GENETIC CORREMWON AND RESPONSE 330
SELECWON {N SiMUM’s’EG WPULATEDNS
“s’hesis for the Degree of Ph. D.
MIiEé-HGAN STATE UNIVERSUY
ROBERT JACK PARKER
19%
'fHESls-
This is to certify that the
thesis entitled
GENETIC CORRELATION AND RESPONSE TO SELECTION
IN SIMULATED POPULATIONS
presented by
Robert J. Parker
has been accepted towards fulfillment
of the requirements for
M;— degree in m
* aim) 9. 7119124424,
Major professor
Due 31 October 1966
0-169
GENETIC CORRELATION AND RESPONSE TO SELECTION
IN SIMULATED POPULATIONS
By
Robert Jack Parker
AN ABSTRACT OF A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Dairy
1966
ABSTRACT
GENETIC CORRELATION AND RESPONSE TO SELECTION
IN SIMULATED POPULATIONS
by Robert Jack Parker
The effects of truncation selection of a primary trait upon
the genetic correlation and the correlated response in a
secondary trait were examined. Genetic pOpulations and the
process of selection were simulated through the use of random
numbers generated by a computer.
Selection was made for one of two quantitative traits, and
the correlated response in the other trait was measured in each
generation. The pOpulation was bisexual diploid and the traits
were expressed in both sexes. The size of the population of
parents was 48 in each generation and mating was random, the
number of offspring produced being determined by the level of
selection. Each trait was controlled by 48 loci segregating
independently, effects were equal at every locus, and gene
frequency was arbitrarily set at 0. 5 at each locus in the initial
generation.
Three degrees of genetic correlation, three levels of
selection, and three levels of environmental variation were
simulated. Two models of gene action, an additive model and
a model of complete dominance, were considered. In the model
of complete dominance, the experiment was carried out
1
2
separately for Opposite directions of selection.
The genetic correlation was determined by the number of
loci which affected both traits and was measured each generation
as the product-moment correlation of genotypic values and by
two methods utilizing phenotypic covariances between parent
and Offspring.
In the additive model the genetic correlation, measured as
the correlation of genotypic values in each offspring generation,
remained consistently near its initial level at all levels of
environment when the fraction of Offspring saved as parents was
as high as one-half. When the fraction of offspring saved became
as low as one-fifth, the genetic correlation decreased. A closer
examination of the genetic correlation indicated that at low
selection intensity the genetic covariance between the traits
was maintained. With greater selection intensity, the genetic
covariance decreased, but the genetic variances of the traits
declined proportionately causing the genetic correlation to be
maintained.
Truncation selection caused a decrease in the genetic cor-
relation in those offspring selected to become parents of the
next generation. The amount of reduction depended on the
heritability of the selected trait rather than on the degree of
truncation selection.
3
Estimates of genetic correlation obtained from phenotypic
covariances between parent and offSpring fluctuated markedly
from the true correlation in the small populations simulated.
The correlated response of the unselected trait to selection
of the primary trait agreed closely with response expected
from theoretical considerations.
In the model of complete dominance, the change in the genetic
correlation when selection was by upper truncation followed
essentially the same pattern as in the additive model. When
selection was by lower truncation, the behaviour under selection
Of the genetic correlation conformed to that for the additive model
although the decrease in the correlation at high intensity of
selection was more rapid. As in the additive model, truncation
selection caused a decrease in the genetic correlation in the
offspring selected to be parents whether selection was by upper
or lower truncation. Estimates of genetic correlation computed
from phenotypic covariances between parent and off3pring were
also poor in the model of complete dominance. The reaponse
of the genotypic mean of the unselected trait to selection of the
primary trait in opposite directions was quite symmetrical for
the first few generations but became distinctly asymmetrical
in later generations. At low levels of selection the response was
fairly linear but became distinctly curvilinear as the intensity
of selection increased.
GENETIC CORRELATION AND RESPONSE TO SELECTION
IN SIMULATED POPULATIONS
BY
Robert Jack Parker
A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
’ Department of Dairy
1966
ACKNOWLEDGEMENTS
I wish to express my gratitude to Dr. Lon D. McGilliard
for the consistently sound advice courteously extended
throughout my period of graduate study and for the patient
guidance provided during the preparation of this thesis.
The valuable suggestions contributed by Dr. John L. Gill
are also gratefully appreciated.
I would like to acknowledge the friendly counsel of my
fellow graduate students, Arthur Dayton and William Wunder,
who helped make the period of graduate study more enjoyable
and rewarding.
I am also grateful for the financial assistance provided by
the Department of Dairy in the form of the C. E. Wilson
Research Fellowship. ,
ii
TAB LE OF CONTENTS
Page
Acknowledgments................... ii
List of Tables 0 O O O C O O O O O O O O O O O O O O I 0 v
ListofFigures.................... vi
Introduction ............. . . . . . . . . . 1
ReviewofLiterature................. 4
Methods and Procedure . . . . . . . . . . . . . . . 30
The Experimental Design and Parameters
Simulated O O O O O I O O O O O O O C C C. O O O O 30
The Structure of the Initial Population . . . . . 33
Simulation of the Genetic Correlation . . . . . . 37
The Mechanics of Simulation . . . . . . . . . . . 44
Results and Discussion . . . . . . . . . ...... . 58
The Additive Model . . . ..... . . . 58
The Effect of Selection on the Genetic
Correlation O O O O O O O O O O I O O O O O O 59
The Genetic Correlation in the Truncated
Distribution . . . . ...... . . . . . . . 78
The Estimates of Genetic Correlation from
Phenotypic Covariances between Parent
and Offspring O O O O O O O O O O O O O O O O O 80
Correlated Response to Selection . . . . . . 85
The Model of Complete Dominance . . . . . . . 96
The Effect of Selection on the Genetic
Correlation................. 97
The Genetic Correlation in the Truncated
Distribution................. 126
iii
Page
The Estimates of Genetic Correlation
obtained from Phenotypic Covariances
between Parent and Offspring . . . . . . . . 128
Correlated Response to Selection . . . . . . 132
Application of Results and Suggestions for
furtherResearch................. 146
Summary and Conclusions . . . . . . . . . . . . . . 149
Literature Cited 0 O O O O C O O O O O O O O O O O O O 156
iv
LIST OF TAB LES
Table Page
1. Deviation of the genetic correlation by parent-
offspring covariances from the product-
moment correlation of genotypic values.
(Additivemodel)................. 83
2. The response in trait X and correlated response
in Y at three levels of genetic correlation,
measured as per cent of selection goal
achieved at the 15th and 30th generation
(Additivemodel)................ 94
3. Deviation of the genetic correlation by parent-
offspring covariances from the product-
moment correlation of genotypic values
(Complete dominance - upwards selection) . . 130
4. Advance possible in the genotypic mean of
X and Y by different methods of selection
and at different degrees of genetic correlation . 136
5. The response in trait X and correlated response
in Y at three levels of genetic correlation,
measured as per cent of selection goal
achieved at the 15th and 30th generation.
(Complete dominance, selection by upper
truncation.)...................141
6. The response in trait X and correlated response
in Y at three levels of genetic correlation,
measured as per cent of selection goal
achieved at the 15th and 30th generation
(Complete dominance, selection by lower
truncation.)...................143
Figu
1.1
-m
1.2
1.3
2.1
2.2
2.3
2.4
LIST OF FIGURES
Figure
Page
1. 1 Change in genetic correlations at three
levels of selection when hi - 0. l
(additive made 1) O O O O O O O O O O O O O O
l. 2 Change in genetic correlations at three
levels of selection when hk- 0. 4
(additive model) . .
l. 3 Change in genetic correlations at three
levels of selection when hi- 0. 7
(additive madel) O C O O O O C O O O O O O C O 0
63
Z. l
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0. 8 and hi: 0. l (additive model) . .
67
2.2
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0. 5 and h'xIBO. l (additive model) . . .
68
2. 3
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b80. 2 and hk=0. l (additive model) . . . 69
2. 4 The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 8 and his: 0. 4 (additive model) . . .
70
2. 5
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 5 and hy- 0.4 (additive model) . . . 7l
Fi
2.6
2.7
2.8
2.9
3.1
3.2
3.3
4.1
4.2
4,3
5.1
Figure
2.6
2.7
2.8
2.9
3.2
3.3
4.2
4.3
5.1
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-0.2 and Iii-0.4 (additive model) . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 8 and bah-O. 7 (additive model) . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 5 and hi-O. 7 (additive model) . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0.2 and hi-o. 7 (additive model) . . .
Mean genetic progress at three levels of
selection when hit-0. l (additive model) . . '.
Mean genetic progress at three levels of
‘ selection when hit-0'4 (additive model) . . .
Mean genetic progress at three levels of
selection when hk-O. 7 (additive model) . .
Change in genetic correlations at three
levels of selection when hk-O. 1 (complete
dominance). Selection by upper truncation . . .
Change in genetic correlations at three
levels of selection when h'x- 0. 4 (complete
dominance). Selection by upper truncation . .
Change in genetic correlations at three
levels of selection when h); '-0. 7 (complete
dominance). Selection by upper truncation. . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 8 and 11* =0. 1 (complete dominance)
Selection by upper truncation . . . . . . . . . .
vii
Page
72
73
74
75
87
88
89
98
99
100
103
FiguI
5.2
5.3 '
5.4
5.5
5.6
5.7
5.8
Figure
5.2
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 5 and Iii-0. 1 (complete dominance)
Selection by upper truncation . . . . . . . .. . .
5. 3 The relationship between the genotypic
5.4
5.5
5.6
5.7
5.8
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 2 (and h);- 0. 1 (complete dominance)
Selection by upper truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic correlation
at three levels of correlation when
halo. 8 and hi. 0. 4 (complete dominance)
Selection by upper truncation . . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 5 and his: 0. 4 (complete dominance)
Selection by upper truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 2 and hi: 0. 4 (complete dominance)
Selection by upper truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b .0. 8 and hi- 0. 7 (complete dominance)
Selection by upper truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when bBO. 5 and h;‘=0. 7 (complete dominance)
Selection by upper truncation . . . . . . . . . .
viii
Page
104
105
106
107
108
109
110
Figu
5.9
6.1
6.2 <
6.3 (
7.1 '.
7.2
7.3
Figure
5. 9 The relationship between the genotypic
6.1
6.2
6.3
7.2
7.3
7.4
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 2 and hi- 0. 7 (complete dominance)
Selection by upper truncation . . . . . . . . . .
Change in genetic correlations at three
levels of selection when hit: 0. 1 (complete
dominance) Selection by lower truncation . . .
Change in genetic correlations at three
levels of selection when hkao. 4 (complete
dominance) Selection by lower truncation . . .
Change in genetic correlations at three
levels of selection when hiBOo 7 (complete
dominance) Selection by lower truncation . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0. 8 and hk= 0. 1 (complete dominance)
Selection by lower truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0. 5 and hi=0. 1 (complete dominance)
Selection by lower truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 2 and hit. 0. 1 (complete dominance)
Selection by lower truncation . .. . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 8 and hi- 0. 4 (complete dominance)
Selection by lower truncation . . . . . . . . . .
ix
Page
111
113
114
115
117
118
119
120
Figu
7.5 '
7.6'1
7.7 'I
7.8".
7.9
8.1
8.2
Figure
7.5
7.6
7.7
7.8
7.9
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0. 5 and hk-O. 4 (complete dominance)
Selection by lower truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when halo. 2 and hit: 0. 4 (complete dominance)
Selection by lower truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b=0. 8 and I a 0. 7 (complete dominance)
Selection by lower truncation . . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 5 and bk -0. 7 (complete dominance)
Selection by lower truncation . . . . . . . . .
The relationship between the genotypic
variances, covariance, and genetic cor-
relation at three levels of correlation
when b-O. 2 and hk-O. 7 (complete dominance)
Selection by lower truncation . . . . . . . . . .
Mean genetic progress at three levels of
selection when h," -0. 1 (complete dominance)
Upper four curves indicate selection by
upper truncation, lower four selection
bylowertruncation. . . . . . . . . . . . . . .
Mean genetic progress at three levels of
selection when bk -0. 4 (complete dominance)
Upper four curves indicate selection by
upper truncation, lower four selection
bylowertruncation. . . . . . . . . . . . . . .
Page
121
122
123
124
125
133
134
Figure
8.3 M
cr‘r-‘P-ezn
Figure Page
8. 3 Mean genetic progress at three levels of
selection when 1132-0. 7 (complete dominance)
Upper four curves indicate selection by
upper truncation, lower four selection
bylowertruncation............... 135
xi
"There is no more common error
than to assume that, because
prolonged and accurate mathematical
calculations have been made, the
application of the result to some
fact of nature is absolutely certain. "
A. N. WHITEHEAD
Th
the eff
knowle
genetic
of the ;
reapon
measux
maxim
10 the .
and to
corre]
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Corre
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Phen
INTRODUCTION
The improvement of economic traits in livestock depends upon
the effective use of genetic variation. Pertinent to this is a
knowledge of the relationships among the traits including the
genetic and environmental correlations among them. Knowledge
of the genetic correlation among traits is necessary to predict the
response to selection of traits not directly selected and to combine
measurements on different traits in selection indexes to secure
maximum improvement. Predictions of this type are valid only
to the degree that the estimate of the genetic correlation is valid
and to the extent that selection itself does not modify the genetic
correlations .
There has not been enough study of genetic correlation and
correlated response to selection to allow conclusions to be
drawn about their behaviour under selection for questions such
as to what extent the correlation can be changed by selection,
over how many generations the correlated reSponses continue,
or what is the total correlated response when the limit of
selection is reached.
The introduction and rapid development of Monte Car lo methods
in recent years has provided a tool for the study of population
phenomena in a more detailed manner than has been possible with
1
2
either the techniques of mathematical genetics or laboratory
studies with biological populations.
The concepts underlying the use of Monte Carlo techniques ,
the use of some mechanical device to create simulated
observations, are not new and may be considered as old as
probability and statistics. The Monte Carlo method came into
use during the 1940's to identify statistical procedures for
Obtaining numerical estimates for problems in nuclear physics.
With the introduction of high speed computers, interest in the
theory and application of Monte Carlo techniques greatly increased.
The applicability of Monte Carlo procedures to quantitative
genetics arises from inheritance having a relatively simple
probabilistic basis , and Monte Carlo methods involve the
simulation of probabilistic mechanisms. Thus, through intelligent
simulation of these basic genetic mechanisms , additional insights
into their consequences for various situations becomes possible.
Yet, it should be stressed that the simulation has to be based on
our present theory of biometrical genetics and the results Obtained
can only be studied in the light of existing theory. The procedures
cannot be expected to increase our knowledge of the basic,
mechanisms themselves. The major contribution may well be to
emphasize and clarify points which should have been recognized
3
previously but which have been overlooked or considered less
important than they should have been.
This investigation was to examine the effects of the
intensity of selection and the environmental variation upon the
behaviour of the genetic correlation and upon the correlated
response of traits not selected to selection.
REVIEW OF LITERATURE
A change in other traits not under selection when traits under
selection are modified has been observed for some time. The
statement by Darwin (1875) indicates that he had noted the
importance of correlated variation: "Hence, if man goes on
selecting, and thus augmenting, any peculiarity, he will almost
certainly modify unintentionally other parts of the structure ,
owing to the mysterious laws of correlation. " In the study of
such correlated response the genetic correlation between the
traits plays an important role in determining their pattern under
selection.
The most important underlying cause of genetic correlation
appears to be pleiotropy, a gene affects two or more traits,
the segregating gene causes simultaneous variation in the traits
it affects. Other possible causes of genetic correlation are
usually considered to be minor or transient. For example,
according to Lush (1948), linkage can be an important cause
only in a p0pulation where either the coupling or repulsion phase
of the double heterozygote is far more abundant than the other.
Such a condition would persist for only a few generations after a
cross because in a freely interbreeding p0pulation, the coupling
4
5
and repulsion phases of the double heterozygote tend rapidly
to become equally frequent. For a second example, Lush
suggests that an apparent genetic correlation could be caused
by different intensities or different directions of selection in
non-interbreeding sub-groups of a p0pulation. If the whole
population were studied as a unit without regard to the sub-groups,
the differences between groups could create a genetic correlation
in the population, although there would be no genetic correlation
within each sub-group considered separately.
The quantitative aspects of genetic correlation were presented
by Hazel (1943), who developed a statistical technique to estimate
genetic correlation based upon the fundamental formulations of
biometrical genetics of Fisher and Wright. The technique of
estimation was based on the resemblance between relatives
similar to the method used in the estimation of heritability.
However, instead of the components of variance of one trait, the
components of covariance of the two traits were computed. In
general, the more closely the animals are related, the smaller
should be the sampling error of the estimate. Sufficient care is
needed to avoid correlated environments of the individuals .concerned.
Estimates of genetic correlations obtained by covariancebetween
relatives have not been precise, however, and are usually subject
to rather large sampling errors.
Reeve (1955) presented a method to estimate the sampling
variance of the genetic correlation coefficient between two
traits in large samples where the correlation is estimated from
the four parent-offspring covariances for the two traits. The
variance was expressed in terms of the heritabilities, genetic
and phenotypic correlations between the two traits. The variance
was the same whether the arithmetic mean or the geometric mean
of the covariances involving both characters was used in calculating
the genetic correlation.
Robertson (1959) develOped a measure of the sampling variance
where the genetic correlation is estimated from variance and
covariance components for the two traits within and between groups
of relatives. He presented formulae for the Special case in which
the two traits havelthe same heritability. Since the standard
errors of the two heritabilities appear in the formulae, an
experiment designed to minimize the sampling variance of an
estimate of heritability should also have the optimum structure
for the estimation of a genetic correlation. An attempt was made
to suggest the form of the more general solution where the two
traits have different heritabilities.
Using a different approach, Tallis (1959) presented a general
solution which reduced to that presented by Robertson when the
7
two traits have equal heritabilities. The formula develOped by
Tallis holds for estimating the sampling variance of a genetic
correlation estimated from an analysis of variance and covariance
provided the estimate of heritability of neither trait is zero and
the number of offspring per sire is constant. A general solution
has also been described by Mode and Robinson (1959) for genetic
correlations estimated from components of variance in a random
model with equal sub-class numbers nested four ways.
Van Vleck and Henderson (1961) presented a procedure for
obtaining empirical sampling estimates of genetic correlations
obtained from parent-offspring analysis. Sampling variances of
these estimates were then compared with the theoretical variances
derived by Reeve (1955). They found that for sample sizes of 1,000
or more, the approximate formulae of Reeve for the variance in,
large samples agreed. For smaller sample sizes (500 or less)
the approximations were not close unless the heritabilities of the
traits were high. In fact, when the sample size was 100 or less,
the approximations were very misleading. Van Vleck and
Henderson concluded that for estimating genetic correlations,
at least 1,000 sets of observations are needed to obtain reasonable
estimates of the sampling variance. Even then the sampling
variances may be too large for the estimates to be of use,
especially if heritabilities of the traits are low. Heritability
8
plays a dominant role in determining the sampling variances
of estimates of genetic correlation.
Scheinberg (1966) showed the approach suggested by Tallis
(1955) could be generalized to estimate the sampling variance
of the environmental and phenotypic correlation coefficients
as well as that of the genetic correlation coefficient estimated
from analysis of variance and covariance. A general formula
was developed for the estimated variance of the correlation
coefficient from which the sampling variance of any one of
the three correlation coefficients could be easily obtained by
proper substitution for two sample variables.
These preliminary discussions indicate that mOst estimates
'of genetic correlation in economic traits are of doubtful
reliability and, moreover, that present methods of estimating
the sampling variance of the coefficient are also of questionable
value except under special circumstances.
Selection applied to one trait generally results in correlated
changes in other traits not under selection. This "correlated
response" depends primarily upon the genetic correlation. Yet
there has been little research reported on whether the theoretical
treatment of correlated response to selection in terms of the
genetic correlation is adequate to explain the responses realized
in experimental results.
9
Falconer (1954) reported an experiment with mice in which
two-way selection under different environmental conditions
was for body weight at six weeks of age in one pair of lines and
for tail length in another pair. The test of adequacy of theory
came from a comparison of independent estimates from each pair
of lines of the genetic correlation between body weight and tail
length. Agreement between the estimates was expected to show
that the theory upon which the estimation of genetic correlation is
based would account fully for the correlated responses observed
in the experiment. Falconer found reasonable agreement between
the two estimates but concluded that the closeness of the agreement
should not be emphasized since the estimates had rather wide
fiducial limits.
Reeve and Robertson (1953) selected for wing and thorax length
in Drosgahila melanogaster and found good agreement between the
estimates of the genetic correlation in the base population and the
correlated responses obtained when either of the two was selected
separately. Their results were based upon fifty generations of
selection. The genetic correlation between the two traits was high,
however, (0.70), and there is some suggestion, (Clayton EL,
1957) , that the magnitude of the genetic correlation affects the
accuracy of the predicted response, accidents of genetic sampling
10
in the correlated trait making the response unpredictable at low
levels of genetic correlation.
Clayton e_t_a_l. (1957), in the third of three papers devoted to
an experimental check on quantitative genetic theory, also studied
correlated response in Drosophila melanogaster. In their study
the genetic correlation between the primary trait (abdominal
bristle number) and the secondary trait (sternopleural bristle
number) was small although positive (0. 05 to 0. 10) in the base
population. Moderate agreement with predicted correlated
response was observed in the early generations while inbreeding
was quite low. The correlated response became entirely
unpredictable with further selection in later generations. These
workers concluded that if the genetic correlation is low, to
measure it by correlated response is unwise unless the inbreeding
each generation can be kept at a very low level; and that careful
experimental design is required to estimate genetic correlations
from correlated responses.
Very little is as yet known about the effects of selection on
the magnitude of the genetic correlation. Lerner (1958) presented
a simple theoretical model suggesting that the genetic correlation
between two traits would eventually become negative if selection
were applied to both traits simultaneously. Those alleles which
affect one
eventually
alleles hat
eliminated
segregatin
two traits,
Lush (194::
effects of
While .
investigati
Friars e:
g€netic CO
for improx
°°rrelatio
danger of
population
out of the
of thes e w
but [1011.81
Provided t
nine Year 51
led the am
t°WardIIn
ll
affect one trait alone or both traits in a plus direction would
eventually become fixed under selection for both traits while those
alleles having a negative effect on one or both traits would be
eliminated. The net result of selection would be to leave
segregating only those alleles which have Opposite effects on the
two traits, thus, resulting in a negative genetic correlation.
Lush (1948) makes essentially the same point when discussing the
effects of selection on genetic correlation.
While this theory seems sound, there have been few experimental
investigations to study the effects of selection on genetic correlations.
Friars e_1_:_a_l; (1962) reported changes over time in estimates of
genetic correlations between traits under simultaneous selection
for improvement in poultry. Trends in the magnitude of genetic
correlations over years within the same population pointed out the
danger of comparing estimates of genetic correlation from one
pOpulation to another. Negative time trends occurred in sixteen
out of the eighteen sets of genetic correlations estimated and six
of these were significant. The remaining two sets showed positive
but non-significant time trends. Fairly good evidence was thus
provided that the genetic correlations were declining over the
nine years of this study. The consistency of the negative trends
led the authors to conclude that selection rather than progress
toward linkage equilibrium was probably the cause. They suggested
that the a
decrease
theory pl
While
quantitati
genetic c
to the the
variable -
coefficier
With ref e
admissim
score Obt
all indivh
rejection;
usually a“
Clearly t}
t1”llllcated
those Sco:
that Such
scores on
“the two
the truhce
°rigina1 p
12
that the additive portion of the genetic covariance could have
decreased through selection which would lend support to the
theory proposed by Lerner (1958).
While there is limited experimental evidence in the field of
quantitative genetics concerning the effect of selection on the
genetic correlation, some attention has been paid in other areas
to the theoretical consequences of truncation selection of one
variable in a bivariate normal distribution upon the correlation
coefficient. Aitken (1964) presented a treatment of the problem
with reference to testing procedures used in determining
admission to educational institutions. One variable was the
score obtained in admission tests, which were administered to
all individuals in the population to decide on admission or
rejection; the other variable was score on achievement test,
usually administered at a later date only to those admitted.
Clearly the distribution of scores on admission tests had been
truncated prior to administering the achievement test since only
those scoring high in the former were admitted. Aitken suggested
that such truncation will change the marginal distribution of the
scores on achievement tests except in the case of independence
of the two variables. The correlation between test scores in
the truncated portion of the p0pulation will differ from that in the
original population depending upon the degree of truncation
exerted.
in the um
truncated
degrees c
portion.
somewha'
argument
in biologi
Mante
Similar ti
betWeen t
Performa
aCtUally (
Candidate
Again a r
general P
the restr
varianee
in the 1111:
correlati
ammon
”Man.
13
exerted. A method was presented to determine the correlation
in the underlying pOpulation from the correlation observed in the
truncated distribution, and values were tabulated for various
degrees of truncation and levels of correlation in the truncated
portion. While this treatment was applied to a phenomenon
somewhat divorced from genetic correlation, similar theoretical
arguments would apply to truncated selection for a single trait
in biological pOpulations.
Mantel (1966) also discussed the problem from a standpoint
similar to that of Aitken, again using the example of the correlation
between tests to determine admittance to a school and subsequent
performance. It was stressed that the correlation which is
actually observed is that within the papulation of successful
candidates rather than within the population of all candidates.
Again a method is described whereby the correlation in the
general pOpulation can be ascertained from the correlation within
the restricted population. All that is required is the ratio of the
variance of the truncated variable in the restricted portion to that
in the unrestricted pOpulation. Conversely, the expected
correlation in the restricted portion, .caused by truncation .
selection, can be determined if the correlation in the unselected
papulation is known.
Both A‘
restricted
in the unre
the problei
selection,
exPected tc
selected pc
grout) is m
change Can
estimeted
involved,
traits in th
Selected g1
0f indiyidu
rePeated f.
“uremic
of seleCtic
PractisEd.
have uPOn
Anothe
correlate<
°°rrelate,
14
Both Aitken and Mantel indicated that the correlation in the
restricted portion will be considerably weaker than that observed
in the unrestricted population. If these findings are related to
the problem of genetic correlation in animal populations under
selection, linear truncation selection of one trait could be
expected to cause a decrease in the genetic correlation in the
selected portion of the population. However, when this selected
group is mated to produce the next generation of individuals , what
change can be expected in the genetic correlation when it is
estimated in this new generation? There are three correlations
involved. Firstly, there is the genetic correlation between the
traits in the initial population; secondly, the correlation in the
selected group; and finally, the correlation in the new generation
of individuals produced by this selected group. This cycle is
repeated for each generation of selection. The magnitude of the
correlation in any population or sample will depend on the stage
of selection and probably also on the type of selection being
practised. The nature of this effect that selection is likely to
have upon the genetic correlation has not become entirely clear.
Another problem associated with genetic correlation and
correlated response to selection is that of "asymmetrical
correlated response", discordance of the pattern of correlated
response with expectation. For example, the same pattern of
response i
when selec
reSponse i
comparabl
selected.
frequently
Falcon
10W planes
under two
traits. Ge
generation
Falconer e
Parameter
ASYmr
by Ben an
selectiOn !
different a
responSe i
selectiOn ‘
Poultry 0v
selected 1'
weight in '
genetic co
15
response in the correlated trait might reasonably be expected
when selection is made in opposite directions. Also, the
response in one trait on selection for the other should be
comparable regardless of which of the correlated traits is
selected. These expectations of correlated response have
frequently failed to develop in experimental data, however.
Falconer (1960) selected mice for growth rate on high and
low planes of nutrition where the same measurements made .
under two different environments were considered two separate
traits. Genetic correlations observed were equal in early
generations but were markedly different in later generations.
Falconer attributed this asymmetry to changes in the basic
parameters due to selection applied.
Asymmetry of genetic correlations also has been observed
by Bell and McNary (1963) and by Yamada and Bell (1963) when
selection was applied to Tribolium castaneum under two
different environments. Siegel (1962) also found asymmetrical
response as measured by realized genetic correlation when
selection was made for body weight and for breast angle in
poultry over four generations. Nordskog and Festing (1962)
selected in both high and low directions for body weight and egg
weight in poultry and observed asymmetry of the realized
genetic correlations between body weight and egg weight when
either the '
was consici
the asymni
metwotra
ClaytOi
observed a
in Dr050pl
and decrea
increase ll
Perceptibll
0V6!“ twent
Phyanine
gehetic C0
The fr
have been
Other than
Bohren 6t
\
COrrelatECI
611% using
results 0h|
cOrrelatet
authors I
16
either the direction of selection or the trait being selected
was considered. Siegel and Nordskog and Festing attributed
the asymmetry to differing genetic variances or heritabilities for
the two traits.
Clayton §_1:__a_._l. (1957) ,in their paper on correlated response,
observed asymmetry in response of sternopleural bristle number
in Drosophila melanogaster when selection was made for increased
and decreased abdominal bristle number. They found a marked
increase in sternopleural bristles in all the high lines but no
perceptible change in the low lines when selection was continued
over twenty generations. They concluded that genetic drift may
play an important part in the correlated response when the
genetic correlation is low.
The frequency with which asymmetrical correlated responses
have been found does suggest, however, that some mechanism
other than genetic sampling is affecting correlated response.
Bohren£t_a_l_. (1966) made a detailed study of asymmetric
correlated response to selection using algebraic methods and
also using a computer to simulate selection experiments. The
results obtained by both methods indicated that asymmetry of
correlated response is to be found quite frequently. In fact, the
authors suggest that to find symmetry in an experiment might
be more an
programme
generation
in genetic \
proceeded.
of gene eff
contributio
negatively .
other than
0f correlat
POSSible wj
0f the gene
Prediction
than for th
°fcorre1at
pal'ameter
The pa
included a]
l‘es‘POI‘ise t
been Consi
quantitativ
17
be more surprising than asymmetry. The computer was
programmed to calculate the change in gene frequency from
generation to generation, and from this the expected changes
in genetic variances and covariance were calculated as selection
proceeded. The procedure was carried out with several models
of gene effects and gene frequencies. Probably the most frequent
contribution to asymmetry in practice will be from loci contributing
negatively to the genetic covariance and having gene frequencies
other than 0. 5. The authors suggest that accurate prediction
of correlated response to selection over many generations is not
possible without prior knowledge of the composition and magnitude
of the genetic covariance. The validity of existing theory for the
prediction of correlated responses is likely to be much poorer
than for the prediction of direct responses. Indeed, predictions
of correlated response probably should be based on the genetic
parameters estimated each generation.
The paper by Bohren fl. is of special interest because it
included an investigation of genetic correlation and correlated
response utilizing Monte Carlo techniques. There has, of course,
been considerable investigation of various aspects of genetic
theory by the Monte Carlo method since its introduction to
quantitative genetics by Fraser (1957).
Fraser I
simulate gel
genotypes al
offlm genet
aho explai]
mdmrfim
effects, se,
In a fu:
prOgress UI
hnkage, 5,
P3P“ were
PTOVision b
r: 0.5 to
Complete 1i
effect on tt.
recombinat
seven and t
complete d
this first p
Barker
SYStEmS Us
(1957.). It
18
Fraser (1957a) discussed the use of a digital computer to
simulate genetic processes and the binary representation of
genotypes and the use of logical algebra to allow the identification
of the genetic nature of an individual at each locus. The author
also explained methods for the determination of phenotypic value
and for the simulation of inter-locus interactions, environmental
effects, segregation and selection.
In a further paper, Fraser (1957b) reported on the rates of
progress under varying intensities of selection and tightness of
linkage. Several of the methods discussed in the introductory
paper were used to simulate a genetic system of six loci, with
provision being made to vary the recombination between loci from
r = 0. 5 to r = 0. O, that is, from independent assortment to
complete linkage. Linkage was shown to produce no qualitative
effect on the rates of advance at values greater than 0. 5 per cent
recombination. The limitation of the number of loci to less than
seven and the lack of provision for dominance relations other than
complete dominance were considered to be the major defects of
this first program.
Barker (l958a,b) continued the study of simulated genetic
systems using basically the same method as described by Fraser
(1957a). In the first paper, selection between two autosomal alleles
at four sta
was tested
had been r
selection b
third chro:
agreement
the second
Dr050phila
show clear
locus was
Fraser
in a furthe
and discus
The consid
(Fraser 1
leation at
°Perate in
gellOtype u
The last p;
“the prev
”product,
Selection a
19
at four stages of the life cycle was simulated, and the program
was tested by simulating two experiments with Drosophila which
had been reported previously in the literature. The first of these,
selection between ST and CH chromosomal arrangements on the
third chromosome of Dr030philapseudoobscura, provided close
agreement between the simulated and experimental results, while
the second experiment, selection between wild type and glass in
Drosophila melanogaster, did not. Nevertheless, the study did
show clearly that selection between two alleles at an autosomal
locus was possible with automatic digital computers.
Fraser (1960a) continued the discussion of Monte Carlo methods
in a further paper in which he re-emphasized the procedures used
and discussed the effects of linkage, dominance, and epistasis.
The consideration of epistasis was continued in yet another paper,
(Fraser 1960b) where he showed that while selection will lead to
fixation at a slow rate in a simple additive genetic system, it will
operate in complex epistatic systems to modify the relation of
genotype to phenotype, the relationship becoming a sigmoid function.
The last paper of the series (Fraser, 1960c) was a direct extension
of the previous paper on epistasis and considered the effects of
reproductive rate and intensity of selection on geneticstructure.
Selection against phenotypic extremes can producea degree of
genetic canalization which is more restrictive than that indicated
by the limit
degree can
These 2
the groundv
in the field
important t
models can
short time,
before CXPE
Sl’dney seri
investigatic
from the p1
"Monte Car
Martin
P
I
a Study dES
prOgreSS 0
Testing ind
seleCtion
and Can re
cases 1653
Baker
and select;
those in P0
20
by the limits of selection, showing that canalization of a rigid
degree can be caused by loose selection.
These seven papers made a valuable contribution by providing
the groundwork for the develOpment of the Monte Carlo method
in the field of quantitative genetics. This method furnishes an
important tool, readily available to the experimenter. Genetic
models can be devised, programmed, and tested in a comparatively
short time, permitting the examination of theoretical consequences
before experiments with biological organisms are planned. The
Sydney series of papers provided the impetus for a number of
investigations in the United States during the early 1960's, mostly
from the project supported by the National Science Foundation,
"Monte Carlo Studies of Genetic Selection," at Iowa State University.
Martin and Cockerham (1960) applied Monte Carlo techniques in
a study designed primarily to explore the effects of linkage on the
progress of small populations evolving under mass selection. The
results indicated that tight linkage can slow down progress from
selection when the populations are initially in linkage equilibrium
and can result in the fixation of some unfavorable alleles. In some
cases less intense selection can lead to more progress.
Baker and Comstock (1961) , on the other hand, found that linkage
and selection produced genetic means which were just as high as
those in populations where selection was practised with no linkage,
at the 5am
on the effe<
the differs:
simulation
Quresh
effects of f
All possibl
linkage, 5
Simulated,
geDOIYPic
21
at the same level of environmental variance. These two papers
on the effects of linkage on genetic progress in finite populations
under selection did not agree entirely on the significance of low
recombination values in retarding genetic advance, although
the differences could be due in part to differing parameters and
simulation procedures.
Qureshi (1963) reported a Monte Carlo study to explore the
effects of finite population size and linkage on response to selection.
All possible combinations of three levels each of population size,
linkage, selection intensity, and environmental variance were
simulated. The report considered changes in the genotypic mean,
genotypic variance, and number of loci fixed for each of thirty
generations for additive gene action only. The initial response
to selection with no linkage conformed closely to predicted values ,
and the size of the papulation affected the rate of response strongly
at low intensities of selection. The effects of population size were
also appreciable at high intensities of selection when linkage was
present, linkage interacting with selection in rate of reaponse when
population size was large. The number of generations required to
reach the limit generally increased with linkage since the genotypic
variance was being conserved and the response was slowed down.
In general, delayed response due to moderate linkage was accompanied
by a lowering of the total genetic advance. Little evidence was
found that
generatior
In a 81
of the effe
to mass 3
conditions
of the de
0f the hor
genes and
generatio
Populatioi
Caused a
mOderate
entire 1y 0
°Verdomi
aplmenti
the rate .
selectiOU
30 getter
size Was
genel‘al’
to the ef;
gene act
22
found that higher limits were attained when the number of
generations to fixation was increased with linkage.
In a subsequent report (Qureshi 1964) the investigation
of the effects of finite p0pulation size and linkage on the response
to mass selection was continued for dominance. Two special
conditions of dominance were considered, complete dominance
of the desired gene and overdominance when the genotypic value
of the homozygotes is equal. In complete dominance of desired
genes and with initial gene frequency of 0. 5, response over
' generations was negative under tight linkage except when the
papulation size was as large as 64. Intense selection apparently
caused a positive response in small p0pulations only under
moderate linkage. The fixation of undesirable genes was almost
entirely due to population size and linkage. In the case of
overdominance, the total response over 30 generations was also
apparently due entirely to population size and linkage although
the rate of response was evidently affected by intensity of
selection. The predicted plateau in the genotypic mean over the
30 generations was observed in overdominance when p0pulation
size was large and recombinations among loci was high. In
general, the effect of selection intensity appeared to be additive
to the effects of papulation size and linkage under both models of
gene action. A strong interaction between papulation size and
linkage we
and fixatic
Gill in
population
—_—__-—_
variation 1
addition, I
three stan
overdomii‘
Four level
leVEIS res
0. 50, and
Were Simu
nine mode
. l
bemg deri
replicatioi
POPUIation
Variatmn. I
each Parai
EEneratiOr
In the
finite ‘
POPL
The {Om i
Under the I
23
linkage was consistent with respect to both response to selection
and fixation of loci.
Gill in 1965 presented a series of papers on the effect of
pOpulation size, linkage, selection intensity, and environmental
variation upon genetic change in simulated populations. In
addition, nine different models of gene action were considered,
three standard non-epistatic models, additive, dominance, and
overdominance, together with six different epistatic models.
Four levels of environmental variation were simulated. These
levels resulted in heritabilities, in the broad sense, of l. 0, 0. 75,
0. 50, and 0. 25 in the initial generation of progeny. Populations
were simulated for each of 16 runs associated with each of the
nine models of gene action, the content of each parameter set
being derived from the orthogonal array of a 1/16 fractional
replication of a 44 factorial design. The four factors were
population size, linkage, selection intensity and environmental
variation. Selection was by upper truncation of phenotypes , and
each parameter set was continued over 30 non-overlapping
generations or until fixation occurred at all loci.
In the first paper of the series (Gill 1965a), the effects of
finite population size on advance from selection were considered.
The four population sizes simulated were 8, 12, 16, and 32 parents.
Under the conditions of complete dominance, the critical size
of a simulat
extinction of
while pOpulaI
loss of allel
total p0pulat
size on the 1
of selection,
of variation I
01' both. In I
theory,
In a SCCC
in his simul
ngress wh
f°rmulation
infinite Popll
C0nform We;
of ”Stricteg
generaIlOns
Widenh Whe
selection 80
considerablé
The authol. (
24
of a simulated population with respect to prevention of random
extinction of desired alleles was between 16 and 32 individuals ,
while pOpulations of 30 or more were needed to prevent random
loss of alleles when overdominance existed and 1/8 or more of the
total population was selected as parents. The effect of population
size on the mean was of major importance, relative to the force
of selection, only in populations possessing considerable amounts
of variation due to dominance effects, their epistatic interactions,
or both. In general, the results conformed rather well to existing
theory.
In a second paper, Gill (1965b) discussed the results obtained
in his simulated populations in comparison with hypothetical
progress which could be predicted utilizing the mathematical
formulation derived by Griffing (1960). Predictions based on
infinite population size, one of Griffing's assumptions, did not
conform well with realized response in more realistic populations
of restricted size. The futility of predicting for more than a few
generations without a re-evaluation of genetic parameters was
evident, whether predictions were linear or asymptotic to the
selection goal. Random genetic drift, as well as selection, had
considerable influence in changing parameter values rather quickly.
The author did stress, however, that the rate and magnitude of
change obse
observed in
Gill (19¢:
effects of in
of finite pOp
selection in‘
|
91089“ PO?
sizes of pa!
ranging fro:
population I
1. 65 standa
unselected
of 0.005, 0
each chrow
for all adja
run. In the
f3Ci0rs s an
allEles OCC
011 lOtal reg
as intense
genetic me
Intrit Was
drift Could
25
change observed in natural populations may differ from that
observed in simulated p0pulations.
Gill (1965c) in the third paper of the series considered the
effects of intensity of selection and linkage on the genetic progress
of finite populations under each of the nine genetic models. The
selection intensities specified were 1/2, 1/4, 1/6, and 1/8 of the
progeny populations. These, when combined with the four specified
sizes of parent populations, determined progeny p0pulation sizes
ranging from 16 to 256 in number and corresponded to selected
population means which were expected to be 0. 8, l. 27, 1. 5, and
l. 65 standard deviations, reSpectively, above the mean of the
unselected pOpulation. To simulate linkage, recombination values
of 0. 005, 0.05, 0. 2, and 0. 5 were applied to the adjacent loci on
each chromosome, with the probability of crossover being uniform
for all adjacent pairs of loci on the same chromosome for a given
run. In the populations with complete dominance, complementary
factors, and duplicate factors, little or no fixation of undesirable
alleles occurred at any level of selection, suggesting that the effect
on total response to selection should be small even with selection
as intense as 1/6. Selection was effective in advancing the
genetic mean in those populations in which the genotype of highest
merit was homozygous even in small populations where random
drift could be expected to cause fixation of some undesirable
recessive
were opti
against tl
The d
genetic n'
pupulatio
In p0pula‘
addition t
genotypic
generatio
linkage (1
however,
Was Cons
disequill}
YOUng
genetic a
large po
Unselect,
correspo
Per Cent
trait und
each 10c
26
recessives. In small pOpulations where heterozygous genotypes
were Optimum, however, selection was evidently rather ineffective
against the opposing pressure of random genetic drift.
The different levels of linkage simulated had little effect upon
genetic merit, gene frequency, or fixation even in the smallest
p0pulations, except during the first few generations of selection.
In populations selected for intermediates , linkage disequilibrium in
addition to inbreeding appeared to bias estimates of components of
genotypic variance - probably the dominance portion - for many
generations , selection for the heterozygote evidently maintaining
linkage disequilibrium. Under conditions of complete dominance,
however, bias in the estimation of components of genotypic variance
was considered to be due to inbreeding rather than to linkage
disequilibrium.
Young (1966) has also used a high-speed computer to simulate
genetic advance in p0pulations under selection. In this study were
large populations of 1,000 individuals per generation in each
unselected pOpulation. Three intensities of selection were used
corresponding to selection as parents of the best 80, 50, and 10
per cent of the individuals of each sex from each generation. The
trait under selection was controlled by ten loci with two alleles at
each locus, the initial gene frequency being 0. 5 for each allele
at each locus
In addition, t
in the "narrc
0.5, 0.2, an
recombinatic
complete dox
was under se
the first of u
gene action \
Under th
aldvances am
estimated in
Prediction 0
Selection int
Under th
overestimat
and underes
IEIEEmem V
decline in a,
selection We
The effect 0
27
at each locus and the initial population in linkage equilibrium.
In addition, three heritabilities , 0. l, 0.4, and 0. 9, measured
in the "narrow sense" and three probabilities of recombination,
0. 5, 0.2, and 0. 05 were simulated, the ten loci forming a single
recombination unit. Two models of gene action, additive and
complete dominance, were discussed, and each parameter set
was under selection for 30 generations. The paper is evidently
the first of what will be a series and results from other models of
gene action will be presented in later communications.
Under the additive model, agreements between the realized
advances and the expected advances predicted from parameters
estimated in each generation were in most cases very close.
Prediction of genetic advances was slightly less accurate when high
selection intensity was applied to lowly heritable traits.
Under the dominance model, predictions were less accurate,
overestimating genetic advance when selection pressure was high
and underestimating it when selection pressure was low, although
agreement was fairly close under low selection pressure. The
decline in additive genetic variance was rapid in both models when
selection was intense and particularly at high levels of heritability.
The effect of linkage on this decline was small although tight
linkage tended to accelerate the decline in the additive model during
the initial generations but had the Opposite effect in later generations.
28
Linkage apparently had no appreciable effect on genetic advance
in these large pOpulations, and no fixation of undesirable alleles
was found even at high intensities of selection, again probably due
to the large size of the pOpulations simulated.
A genetic model for correlated reSponses has been described
by Bohren e_:_t_a_i_l_. (1966) in a paper previously discussed in this
review. Expected values of these correlated responses were
obtained for each of nine generations of selection. Four different
types of loci, A,B,C, and D were considered in the model, gene
effects being additive in each case. Locus A affected the first
trait only, having no effect on the second, while locus D affected
the second trait only. Loci B and C affected both traits, the
former making a positive contribution to the covariance, that is,
affecting both traits in the same direction; and the latter making
a negative contribution to the covariance. The computer was
programmed to obtain the expected gene frequency at each locus
for each generation. The new gene frequencies were then used to
calculate the genetic covariance, the genetic and phenotypic
variances, the mean of each trait, and the standardized correlated
response for each generation when selection was on either of the
two traits. Environmental variance was set equal to the genetic
variance in all runs when all gene frequencies were one-half,
giving initial heritabilities of both traits close to one-half in every
case. The!
ofgene effe:
The pur;
to asymmet:
the relative
positively 3.1
selected, wi
practice prc
covariance ;
The for
considerablv
highlights tl
correlation1
the depends
0f the liter 5
basic Simpl
an approac}
resPOnse ,
Whflethe tr
I
a new aveni
omerinVes
situatmn.
29
case. The entire procedure was carried out for several models
Of gene effects.
The purpose Of the investigation was tO study conditions leading
to asymmetric correlated responses. Asymmetry resulted when
the relative change in gene frequency at the loci contributing
positively and negatively to the covariance depended on the trait
selected, with the most frequent contribution to asymmetry in
practice probably coming from loci contributing negatively to the
covariance and having frequencies other than 0. 5.
The foregoing review, while indicating that there has been
considerable discussion Of and interest in genetic correlation,
highlights the paucity Of reliable information on the nature of the
correlation, its behaviour under selection, and the behaviour Of
the dependent and important correlated response. An examination
Of the literature on the development Of Monte Carlo methods in
quantitative genetics research leads to the conclusion that the
basic simplicity and applicability Of the techniques might provide
an approach to the problem Of the effects Of selection on correlated
response. Despite the shortcomings Of the simulation method and
while the true situation might still remain undiscovered, at least
a new avenue, might be Opened tO the problem which could prod
other investigators to seek alternative pathways tO clarify the
situation.
The Experir,
The ma):
of degree of
trait upon th
correlated r
selection. 1
combinatiOn
numerous 11*;
different 1m,
Correlation
0f self3Cti0n
of traits 0011
and Cost. hO
arbitrary nu
by the inVes
hopefully, 1)
most fruitfu
The fact
invesfigatio
of each whll
bounds . are
METHODS AND PROCEDURE
The Experimental Design and Parameters Simulated.
The major Objective Of this study was to investigate the effects
Of degree Of heritability and Of truncation selection Of a primary
trait upon the behaviour Of the genetic correlation and the
correlated response in a secondary trait in populations under
selection. A completely comprehensive study could embrace all
combinations Of a large number Of different factors, including
numerous models Of gene action, interaction, and correlation;
different levels Of environmental variation, genotype-environment
correlation and interaction; and various methods and intensities
Of selection for one or both traits under consideration. The number
of traits could also be increased beyond two. Consideration Of time
and cost, however, quickly limits the size Of any such study tO an
arbitrary number Of factors and levels thought to be most important
by the investigator. The results Obtained in this initial study should,
hOpefully, prod the researcher in the direction Of the potentially
most fruitful avenues Of inquiry to be explored in later research.
The factors most important for the purposes Of the present
investigation and the levels allowing for a wide range Of effects
Of each while containing the size Of the experiment within reasonable
bounds, are given below.
30
by
inf:
res
ii.
iii.
the
iv,
the
31
i. Two quantitative traits X and Y with direct selection
by upper or lower truncation on the phenotype Of the
individual, for X alone. Y is not selected, but correlated
response is Observed.
ii. Three degrees Of genetic correlation, 0.25, 0. 50, and
0. 75, between X and Y in the initial generation Of Offspring.
iii. Three levels Of selection, 20, 50, and 80 per cent Of
the Offspring each generation.
iv. Three levels Of environmental variance, V(E), for
X and Y, relative to the expected additive genetic
variance in the initial generation Of Offspring, V(Ca). The
_ weal
_ V(Ga)+V(E)
was equal to 0.1, 0.4, or 0.7. When all Of the genetic
levels were chosen in such a way that h'
variance is additive, h' is a measure Of heritability in
the "narrow" sense. When genetic variance other than
additive is present, h' will be greater than heritability
in the "narrow" sense.
The four factors, genetic correlation, intensity Of selection
for X, and environmental variation Of X and Y, each at three levels,
were considered in all combinations , and each treatment combination
or parameter set was replicated. These factors and levels provided
81 treatment combinations in a 34 factorial experiment which,
when replicated, resulted in 162 parameter sets. The factors
32
and levels simulated are shown below where b is the fraction of
the offspring becoming parents each generation, rG is the
genetic correlation between the two traits in the first generation
of offSpring, and h'x and h'y represent the levels of environmental
variation of X and Y, respectively:
LEVELS
11. 1 2
b 0.80 0.50 0.20
rG 0.25 0.50 0.75
FACTOR
bk; 0.10 0.40 0.70
by 0.10 0.40 0.70
The experiment was conducted separately for each of the
following two models of gene action:
a) Additive model in which the contributions to the
genotypic value were 2, 1, and 0 for the-0+, + -,
and -- phases, respectively, at each locus. Selection
was for the desirable allele.
b) Model of complete dominance in which the contri-
butions to the genotypic value were 2, Z, and 0 for the
+1., +-, and -- phases, reapectively, at each locus.
In this case selection was in both directions, upwards
for the dominant allele and downwards for the
recessive allele.
These models provided three separate experiments resulting
in a tnta
The Strii
Speci
carries u
importam
The p!
the two Q"-
Since the 1
parents ra
0f Parents
combinatic
And 24 {BIT}
Parents Wa
PTOVide lexy
ineach gen
prOduced El
intensity Wa
The
with replace
sex of Whic}
i
.01‘ the p05 9
in
any gene,
and {Ema 1e
33
in a total of 3 x 162 or 486 parameter sets.
The Structure of the Initial POpulation.
Specification of the basic structure of the initial pOpulation
carries with it necessary assumptions which are of sufficient
importance to require discussion in some detail.
The population in this study was the bisexual diploid type, and
the two quantitative traits X and Y were expressed in both sexes.
Since the size of the population was related to the number of
parents rather than to the number of offspring produced, the number
of parents was held constant each generation for all treatment
combinations. The parents were limited to 48 individuals, 24 males
and 24 females, and the number of offspring produced by these
parents was determined by the selection intensity desired. To
provide levels of b, the fraction saved, of 0. 80, 0. 50, and 0. 20
in each generation, 30, 48, and 120 male and female offspring were
produced giving 60, 96 or 240 offspring each generation. Selection
intensity was equal in the two sexes.
The selected parents were mated at random by sampling
with replacement, and each mating produced one offspring, the
sex of which was specified alternately. This procedure allowed
for the possibility of both full-sibs and half-sibs among the offspring
in any generation. Sampling without replacement from both male
and female parents could have been done and could have allowed for
an equa
an aqua
for natu
conforn
Eacl
provide
generati
an unnec
The
the aims
With the
loci cont
W the st
loci invo
easill’ be
The c
DATA 36
With lax-g.
and which
Word in t}
bits of da
handling (
34
an equal number of progeny from each selected parent. But,
an equal number of progeny per parent is an idealized situation
for natural finite populations and sampling with replacement
conforms more closely to the situation in natural populations.
Each parameter set was continued for 30 generations to
provide sufficient Opportunity to observe a selection limit. The
generations were non-overlapping, overlapping generations being
an unnecessary additional complexity.
The genetic structure of the base population has to vary with
the aims of the investigation being conducted and, to some degree,
with the peculiarities of the computer available. The number of
loci controlling the genotype, for example, is likely to be limited
by the storage capacity of the computer. Clearly the number of
loci involved in most quantitative traits in farm animals could not
easily be simulated.
The computer system available for this study was the CONTROL
DATA 3600, which is a general purpose digital computing system
with large storage capacity and exceedingly fast data transmission
and which is efficient in solving large scientific problems. Each
word in the storage module has a 51 bit structure made up of 48
bits of data and three parity bits, thus allowing for the expedient
handling of a 48 bit data word. Magnetic core storage of 32,768
of these 48 bit words is available. For these reasons the number
of loci .'
meant t
represe
Thus, fc
individu.
mpulatit
No li
consider
at all 10c
EEDe inte
environm
Set at 0. 5
heterozy;
Changes 1"
AS sta
additive a.
OI gene ac
Symmetric
genotl’pic 1
wh .
m the QEnr
35
of loci affecting each of the two traits was specified as 48, which
meant that two 48 bit words could be conveniently used to
represent the genotype of each trait, simulating two chromosomes.
Thus, four words were required to store the genotype of each
individual, and 4 x 48 or 192 words were assigned to the parent
population.
No linkage was specified in the genetic structure; all loci were
considered to be completely independent and the gene effects equal
at all loci. Further restrictions were those of no inter-allelic
gene interactions and no interaction between genotype and
environment. Gene frequency at each locus was arbitrarily
set at 0. 5 in the initial generation by simulating complete
heterozygosity at each locus in the base population to allow for
changes in gene frequencies in either direction.
As stated previously, two different modes of gene action,
additive and complete dominance, were simulated with the mode
of gene action the same at all loci for a particular run. For the
symmetrical additive model, where the contributions to the
genotypic value at each locus were 2, l, and 0 for the++,+ -, and --
phases, respectively, the genotypic value for each trait was Zn1+nz
where n1 is the number of++phases and n2 the number of+- phases
in the genotype. With independent assortment and q the gene
freqnenc
genotypic
was ant
In this ca
the expec
were 48 a
1n the
to the ger
+-, and -.
The exPe‘
(KEmptho
resPemixn
the first E
reSPeCtiK-w
Was made
varianCe (
LeV'els
I"native tC
gEHEratiOt
heritabilit
additive m
l
EVEIS r831
36
frequency of the plus gene the same at all loci, the expected
genotypic mean was an and the expected genotypic variance
was 2nq(l-q) where n is the number of loci affecting the trait.
In this case, with 48 loci affecting each trait and q equal to O. 5,
the expected genotypic mean and variance in the initial generation
were 48 and 24 respectively, under the additive scheme.
In the model of complete dominance where the contributions
to the genotypic value at each locus were 2, 2, and 0 for the++,
+-, and -- phases, respectively, the genotypic value was 2(n1+nz)
The expected genotypic mean and variance in the initial generation
(Kempthorne, 1957) were then an (2-q) and 4n [2q(l-q)3+ q2(l-q)2] ,
respectively which resulted in an expected mean and variance in
the first generation in the population simulated of 72 and 36,
respectively, for each trait. With q- 0. 5 the genotypic variance
was made up of additive genetic variance of 24 and dominance
variance of 12.
Levels of environmental variation were simulated
relative to the expected additive genetic variance in the first
generation of offspring to produce the desired degrees of
heritability in the "narrow" sense of 0.1, 0. 4, and 0. 7 in the
additive model. In the model of complete dominance these
levels resulted in heritabilities of 0. 095, 0. 33, and 0. 52,
respectively. The environmental component was assumed to be
indepf’nd
Hence, t
was dete
was a p!"
unit vari:
standard
heritabili
investigat
60, and 3‘
produce t}
therefore,
environme
”Ormal de'
1'equired P
heritabilitx
Order to a]
lamb upot
Siflee t
parametEr
37
independent of the genotype and constant over generations.
Hence, the phenotypic value of each trait in each individual
was determined by adding xci to the genotypic value where x
was a properly generated normal variate with zero mean and
unit variance and Ci was a constant designating the environmental
standard deviation required to produce the desired degree of
heritability. For the three heritabilities simulated in this
investigation, 0. l , 0. 4, and 0. 7, phenotypic variances of 240,
60, and 34. 3 were required. The constants, Ci' required to
produce the environmental variances of 216, 36, and 10. 3 were,
therefore, 14.697, 6.000, and 3.207, respectively. These
environmental standard deviations multiplied by a random standard
normal deviate and added to the genotypic value resulted in the
required phenotypic variance to produce the desired degree of
heritability in the initial generation in the additive model. In
order to allow study of the effects of the different environmental
levels upon the change in the genetic parameters simulated, no
attempt was made thereafter to keep heritability constant over
the thirty generations .
Simulation of the Genetic Correlation.
Since the genetic correlation was clearly the most important
parameter simulated in this study, the method of simulation and
38
its justification will be discussed in some detail. That the
cause of the genetic correlation was attributed solely to
pleiotrOpy should be stressed. The degree of correlation
arising from pleiotropic gene action expresses the extent
to which the two traits under consideration are influenced by
the same genes, and the resulting correlation is the overall
effect of all the segregating genes that affect both traits. All
of the genes affecting the two traits affected each one in the same
direction, thus making a positive covariance. Other systems
could have been simulated, some genes affecting one trait in
one direction and the other trait in the Opposite direction making
a negative contribution to the covariance and resulting in a
genetic correlation which could vary from -1 to +1. Limitation
of the size and scope of the present study prevented the simulation
of negative genetic correlations or of zero correlations although
these could be interesting parameters for later investigation.
In this investigation the genetic correlation was determined
by the number of loci which had an effect on both traits. As 48
loci affected each of the two traits, the number of these 48 which
were shared by the two traits determined the degree of pleiotropy
and of genetic correlation. To produce genetic correlations of 0. 25,
0. 50, and 0.75, the number of loci in common was set at 12, 24,
and 36, respectively. The remaining loci of the 48 affecting
39
each trait affected each trait independently. The table below
might illustrate the method more clearly:
Loci A Loci B Loci A,
Trait X + + 0
Trait Y 0 + +
No. of Loci m n m'
There were three different types of loci in the genetic
system. Those in group A affected trait X only and had no
effect on trait Y; those in group A' affected trait Y only and
had no effect on trait X; those in group B affected both traits X
and Y in the same direction, the magnitude of the effect being the
same for both traits. The total number of 48 loci affecting each
trait, was made up of n loci which affected both traits plus m or
m' loci which affected only trait X or trait Y, reSpectively.
Thus, when the genetic correlation was 0.25, n-12 and mum'u36;
when the genetic correlation was 0. 50, n-24 and m-ml-24; and
when the genetic correlation was 0. 75, n-36 and m-m'IIIZ. Clearly,
if the number of loci affecting each trait were not the same, m
would not be equal to m'. In this case, however, m was always
equal to m'. The genetic correlation was then simulated simply
as B. . The genotype of trait X, Gx, was determined by the
n+m
loci in groups A and B, and the genotype of trait Y, Gy, was
40
determined by the loci in groups A' and B. In this simulated
population, Gx and CY were obtained in each generation for
each individual. The genetic correlation was measured in each
generation as the product - moment correlation between the
genotypic values thus:
IO = cov GxGY
lthx) V(GYl
where rG is the genetic correlation, cov GXGY is the covariance
between the genotypic values, and V(GX) and V(Gy) are the
variances of the genotypic values.
For the additive case the genetic correlation as measured
in this way is simply fi—m' or the ratio of the number of loci
which the two traits share to the number of loci affecting each
trait , as follows:
GX a GA + GB and CY “GA. + GB since the loci are independent.
- cov(GA+GB)(GA'+GB)
' ' r0 - JV(GA+GB) V(GA'+GB)
covGAGA'+ covGAGB + covGBGA' + V(GB)
vatGAl+VtGB)4-2cov GAGE] [V(GA')+V(GB)+2covGA'GB]
But since all loci are independent and the effects are equal and
additive at each locus, all covariances are expected to equal zero.
41
V(GB)
- /[V(GA)+V(GB)] ENGA') +V(oB)]
.I’G
Under the assumptions of the model, these variances
can be written in terms of the number of loci, gene
(+ +)-(--)
2 at each locus.
frequency (q), and the effect D -
.°.rG _ 2nq(l 'qlDz ,
/t2mq( 1 'Q)D2+ 2nq( 1 -q)DZj[2m'q(l -q)Dz+2nq(l -q)D2]
But, gene frequency and D are equal at all loci
.JG a n ;
J(n+m)(n+1n')
and since m-m' in this case,
n
n+m
.'.rc,- or the ratio of the number of loci affecting both
traits to the total number of loci affecting each which was the
method of simulating the genetic correlation.
The genetic correlation was also measured each generation
by the method proposed by Hazel (1943) utilizing covariances
between phenotypes of parent and offspring. Two variations
of Hazel's method were used to allow comparison of the
accuracy of the methods. The two methods were:
a) I; - (covapPyo).(covP:prxo)
G (covapro) . (covapPyo)
42
b) s’G a (COVPXPPYO)+(covapro)
ZMOVPXPPXO). (covapPyo)
where Pxp - phenotypic value of trait X in the parent
Pyp - H n u trait Y n n n
Pxo - " " " trait X " " offspring
Pyo I H H n trait Y n u it
Both of these methods reduce to a measure of ~9—
nm
for the case of equal and additive effects , independence, and
equal gene frequency at all loci. This is done as follows for
method a) for example:
Let pr — genotypic value for trait X in the parent
Gyp - H H H trait Y H u n
Gxo - " " " trait X " " offspring
Gyo - H H H trait Y n n n
and Exp =3 environmental contribution to trait X in the parent.
Eyp :- environmental contribution to trait Y in the parent.
Exo =- environmental contribution to trait X in the offspring.
Eyo I- environmental contribution to trait Y in the offspring.
2 = (cov PxpPyo). (cov Pprxo)
G (cov Pxpro). (cov Ppryo)
H)
[cov( pr+Exp) ( Gyo+Eyo) ] [ cov( gyp+Eyp) ( Gxo+Exo)]
lcov(pr+Exp) (Gxo+Exo) ][ cov(Gyp+Eyp) (Gyo+Eyo)1
since all covariances between G and E are expected to equal zero.
43
= (COV prGyoHcov GypGxo)
(cov prGxo)(cov GypGyo)
=[Covchfi'GBptchio+GBo) ] [COViGAvp+GBp)(GAo*GBo) ]
[cov(oAp+GBp)(GAo+ G30) ] [cov(GA,p+GBp)(GA.O+GBO)]
and since all loci are independent
_(cov GBpGBo) (cov GBBGBO)
-(cov GApGAo +cov GBPGBOHCOV GA,pGA,O+cov *GBpGBo)
GBpGBo/V(GBP)V(GBO) erBPGBOfimemeO)
But rGBpGBo= WGAPGAO rGA'pGA'o = 1/2
and won) =- vao) and V(GAP) - V(GAO) - V(GA.p) -V(GA.01
.- 3%; V(GB) V(GB)
[web ch) ][V(GA)+V(GB)]
...?G V(GB)
V(GA) + woB)
which in terms of n, the number of loci, gene frequency and
D is equal to
anU-qm2
2nq( l -q)D2 + 2mq( l -q)D2
n o o c o
a .... Wthh again measures the genetic correlation as
n+rn
it was simulated due to pleiotropy. The same solution can be
44
obtained for method b) in which the arithmetic mean of the
covariances is used in the numerator rather than the geometric
mean, in computing the genetic correlation.
The discussion above has explained the method of simulating
the genetic correlation, the justification for this method, and
has shown that the usual method of computing the genetic
correlation in economic species is also theoretically an
adequate measure of pleiotropy.
The Mechanics of Simulation.
In this section the logic of the program developed to
simulate the population will be described. A detailed discussion
of the structure and genetic properties of the population has
already been presented as has a short description of the
computer which was available for the study.
A feature common to all types of investigation involving
Monte Carlo methods is the use of pseudo-random numbers,
which, although truly random only conceptually, have fulfilled
as many criteria of randomness as possible. A library program,
RANF, was available at Michigan State University for the
generation of uniformly distributed pseudo-random numbers.
Repeated use of RANF generates a uniformly distributed
45
sequence of random numbers in either fixed or floating point
format. If floating point is used, the numbers range from 0
to less than 1.
The random numbers are produced by the standard
multiplicative congruential method. The derivation of the
multiplicative method used is of the form
Xi+ l—Xik (modm)
where X is any odd number
)\"'52 x 15
47
m—2
These parameters have been shown to satisfy the sufficient
conditions for a sequence of maximal period. The period
45
for this generator is 2 in the computer which was used.
Tests for accuracy, indicated by Rotenberg (1960), were
performed and the results agreed very well with the theoretical
distribution.
The procedure was modified by Rotenberg (1960) to the form
Xi + 1 .. (23 +1)xi+ C with a_>__2 and C odd.
In the random number generator available a: was set equal
to 10 and C equal to 101, making A-210+1 —1025 and
Xi + l - 1025Xi +101
The method used has passed many tests of randomness
including a test of the frequency distribution of the random
46
numbers , a test of the frequency with which a number of
a certain magnitude was followed by a number of another
certain magnitude, a test of the frequency distribution of
the length of runs of numbers either above or below the
mean, serial correlation tests, and others.
When the genotypic value for each individual was
determined for each of the two traits, an environmental
contribution had to be added to this genotypic value to provide
the phenotypic value for each of the traits. The determination
of the environmental contribution required the generation of
a standard random normal deviate which when multiplied by
a constant representing the desired environmental standard
deviation, provided the random environmental contribution to
each phenotype. Such environmental contributions should have
the desired environmental variance to give the desired degree
of heritability of the trait. For example, for the additive
model, where the expected additive genetic variance is 24
in the first generation of offspring, suppose the heritability
of trait X is to be simulated as 0. 4 in the first generation.
- Then, an expected phenotypic variance of 60 is required which
means that environmental contributions to the phenotype of
each individual for trait X should have a mean of zero and
variance of 36. Thus, the environmental contribution is
47
required to be of the form 6x, where six represents the
required environmental standard deviation and x_ is a N(0, 1)
random deviate.
There are numerous methods available for the generation
of random normal deviates and, indeed, generation is not
really necessary since tabulated values can be stored in the
computer. But, because the generation process was
relatively simple, and because a very large number of deviates
were required in this study, the deviates were generated as
needed.
The general procedure used has been described in detail
by Gill (1963). A specified number of uniformly distributed
random numbers in the range -1-29
‘
h- .
~~~~~ ”‘.--------_----—-----d
a . ’
----_ - ’a
01 >0
‘— v 1'
0 5 10 15 20 25 30
Generations
Figure 2. 1 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when b-O. 8 and hi .. O. l (additive model).
Var
Cov
Var
Cov
Var
Cov
68
4 ,10
u!
_-15
\
a‘()!\T.§::::-o-o‘.~
t.-- .... ....‘.N‘ - - - - LlwrG.o. 75
‘ ----.~‘-~~~ - -..”.N .....
-- -------- ~ 00.. -------- P ...-Q
[1.1 ~~~-- “ £
----- ------------ c-("as
4
0. O
( JD
364
t b'1s
i
.“Lo- - - -.-
0‘...‘0.0;0‘:00000m....... " .—.~ ‘ .‘wrG‘OO 50
J .-.- §=:::- o. .0 so... ..‘I
'11---‘- ------------ ---- - - --‘”2‘
--~- ....... —---- --.l
4
0+ do
J __ rG 5|.O
3,, ...... cov ny
-O-.-.-.- var OX ..15
J
“- var Gy
m”, - Q -. ,.o"'°-.
t °'°-*’-°fl°.~nmm...~n:'_".:_ ......3 ...—.~, “‘0 rG'0° 25
. .. .- ..... ~.'.. ‘ .. ..
‘1‘/\_ vvis
dt‘“ --—_ -------------- “‘~‘~‘ 'a—a-a—at
“"— 5°
0‘
0 5 10 I5 20 25 30
GENERATIONS
Figgre 2. 2 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when bao. 5 and 15; a0. 1 (additive model).
69
4 ,1.0
as.
‘ L015
J ‘
24A”... -0: - ~.\
var ‘4 0000.. ..‘.‘.~.......... P.” rG-o. 75
Cov b------. ‘.~.;.:Ooooooo...
l ~~~~~~~~ ‘I~ ~'.Io.oo 0.... .
“J --.‘ ‘~ ..~ 0‘.~ ~ . °OO.o0-L01‘
4 ‘§‘---- -.--~.-.-
4 ---“
0 'O
‘ FLO
367
#15
‘*1 -.-.-. °'°"'OOOoooooo y
Var .h'°"""'--:.I.‘3:.~w:-°""°"°'-..... y“ rG-O. 25
COV '~.‘. .
.t‘ ____ ~ ‘ -.- 5'15
4..---- ~ -.fi
---_------ — - - _ -——--___--_ _ _-_- ...—-..---u-u-ul
0t ’0
0 5 10 15 20 25 30
GENERATIONS
Figure 2. 3 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when buO. 2 and h; 1- 0. l (additive model).
70
, ,|.O
36.
i — .15
4 fi/V
' “he...
Var . F‘s... Fmrc'o'?!’
Cov .F“-.. '----------.-._.............§,__.m.wu~ - —-:':::':."'=--'.r.-- -:.-.:
‘1t ~~--------------~-~ "-—-----d
. ~~...—-"' MS
1
ol 9
J pI.O
364
..‘75'
Var Frat-0. 50
Cov
.o‘lS
OJ .0
rG ,1.0
6.
a -------- cov ny
......... var Gx _, .15
‘ ............... var G
var 1 .~°‘o~o‘0-.~ ........“Oouoooonoooouou oooooo...oo-.0°“'.o..... i-C’erO, 25
COV 4 .‘N- ._.—-—--°"’ ---------------- ~"o‘.::‘1.~
‘1‘ /N 15
d\ ‘-—-~ ..
0‘ LO
0 5 10 1'5 20 25 30
GENERATIONS
Figure 2.4 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when b=0. 8 and bi = 0. 4 (additive model).
71
_1.o
3‘4
‘ 75
4\
2‘4“,” \
Var , Q'~'.';'-..._ :‘br C3,-0. 75
cov Jh“‘~~ ‘.‘~..:-.;.: -. ...: ~ '00. .000
‘~‘--- -0-.- ~.:: 000 a... ..
‘ti -------~--- ~ ~'~o~.~ .....'oo. ”a.
4 ----~-~--.:-.:
O 4 ID.
d .10
36.
‘ ‘ L076
Var -"°rG-0. 50
Cov
.. M”
. O ‘ . '0
J 1.0 ..I.O
a.
4 xy
-o-e-e-o- var Gx HTS
“4 . . ... .. 0.0.0.000000000 var GY
J‘:ooeoo 00........-.
Var \.‘.‘ Huh-..." cSDerO.25
COV .‘°\—o-._ ......H.H""."°"-o.....
J - - o‘.‘ ° 00 e-
"- E‘< ..... ... "as
Jb-----------‘----.‘§‘~~‘~ -.-'~ a-..
o ‘ §-- ...... ~_----.-‘------..+i°
0 5 10 15 20 25 30
GENERATIONS
Figgre 2. 5 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b a 0. 5 and I; n 0. 4 (additive model).
72
1 .1.0
36 ,
. 5'15
4\ ‘A
31.4.. \
s,‘ .m
Var L "K * rG-O. 75
C‘.
COV ‘ ‘\~\ ~'~.‘:: ' .. ......
'2‘ “‘ ~ ~.~.~. '0 ...
----~-- .~o‘ N". 7'15
4 -‘~‘~ ~.‘ . .. . o.
“~~~ .-.‘. o ”a:
1 -------.~‘- - -
0. -'-------.-dlua-a-a_ O
+ . .L0
36.
1 .915
344‘...... n I
Var ‘-\ _.sorG=O. 50
\. '-
4 —- rG "'0
3‘” ----- - cov ny
‘ ........ —var ex ..7:
241.,\......,,... ----var C}y
Var . x,‘ i‘r :0. 25
.\ G
GENERATIONS
Figure 2. 6 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when b-O. 2 and 15; a O. 4 (additive .model).
73
« 1.0
36.
J ..15‘
2%»."g.
Var . ‘\-- --~'° 1‘50. 75
COV K~ ““ .\ ........... ~°~°~0~._ ;.“:...:.:::.—°':"° 000...... .
12.4 ~~----.._.____ __ ____ -~.-,_.:-_--.-::..........
" " ------ --____ ____:'_"_;: as
J
0 L0
. .m
“4
_.15
J
fibre-0. 50
(.15
04 o
4 —— rG 1.0
36.
-------- cov ny
j —.-.-._._ var Gx 5'7;
144..-- ... " ° ........... var Gy
Var a '~.o~.‘. °.°o....”. piercfi). ZS
Cov K.‘ ‘ ..."°"°-. . .. ......,..-.9..,........ ..
\._______‘-__‘ .4
'1. d .~."‘.~
4 i was
0 ‘ --- --------- ~ ----------- ‘50
0 5 '10 1'5 20 25 30
GENERATIONS
Figgre 2. 7 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when b-O. 8 and hi ...0. 7 (additive model).
74
J .1.0
(
~
~
5‘1 ‘ ‘. ......
~~-- ..~ . ...
--- ~ 0 . ."°ou ’ 15-
‘- - ~. ... ....
i s- -O—.-. 900.000....
~‘~ ‘. '00...
h------~ " ‘00."
t --- ‘0
~~°~.
N_~O~:.-
-
.1 ~50
4 1.5.0
as
Var ‘\" . ..corcao. 50
Cov t - . .
'20.
.as
O 4 -------°
1 rG 1.10
------- cov ny
-0-e-o-c— '15
var Gx t
Z‘iK.......'....o-o-"‘°..°o.“... '0-00 ... 0.00000 var GY
Var ) ~.\ .. ring-30.25
Cov '\ H
)
r15
1.0
0 5 10 15 20 25 30
GENERATIONS
Figgre 2. 8 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when beO. 5 and h): a O. 7 (additive model).
75
4 .10
36.
J .375
24‘
Var . --sorG,0.75
Cov .
I1...
..15'
O 4 -0
p10
“4
J v75
4
Var {513620. 5
Cov
‘ '°----............. ......e.....................l’°15
O . ~~~~“- :4:.°-.'='.".':.'.= '_-:=.scam-Iantq_°
1 —— rG .1.0
36 - G
----- COV XY
J ...—0... var Gx “15
“‘Q.. ......o....... var GY
Var 4 \ vases. 25
Cov . \.\
“r25
0
0 5 10 15 20 25 30
GENERATIONS
Figure 2. 9 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b=0. 2 and 1;; =0. 7 (additive model).
76
generations of selection, and the genetic correlation remained
near its initial level in every case. The genetic covariance was
also conserved at the remaining two levels of environmental
variance when b - 0. 8 (Figures 2.4, 2.7). Thus, when level
Of selection was low, the genetic covariance remained quite
high over all levels Of environment, the greatest decrease
being at low environmental variance (h'x- O. 7) when genetic
covariance was about halved over the 30 generations of selection.
(Figure 2. 7)
With increasing level of selection there was a greater
decrease in the covariance. At (b =- 0.5), for example, the
covariance clearly decreased at all levels of heritability while
the correlation remained high and only decreased noticeably at
h'x - 0.7 (Figures 2. 2, 2. 5, 2. 8). For initial rG of 0.75 in
Figure 2. 2 the genetic correlation was still 0. 7 at the 30th
generation while the covariance had dr0pped from 18 to almost
9 or by about 50 per cent. The genetic variances of the two
traits had also declined in proportion to the covariance causing
the correlation to remain high.
When the selection intensity was high (b = 0. 2) , the genetic
covariance quickly declined at all levels Of environmental
variance (Figures 2. 3, 2. 6, 2. 9). In Figure 2. 6 when initial
rG was 0.75, the genetic covariance had already dropped from
77
18 to about 2 by the 20th generation while the genetic
correlation had only decreased from 0. 75 to about 0. 55. In
general, at low levels Of selection both the genetic covariance
and variance were maintained over the 30 generations Of
selection. At higher levels Of selection, however, there was
a distinct downward trend in the genetic covariance; but
because of an accompanied decrease in the denominator of
the correlation coefficient, the genetic correlation remained
quite near its initial level. Only when both selection intensity
and heritability were high, was the correlation coefficient
decreased markedly, and this decrease mostly came suddenly
' after the 15th generation Of selection. This sudden decrease can
be clearly seen in Figure 2. 9 where the genetic correlation,
initially at 0. 75, was still close to 0. 50 at the 15th generation
despite a rapid decrease in the genetic covariance. By the
20th generation, however, the genetic correlation had dropped
almost to 0. 1 at a time when the decrease in the genetic
covariance was leveling out.
A reduced correlation could, in fact, be due to an increase
in genetic variance rather than a decrease in genetic covariance.
An example can be seen in Figure 2. 1 when the initial genetic
correlation was 0. 5; there was a small decrease in the genetic
correlation between the 10th and 15th generations. Yet, the
78
genetic covariance increased somewhat in the same period,
the reduction in the genetic correlation being due tO an increase
in the genetic variance Of the selected trait. Thus , the genetic
correlation does not necessarily provide a reliable estimate Of
the genetic covariance.
The Genetic Correlation in the Truncated Distribution.
The effect Of linear truncation Of one variable on the marginal
distribution Of a correlated variable has been discussed
previously. In general, the conclusion was that the correlation
Observed within the sample Of individuals selected as parents
will be lower than that Observed within the population of all
Offspring. The theoretical treatment of this problem by Aitken
(1964) and by Mantel (1966) has already been reviewed and was
concerned solely with the phenotypic cOrrelation between the
variables. Whether the same effect would hold for the genetic
correlation has been examined by measuring the genetic
correlation each generation in those offspring selected to be
parents Of the next generation. The phenotypic correlation
is a function of both the genetic correlation and heritability
and also of any environmental correlation between the traits.
Thus a reduction in the phenotypic correlation between the
variables might not necessarily mean a reduction in the genetic
79
correlation.
Figures 1.1, 1.2, and l. 3 show the genetic correlation
both in the complete offspring generation (solid line) and in
those offspring selected as parents or in the truncated
distribution (broken line). Clearly, truncation has caused
some decrease in the genetic correlation. This decrease is
apparently a function of heritability rather than of degree of
truncation selection. When the environmental variance was
high relative tO the genetic variance (h'x — 0. l) , there was
some tendency for the genetic correlation in the selected groups
to be lower. The difference, however, was quite small and
not consistent. Again at h'x- 0.4 (Figure l. 2) the difference
between the two correlations did become larger, but neither
level of selection nor initial degree of genetic correlation had
any appreciable effect although the difference did seem rather
more consistent when initial rG was 0. 75 than otherwise. When
the environmental variance was high relative to genetic variance
(Figure l. 3), the effect of truncation selection on the genetic
correlation became considerably greater. There was a more
consistent reduction in the genetic correlation, and the reduction-
was greater. Again there was little effect Of level of selection
on this decrease. The results shown in Figure 1. 3 correspond
most closely with those expected in the phenotypic correlation
80
since the expectation of the phenotypic correlation approaches
the genetic correlation when heritabilities are high.
The effect of truncation selection of one variable is to
reduce the Observed genetic correlation between it and another
variable and should be kept in mind in selection practice. The
amount of this reduction depends markedly on the heritability
Of the selected trait rather than on the intensity Of selection
practised. A more detailed examination of this problem,
including the effect Of selection on the environmental and
phenotypic correlations and on the heritabilities as well as
on the genetic correlation, should be carried out to clarify
the effect Of selection on all Of these parameters and on their
interrelationships. The statistics required to examine these
relationships were available, but a detailed investigation
was considered beyond the scope Of the present study.
The Estimates Of Genetic Correlation from Phenotypic
Covariances between Parent and OffSpring.
In addition to measuring the genetic correlation from the
product-moment correlation Of genic values, two separate
estimates were Obtained from the covariances between phenotypes
Of parent and Offspring. The two methods were:
81
a) 9G = (covapPyO) . (covaprO)
(covapro) . (covapPyO)
b) 4’0 _.__. (covPXpPyo)‘+(covapro)
2/(covapro) . (covapPyo)
where Pxp -phenotypic value Of trait X in the parent
Pypa H 11 11 trait Y 11 11 11 i;
Pxo= " " " trait X in the offsPring ]
PYO = 11 11 11 trait Y 11 11 11 ,1
Since the number of observations upon which the genetic
correlation is estimated is known to have considerable effect
on the precision of the estimate, it should be emphasized at
the outset that the sample size available here was small and
varied with the level Of selection practised. The number Of
Observations on which the estimates were made was 60, 96,
and 240 when the level of selection b was 0. 8, 0. 5 and 0. 2,
respectively.
Results Of both methods were extremely erratic and were
almost impossible to interpret. Extreme selection Of parents
could be expected to bias the correlation, and extreme selection
occurred here when the number Of Observations was largest.
When level Of selection was low, the number of Observations
was small resulting, in both cases , in unreliable estimates.
82
Lush (1948) has suggested that where sampling errors are a
major concern, as when the volume of data is small, it may
be better to use an arithmetic mean Of the two covariances in
the numerator rather than the geometric mean. Thus, although
both methods of estimation gave'completely unintelligible results,
a few examples Of results when the arithmetic mean Of the two
covariances in the numerator was used are presented.
The results are in Table l and are the deviations of the
genetic correlations measured by parent-offspring covariances
from the product-moment correlations Of genic values. Only
two levels of selection were considered, b=0. 8 and b -0. 2,
in an attempt to detect any difference due to number Of
Observations. In addition, five different combinations of
environmental variances are presented to examine as wide a
range Of these effects as practicable. Sample estimates are
given for the first replicate for generations 2 through 6 and
then for every fifth generation thereafter. To distinguish any
pattern in the results is futile. Most estimates fluctuate
markedly and apparently randomly from the true correlation.
In fact, it is rather rare tO find an estimate within 1 0. 20
of the expected correlation. In general, however, there is
some tendency for the correlation to be considerably under-
estimated. Of the 300 deviates shown in Table l, 179 were
83
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35- 555- 35- 555 $5 57735- 555- 35 $5 55 55 N5 ..
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on 3 em m5 2 e m e m N 5.5 xi 5 on
zofi0
3 I j’ I U
0 5 10 15 20 25 30
GENERATIONS
Figure 5. 2 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when bro. 5 and
Selection by upper truncation.
' - 0.1 (complete dominance).
105
var “ ‘. ... 9 e. rG'Oe 75
l ‘\‘ ‘9‘. . e .. i‘o
COV ‘~ ‘ ~.‘ ~ .' e
1 §~-~---- O-.- -. ~. ~ . .-
ml ..----- ‘0 so ..
4 1.6 ~ . .-.-..“w1‘
‘ -s-e-s-o-var Gx
o d e seen-var G ,0
Y
. “.0
new”...
4 \o ........ e.
‘o\. .... 0......” p.1‘
d ‘4‘ "-
o‘.‘
1“ .\\ .....90o.00se
Var .\ -~.~. Lrpzo. 50
L. ‘0‘... ceases-.....u....~
COV'. ~~- ‘————__:;:»- —e~.______
‘1. \ ‘.~"s~.
0‘. . . . . 5.1;
OJ .0
l ”.0
“J'Vé‘fioseuo
.‘. Queue........
i \ ‘ ".... e. 15
J o‘. .7 "a
\.
2M \ x.
Var l "m‘. EMS-0.25
COV . ‘."\.‘
n‘ s.~-~._.-.-
'-"‘---.-.-.r15
l" """ “"~-~---..- \
J ~~~--~-----.
--_- - - --.-- ---_--4
e l o
o 5 16 f5 20 25 30
GENERATIONS
Figure 5. 3 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when b n 0. 2 and h; - O. 1 (complete dominance).
Selection by upper truncation.
Ii- ..1 in»! M.
".42.
106
4 .LO
3. .l—nnsflhfllmtfdgfi
Q3”
‘ A ‘135
J—------‘-\
1‘4 ..9'00. .0... ....... "0
~‘ .‘0-0-e-s-e;:. ......“-
Var . ‘5‘ "‘°"°-~-- -~ —.-....._ '“l‘G-O. 75
s‘ '
Cov 4 \“‘-‘----¢----------~ ---.q
all r6 r1‘
‘ -..—...— cov Gx
‘ -e-o-e- var G
e l x lo
soeeoeseso var Gy
‘ ”.0
“4L", -'
C'.‘. o.
J O..~.~. ....
'Oe-o‘e~ ... Ts
( ."\ ~ ' "
1’J ‘-o-e~e-e‘.~.~ .q
Var ,; ‘.‘ y QrG’o. 5
COV ~~ ~ ~. -.- -.- ...
~~~~_---.---o--
'1 ----..-
-~--- .. fi—Q—O--‘-----“-‘L'ls
O 4 50
‘ ,'.0
a ‘Wtazooeeeeeeeeee...
0‘. 90.0.0.... ..........
J ~‘uo‘ . ”... .... "...... .15
.‘s ....."Oe. .0009...“
‘ "§.~ ...... °.
“4 ‘~a
var ‘0‘ -—-._‘_ ‘o-e-s§ ‘ ‘WG’O‘ 25
COV .l -.-“
'
‘JA \ "1‘
‘1'- “~~ ------- -~~~ —
Q~~-_-----_-- ...... d
4
o 4 L0
0 5 10 15 20 25 30
GENERATIONS
Figure 5. 4 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when baO. 8 and hi 4- O. 4 (complete dominance).
Selection by upper truncation.
107
J 5.40
as 1‘
'~§§.
a; ‘7‘
dk ‘ ‘11: 6 so. I s . \
“, ‘\“ ‘~ ~-~.... ...... ..
Var . \~‘ -‘s.‘ "*n... ,wrG-IO.75
‘ ------ - . .....ess
cov ‘ ~“~“s‘ '0 usssssss...
~ ‘3- .. . "'.Ooo
.‘ ‘ ~Q-- .~9~I~. ..."-
rG -~~ .N ’5
‘ -~~~~;-O- - - - -
- ..... - var Ox 1-
O 4 O
............ var GY
J HO
3‘.
N s
3‘s.sss.....
‘ \.\. .....'.'s.ssss V‘s
‘ \.‘ .........
.‘. ....9s
I‘J ‘.‘.‘ ..O~OOOOO...
var ‘ .‘.‘ sssssss sss ssssssss... ..... bwrG.O. 50
cov 5“ s .....‘ls
. s~~~ ‘~‘ v \
'1 ‘------ ‘- - --o- -
--~‘~ ‘.~.~ h.“
4 ~~~~~ .-.-.‘s
04 +0
.10
ulk:"¢ss
‘0 . 00s.
J ‘.‘.\ .. .......“°ss..ss s s sss p.15
‘ .\. ssoeossssssssssss~
t" ‘0‘ ‘.‘ s... ...“
Var . ‘\-\ L-firG'O- Z5
‘.‘.
Cov . 7"»-.-
‘ta A ~ .~." ‘ . L1‘
\ ‘s‘t
1§--------- ‘%
C 4 '0
O 5 10 15 20 25 30
GENERATIONS
Figure 5. 5 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 5 and hi :- 0. 4 (complete dominance).
Selection by upper truncation.
108
‘ no
3..
.4 L15
4
8".
var , .flr so. 75
\\ G
Cov 4
"s . .......O.s~..0000000. 115
4 ; . - Cohosssssssssssssss0+
J ...-.0 Cov ny ~‘-~:;-:.°-0 .-.-.-
o J -. -.-.-var Gx -..--. - ----..:>°
.......... .var (:3,
J ,I.°
sol"
4 "I‘m.
\ .15
J \.\ ....
"d ‘0‘ ...'e
Var .\. .....ssssssssssssss... rwrG‘o' 50
Cov ‘~“‘ \ ... ......
~“‘ \\ . ...... o.....ou...........~
“4 “‘~‘ ~.~.~.~.~.\ ‘2‘
4 §‘-------:-~.~. . \
0‘ ~----‘-“‘-:‘-".:::l0
4 .IO
864
J‘\... .
x, “1‘
J \.
z“ \\ - - ””‘mml
Var ‘ \‘ 5“ 1.6.0. 25
\.
COV J \N.
.\
‘14 \ ‘.‘. r26
~~~~ ~°\.
~‘~‘ W
0‘ ‘------—__--------..t':.l::'.'::.:::io
0 '5 10 15 20 25 30
GENERATIONS
Figure 5. 6 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when halo. 2 and hi s 0. 4 (complete dominance).
Selection by upper truncation.
109
J fl.°
“J
N ..
‘ “0".:--....
.~.‘ ~ .... O -A H16
4 “ ~ \
HQ ‘0‘. W .......
1‘4 \~~“ ‘s-~.~.~ ...... .. .............
\~ ----- - 0‘. ............. .
Var 4 ".~~~~~ "~ -.- ~ ~ ....... “para”. 75
COV ‘ ...N--------.- ~ - ~ - - ~
't -----‘-- a4
‘ -----~p‘z‘
1'0
d
------cov Gx
‘ ------ ..var C}x
° ‘ ............. var GY *0
d ”'0
3““:m.....
d ‘g ‘2‘:;:: o . ...“ b'”
d 0 ‘.~.‘ 0. Q
1‘+ ~ ~.~ Oo............. +
Var . “-~.- ~ m rG-o. so
COV +_~-"—----~~- ~ ‘0- - - __$
‘ ~ ‘~. ~'-.~
1‘ ~~~~~¢ ------ a- “as
04 lo
a ...o
3‘4h:ouoo
‘.
\O
\K. . “75
l “\.
1‘4 .~ ~. - ~ .~ ..
Var J -~..~ ~,~. "‘3 I'G-O. 25
Cov 1 ~ ‘ ~.-._ _
I14 -—.— —. ~ ... ‘1‘
4%
J ‘ ---------------------------------- _ ______ .
O4 $0
I I T I U
0 5 10 15 20 25 30
GENERATIONS
Figure 5. 7 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when buO. 8 and hi .- 0. 7 (complete dominance).
Selection by upper truncation.
110
J “.0
364‘
J "‘m
\;-. ,JS
4 ‘-q-
s
14. ‘\‘\ \ \ ‘ ~o......\
var J ‘\\s \.\'s .....“o. \h”rG.o. 75
00" “~‘;:-‘.‘ ..."~.
m __ r \~;-.:;\.~ ... . . -.....19 H}
to '
m-
t: -
..10
..SOrGIO. 50
.15
Lo
4 n'o
are...
A .‘. .
\I\ .00. I."
4 “.
u. \'~.‘
Var J KN marcuo. 25
‘\
Cov J '\ .‘_
"H ‘.\.‘0§-§
a ------ “-"‘“‘-—-----"‘.o
'0 '5 i0 i 5 ab 25 30
GENERATIONS
Fi re 5. 8 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b - 0. 5 and h; a 0. 7 (complete dominance).
Selection by upper truncation.
lll
J JD
364
‘ L575
1‘4
var ..30 rGuO. 75
Cov
ut 4 .13
O 4 'o
"Hon-""VEI' GY
‘ JD
16 «H
J .\ .........
o\ I...~ ’1‘
i '\
w \.
\
var ‘ .\ 000.....0..... 5-“rG.00 50
cov +K\ .... . ... ......“..°.ocee
s . "n. "
“ ‘0 “......
a. \\ ~ \ c
I
J “‘~ \ ~ ~ \ 5
( “‘\---::;~.- - -.-
. i - ...... ‘2 :.:_-:_:_-_-:.'.= :94 o
4 HO
91k:....
4 \,
\ p.15
4 .\.
"4 \°\ ‘00....4...oo....-00"0000.
Var 4 ~\ “use real). 25
cov 4 ‘.\
GENERATIONS
Figure 5. 9 The relationship between the genotypic variances ,
covariance, and genetic correlation at three levels of
correlation when b r 0. 2 and hi I O. 7 (complete dominance).
Selection by upper truncation.
112
expected from selection for a dominant allele, the change in
gene frequency by selection becoming more difficult as the
frequency of the recessive gene becomes less. For example,
in Figure 5. 9, top graph, the genetic variance had decreased
from 36 to about 6 by the 15th generation of selection, yet did
not reach zero in any case over the last 15 generations of
selection. During this time dominance of the favoured gene
was actually a hindrance in changing gene frequency because
of the abundance of the favoured gene in the population.
Essentially the same conditions are acting to maintain the
genetic correlation as in the additive model.
As previously noted, a further experiment was conducted
under the complete dominance model. Rather than to select
upwards for the dominant allele, selection was by lower
truncation for the recessive allele. All other assumptions
and conditions were identical to those for the first experiment.
Figures 6.1, 6.2, and 6. 3 show the change in the genetic
correlation over 30 generations of selection. The behaviour of
the genetic correlation measured in the unselected offspring
(solid line) conformed closely in most cases to that already
observed for the additive model and in the complete dominance
model when selection was by upper truncation. When selection
level was high, however, the decrease in the genetic correlation
113
"J
b-0.Z
. 00+
'1“ “.---..-- -- ...-I... " '-
“‘ "’- ‘ -
b-0.5
rG.
OJ
l0.
—— rG (all offspring)
------ rG (selected offspring)
0.
V T
o 6 16 1'5 20 25 30
GENERATIONS
Fi ure 6. 1 Change in genetic correlations at three levels
of selection when 1),; a 0. 1 (complete dominance). Selection
by lower truncation.
114
L04
b-0.Z
b-O.5
"d-----—--—-- - --
-- '-
.15.?
~41. ~q-“-— -—----~-------‘--
‘b'O. 8
‘~~~/" rG (all offspring)
OJ ----..rG (selected offspring)
o 5' io 1'5 so 25 3o
GENERATIONS
Figure 6. 2 Change in genetic correlations at three levels
of selection when lg II 0. 4 (complete dominance). Selection
by lower truncation.
115
"o4
l.04
13:0. 5
b=0,8
rG (all offspring)
o. --..-.rG (selected offspring}
o 5 10 15 20 25 30
GENERATIONS
Figure 6. 3 Change in genetic correlations at three levels
of selection when lg = 0. 7 (complete dominance). Selection
by lower truncation.
116
was more rapid and reached zero by the 25th generation of
selection when heritability was high (Figure 6. 3).
Figures 7. 1 to 7. 9 present) the genotypic covariance along
with the genotypic variances of the two traits. When level of
selection was low (b-O. 8), the genotypic covariance and variances I.
of the two traits were maintained at near the initial level
93"‘35' “it? {5‘ .-. 'x" ‘
especially when environmental variance was high (Figure 7. 1)
As level of selection increased to b-O. Z, the decrease in the
genetic covariance became quite rapid regardless of the level
of environment, although the decay was more extreme when
environmental variance was small (Figure 7. 9). The covariance
and variance were maintained at a fairly high level for the first
five or, at most, first ten generations, and then the decrease
became very rapid and curvilinear through generation 20 after
which they levelled out again through the 30th generation.
The shape of the curves showing the decay in the genotypic
covariance and variance of the selected trait were quite different
for selection by lower truncation from selection by upper
truncation. When selection was by upper truncation, a rapid
decrease occurred in the early generations of selection while
lower truncation selection did not change the magnitude of the
genetic variance or covariance greatly until after the 5th generation,
and then the change became quite rapid. These observations
117
4 JD
’0’.~.
u4-me,.,3.._,.,..;ffif.:....u- .::..‘.~.
J 0.0.5.0... 0.....‘.~ - — - ‘ -.
I ....“ oooooooooooooooooo . . ...O 03%.; m”
Q ,i”.‘~‘ w "’E'.— u ‘—
1‘4h ----qu" ~‘§‘§- .4
var J ----n-------"‘—‘ Marc-o. 75
Cov ,
'14 rG Lu.
J
J ------ cov G
X
...... - var G
O . Ox ’0
............ var
Y
. “ID
“in.“ ‘00. ‘ 0’0
-0- too... .... . .’.
J ‘ 'Q.‘.‘ :;..}“..’OOOO.. .h '.n
4 ‘0’ s .0. ----- -‘
144
Var arc-0. 50
COV Jp—-—-------~ ‘-’- -------.~~~-~ J
I .4
1 ~25
d
O... .0
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4
O4 .0
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GENERATIONS
Figure 7. l The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 8 and h; a O. I (complete dominance).
Selection by lower truncation.
I.
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0 5 IO 15 20 25 30
GENERATIONS
Figure 7. 2 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 5 and h' n 0. I (complete dominance).
Selection by lower truncation.
119
,u
v75
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GENERATIONS
Figure 7. 3 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when bro. 2 and hi I! O. 1 (complete dominance).
Selection by lower truncation.
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Figure 7. 4 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 8 and h," . O. 4 (complete dominance).
Selection by lower truncation.
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GENERATIONS
Figure 7. 5 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when bro. 5 and h' 4- 0. 4 (complete dominance).
Selection by lower truncation.
122
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GENERATIONS
Fig‘gre 7. 6 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 2 and h; a. 0. 4 (complete dominance).
Selection by lower truncation.
123
4 4.0
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0 5 IO 15 20 25 30
GENERATIONS
Figure 7. 7 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 8 and hi 2 0. 7 (complete dominance).
Selection by lower truncation.
fl
124
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o 5 10 15 20 25 30
GENERATIONS
Figure 7. 8 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when b-O. 5 and h; 4. 0. 7 (complete dominance).
Selection by lower truncation.
125
u». --\
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Cov J
IIJ r
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as
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GENERATIONS
Figure 7. 9 The relationship between the genotypic variances,
covariance, and genetic correlation at three levels of
correlation when bro. 2 and hino. 7 (complete dominance).
Selection by lower truncation.
126
follow quite logically from the theory of the rate of change in
gene frequency under selection when dominance exists. These
circumstances have been discussed in detail by Lush (1945).
An interesting observation in the results, especially in the
dominance model, was the close agreement between the change
in the genotypic covariance and the change in the genotypic
variance of trait Y, the correlated trait. A distinctly similar
pattern of response can be noted in every treatment combination
(see, for example, Figures 7. 6 and 7. 9). This similarity of
response to selection is expected, however, since each is
simply measuring the reduction in the genotypic variance in
the loci of type B, those loci which affect both traits in the
same direction. That the genotypic covariance is a measure of
the genotypic variance of the common loci has been shown
previously; all other covariances are expected to be zero. Also,
the only loci affecting trait Y which are under selection pressure
are those same loci which are shared. Thus, the decrease in
. the genotypic covariance and in the genetic variance of Y are
both a function of the change in gene frequency at the pleiotropic
loci.
The Genetic Correlation in the Truncated Distribution.
The effect of linear truncation of X on the genetic correlation
127
again has been graphed for both experiments in the complete
dominance model. The broken lines in Figures 4. l to 4. 3 and
in Figures 6. l to 6. 3 represent the genetic correlation in the
group selected to be parents for upwards and downwards
selection , respectively.
.55 I"
‘Q
As in the additive model, truncation selection caused some
decrease in the genetic correlation. When selection was by
upper truncation, the amount of reduction in the genetic cor- ‘ 5.:
relation was a function of both level of heritability and
selection. The amount of decrease seemed greater than was
observed in the additive model. The magnitude of the initial
genetic correlation was also affecting the amount of decrease,
a larger and more consistent reduction resulting when the
initial correlation was 0. 75 than when 0. 25. When level of
selection was high (b - 0. 2) and environmental variance was
low (Figure 4. 3), the amount of decrease became very large
when the initial genetic correlation was 0. 75. In fact, the
genetic correlation in the selected group was generally about
0. 2 less than the correlation in the whole offspring generation.
Again, however, when intensity of selection and heritability
were both low, the reduction in the genetic correlation was
fairly small, which can be expected when 80 per cent of the
offspring are selected to be parents.
128
When selection was by lower truncation (Figures 6. l to 6. 3)
similar results were obtained, the amount of reduction
increasing as heritability increased. As observed in the additive
model, level of environmental variance was more important
than level of'selection in producing a decrease in the correlation
in the selected group.
The Estimates of Genetic Correlation Obtained from Phenotypic
Covariances between Parent and Offspring.
In the model of complete dominance two separate estimates
of the genetic correlation were obtained from the covariances
between parent and offspring phenotypes. However, because
of the erratic results obtained in the additive model, the estimate
for the complete dominance model using the geometric mean of
the two covariances in both numerator and denominator was
rejected if the two covariances in the numerator or denominator
were of unlike sign. This condition occurred in the majority of
cases.
In general, the results obtained using the arithmetic mean
of the two covariances in the numerator were equally as poor
as those previously obtained for the additive model, whether
selection was by upper or by lower truncation. Since to present
all the results for both types of selection would be of little value,
129
a sample of results has been presented only for selection
by upper truncation. These are shown in Table 3 as
deviations of the genetic correlations computed by parent-
offspring covariances, from the product-moment correlations
of genotypic values. The same group of treatment combinations
and examples for the first replicate in selected generations as
for the additive model are presented. The asterisk beside some
estimates indicates that the two covariances within the numerator
and within the denominator were of the same sign. More
estimates had covariances of unlike sign than not. In those
cases where the correlation could be computed by the geometric
means in both numerator and denominator , seldom did the
correlation by geometric mean agree with that by arithmetic
mean.
Table 3 indicates that an interpretation of the results would
be unwise, apparently random fluctuation prevents observing a
predictable pattern. There is, however, the same tendency
for the correlation to be underestimated as in the additive model.
Of the 300 deviates presented in Table 2, 194 were negative,
and the overall average deviation was - O. 40. When the level
of selection was low and the number of observations small,
the average deviation was - 0. 37 while at high intensity of
selection the average deviation was - 0. 43. These averages
130
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