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MEASURiNG YEARLY
ENVIRONMENTAL DIFFERENCES
WITHIN A HOLSTElN HERD

Thesis for £219 Degree of M. S.
MICHIGAN STATE COLLEGE

Len fl. ffisfiiiliard
594?

' I -' '-\ D 7N1“, l"."' 77!?) ‘. ,' ". 'vs‘ - ."v' ‘ ,~, ' “‘- I .‘I ’ ' ‘fy—“P:

IfiES|$_ . .

This is to certify that the

thesis entitled

"Measuring Yearly
Environmental Differences
Within A Holstein Herd"

 

presented [)9

L011 D. McGilliard

has been accepted towards fulfillment

of the requirements for

~__Mg_.S‘gflsdegree i1|_ch1,i.ry _, ,_

”Q : ./’:2 ”2/,
<<E§E3z/i/%:§:;:;2Q242/¢2:(/£L

Major professor
Date—Beam}: 41374947 _

M-795

 

.51.. .40

‘ I45). 0

MEASURING YEARLY ENVIRONMENTAL DIFTFEENCES

WITHIN A HOLSTEIN HmD

by

Lou D. McGilliard

1947

MEAwRING YEARLY ENVIRONMENTAL DIFFERENCES

WITHIN A HOLS'I‘EIN HERD

Lon D. MoGilliard

A THESIS

mbmitted to the School of Graduate Studies of Michigan
State College of Agriculture and Ipnlied Science
in partial fulfillment of the requlrmente
for the degree of
MASTER 01' SCIENCE

Department of Dairy

1947

15141515

 

 

ACKNOWLEMIENTS

For their invaluable assistance and suggestions in the
develoPmmt of this project and in the preparation of this paper,
the author wishes to express his sincere appreciation to 11-. Earl
Weaver, Head of the Department of Dairy, and to 11-. Ronald E.
Nelson, Associate Professor of Animl Husbandry.

Gratitude is also expressed to a. Leo Kata, Associate

Professor of Mathematics, for his aid in the statistical treatment

of the problem. '

TABLE OF CONTENTS

Introductionnnu...........................................
mean! of Environmental Determination........................
Source of mta...”..........................................
Plan I - Paired Year Comparisons.............................
Plan II - Direct Comperieons.................................
Plan III - Adjusted Mean Differences.........................

commrison Of Plan.00000.0......0..COOOOOOOOOOOOOOOOOOOOOOOO.

WOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOQ

Page

10

28

37

INEOID'CTI ON

he prole of determining the extent of distortion that varying
environmental effects produce on the true hereditary pichire in ail]:
production of dairy cattle is extremely useful yet particularly difficult.

ihe usefulness of a mocessful study of this problea is almost un-
limited. listaking the effects of environment for the effect of heredity
is generally the largest barrier to the breeder's rapid attainment of his
goal. If variable environmental effects upon milk production could be
equalised. the breeder would the: have a clear picture of the value of
his breeding program. one unobscured by outside influences. He would have
a reliable estimate of his animals to aid in selection and mating toward
the improvement of his herd. By the sue theory he waild be able to place
a more reliable index upon his herd sires, one which would be of greater
use and reliability. me whole course of a breeding program would be
clarified were it not fa‘ the confusing effects produced by non-hereditary
factors.

In dairy cattle the gena‘al ori terion of productive ability is the
amount of silk or the mount of butterfat ehich a cow is able to produce
in a givm period of time. This quantity is used not only to rgresent
the potentialiv of the individual cow, but it is also incorporated in the
index by which the value of her she is Judged.

The amount of milk and fat a cow can produce in a lactation period
depends upon a multitude of factors. These factors my be divided into two
general clause. hereditary factors and environmental factors. The heredi-

tary factors include all genetic effects shile all else not inherent is

-2-

grouped under mvironment. So that the issue not be confused. the
assumption is made that non-additive environmental interactions are
non-existent.

Various estimates of the heritability of fat prediction have been
made. Among these there has been some variation but in general the esti-
mates are similar. the usual figure is below 43 per cent for the heri-
tability of fat promotion. lush (2) mentions that intrasire regression
of dnighter on dam have yielded values of around 0.15 to 0.30 for herr-
tability of differences in fat production between cows in the same herd
where each cow is represented by one record. ‘Ihe remain ing variance of
fat production is then grouped under environmental effects. It is easy
to imagine how we effects of environment could be confused with heredity
and complicate breeding and selection operations greatly.

In general the observed variance may be described by the equation,
a: + 0;: a: where c: represents the variance due to heredity and O:
is variance due to environment. Obviously a change in one of the first
members produces a corresponding change in the opposite direction in the
other. Thus by a decrease of G: an increase in 0': results. Conversely,
if a greater proportion of observed variance is due to environmental varia-
tions, variance due to hereditary differences will be lessened. In order
to increase the value of CT: . it is necessary that 0’: be redioed to a
minim. If variance due to differences in heredity can be made to consti-
tute fine attire observed variance. breeding and selecting processes will be
such simplified. ihe primary purpose of this study is to attempt to deter-
mine a method whereby environmental variance can be radioed to a mini-in to

produce a corresponding clarification of hereditary variance.

-3-

There are two methods possible for reducing the o: , the physical
control of environment and statistical correction for environmental effects.
'ihe first of these would require testing all animals under standard condi-
tions to reduce a: to zero. Obviously complete physical control of
environment is an impossibility. Eva). partial control may be very imprac-
tical.

Statistical control sense a little more plausible under present con-
ditions. aich control would consist of using correction factors to allow
for unusual conditims varying from standard or to equalise the conditions
under iiidi two animals may have been. The breeder performs this operation
to some extent mentally when he knows and allows for the fact that one
eninml performed under more adverse conditions than anotha' or when he
recognises that his feed was inferior quality one year, or that one cow had
a short dry period. or an abortion one lactation. The range of corrections
would run from this type up to very complicated computations involving too
mach labm' for the accuracy involved. It is not unreasonable to believe
that some of these corrections might actually indice added error, but
ordinarily they should be usable if they are more often correct than wrong.

Some correction factors in the intermediate range are about in use
in dairy work. Examples of these are the corrections for lactation length,
for the Requency of milking. and for age. the latter being the manire equiva-
lent correction. Another example might be the separate prediction classes
for animals meeting certain calving requirements. 'ihese are all corrections
to aid in equalizing environmental effects. and some reliability is attached
to the. It is true they are composed from avcages and my be radically
incorrect for some individuals. but these errors will tend to cancel each

other when numbers of individuals are involved.

 

-4-

An attmpt was made by Johannson (l) to correct for the total
environmental differences between years in a dairy herd by a relatively
simple method which will be descri bed fully later in the paper. his
method was again presented along with a complicated statistical method
for which there was a test of significance by Nelson (3). 'me results
of the two methods agreed: therefore. the simplier was selected as the
desirable methOd.

'Ihis papa‘ will present sevm'al different methods which might be
used to remove the environmental differences between years by the calcu-
lation of correction factors to be applied to herd averages. lhai
corrected. the herd average should present a picture of the decline or

improvement in hereditary merit of the hm‘d.

'IHEORY 01‘ H‘VIRONMENTAL WINATION

the principle involved in the determination of environmental
effects is that any individial possesses an inherent capacity for per-
formance. whethm' each individual reaches the full extent of its pro-
dictive capacity is limited 1y all factors other than inherent, in
otha' words enviromental. The individual has his ability to produce
at a certain level under optima conditions. by variation from Optimum
involves a rediction in performance below snximnm.

Unda' fliis theory if we cmsider the production of a cow during
one lactation and assume that to be her capacity for production. than any
other of her lactations under the some conditions should be equal. Any
variation between them should be die to one or more of the effects which
shall be classified as environmental.- If these differences between lactar-
tions take the form of corrections to be applied to the herd averages
in their respective years. then the corrected herd averages will be equal
as far as environmental differences are concerned: that is. all environ-
mental differences will be removed. Remaining differences are those the
to genetic merit. Any fluctuation of average sould demonstrate hereditary
improvement or decline in the herd. It is not impossible to believe that
this migt be carried further and this fluctuation pinned to specific

sires perhaps as an aid in formulation of sire indices.

SCIJ'RCE OF DATA

ihe data for this problem was drawn fro: the records of the
"ichigan State College miry Department. Since his early nineteen
tundrsds a Holstein herd has been maintained as a part of the college
dairy herd for which fairly complete records have been kept. especially
in regard to production. Althougi five dairy breeds are maintained by
the department. the Holeteins wce selected for this study because of
the larger number of animals.

The minor had occasiea recently to combine all records of the
Holsteins in the college herd: these included breeding, disposal.
calving. and production records. 'Ihe production records for this wa-k
were drawn hon this information.

At first it was planned to conduct the study from 19:!) to 1946.
The year. 1930, was selected as a starting point since it was prior to
this year that the dairy herd bee-so seriously affected with contagious
abortion. Luring this and the period immediately precedixg 1930 all in-
fected animals were removed fro: the herd. In 1929 and 191!) the nonre-
actors were moved across the river to a new barn. All reactors remained
behind and were eliminated. Only a very few of the original herd were
moved to the new barn. Beva'al purchases were made to replenish the
herd. these anisals altering the herd in 1930.

The first plan was to include all animals is the herd in 1930 and
to drOp back before 19:!) to pick up w records which the transfer cows
had made prior to then. This plan was reJected when it was observed

that thee were too few transfer cows, and also because of the couplets

-7-

change of conditions with the move to the new barn. Consequently, 1930
was chosen as the first year.

After all of the records were collected. it becase apparent that
the prediction of the purchased cows was considdrably higier in 1930 than
the prediction ef the transfer group and was also higher than any subse-
quait lactations of the purchased cows. An investigation as to the
case revealed that the better cows were milked four times a day inl930
while some were milked partly it and partly 3:: others were milked 2x
and 3: entirely. However. there sealed to be no record of the exact
number of milkings of each frequency for each cow. Also some cows were
milked by hand and some, by machine. Starting with 1981 and contimiing
to now all cows were milked three times a day. In view of this, it was
decided to use 1931 as the starting year: consequently, records of 1931
to and including 1946 wu'e usad.

bring this entire period all can in the had were milked three
times a day, and the milk weights were recorded for each milking. llonthly
fat tests were made. and butterfat production was computed.

Over this span of years the herd has bed maintained largely by
replacaents bred and raised by the college. Very few animals have been
intromced from outside sources. 'mere presumably has not been too
mch selection and culling although as will be shown later. closer culling
was practiced than was supposed at first. General condiii one were thought
to have varied sometdiat but not extremely. 'ihe same herdsman was in charge
for the entire period.

In the compilation of records. it was obsa'ved that most lactations

were longer than 305 days. iiile some were shorter. ‘11 records of less

-3-

than 270 days were eliminated althougi there may be some question as to
whether they should be as they say have been shortened by some environ-
mental effect. The best plan would be to deal with them separately as
they are not comparable with the 305 day records, and they may induce
considerable error in calcuhtions if included. All lactations of more
than $5 days were rediced to 335 days by interpolating in the month in
which the 305th day occurred and totaling with the preceding monthly
amounts.

Inch record was included in the year in which that lactation was
initiated. the year being the calendar year. Actually some of the lacta-
tions recorded as a part of one year may have taken place mostly in the
following year: for instance a lactation beginning in December of 1931
was recorded as 1931. but nine months of the production really occurred
in 1932. It is unavoidable that some overlapping of records should occur
with a system mch as this where cows are freshening every month of the
year. and, unfortunately. this overlq of records masks the definite
demarcation of yearly effects.

Since, as it was mentioned previously, it is necessary that all
records be on an equivalent basis to remove the age effect, the records
were all converted to ”mature equivalente.' 'nie factors used for con-
version were the same as those furnished ly the IBM for field use. The
smallest division was three months: furtha' accuracy was unnecessary
since the entire problem is an approximation. mat conversion left the
records in working form as 3:. EDS-day, 11.1}. records. There is always a
question as to the reliability of mature equivalent records. which argu-

ment belongs not to this discussion. It nmst be assumed that mature

equivalent records are sound and reliable for this stuck is based upon

-9-

that assumption which is the foundation of the theory of environmental

correction.

-10-

PLAN I - PAIRED YEAR COMPARISONS

lIiiis is the system mentioned earlier as having been advanced by
Johannson (1) and tested by Nelson (3). Nelson found similar results
were obtained by this as by a more extensive plan and concluded that
he simplicity of the method made it more desirable than the longer
method.

This plan is developed under the basic assumption that all of
any cow's lactation records of butterfat production if equal in length
and frequency of milking, and if converted to remove age differences,
will be etpal under identical environmental conditions. In other words
if a given cow produces 600 pounds of butterfat on 3:, EDS-day, mm. in
1931. her record in 1932 on this same basis and under identical condi-
tions should be 600 pounds. Likewise, her 1933. 1934, and 1935 records
should be the same. lhe inherent ability of the individual is determined
by the genetic makeup of that individual and is not variable within the
individual. If, however. there is some variation between different lac-
tations of the same cow. some factors or conditions othm' than inherent
rust have been present to alter the production. 'lhese are grouped under
the title of environment, mid it is these differences which are to be
determined.

'ihe actual opm'ation of this plan is very simple. It consists of
grouping the 11.1». records of cows with lactations in two successive years
in their respective years and calculating the mean afferencs between

the production of the two years. 'l'he process would be:

2: (XLG - xtm)
N

--11-

where xlu) equals the promotion of a cow in the first of two
succeeding lactations: 18(2) equals the production of the same cow
in the atcceeding year: and N is the amber of comparisons. For
example, from the theoretical data in Figure 1, the differences are
calculated in Figure 2. lbs result indicates that the environment of
year, 1932, decreased the production of the same cows an average of

100 pounds under 1931 while 1933 was 100 pounds better than 1932.

 

 

 

 

 

 

 

 

: 1931 : 1932 2 1933
: 2 z
: s :
Cow A : 600 z 500 : 600
s z :
Cow B : 500 : 400 :
Cow C : z 600 x 700
z : :
Figure I. THEGKETICAL BUTTmFAT RECORDS
: : z: :
: 193i : 1932 :: 1932 s 1933
: : a: :
z x z: :
Cow A : 600 z 500 :2 500 : 600
x : z: :
Cow B x 500 z 400 : x
x x z: :
Cow C : z x: 600 : 700
: x : :
s : : 2
To tal : 1100 x 900 x z 1100 : 1:130
A : 2 : z
c : : :
2(1‘10)’ x1(2) "200 : ~100 :: +200 I f 100
N 2 2

 

 

FIGURE II. CCMPUTATION 0F DWIRONMENI‘AL DIFFERENCE FEW DATA IN FIG. I.

-12..

his part of the procedure agrees with the methods set forth by
Nelson and Johannson. Some difference lies in the use of the factor as
a correction to be applied to the herd average. Johannson developed a
multiplicative factor to be used in the correction of the hcd averages.
Nelson's method was to select a base year. to determine a relative devia- -
tion of each year from the base by the addi tion of deviations from year
to year, and to use these deviations as corrections for the herd averages
by reversing the signs. his is the procedure used in this study. The
first year, 1931. was selected as the base year only because the use of the
initial year simplifies the process in that the additive correction proceeds
only in one direction. However, it miglt be desirable to use the highest
plus year for the base with all corrections figred in comparison with it.
but the same relative result is obtained eithc way.

The results of this first plan are shown in 'Dable 1. lbs column
of sums of differaicss. (6) is a cumulative difference for each year by
tich each year is placed on the same basis. in this case by comparison
with 1931. lbs herd averages shown are the average records of only the
cows completing the full lactation started in each particular year. They
include a few indivi dials who completed only one lactation and were not
used in the evironmentel difference calculations. ihe corrected average
column (8) contains the herd average for each year after the calculated
environmental difference has been applied to the herd average.

It is obvious that a very distinct trend toward the negative
exists in the data presented. Only four positive differences show, and
those are not great enough to counteract the negative trend. Column 6 of

the table duonstrates that the swing to the left is extreme and much out

-13..

PAIRED YEAR COMPARISONS

PLAN I-A.

TABLE I.

6

5

8
8

4

3

 

d
m
u n
rA
r
.w
dd
PM
.u
r...
00..
.mmm
O
unw
3’1
.1
O.
f
.7.)
m3
.m n
u a
CV.
r
P
m»
”a
6
may.
f
08
mm
N
w

:(Cum. 5):

: (7) - (6)

Av.

AV.

 

511 511

O

1931 8

586

507

z - 79

-79

481

560

1
1

656

570

533

54)

652

561

:-91

559

564

2
1

669

: 666

: -103

-12

536

548

-109 8 -212 506 718

4'70

579

528 717

: ~189

+23

524

501

"a "w “a «w “W mm

: -196 529 725

7

558

565

745

756

.. .. .3 .0

567

565

8 ~179

+27

.0

622

595

9
3

: -191

-12

.. .0

564

576

.0 x

.0

762

527

z -235

795

565

: ~230

806

z 523

: ~283

818

530

: -888

508

513

o
1

ml
936

:..::

56

.0 .0 2 8

278
-293

00......

+10
-15

.0 .0 .. g

533
555

04.
1

.C...’..:..::::::..::

m. n. a. ma u" a. a"

 

lotal No. of Records - 114

 

ef preportion. his is further unphasised in column 8. It is entirely un-
reasonable even to presume that the merit of the herd increased from 511
pounds of butterfat in 1931 to 936 pounds in 1946 as indicated or that the
environment was as extremely limiting in the last years. Some factor seems
to be working in the data to produce this severe irregulari ty.

When this negative swing was noted, the first possible explanation
to come to mind was that the nature equivalent factors were unsuitable for
this population. Since the comparison between each two years moves pro-
gressively forward. with each comparison young individuals enter into the
first year average while the next year they are one year older. met
might indicate that the conversion factors were too high for the younger
ages since such a situation would produce the negative trend. ibis is one
of the cri tici one of mature equivalent factors usually offered.

An effort was then made to check the suitability of the 51.3. factors
used for this pepulation by computing factors from the records of the animals

used in this study. 'Ihe results are shown in Table 2.

TABLE II. COMPARISON OF MA'lURE EQUIVALENT CONVERSION FACTORS

 

 

: : 2 : z : 3 z x :
Age :2-0:2-6:3-0:3—6:4—0:4—6:5-O:5-6:6-026-6:7-0
: x : x a z : e z s 2
z z z : x x x x : : :
No. of z z : z : x- : : x z :
Conparisons : 9 : 5 : 7 : 9 : lo : 9 z 10 : 7 x 10 : 10 z 10
z x : z x z x x 8 z :
Computed 11.1}. :1.448:l.241:l.351 :1.l74:1.210:l.189:1.0:59:1.095:1.000:l.000:1.000
: 8 x x z x x : : x :
Factors : x z x z x x t 2 x x
1311 M.E. : z : z z z z x : z :
Factors :1. 377:1. 275 :l.203:1.131 21.077 21.035 21.017 :1.006:1.000 31.000 :l.000
: z 2

 

-15 ..

In spite of the small numbers used. these computed factors seem to
be fairly comparable with the ones used: if anything. they do not make a
great along: correction for the younger individuals. This evidence
lessened the possibility of unsuitable mature equivalent factors influencing
the results markedly.

The second possibility suggested was that selection may have been
more rigid than was supposed. thus causing the observed trend in the results.
The theory behind this is when one record is used as the indication of the
level of an individual's producing ability. the next record can be expected
to regess toward the population average. If selection has been practiced
to any extent in the herd. the selected individuals will be above the herd
average: consequently. their aibsequent lactations will tend to regress
toward the average of the herd.

A check on the selection employed can be made quickly by a comparison
of the records of cows retained in the herd with those removed from the had.
me herdsman and others connected with the sanagemait of the herd insist that
there has been practically no selection, that the problem of maintaining
sufficient numbers in the herd has been difficult. Table 3 illustrates that
the average production of the cows retained in the herd as compared with the
herd average for each year is consistently higher. In some of the years the
better cows passed out of the picture. pulling the retained cow average down.
but especially in 1931. 1932. and 1938 was there a great difference between
cows retained and cows removed.

'lhis evidence makes selection a likely factor in producing the bias
in the correction factors. An error introduced in the early years is carried
through to the last year plus any errors introduced in the intervening years:

consequently, the end year accumulates all of the errors. The next question

is what can be (has to eliminate sale of this error.

TABLE III. .CWPARI SON 01" AVERAGE BUTTBBFAT PROHJCTION 0F COWS RETAINED
IN THE HERD WITH THE MD MIRAGE

’.
1

 

 

: : : : : : : : : : : : : : : :
:1981: 32: 33: 34: 35: 36: 37: 38: 39: AD: 41: 49: 43: 44: 46: 46
: : : : : : : : : : : : : : : :
: : : : : : : : : : : : : : : :
No. Cows : 22: 17: 17: 13: lo: 12: 15: 9: 13: 14: lo: 12: 1:5: 15: 13: 9
Herd Av. : 611:507:570:561:566:506:528:529:567:565:52’7:565:523:530:523:643
: : : : : : : : : : : : : : : :
No.3etained: 13: 9: 14: lo: 4: 9: 8: 6: 8: lo: 5: 9 : 10: 12: 4: -
Av. : 547:541:569:558:579:501:546:595:676:552:546:565:515:539:555: -
: : : : : : : : : : : : : : : :
No.Removed: 9: 8: 3: 3: 6: 3: 7: 4: 5: 4: 5: 8: 2: 3: 9: -
Av. : 458:469:574:575:558:521:509:449:553:598:507:566:569:494:509: -
: : : : : : : : : : : : : : : :
Retained Av: +36:+84:- 1:- 8:+l:5:- 5:+17:+66:+ 9:-13:+l9: O :- 8:+ 9:+22: «-
Minus Herd : : : : : : : : : : : : : : : :
Aver-age : : : : : : : : : : : : : : : :

 

According to lush (2) the repeatability of yearly fat production
considering only cows in the same herd is somewhere between 1/3 and 1/2.
and he suggests the use of the general figure. 2/5. although a higher
estimate might be Justified in herds with standardised management and
relatively complete environmental corrections. This repeatability (r) or
correlation coefficient between records ends by the same cow is the part
of the total variance between corrected records which is due to permanent
differences between cows.

here the repeatability of production is low, the use of one record
as the indication of the merit of the cow is not reliable. To estimate

the probable producing ability of a cow. Lush offers the equation:

 

Probable producing ability I f“: + nr 1 (cow's record)
1.2.!— x (herd average). Using one record and the

1 - r 4 nr
repeatability factor. 2/5/. the equation becomes: Cow's ability : 2/5

(her average) 4» 3/5 (hard average). This equation seems useful in this

case wha'e all known environmental effects have been corrected. and thaw

is still evidence of low repeatability of production.

It was decided to try applying this regression factor to the
study to determine if it would bring the corrected herd averages within
a reasonable range. The factor was applied to the first year of each
comparison, the reasoning being that on the basis of the one record and
a rQeatability of production of 2/5. the second record could be pre-
dicted at 3/5 closer to the herd average than the first. Any remaining
difference or the difference between the predicted and the observed amount
would then be due to temporary environmental effects. The results of
this study are shown in Table 4.

This apa‘ation reduced the extreme negative trend shown in the
results of the first table and produced several more plus differences
between years. but in comparison with the conditions of 1931. all of the
years are still poorer. It is very diffimlt to believe that in 1946 con-
ditions were so inferior to those of 1931 as to redice production 150
pounds per cow. Of course. the number of cows in the study is va'y small
and the discrepancy may lie in that since. as will be shown later. there
is more variance between cows wi thin years than between years.

Johannson (l) and Nelson (3) obtained fair results with this method
in their studies. There was no mparent selection practiced in the horde
they studied. and no regression factor was used. Ihe method perhaps has
possibilities: it is a simple and short means of estimating environmental
effects from year to year. but lists as its disadvantages that it does not
use all records and the lack of a test of significance to determine if
there are actial environmental differences for which a correction need be

made. Where selection is practiced as occurs in nearly all herds excqpt

RERESSION CORRECTED PAIRED YEAR COJPARI SONS

PLAN I-B.

IKBLE IV.

8

Corrected

8
2

7

8
8

6

8
2

5

4

3

2

....

1

 

Av.

am .
mum.
A
u...
0.15
f a
new
“.1.
mm
a.
we
Mil
um...
HYL
:...n
ma
“.3”...
C
O O
awe
o...
0'
V
mm

Year

 

: 511

511

0

1931

557

.0

: 507

- 5O

.0 .0

481

531

1
1

607

570

-37

+ 13

533

520

607

561

- 46

559

568

2
1..

632

: 566 :

- 66

536

556

673

-167 : 506

-101

470

n
.5

: 528 : 675

-147

504

5

4.

9

661

+ 15 : -132 529

558

543

....

567

622 + 67 : - 65

555

564

n
so

- 51 : ~123

509

w
a5

+ 16 : -107 565 672

551

5
so

3 -160

- 53

512

..

$ .
5

: -166 530 696

6

508

.4
m

.0 .0 .. .. .. Q. .0 .. .. .0 .. .. ..

5

8

5

9

0
1

: 523 677

: -154

+ 12

543

O
1

793

643

: -150

540

536

......

4.

.z....:..-............:........:......:................

”mugwmmwmu

a

mafia

 

Total No. of Records - 114

 

-19...

experimental. another variable, regression. enters in and may induce more
error. The method should provide a very rough approximation of the year

to year environmental differences within a herd. but the reliability in

this case is questionable.

- a) -
PLAN II - DIRECT COMPARI S3318

One disadvantage of the first plan was that it did not make full
use of all the records of a cow. For instance a cow produced in two
consecutive years. skipped the third year for some reason. and then had
a lactation in the fourth. 'lhe lactation in the fourth year had to be
discarded because the only comparison usable was between consecutive
years.

Another feature of Plan I which seemed as if it might be improved
was that by making comparisons of only two consecutive years and than
basing all corrections on the first year. the final year's correction
was an accumulation of any errors made in previous corrections. I

The attenpt to improve these defects yielded Plan II. The solution
seemed to be in developing a method of trying all of an individial's
lactations to her first. If all of a cow's lactations were compared
directly to the first year. that would correct both of the above situations.
With a direct comparison to the base year. records following "blank " years
would then be usable making all records usable. Also direct comparisons
would destroy the cumulative error in the final year.

Direct comparison was the answer to the problem except that no cow
produced the full fifteen years. One group produced for a certain period
of years. dropped out. and a new group prochced from there. This was
progressive. of course. with a few dropping and a few being added each year.
Consequently. there was no direct comparison of 1946 to 1931: in fact, the
only direct comparisons with 1931 were the years up to 1937. The year. 1932.
also could be compared with all years to 1937. Year 1933 was comparable with

all years up to 1939. The question then arose as to how these direct com-

-21..

parisons could be linked together and related back to 1931.

he first attenpt at linking these direct comparisons is shown
in Table 5. Ag before. 1931 was used as the base year. Line 1 shows
the comparisons of the average prediction in 1931 with the average
production of the same cows in each subsequmt year. 'Ihe encircled
numbers indicate the number of cows from the original year who had
records in each particular year. In the line l'ahde'd"differences'l the
average of all differences for the year is taken and used as the en-
vironmental difference between that year and 1931. Of course the only
comparison available for 1932 is the direct comparison with 1931 which
is - 79. This figure is used as the value for 1932.

Line 2 consists of direct conparisons with 1932. Since the
yearly averages are now compared with 1932. 1932 becomes the base in
this case. A, has already been determined. the representative value
for this year is - 79; therefore. the difference between 1932 and 1933
is applied to this base to lake the value indirectly the difference be-
tween 1933 and 1931. Likewise the acmal difference between each
succeeding year and 1932 is added to - 79. and the sum is recorded in
its respective column. For ample the actual difference between 1932
and 1933 is - 7. applied to the base value of 1932. it gives the value.
- 86. which is recorded under 1933 to indicate by this comparison 1933
was 86 pounds worse environmmtally than 1931. The actual difference
betwaen 1932 and 1934 is + 57, applied to the base - 79, it gives a
value of - 22. This process carries the comparison back to 1931: so

each figure shown is the compari son with 1931.

 

 

 

 

 

 

 

 

 

 

 

 
  

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

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-22..

Since all of the differences are alike in that they are the result
of comparison with the first year. they may be combined in an average to
produce a representative value for each year. As - 79 was the value of
1932. so the eight comparisons of + 5 may be averaged with the eight of
- 86 to obtain a base value of - 41 for 1933. This figure is used as
the base for the direct comparisons with 1933 given in line 3. In the
same manner a direct comparison between 1933 and 1934 yields - 5 which
is applied to the 1933 base, - 41. to give - 46. The final value for
1934 in relation to 1931 is the weighted average of the direct comparison
with 1931. the indirect comparison through 1932. and the indirect compari-
son through 1933. This same general procedire is used right on to the
last year.

The values in the difference line are then the environmental
diffa'ences between each year and 1931. They are. as explained before,
averages of all the comparisons between that year and every preceding
year. each referred back to the base year. 1931. If a comparison between
any two particular years is desired, it may be obtained by taking the
difference between the values for those years.

A few questions arose in working with this procedire. One was
whether a straight average or a weigh ted average would be more desirable
in calculating the base values for the various years. Ordinarily a
weighted average would be the only method. but in this situation there
was a question of whether a greater number of records was more valid than
a direct comparison. For example. in 1937 the direct comparisons with
1931 totaled 2. The comparisons of 1937 with 1936 and indirectly to 1931

totaled 9. In a weighting system based on frequencies the latter is

weithed 4 1/2 times the first. Siould the 9 indirect comparisons be
weighted 4 1/2 times 2 direct comparisons where a direct comparison is
certainly more valid than an indirect? It was finally decided that the
average weighted as to frequency. in the absence of a method of deter-
mining Just what weight each valne should receive. would be more nearly
correct.

A second question was whether the regression factor was applicable
in this procedure. The first method was a straight comparison between
each two years. but in this case all records were used and averages of
the comparisons were used. It might seem that in the use of averages
the reyession value might not apply. but on the other hand every com-
parison is between only two lactations. and if the first varied from the
herd average, the one following could be expected to be newer the herd
average. In this case correction for regression nigit be in order.

Table 6 shows the results obtained when correction fa' regression
was applied. The regression factor was used on the first year of eadi
compari son to predict the probable production for the following years,
and differences of the actual records from this probable value were taken.
no differences show the same general trend. as nigit be expected, but
they are smaller.

The third consideration was that this qsta utilized the same
records in several different conparisons, these comparisons being averaged
to provide a value for each year. For example, 6 of the 8 records used in
comparisons of 1933-1931 are used again to compare 1933 wifii 1932: hence
these 6 records in 1933 are compared with the base year 6 times directly

and 6 times indirectly to constitute 12 of the 16 comparisons making up

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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-25..

the average value for 1933. Likewise 4 of the 6 records in 1934 compared
with 1931 are used in comparison with 1932 and all of them are compared
with 1938 causing comparisons of those six records to appear 16 times in
the 24 composing the yearly average.

In the analysis of data it is desirable to utilize all informa-
tion to the fullest extent, but ordinarily there should be no repetition.
Ibis repeated use of records may be Justifiable in the manner in which
they was employed here, but an investigation was made to determine the
effect of eliminating this repeated use of records in various comparisons.
Table ’7 gives the results of this work. The only lactations used in the
comparison with each base year were records of cows initiating their pro-
duction in the hard that particular year. “he records of cows producing
in the herd appear only in comparison with their first year of lactation
in the herd. Their records are used in no otha‘ direct comparisons ex-
cept with their first year of lactation: hence they appear only once in
a yearly average. For example, cows entering the herd in 1982 have com-
parisons only on line 2 in which each subsequent record of those cows is
compared with the 1932 record to compose differences entering into each
respective yearly average. 'Ihose entering in 1933 show only in comparison
with 1933. End: individial record appears only once in chari son. These
results again show a trend which is sanewhat similar to the previous methods.

Table 8 is the final result of this method. It embodies the
utilization of all records corrected for regression with a reduction of
cumulative error in the last year, and without repetition of records in com-
parisons. As for the first method, there is no test of significance for

this method.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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PLAN III - ADJUSTED MEAN MITRENCES

'lhe last method is a relatively complicated system requiring con-
siderable calculation. but carrying a test of significance. It is offered
as a possibility for more exacting environmental determinations and as a
check on the two preceding plans.

Nelson (3) found the least squares method of fitting constants as
shown by Yates (5) to be applicable to similar data to these in environmental
difference determinations. The method was satisfactory except for the ex-
tremely lengthy calculations.

Since these data were similar in being adaptable to a tw0 way table
of classes containing disprOportionate numbers, it was decided to use the
least squares method of fitting constants to analyze these data. However, an
alternate method was offered which seaned much simpler and was adapted for
this trial.

he method used is that described by Patterson (4) as a method of
adjusting which when applied to data with unequal subclass numbers, makes it
possible to obtain a sum of squares for each source of variance that is free
of the influence of the other effect. Likewise adjusted border means may be
calculated by the method of adjusting, such means being devoid of the con-
founding effect produced by disprOpor tionate subclass numbers. A complete
discussion of the method may be had by referring to the original article.
'lhis paper will be confined principally to a description of the actual pro-
cess involved.

'Ihe problm is to adjust the data so that the adjusted year means

will be devoid of any effect of group. 'Ihe first Operation is that of

-29-

setting the data up in a two way table mch as that shown in Table 9. The
data are divided into cow groups as the first step. A cow group consists

of all individuals having identical emer iences in that they have lactations
in the same years. For instance Cow Group 1 contains 5 cows and the only
cows in the herd who have records only in 1931 and 1932. In Cow Group 2 is

. one cow, the only one with records in 1981,1982. and 1983. Cow Group 3 con-
sists of 2 cows having records in 1931,1932, 1933, and 1934. This continues
to Cow' Group 40 which contains 2 cows with lactations in 1944, 1945, and 1946.
Altogether there are 40 cow groups containing 183 individual records. After
the division is made according to cow groups, the records of each cow group
are placed in their respective years. Thus we have subclasses divided on
the basis of cow groups and years.

A simplification of the data thus set up may be provided. If a
mean is taken for each cow group. and these means compared with the mean of
the entire data. the differences show how much each youp is above or below
the population average. If the individuals within each group are corrected
so that each group mean is equal to the over-all mean, then there are no

differences between groups. This process is shown ‘w be following equation:

(1) x“ - x1 + x = A”
where I“ is the i th cow in the Jth row or column. 11 is the mean of the

Jth row or column, i is the grand mean. and A” is the adjusted‘ith cow in

the Jth row or colnmu. Then :

(2) ZULU-15+ DH:
".1
his does not affect the variability within subclasses: hence it is necessary

 

 

to correct only the means of the subclasses:

 

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hum» .. .m.mwm . u a . a n u . . u . n . . somebfinmw. H

v a new: “mom." ”mom.” 3va ”Q5." $de .32.. ”QmH 3an «mama .ban 3me Emma 3an ”mam... .Nmmu :an ”ducky

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359mm ho Hag H43 05 was” .5" fig.“

-31-

_ (3)11J-XJ+X=A1J
x” being the mean of the ith subclass in the jth row or column and 113.
the corrected mean of the ith subclass in the jth row or column. This
is the process of adjustment to be used in the analysis.

'lhe adjustment process is useful in this case where the sub-
class numbers are unequal. When subclass numbers are disproportional,
the differences between border means are not true estimates of the
parameter differences because the differences are determined not only
by one classification but include some of the effects of the otha' classi-
fication. For example, the differences between year means contain some
of the cow group differences: likewise group mean differences include
some of the effects of year mean differences. The object then is to
remove the effects of group mean differences from the year mean differences
which may be done by adjusting according to the final equation.

me first step is to estimate the sun of squares (he to year
means. he year mean differences contain effects of grozp because of dis-
proportionate subclass numbers: hence if all group means are corrected to
the grand mean, the group effect will be removed from the year differences.
'lhis adjustment is shown in Table 10. All group means are adjusted to the
grand mean so no difference remains between them. The sum of squares of
years means with group effect removed is than 89,700. 'ihis figure is less
than it diould be because the removed group differences contained some year
effects due to disproportionate subclass numbers: therefore an adjustment
in the other direction must be made to remove these effects.

When year differences are renoved by readjustment. the group

 

 

 

 

 

 

 

 

 

 

 

 

 

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-33..

differences will no longer be zero, again due tounequa'l subclass numbers.
The sum of squares is then calculated for group means yielding 8.200.
This is a portion of the year sum of squares removed in the first adjust-
ment for group and is added to the first sum 89,700 to give 97, 900 as a
sum of squares for between years.

Again readjustmmt must be made for group effects, and the
weighted sum of squares calculated. As this procedure continues. each
readjustment recovers a portion of the between year sum of squares lost
in the first adjustment. The total of all the sums of squares is the
sum of squares of between year means. his total consists of the con-
tributions from both sets of border means. Continued readjustments
result in smaller and snaller sums of squares, and they should cease
when the adjustment results in only a fraction of the initial aim of
squares.

The sum of squares for between years. in this case after 5
adjustments, is 89,700 + 8,3)0 + 1,200 + 1,600 + 700 or 101.400.
Further adJu stment in this direction is unnecessary because the part of
the year differences unrecovered is small since the last sun of squares
is only a fraction of the initial sum of squares.

The variation between years has been determined: the next
step is to estimate the sum of squares for cow groups. This is accomplished
by similar adjustments except that the first correction must be made to
remove the effects of year this time. he same process follows and yields
a sun of squares for between cow groups of 848,900 from 4 adjustments com-
posed of sums of squares. 801.000 #- 36,000 + 4,900 + 2,000.

'ihe sums of squares for between years and between cow groups

-34-

having been determined. interaction can be calculated. The assumption
is made that the sum of squares of the subclass means they are adjusted
for border effects is an efficient estimate of the variance the to inter-

action. The combined results are shown in Table 11.

TABLE II. ANALYSIS OF VARIANCE OF YEARLY DIFFERENCES IN PRODUCTION

 

“'3'

 

Source of Variance 8 D/l' 8 an of Squares : Mean Square
1 i i
Between Subclasses 8 157 : 1,610,0(1') 8 10,300
Between Year Means : 15 8 101.400 : 6.760
Between Cow Group Means : 39 : 843,900 : 21,638
8 3 8

r 2- 6.7505,.” = 1.048

lbs analysis of variance shows no evidence of significance in
year mean differences. This may be due to the small mmb er of individuals
or the scattering and numerous subclasses. but no evidence of significance
is demonstrated in the yearly environmental differences in this herd. One
difficulty lies in the fact that there is more variance within year than
between years.

Although the yearly diffa'ences in these data may not be signifi-
cant. the rest of the method will be presented to compare with the two plans
already presented. from the adjustments already made the adjusted year
means. relatively free of group effect. may be determined. This consists
of adding to the gand mean the correction factors or the differences be-
tween the succeeding group adjusted year means and the grand mean. Con-
tinually diminishing differences are obtained with each adjustment. For

the purpose of this study the total of the differences between each group

-35-

adJusted year mean and the grand mean may be used. The results of this
process are shown in Table 12. From these the yearly differences can
be computed by taking the differences between the total corrections of
the years in question. To be on the same basis as the results of the
previous trials. differences are taken between 1931 and each succeeding
year.

As shown, this method has the advantages of utilizing all
possible data and it has a test of significance. 0n the other hand,
it requires considerable computation especially considering that at best

the result is only an approximation.

~86-

 

 

TABLE III. ENVIRONMENTAL DH’FRENCES DETERMINED BY ADIUST.HETHOD

; Adjustments ; * : Environmental
3 1 ' 2 3 3 ‘ Total : Difference

1931 g + 20.9 8 + 3.0: +2.2: + 25.1 : Base

1932 i - 21.8 : + 0.9: +2.2: - 18.7 - 4.5

1933 : 4- 5.0 : + 8.18 +1.8: + 9.9 2. - 15

1934 :4' 15.3 ;+ 3.4: +1.6; 4 20.3 ; '- 6

1935 : + 38.5 : + 6.98 +2.1: 4 47.5 i + 21

1936 i - 38.8 : ~ 5.7! -0.9: - 45.4 g - 72

1937 ; -17.5 : -- 4.92 “1.6: - 24.0 E "' 50

1988 i- 7.4 E- 4.3: .2.0: ~ 13.7 § - 40

1939 E + 31.0 E + 1.6: 4.6: + 32.0 1 + 5

1940 :4 15.3 :4 2.5: +0.5: 4 18.3 : ~ 8

1941 i - 24.2 ; + 1.9: +0.8: - 21.5 : - 48

1942 i + 17.6 Q + 1.3; -o.7§ + 18.2 Q - s

1943 ; - 17.1 : - 3.2: -1.7; - 22.0 : - 48

1944 i - 39.4 ; - 3.3: “2.0; "' 25.7 i - 52

1945 ;- 1.5 :- 4.9: -2.7: - 9.1 ; - 35

1946 ; + 35.4 ;+ 3.1; 43.9: + 38.6 i 'I' 13

*m of group adju
goup adjustments.

sted year mean - grand mean differences for three

8

2

COWARI SON OF PLANS

Three methods have been presented for determining the yearly
environmental differences within a hard. It might now be desirable to
compare these methods as to their possible merit.

Each of the methods was used to calculate the yearly environ-
mental differences in the M. S. C. Holstein herd over a period. of 16 years.
To be placed on an equivalent basis. they were conputed in relation to
the base or first year, 1931. Table 13 contains the results of the three
plans including 5 variations of the second plan. It met be remembered
that the number of records was small which limits the validity of any of
the calculations. and the analysis of variance gave no indication of
significant year differences in average production.

The adjusted means method supposedly gives the most likely
correct values for yearly environmental differences although the devia-
tions are a little smaller than they might be if flue adjustments had been
carried further. Since this method should be reliable. it is used as a
standard for comparison with the less elaborate systems.

I-A. the paired year comparison, is apparently completely out
of preportion in this emu. There is an extreme accumulation of negative
values throwing the comparison very Inch out of line. 1-3. likewise, is
not well in agreement with the otha' results. Why this systen should
produce such an increasing negative value still is not apparent.

II-A. II-B. and II-C are the variations of the direct comparison
plan utilising all the comparisons with every year. As was stated earlier,

there is some question as to whether the weighted average was really

Justified in this case where a direct comparison is more reliable than an

m.

ENVIRONMENTAL DIFFERENCES DETERMINED BY VARIOUS METHODS

TABLE XIII.

..

 

II-D: II~3: III

II-A : II-B : II-C :

I-B

I-A

 

Base Year

1931:

16

0000000000

w

6

.0 .. 8 .0 o.

9
7

8
5

w

n
.9
9
7

3

a

9
7

n.

my

1932: -

'79

1988: -

86

8

1934: -

6

36:+

~22:-

46

91

141+ 10

1935: - 103 z -

+

5

548+

218+ 11

66

5%

.. .. z .0 ..

5
5

1936: - 212 i - 167 :

1937: - 189 8 - 147 8

'71:- 55

:-1048-

8

54:-

--98

1938: - 196 : - 182 :

1

47:-

-988-1208-

6

4.

63

18+ 25:+

-42:+ 198-

65

598-

8

19408 - 191 8 -

11:4» 20:+ 41

19418 - 235

-116

:-123:

:9‘...’

9
3

+

4
1

m

4.
3

‘38::

8
8
8

194.2: - 230

1943: - 283

19448 - 288 8 - 166 8

52

75

-1338-1288-

96:-66:-328-118 35

-119 : -

1945: - 278 : -154 8

8 8 8

218+ 288+ 218+ 698

+13

19468-2938-1508

8

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I

II-E Direct Compari son

II-A Direct Comparison (Weignted)

(Regression Corrected)
(Non-Diplicating)
AdJus ted Means

II -B Direct Comparison (Unweigited)

III

 

indirect comparison, but if it is assumed that the indirect comparisons
are reliable, the weighted average is desirable. II-C is theoretically
the best estimate of the three as it includes the weigIted averages and
is also corrected to counteract the effects of the expected regression
from a single record toward the herd average in succeeding records. It
will be noted that it is in general agreement with the "adjusted mean“
values. The trend in direction in every year except 1937 agrees, or
each year is + or - in respect to the preceding year in both results.

II-D and II-E are the products of the same type of calculation
as the preceding three methods, but each individual is used in only one
comparison rather than being used repeatedly as in the preceding types.
II—D is the straight weighted average calculation while II-E is corrected
for regression. In both of these the general picture is similar to the
"adjusted means" result.

One point demonstrates itself fairly well in that the regression
corrected figures are drawn closer to zero deviation than the uncorrected.
Where negative differences predominated in the uncorrected figures, the
regression correction pulls them down until positive differences appear
also. 'Ihe regression corrected figures of all methods correspond more
closely to the "adjusted mean" results than do the uncorrected.

If the "adjusted mean" differences are assumed to be the most
reliable, the most closely agreeing would be method II-C or II-E. The
numbers of records are so small, however, that no definite statement
should be made concerning the best system especially in view of the fact
that no significant difference between years was evident.

It seemed that it might be interesting to apply these differences

-40..

as corrections to the actual herd averages each year to see what picture
they night prodice. Inch difference was applied to the herd average of
each respective year as a correction or with its sign reversed. If the
yearly environmental difference was positive, the year was better than
the base year: hence, that difference must be deducted from the herd
average to make all yearly averages equivalent. The results of applying
I-B and II-l are shown graphically in comparison with III and the actual
herd averages in Figure 3: II-C and II-E are compared in Figure 4.

With the assumption that III is correct, its curve waild indi-
cate the genetic trend of the herd. By means of the conputation of Plan
III nearly all environmental differences are removed frm the herd average
leaving only the heredity differences. This then gives the picture of
the success of the breeding and selection operations carried on by the

breeder in respect to butterfat production.

 

 

WY

'ihe mistaking of environmental effects for hereditary effects
is one of the h'eeder's largest barriers to the rapid improvement of his
herd. the removal of the confu sing effects of environment is highly
desirable and can be accomplished with greater practicability statistically.

Various methods of equalizing yearly environmental differences
were presented, all of these being based on the theory that under constant
environmental conditions, an individual cow should produce the same amount
of butterfat each lactation. Any variation from this even production
should be due in varying environmental effects.

The mature equivalent, 3X, 305-dsy records of all cows in the
Holstein herd at 11.8. C. with more than one lactation since 1930 used to
compare the various methods of determining the yearly environmental differ-
ences in the herd.

The first method was a comparison of the records of cows in one
year with the records of the same individials in the succeeding year. his
method is simple and brief, but it has no test of significance: any errors
may be eumlative with the use of a base year: and not all records can be
utilized where there is a skipping of years. In this particular case
selection in the herd affected the results for which a regression correction
was made.

The second method was characterized by the use of direct comparison
of records of cows in each year with the records of the same cows in a base
year where possible and indirect comparison elsewhere. Weighted and

straight averages for computing difference values and regression corrections

-44..

were utili zed and compared. Another variation was presented in which
duplications of comparisons were avoided and the comparisons of records
of the same cow were used only once. This general method is slightly
more complicated than the first method, but it has the added advantage
of utilizing all available records of cows with two or more lactations.
There is still no test of significance.

The final method was a lengthy process of adjustment of border
means. The computation is the limiting factor in this method. It does
have the advantage of utilizing all of the records, and there is a test
of significance to indicate if there is a yearly difference which needs
correcting.

Since there was no evidence of significant difference between
years, and since the number of records was small, few conclusions may be
drawn from the analysis of the methods. They are suggested as possibili-
ties. Fhere selection is employed in the herd as is the umal occurrence,
some correction must be made. A more precise correction than the re-
gression factor used in this study must be determined.

'iha'e is considerable question as to the validity of mature
equivalent conversion factors which may have some bearing on this work.
'Ihese must be correct or the entire basis of the determination is destroyed.
There is some question as to how useful they are, and there is no doubt
but that they may be very incorrect for certain individials. When there
is such a definite doubt about M.E. conversion factors. more errors may be
introduced than are removed. Probably the only solution to this is
to use only the actual mature records of the cows. Obviously this would

require a tremendous number of cows in the herd, more than the average

-5-

herd contains since the makedup of most herds is young cows.

It would seem that the type of environmental corrections dis-
cussed in this paper are, in general fairly impractical at present. If
only records of mature cows in sufficient numbers were available, and if
it could be determined.what.their likely production might be to eliminate
the regression factor or if they were unselected animals, there might be
encouragement for the use of one of these methods. At present the value
obtained from the results may'be overbalanced.by the error and.the compu-

tati on involved.

(1)

(2)

(3)

(4)

(5)

L ITEA’IURE CI TED

Johannson, I. "
Undersokningar over avkastnings - resultaten inom
thorsgtrabesattningen under stallfodring och betsesgdng.
Noddelande Fran Kingl. Landtbruksakademiens
Lantbmksavdelning Nr. 138 1-53. 1988
(Cited by n. H. Nelson. See Reference (3) ).

101.1311. Jay Le
Animal Breeding Plans, Third Edition.
Iowa State College Prass. 1945.

Nelson, R. H.
'me Effects of Inbreeding on a Herd of Holstein-Friesian
Cattle.
{thesis for Ph.D. degree, Iowa State College. 1943.

Patterson, R. E.
The Use of Adjusting Factors in the Analysis of Data with
Disprcportionate Subclass Numbers.
Jour. Am. Stat. Assoc.. 418 334-346. 1946.

Yates, F.
'me Analysis of Multiple Classifications with Unequal
Numbers in the Different Classes.
Jour. Am. Stat. Assoc" 29: 51-69. 1934.

 

 

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