.III.xI.o.IIII.o. II

on new IN .III.IIxx .ngIIIIII .II.II IooIIIcw.oI.~nIIII.
I I Io...III>II

I I II I II I .uco. .wa. Inca I oIIca on

I Icaou .II. II I IIIga on o I II
o I II v.0; oIocIu I .oI oIII ou-o cI nous I

Icaou ea c: Ion. a can cI I I. In ”In... Io>ou ouIIIIIIcI I

oo new

IIIcaouIIII. I IIII.IIIIc.cI I cIII.II on new

a I III.IIIIIuoc
Inna. I I~...Iocu.oz I I II I II

~22. I 2.3.3:...

I omI .II. II I «IIzn on
I I II I .—

3()5

new
change

8 2m

._ new
..ucaouv_aox \..aa_.¢.uuo.om..aou.
.._.nvuuo_oa...o¢...>=av..u. + .a_.u..oo¢vvuc.c< . ...u

o..u

.9

A.uc:ou...o¢ \..na_.e.uuo.om..oo¢. +

n._.n.»uo.omv.aouv...«.¢..aou + .._..V.-o¢vvuc.c< . ...g
cusp . ...evuoo.om .uu. a_.n.uuo.om v ._

Apsam +
+ .._.n..uo.om.._.~.uoo.om..an.¢ +
+ ....~.uuo.om-....vuuo_om...~.. .

o .._...uuo.om.=.....¢. . ...m
.ugo; c. .as.uco hon u.»oga tau; ouaLo>u a. 03..) o>.uuoqno .

on new
.—.vu.wogauuodc.oe._.v—o>am + .2am u .nnm
p + __ u .—
a 3 .pa. __ v o..;: oo
o u .—

one—acou

on new
.— new
.— new
. - an.__vwo>am
cosh A ...uvuoo.ou .ua. .a.__.xco. . .—
cugp . w .ou. Aor-a.__v—o>om . .—
p + __ u .—
.= .»J. ..v o..;3 on
o u .—

or

napxavuc_ - ~
Augos.x.;vuuo.oo coca gouoogo ace; gu.3 «Loam»: ..om .

on new
on new
*— new
a. new
m. new
. + .a.ss.xcou . .a.ssvxco¢
cosh a .a.ssvxco¢ .au. .4.__.xca¢ . .—
cos» a . .aw. nop.s.ssupo>om v *—
cocp n p .aw. .o--a.__VPo>am v a.
p + as a as
a z .pa. as v o..;3 on
__ a as
p o _~ u _.
a : .hd. ._ v o..c3 on
o u .—
nucmxcog 50*.og c. no.» >co goose .

on new
m— new
on new
m. new

.— ucw
— + .a.__.xco¢ u .a.__.xca¢
cog» . .¢.__.—xooc_ .pu. «a.ss.—xouc_ . .—
cogp . F .au. .o..a.aav—u>om v m.
p + as u as
a : .»J. as v o._;3 on
a a sa
— u .a.__vxco¢
cos» n — .aw. nap.a.__VPo>-m v a.
p o __ n .—
0: .pa. ..v o..z3 on
o u .—
uo_goa a:o_>oga one c. uo>au a. «gas—o: scam .

on new
*— new
no new
9 + as n as
A anao-asv.__Vpouaov.oo¢caaavx v
+ Aa.__v-xouc_ u .a.__.—xouc_
:3 uac — 55.3 c. a. .822. “as: ..3

*

306