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dissertation entitled

POWER AND ACCURACY OF DETECTING LINKAGE BETWEEN
QUANTITATIVE TRAIT LOCI AND GENETIC MARKERS

presented by

Zhiwu Zhang

has been accepted towards fulfillment

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POWER AND ACCURACY OF DETECTING LINKAGE BETWEEN
QUANTITATIVE TRAIT LOCI AND GENETIC MARKERS

By

Zhiwu Zhang

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Animal Science

1998

ABSTRACT

POWER AND ACCURACY OF DETECTING LINKAGE BETWEEN
QUANTITATIVE TRAIT LOCI AND GENETIC MARKERS

By

Zhiwu Zhang

Power of testing additive effects of quantitative trait locus (QTL) linked to
genetic markers and accuracy of the parameter estimators from a QTL and
polygenic additive mixed model were evaluated through simulations. The
underlying conditions included QTL location, additive genetic variance due to
QTL linked to markers, additive genetic variance due to polygenic effects
unlinked to markers, and residual effects.

Granddaughter population designs were simulated under a mixed model
with additive polygenic and QTL effects for various combinations of factors at
various levels. Items of interest included: selection schemes, marker intervals,
number of daughters, magnitude of QTL effects and heritability of traits. Sons in
the design were selected by 4 alternative schemes of random, disruptive,
truncation and stabilized. Fifty replicates were generated for each of the
populations and analyzed separately by restricted maximum likelihood.

Estimates of QTL location, variance due to QTL, and polygenic and

residual effects were unbiased in unselected populations. In fact, estimates of

QTL location were unbiased in populations under all selection schemes.
However, estimates of variances due to QTL, polygenic and residual effects were
biased in populations under nonrandom selection schemes. Magnitude of the
biases were dependent on marker intervals, numbers of daughter, magnitudes of
QTL effect and heritability levels.

Selection schemes had a significant influence on the power of testing
linkage between genetic markers and QTL using restricted maximum likelihood.
Disruptive selection generated higher power than random selection, whereas
truncation and stabilizing selections have less power than random selection.

Power of test using restricted maximum likelihood also depends on the
number of daughters of each sire, marker intervals, magnitude of QTL effect and
heritability. Power of test was higher with more daughters per sire, smaller
marker intervals, larger magnitude of QTL effect, and higher heritability. The
magnitude of the differences due to changing a factor was larger when power

was less saturated by other factors.

A Dedication to My Parents

ACKNOWLEDGMENTS

This research reflects the efforts of many. My major professor, Dr. Ivan
Mac, is acknowledged for his role as a mentor and friend without whom I could
not have succeeded. Dr. Dennis Banks and Dr. Rob Tempelman are recognized
for their assistance in the study. From the Department of Statistics, gratitude is
extended to Dr. Raoul LePage for his invaluable advice in the interpretation of
the results.

I thank Dr. Fernando Grignola for providing QTL analysis program and
comments for the study.

Appreciation is extended to Dr. Wejun Zhao for his role in the initiative of
my program at Michigan State University.

I thank my office mates and fellow graduate students, Charlie Chang,
David Norris, Morten Rye, Thomas Mark Lars Nielsen, Ole Pedersen, Mara
Preiler, Renee Bell and Kadir Kizilkaya for their help and friendship.

A final note of recognition is extended to my wife, Wanling, and my son,
Jingwei. My wife always understands me and always loves me. My son does not

always understands me but always loves me.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................... ix

LIST OF FIGURES .............................................................................................. x

GENERAL INTRODUCTION ............................................................................... 1
CHAPTER 1

LITERATURE REVIEW ........................................................................................ 5

Introduction ................................................................................................ 5

Resource Populations ................................................................................ 9

Line Crosses ................................................................................... 9

Outbreed Lines .............................................................................. 11

Statistical Methods ................................................................................... 1 3

ANOVA .......................................................................................... 1 3

Regression .................................................................................... 14

Maximum Likelihood ...................................................................... 15

BLUP Based Methods ................................................................... 16

Sampling Strategy .................................................................................... 20

Random Selection ........................................................................ 20

Disruptive Selection ...................................................................... 20

vi

Truncation Selection ..................................................................... 21

Stabilizing Selection ..................................................................... 22
Marker Assisted Selection ....................................................................... 23
Genetic Markers ............................................................................ 23
Advantage of MAS ........................................................................ 25
Limitations of MAS ........................................................................ 26
CHAPTER 2
METHODOLOGY ............................................................................................... 27
Population ................................................................................................ 27
Simulation of Genotypes .......................................................................... 30
Simulation of DYDs .................................................................................. 30
Selection Schemes .................................................................................. 33
Statistical Analysis ................................................................................... 34
Experimental Design ................................................................................ 36
CHAPTER 3
BIASES lN GENETIC PARAMETER ESTIMATES FROM A POLYGENE AND
QTL MIXED MODEL UNDER GRANDDAUGHTER DESIGN. ................ 37
Abstract .................................................................................................... 37
Introduction ...................... - ........................................................................ 38
Materials and Methods ............................................................................. 40
Results and Discussions .......................................................................... 45
Conclusions and Implications .................................................................. 54

vii

CHAPTER 4
POWER OF DETECTING MARKER ASSOCIATED QTL EFFECTS IN

GRANDDAUGHTER DESIGN ................................................................. 56
Abstract .................................................................................................... 56
Introduction .............................................................................................. 57
Materials and Methods ............................................................................ 60
Results and Discussions .......................................................................... 65
Conclusions and Implications ................................................................... 77
GENERAL CONCLUSION .................................................................................. 78
APPENDICES
A - Expectation of Heritability Weighted DYD .......................................... 79
B - Input and Output of Grignola’s Program ............................................. 82
C - Flow Chart of Pedigree ....................................................................... 83
D - Flow Chart of Subroutine of Crossover .............................................. 84
E - Flow Chart of Subroutine of Randasign .‘ ............................................ 85
F - Flow Chart of Subroutine of Phenotype ............................................. 86
G - FORTRAN (90) Code of Simulation Program ................................... 87
BIBLIOGRAPHY ................................................................................................. 95

viii

LIST OF TABLES

Table 1 - Median and SE of Bias Rates (%) in the Estimation of QTL Allelic
Variance .............................................................................................. 50

Table 2 - Median and SE of Bias Rates (%) in the Estimation of Polygenic
Variance. ............................................................................................. 51

Table 3 - Median and SE of Bias Rates (%) in the Estimation of Residual
Variances. ........................................................................................... 52

Table 4 - Power from Populations Subjected to Alternative Selection Schemes
with Different Levels of Heritability (h2) and Magnitudes of QTL Effect
(v2) ....................................................................................................... 68

ix

LIST OF FIGURES

Figure 1 - Population Structure of Granddaughter Design ................................. 28
Figure 2 - Pedigree of 20 Sires in Simulated Granddaughter Design Population29

Figure 3 - Genetic Marker Loci and one QTL on an Autosomal Chromosome ..31

Figure 4 - Estimates of Polygenic, QTL and Residual Variances ....................... 48
Figure 5 - Profiles of Test Statistics ................................................................... 66
Figure 6 - Comparison of Powers between Random and Truncation Selection
Schemes ............................................................................................ 69
Figure 7 - Marginal Effects on Power of Test with Random Selection ............... 70

Figure 8 - Effect of Number of Daughters on Power of Test under Truncation
Selection Scheme .............................................................................. 71

Figure 9 - Effect of Heritability on Power of Test under Truncation Selection
Scheme ............................................................................................. 72

Figure 10 - Effect of Marker Interval on Power of Test under Truncation Selection
Scheme ............................................................................................. 73

Figure 11 - Effect of Magnitude f QTL Effect on Power of Test under Truncation
Selection Scheme .............................................................................. 74

GENERAL INTRODUCTION

For many years, Animal breeders have changed the genetic composition
of farm animals through selection without knowledge of the underlying genes
(Bovenhuis et al., 1997). This approach is based on the assumption that a trait is
controlled by infinitive genes each having an equally small effect. Due to the
environmental effects and the segregation of genes from parent to offspring,
accurate estimation of the breeding value of an animal depends on the amount of
records on the phenotype of the individual itself and/or relatives. In general, the
requirement of a large number of records postpones the age at which the animal
can be selected as a parent and therefore, restricts the annual genetic progress
(Bovenhuis et al., 1997).

However, genes with large effects on economically important traits have
been identified. Notable examples are the double muscling gene in cattle
(Hanset and Michaux, 1985a and 1985b), the gene determining halothane
sensitivity in swine (Smith and Bampton, 1977), and the estrogen receptor
genes influence a litter size in swine (Rothschild et al., 1994).

The possibility of finding major genes affecting economically important
traits has been greatly increased due to the development of methods to detect
polymorphism at the DNA level, for examples, RFLP (Paterson et al., 1988;

Visscher et al., 1990; Soller and Beckman, 1990; Georges et al., 1995). Genetic

2

linkage maps of polymorphic molecular markers have been developed in many
domestic animal species. It was possible for the first time to begin the systematic
search for individual loci affecting quantitative traits of economic importance.

Unfortunately, the majority of genetic markers, especially DNA based
polymorphism, are not likely to affect animal performance. They may be located
closely to the genes affecting quantitative traits (Soller and Beckman, 1990).

The detection of QTL linked to genetic markers is a statistical inference. It
is structured around the formal test of the null hypothesis, which proposes that
the recombination rate between genetic markers and QTL is 50%. Experiments
are designed to test the null hypothesis against a alternative hypothesis.
Statistical analysis may either reject null hypothesis, suggesting the existence of
the linkage, or may not reject it, suggesting that genetic markers are not linked to
the QTL.

A statistical interface involve two kinds of errors: Type I, when a true null
hypothesis is rejected, and Type II, when a false null hypothesis is accepted. The
probabilities of committing Type I and II errors are denoted by a and p,
respectively. The probability of not committing a Type II error is called power of
test, which is (1- [3).

Power of test is more important when nonsignificant results are obtained,
since in this case, a valid assertion of those results is only possible if the power is
high. On the other hand, higher power leads to larger sample size in general,

which can be costly. Powers of 0.8 to 0.9 are generally used to accept null

3

hypotheses (Searcy-Bernal, 1994). Therefore, it is crucial to predetermine the
desired power of the linkage test in evaluating candidate markers.

Once marker linked QTL effects are detected, the use of the linkage
between marker loci and QTL provide additional information for increasing the
accuracy of selecting the genetic difference between individuals. Efficiencies
were compared between purely phenotype selection and marker assisted
selection (Kashi et al., 1990; Meuwissen and Van Arendonk, 1992; Meuwissen
and Goddard, 1996; Smith and Simpson, 1986; Stam, 1986; van der Beek and
Van Arendonk, 1996). Marker assisted selection was more efficient than
selection without using marker information in 1) early generations; 2) lowly
heritability traits; 3) large populations and 4) close linkage between marker and
QTL.

This conclusion is based on the assumption that the parameters of
quantitative trait loci are known without error. Simulations demonstrated that
overestimation of QTL variance decreased genetic gain for marker assisted
selection (MAS) over the long term. For an error of 15 centiMorgan (cM) on the
location of QTL, genetic superiority of MAS was reduced by 80% in the first
generation than MAS without error on QTL location (Spelman and Van
Arendonk, 1997b).

The power to detect linkage and accuracy of estimates of QTL parameters
depended on several factors, including statistical method, sampling strategy,
sample size, marker density and magnitude of QTL effects. Powers have been

examined for the statistical method of ANOVA (Weller, 1990a), maximum

4

likelihood (Knott and Haley, 1992b; Le-Roy and Elsen, 1995; Carbonell et al.,
1993; Jensen, 1989; Lander and Botstein, 1989; Knapp and Bridges, 1990;
Elsen et al., 1997) and regression (Moreno-Gonzalez, 1992). Knowledge power
using restricted maximum likelihood remains unknown.

The accuracy and precision of estimates using restricted maximum
likelihood (REML) were evaluated in many studies for additive polygenic models
in unselected and selected populations (Henderson, 1975a; Banks et al., 1985;
Gianola et al., 1986; Beaumont, 1991; Sorensen and Kennedy, 1984). Grignola
et al., (1996b) evaluated accuracy and precision of estimates using REML with
QTL and polygenic mixed models in unselected populations.

The objective of this study was to evaluate parameter estimates from QTL
and additive polygenic mixed models and to examine power of detecting QTL
using restricted maximum likelihood method in combinations of different selection
schemes, marker intervals, magnitude of QTL effect, heritability, and number of

daughters in a granddaughter design.

Chapter 1

LITERATURE REVIEW

Introduction

For many years, Animal breeders have changed the genetic composition
of farm animals through selection without knowledge of the underlying genes.
Recent developments in molecular biology have changed this situation and have
allowed the genes controlling traits or genetic markers linked to the genes to be
identified (Bovenhuis et al., 1997).

Classical animal breeding approaches to estimate the additive genetic
value of an individual depends on phenotypic observations on the individual itself
and/or relatives. This approach is based on the assumption that a trait is
controlled by infinitive genes each having an equally small effect. The action of
individual gene cannot be observed directly and a trait is generally described in
terms of summary statistics such as the heritability.

For most of the traits of interest to animal breeders, differences in
phenotypic observations are determined by both genetic and environmental
differences. Further more, segregation of genes takes place each time genes are

transmitted from parent to offspring. As a result of these factors, accurate

6

estimation of the breeding value of an animal is possible only if a large number of
records on the phenotype of the individual itself and/or relatives are available. In
general, the requirement of a large number of records postpones the age at
which the animal can be selected as a parent and therefore, restricts the annual
additive genetic progress.

However, some genes with a large effect on economically important traits
have been identified. Notable examples are the dwarfing gene in poultry, the
Booroola gene affecting ovulation rate in sheep, the double muscling gene in
cattle (Hanset and Michaux, 1985a and1985b), the gene determining halothane
sensitivity in swine (Smith and Bampton, 1977), and estrogen receptor gene
controlling litter size in swine (Rothschild et al., 1994).

Unfortunately, most genetic markers, especially DNA based
polymorphism, for example, RFLP, are not likely to be the alleles that affect the
performance of animals. But they may be linked to the genes affecting
quantitative traits. Several studies have shown that individual loci affecting
quantitative traits can be detected if linked to genetic markers (Soller and
Beckman, 1990). For example, a genetic marker on chromosome 8 was found to
be linked to a QTL with an additive effect of 3 ova (Rathje et al., 1997).

Once genetic marker, QTL linkage is established, information of linkage
provides additional information for increasing accuracy of selection, especially for
the traits that are difficult to improve when using traditional selection methods
(Kashi et al., 1990; Meuwissen and Van, 1992; Meuwissen and Goddard, 1996;

Smith and Simpson, 1986; Stam, 1986; van der Beek and Van Arendonk, 1996).

7

The earliest usage of linkage between genetic marker and QTL was
demonstrated by Sax (1923). However, it was restricted by the limitation of the
number of genetic markers. Genetic markers are available now due the
development of methods to detect polymorphism at the DNA level (Kashi et al.,
1990). The genetic linkage maps of polymorphic molecular markers have been
developed in many domestic animal species. It is possible, for the first time, to
begin the systematic search for individual loci affecting quantitative traits of
economic importance (Paterson et al., 1988; Visscher et al., 1990; Soller and
Beckman, 1990; Georges et al., 1995).

There are several activities involved in identifying and utilizing the linkage
between genetic markers and QTL. These include recording animals for the
character of interest, typing them for genetic markers, testing for statistical
associations between genetic markers and phenotypic score and, if associations
are found, applying marker assisted selection in breeding schemes (Paterson et
al., 1988). Two statistical inferences are involved in these activities. One is the
power of test. The other is the estimation of QTL location, QTL variance,
polygenic variance and residual variance.

Identification of linkage between a genetic marker and QTL is structured
around the formal test of a null hypothesis (Ho), which assumes genetic markers
are not linked to QTL. Statistical inference is to test Ho against the alternative
hypothesis, which propose that genetic markers are linked to the QTL. Statistical

analysis may either reject Ho, suggesting the existence of the linkage, or may not

reject it, suggesting that genetic are not linked to the QTL at some level of

8

confidence. This statistical inference involves two types of errors: Type I error,
the rejection of a true null hypothesis, and Type II error, the acceptance of a false
null hypothesis. The probabilities of not committing type II is called power of test.

Statistical power is more important when nonsignificant results are
obtained. In this case a valid assertion of those results is only possible if the
power is high.

In order to decrease the probability of making Type II error, power of test
must reach a certain level. Levels of 0.05 and 0.01 are accepted as popular
significance levels for rejecting the null hypothesis, while powers of 0.8 to 0.9 are
required for acceptance of a null hypotheses (Searcy-Bemal, 1994).

Power of testing linkage between genetic markers and QTL depends on
many factors (e.g., statistical procedures, sample size, population structure,
magnitude of QTL effect, recombination rate, heritability of traits, and sampling
strategies). Because of the expense in both genotyping and generating a
experimental population, efforts have been spent in optimizing the design of
experiments for a optimal power.

A quasi-theoretical numerical method can be used to predict power of
detecting QTL from the shape of the multidimensional expected likelihood
surface (Mackinnon and Weller, 1995). However, theoretical calculations of
power are not always empirically possible. Computer simulation can be
employed in the situation.

Once a genetic marker associated QTL effect is detected, it is of interest

to find the location of QTL in the genome, and to determine the effects of the

9

QTL by estimating the phenotypic variance explained by QTL linked to markers.
All of these parameters are required in predictions of the breeding values in
cooperating genetic marker information.

An overview is given for the characteristics of the affecting factors in
identification and utilization of linkage between marker and QTL, including
different resource populations; statistical methods strategies of selective

genotyping and marker assisted selection.

Resource Populations

There are two primary types of data used for mapping a quantitative trait
locus: data derived from linecrosses that include backcross and F2 populations
(Soller et al., 1976) and data from outcross populations (Soller and Genizi, 1978;

Weller, 1990b).

Line Crosses

Most successful QTL mapping efforts described to date have exploited F2
or backcrosses obtained from parental populations divergent for the traits of
interest (Paterson et al., 1988). The main reason is that line crosses generate

disequilibrium.

IO

Linkage disequilibrium between genetic markers and QTL creates
difference on a trait across marker genotypes. One way of introducing linkage
disequilibrium in a population is by crossing lines that differ with respect to their
allele frequencies at marker loci and QTL. Associations between genetic markers
and QTL can be studied by comparing the phenotypic performance of F2 or back
cross individuals with different marker genotype configurations.

ln ideal situation, all F1 individuals are heterozygous for the marker as well
as QTL. There is complete linkage disequilibrium between the marker and QTL in
the F1. All F1 individuals have the same linkage phase. An empirical case of such
a design is the use of inbred lines.

Linecrosses are frequently used in laboratory animals (e.g., mice) and
plants. For farm animals, however, inbred lines are seldom available. In addition,
rearing large numbers of F1 and F2 individuals is possible for some farm animals
(e.g., chicken) but not for others (e.g., cattle) because of the long generation
intervals and costs of the experiment.

Power to detect QTL has been described for crosses of inbred lines
(Soller et al., 1976; Weller, 1986; Jansen, 1993; Lander and Botstein, 1989; Luo
and Woolliams, 1993; Simpson, 1989; Knott et al., 1992b and Darvasi et al.,

1 993).

ll

Outbreed Lines

Linkage disequilibrium in an outcross population between a marker and
linked QTL is more likely within families (Neimann-Sorensen and Robertson,
1961)

In commercial dairy cattle populations, sires often have hundreds or even
thousands of daughters produced by artificial insemination. Thus, a segregating
QTL can be detected by analyzing the progeny of heterozygous sires. Daughters
inheriting the different sire marker alleles should also display a difference for the
quantitative trait.

There is more power to perform this analysis over multiple sires rather
than a single sire. Even if a sire is heterozygous for the genetic marker, he may
still be homozygous for the QTL. Different marker genotypes can not show any
difference on the QTL genotypes.

Further more, if the sire is heterozygous for both marker and QTL loci,
linkage phase between the marker and QTL alleles may be different from
different sire. Thus, analysis should be performed within sires.

The analysis can be performed within paternal half-sib families using
either the daughter design or the granddaughter design (Weller, 1990a). The
basic idea of the daughter design is to trace marker alleles from the sires to his
daughters and to determine whether daughters that inherited alternative sire
alleles differ with respect to the quantitative trait. In the daughter design,

daughters of a sire are scored for markers and evaluated for the quantitative trait.

12

However, in the granddaughter design, the sons of a proven sire are scored for
genetic markers and granddaughters are evaluated for the quantitative trait. In
this latter case, the observations on the granddaughters are used to estimate the
breeding value of the sons. This breeding value has a lower residual variance
compared to a single observations which increases the power of the experiment
(Weller et al., 1990b; Van der Beek et al., 1995).

In granddaughter design, marker associated effects measured in
granddaughter generation will be halved with respect to marker associated
effects measured in the daughter generation. Nevertheless, the standard error of
the contrasts are smaller, so that granddaughter designs may be able to deliver
equivalent power while scoring fewer individuals for the markers, the most costly
part of the program. Also, it may be easier to collect blood or semen samples
from sons of sires, concentrated in Al centers, than from their daughters,
scattered over many farms. Power to detect QTL has been described for
outbreeding populations (Knott et al., 1992b; Bovenhuis and Weller, 1994).

Because of different linkage relationship among sires, outcross
populations (eg. the daughter design) have less statistical power than line
crosses. To detect a QTL with a substitution effect of 10-30% of a phenotypic
standard deviation, it is necessary to determine the genetic marker genotype of
thousands of daughters (Weller, 1990a). With inbred lines, the same power can
be obtained by determining the genetic marker genotypes of less than 1000

progeny (Solleret al., 1976).

13

Statistical Methods

Statistical methods of mapping QTL vary depending the structure of
populations and interpretation of the nature of QTL effect. Effect of QTL can be
considered as either fixed effect or random. This leads to a different choice of
statistical approaches. Fixed effect is only one that t test (Simpson, 1989),
ANOVA and regression methods can deal with, while, maximum likelihood
method, as well as BLUP based methods are suitable for both (Simpson, 1989;
Lander and Botstein, 1989; Cowan et al., 1990; Weller, 1990a; Haley et al., 1994;

Lander and Botstein, 1989; Knott and Haley, 1992a).

ANOVA

ANOVA is performed by contrasting marker genotype effects by (Soller et
al., 1976). This method of analysis yields estimates of marker allele substitution
effects. However, the analysis does not provide any information about the
location of the QTL (e.g., the method cannot distinguish between a slightly linked
QTL with a large effect and a closely linked QTL with small effect). Another
disadvantage of this type of analysis is that some of the progeny cannot be
assigned to one of the two parental alleles. These animals have to be excluded
from the analysis, which results in reduced power. Weller and Wyler, (1992)

evaluated the power of ANOVA in daughter design and granddaughter design.

l4

Regression

Haley and Knott (1992) and Martinez and Curnow (1992) independently
introduced a regression method. Regression is performed on the probability of an
individual having a QTL genotype, given the genotype for the flanking markers.

ANOVA is identical to regression with specific protonts. If the probability
that an animal has inherited a particular QTL allele from its parent is based only
on information from single flanking informative marker, regression is equivalent to
an analysis of variance. The position and the effect of the QTL cannot be
disentangled.

However, QTL location can be estimated by utilizing marker bracket. This
method results in an estimate of the QTL position as well as the variance
explained by genotype contrasts. The probability of an individual having a QTL
genotype, given the genotype for the flanking markers depends upon the location
of the QTL. By moving a putative QTL along the chromosome, the most likely
position of the QTL corresponds to the position with minimum residual sum of
squares (Whittaker et al., 1996).

Once QTL genotype probability by marker genotype is determined,
standard statistical software packages can be used for regression part of the
analysis (Spelman et al., 1996; Weller et al., 1990b; Hoeschele, 1990; Cowan et
al., 1990; and Goddard, 1991). Power of using regression was investigated by
(Moreno-Gonzalez, 1992 and Jansen, 1994a). The distribution of the statistics of

testing marker associated QTL effect was studied (Hyne and Kearsey, 1995).

15

Maximum Likelihood

Weller (1986) developed maximum likelihood methods to detect marker
associated QTL effects. It involves maximizing and comparing the likelihood of
the data under different genetic models to ascertain the most likely genetic
structure. The maximum likelihood of the data under a additive polygenic model
is compared with that under the combined model containing a major gene linked
to markers and additive polygenic component. A significant improvement in the
likelihood obtained by incorporating a major gene in the model provides evidence
for a linkage between QTL and genetic markers.

Maximization is usually with respect to five parameters: mean, the additive
and dominant QTL effect, recombination rate between marker and QTL, and
within QTL genotype residual variance (Weller, 1986; Bovenhuis and Weller,
1994).

Interval mapping is more accurate than a single marker in estimating QTL
location. The advantage depends upon the heterozygosity of the markers and the
position of the QTL within the flanking markers (Darvasi et al., 1993; van der
Beek et al., 1995)

The power of test depends on family size (Knott et al., 1992b), QTL
effect, recombination between marker and QTL (Le-Roy and Elsen, 1995) and

heritability of quantitative trait (Carbonell et al., 1993 and Jensen, 1989).

16

Currently, the most popular analytical method to investigate QTL is carried
out by either regression, or by the maximum likelihood approach. The two
methods yield very similar results (Lander and Botstein, 1989; Haley and Knott,
1992; Martinez and Curnow, 1994), the regression approach applies a more
straightforward test of significance and is programmable using standard

statistical packages.

BLUP Based Methods

Method of ANOVA, regression and ML were developed mainly for line
cross populations. They can not fully account for the more complex data
structures in outcross populations ( e.g. data on several families with
relationships across families, unknown linkage phases in parents, unknown
number of QTL alleles in the population, and varying amounts of data information
on different QTL or in different families). Best linear unbiased prediction (BLUP)
based methods were developed to overcome these drawbacks.

For data that does not contain genetic marker information, BLUP has
proved to be a very flexible method. BLUP can handle data with many
nongenetic effects (e.g. season), with arbitrary pedigree structure, and with
nonrandom mating and selection. Currently BLUP is effectively used for the
prediction of breeding values of farm animals.

The prediction of an animal's breeding value is based on phenotypes of

the animal itself and relatives. When only phenotypes are considered, the

l7

contribution of observations on relatives to an animal's breeding value depends
on the additive genetic relationship, ’19,, the average proportion of genes shared
in common by descent, and the heritability of the trait. The additive relationship
between individuals is formed without knowledge of the actual gene contributed
from parent to its offspring.

Recently, the concept of the additive genetic relationship has been
extended to the gametic relationship where paternal and maternal gametes of an
animal are considered separately. The gametic relationship has been used for
constructing the relationship due to dominance effects and for the analysis of
gametic imprinting effects (Schaeffer et al., 1989). For these reasons it is also
likely to be useful for analyzing data containing information on genetic markers, if
the assumptions of BLUP are reasonably satisfied (Grignola, 1996a).

lnforrnation on an animal's genotype at a marker locus provides
information on transmission of a chromosomal region from parent to offspring. If
QTL are located in the chromosomal region, this information can be used to
obtain a more accurate estimate of breeding values because the inheritance of
alleles at the chromosomal region can be traced more precisely than inheritance
at an unmarked QTL. In this case, the additive genetic value of an animal can be
partitioned into additive genetic value at the marked chromosomal region and the
sum of additive genetic effects of polygenes linked to markers.

Construction of variance and covariance of QTL effects linked to markers
is the key factor to apply BLUP in QTL analysis. Fernando and Grossman (1989)

showed that information on a single marker can be used in an animal model by

l8

fitting additive effect for alleles at QTL linked to a genetic marker and additive
polygenic effects for alleles at the remaining quantitative trait loci. Goddard
(1991) extended the model to include information from more than one marker.
Wang et al. (1995) presented an algorithm for additive relationship matrix
construction without the requirement on the information of origin of alleles.
Meuwissen and Goddard (1996) presented a method in which the covariance
matrix of effects at the marked QTL is approximated. This approximation reduces
the computational requirements.

The linear mixed model equation is greatly enlarged by including QTL
effect linked to markers. The size of the equation can be reduced by use of a
reduced animal model (Goddard, 1991). In this case, effects are only predicted
for animals that are parents. Breeding values and additive QTL effects for non-
parents can be obtained by back solving.

Another way of reducing the size of equation is to link phenotypes to the
total additive effects and link total additive effects to QTL. An animal model
method to reduce the number of equations per animal to one was presented by
Van Arendonk et al. (1994) combining information on marker linked QTL and
QTL unlinked to marker into one numerical relationship matrix. A reduced animal
model version of Van Arendonk et al. (1994) is also developed by Saito and
Iwaisaki (1996).

Hoeschele and VanRaden (1993a and 1993b) indicated that if some of
the animals to be evaluated do not have marker data and do not provide

relationship ties among genotyped descendants with known marker data, the

l9

marker linked QTL equations for such animals can be eliminated. The inverse of
a covariance matrix among total additive polygenic effects and the additive
effects of the QTL alleles can be obtained directly. When only a small fraction of
the animals are genotyped for markers and the remaining fraction do not provide
marker data, the procedure of Hoeschele and VanRaden (1993a and 1993b)
has the advantage of reducing the number of equations to be solved.

All of the algorithms above are based on an unknown parameter, which is
the recombination rate between genetic markers and QTL. One solution is to
maximize the log likelihood at each point at the marker interval by using
restricted maximum likelihood with respect to the other parameters, eg. additive
variance due to allele at QTL linked to marker, additive variance due to alleles at
the remaining quantitative trait loci, and residual variance. The location of QTL
was estimated at the point corresponding the maximum log likelihood over the
entire marker interval (Van Arendonk et al., 1994; Grignola et al., 1996a)

Restricted maximum likelihood (REML, by Patterson and Thompson,
1971) has become the method of choice for estimating the variance components
in animal breeding. The first attempt for estimating position and variance
contribution of a single QTL together with additive polygenic and residual
variance components by REML was undertaken by Van Arendonk et al. (1994)
with a single marker. Grignola et al., (1996a) extended this method to multiple

markers.

20

Sampling Strategy

Most of the studies of detection of QTL and marker assisted selection
assumed that individual sampled are randomly chosen from population. This
assumption is seldom true. In detection of QTL effects linked to markers,
individual with extreme phenotypic values are selected for genotyping to increase
the power of test. In most livestock species, data must be obtained from existing
commercial populations. Usually, such populations have been selected for many
generations toward a desired breeding goal. The impacts of selection on power

and estimates of parameters of QTL effects are discussed in following section.

Random Selection

Random selection assumed that individuals are randomly chosen from
generation to generation. Sampled individuals share the same gene pool with

base population.

Disruptive Selection

To increase power of detecting marker associated QTL effects, animals
for genotyping were selected with extreme phenotypic values. The statistical
power of selective genotyping for the purpose of detecting linkage between QTL

and markers was investigated by Lander and Botstein (1989); Darvasi and Soller

21

(1992). It was found that power to detect a QTL effect is increased by selectively
genotyping individuals with extreme values for the quantitative trait (Weller and
Wyler, 1992; Lin and Ritland, 1996).

Mackinnon and Georges (1992) found that selection by truncation for the
trait of interest significantly reduces the genetic variance of the trait, thus
reducing the power to detect linked quantitative trait loci.

In the absence of selection, the estimates of QTL location, variance due to
QTL and polygenic effects, and residual variance are unbiased by using REML
(Grignola et al., 1996b). In another simulation study, bias of the estimated QTL
effects was less than 2% in the absence of selection (Meuwissen and Goddard,
1997). The selection bias using BLUP based methods are needed to be

investigated for polygenic and marker linked QTL mixed model.

Truncation Selection

Theory indicates the effects of selection can be accommodated by an
appropriate model that includes all data upon which selection decisions were
based, tracing back to the unselected base generation (Henderson, 1975b;
Gianola et al., 1986). Sorensen and Kennedy (1984) simulated several
generations of selection and omitted data from earlier generations. They
concluded that the estimate of the additive genetic variance before selection was
nearly unbiased when their model acknowledged all relationships that developed

in previous generations.

22

However, results in a similar study, Van der Werf and De Boer (1990)
showed that there were small biases in estimates of additive genetic variance of
the base unselected generation in some cases, even with all relationships tracing
back to base generation.

Impact of selection on the estimates under additive polygenic and marker
linked QTL mixed models were also carried out for ANOVA, regression and
maximum likelihood. Biases were found for both the population under going
selection and population with selective genotyping. The bias is a function of
selection intensity and the magnitude of QTL effects. Lin and Ritland (1996)
showed selective genotyping can bias estimates of the recombination frequency
between linked QTL. The QTL effects linked to markers tend to be overestimated
increasingly with decreasing family size and true QTL effect (Georges et al.,
1995). Bias of the estimate of linkage increased when parents were not a random

sample from a population in linkage equilibrium (Uimari et al., 1996).

Stabilizing Selection

Stabilizing is applied when standard production of livestock is required.
The individuals with large or small observation are excluded. Genetic

homogeneous is desirable.

23

Marker Assisted Selection

The usage of linkage between genetic marker and quantitative trait loci
was shown earliest by Sax (1923). However, the low number of genetic markers
available has been a limiting factor for application of marker assisted selection

until the recent discovery of molecular technology.

Genetic Markers

The first molecular markers used were allozymes, protein variants
detected by differences in migration on starch gels in an electric field. Since
19605, this class of markers has been extensively applied to population genetic
problems. This approach was largely replaced by the method of evaluating
variation directly at DNA level in mid of 1980s. A simplest approach is to digest
DAN with a variety of restriction enzymes, each of which cuts the DNA at specific
sequence. When the digested DNA is run on a gel under an electric current the
fragment separated out according to size. The individual bands can be isolated
by using labeled DNA probes that base pair complementarily to particular region
of genome. This approach is called restriction fragment length polymorphisms
(RFLPs). Each RFLP probe generally scores a single marker locus. The marker
alleles are codominant. Herterozygotes and homozygotes can be distinguished.

Another approach is uses short primers for DNA replication via the

polymerase chain reaction (PCR) to delimit fragment sizes. The fragment flanked

24

by primers is amplified. The primers are random short sequence. This approach
is called randomly amplified polymorphic DNAs (RAPDs). RAPDs have
advantages over RF LPs in that a single probe can reveal several loci once. They
also require small amount of DNA.

The number of genetic markers has increased rapidly in recent years. The
advent of techniques to detect molecular variation, beginning with protein
electrophoresis and genetic polymorphism at level of DNA, has increased the
number of markers now. These markers include multisite restriction fragment
length polymorphism (RF LP) haplotypes, variable number of tandem repeat
(VNTR) sequences, and polymerase chain reaction (PCR) based polymorphism
(Mullis, 1990). The huge amount of genetic variability revealed by these
techniques in many agriculturally important species has allowed the construction
of detailed genetic maps evenly spaced throughout the genome (Paterson et al.,
1988; Visscher et al., 1990; Soller and Beckman, 1990; Georges et al., 1995).
Polymorphism at DNA level makes it possible to identify genotypic differences
among individuals at many genomic sites. The use of information on markers is
expected to accelerate genetic progress through increasing accuracy of
selection, reduction of generation interval and increased selection differentials
(Kashi et al., 1990). The advantages and limitations of marker assisted selection

are described in following.

25

Advantage of MAS

Regression and BLUP are currently methods used to integrate phenotype
with marker information. Lande and Thompson (1990) proposed an index
incorporating the phenotype value and molecular score of individuals. The
molecular score is computed from the effects attributed to markers by multiple
regression of phenotype on marker genotype. BLUP, incorporating marker
information, was described by Fernando and Grossman (1989). The total
additive genetic value composed of the prediction of the sum of two QTL alleles
linked to marker and polygene effect at remanding loci unlinked to markers.

The efficiencies were compared between purely phenotype selection and
marker assisted selection (Lande and Thompson, 1990; Gimelfarb and Lande,
1994; Ruane and Colleau, 1996; Hospital and Moreau, 1997). Zhang and Smith
(1992 and 1993) conducted similar simulations of marker assisted selection.
However, they compared MAS not with purely phenotypic selection but rather to
selection based on the BLUP estimate of an individual‘s breeding value. The
common conclusion was that marker assisted selection was more efficient than
selection without using marker information in 1) early generations; 2) low
heritability traits; 3) large populations; 4) close linkage between marker and QTL.

Marker assisted selection can still be effective in populations that have
been highly selected for many generations on phenotype or predicted breeding
values. Georges et al. (1995) demonstrate that loci with considerable effects on

milk production are still segregating in highly selected populations.

26

Limitations of MAS

First of all, the limitation of MAS comes from the nature of genetic
composition of traits. The genetic variance may be controlled by many genes
with small effects even though there may be a few loci with large effects
(Shrimpton and Robertson, 1988). Traditional methods will be effective to identify
the effect of these genes. Further more, due to arising of variance within
populations by high mutability of polygene, traditional methods is effective to
improve whatever previous gains have been made (Hill, 1982).

Another factor limiting efficiency of MAS is the error of QTL parameter
estimates. The QTL parameters in MAS studies have assumed to be known
without error when genetic and economic responses to MAS were estimated.
However, this situation is not generally true. The variance associated QTL effects
was overestimated when the analysis had low power (Wang et al., 1995).

Deduction on selection response can result from the error of the estimates
of QTL parameters. For a simulated error of 15 cM on the location of QTL,
genetic superiority of MAS was reduced by 80% in the first generation. Zhang
and Smith (1993) showed that poorly estimated QTL effects added noise to the

system and reduced selection response in marker assisted selection.

Chapter 2

METHODOLOGY

Data is generated given specified underlying parameters for a marked
QTL and polygene mixed model. The estimates of the parameters are obtained
by analyzing the simulated data using REML. The design of the breeding
populations simulated, the simulation of genotypes on markers and QTLs,
simulation of phenotypes, selection schemes, and the statistical model for

analysis are described below.

Population

The granddaughter design, together with daughter design was proposed
for the purpose of identifying genetic markers associated with QTL (Weller,
1990). Granddaughter design requires the genotyping of sires and their sons at
polymorphic loci and the recording of phenotypes on granddaughters. There are
two generations involved in daughter design: sires and daughters. Phenotyping is
on daughters. Genotyping is on both sires and daughters (see figure 1).

One of the advantages of granddaughter design over daughter design is

that it may be easier to collect blood or semen samples from sons of sires,

27

28

 

 

 

Genotyping Sire, Sire, Sirezo
. l .

Genotyping Son] Sonj Sonloo

Phenotyping I f I

Daughter] ... Daughterso 0, 100

Figure 1 - Population Structure of Granddaughter Design

29

 

 

 

Grandfather of sires GF. GP;
1 l l l I l I
Father of sires F61 F82 F83 FS. F85 F85 F87
lllllllllllllll—lmm
Sires S1 32 33 S4 S5 36 37 Se 39 310 S11 S12 313 314 S15 S16 S17 S13 S19 320

Figure 2 - Pedigree of 20 Sires in Simulated Granddaughter Design Population

30

concentrated in Al centers, than from their daughters, scattered over many
farms. In this study, granddaughter design populations are simulated. Each
population consisted of 20 sires from 9 different grandsires, just as Grignola et al
(1996b). Each sire had 100 sons for a total of 2000 sons. The number of

granddaughters per son is set to either 50 or 100 (see figure 2).

Simulation of Genotypes

Five equally spaced marker loci on an autosomal chromosome are
simulated (see Figure 3). There are five alleles with equal frequency (20%) on
each locus. A single QTL is set to the midway between the third and fourth
marker loci. There are two alleles at the QTL. The gene frequencies of the two
alleles are set to equal (50%). The effect of one allele is set to or, and -or for the
other allele. There is no dominant effect. Therefore, the additive genetic variance

due to QTL is 2012.

Simulation of DYDs

Instead of individual yield on each daughter of son (granddaughter), the
daughter's yield deviation (DYD) of is used for analysis. The values of DYDs can
be generated from each daughters' yield or generated for each son directly. A

model with a mixture of polygenic effects and the allelic effect due to the QTL

31

linked to marker is defined for the phenotypic observations on granddaughters

below. The trait that is recorded is called “daughter yield deviation”.

N
DYDi 131—20134. v3.) + 0.5u, + a,

d i=1

where DYDi is the evaluation on son i; N. is the number of daughters per son; v}j

was the additive genetic effect of one allele at QTL linked to genetic markers of

granddaughter j of son i; vi was the additive genetic effect of the other allele at

QTL of granddaughter j of son i; ui was the additive genetic effect of the

polygene unlinked to genetic markers of son i; and a; was the random residual

effect corresponding to DYDi. The variances of v}j and v3. were both assumed
to be of , which was the additive genetic variance due to the QTL allele linked to
genetic markers. Therefore, the total additive genetic variance was of =2 03 + of .

The ui was generated from a normal distribution with zero mean and a (co)

variance matrix of I03 with of being the additive genetic variance due to

polygenic effects unlinked to genetic markers. The residual effect (8i) was

generated from normal distribution with mean of zero and variance of

fil—(OJS of + of), where of is the environmental variance of the
d

granddaughters’ yield. All covariances between vL, v3, ui , and ei were assumed

0,2

to be zero. The heritability of the yield trait was defined as h2 = 2 a 2
0a “I" 0'e

 

32

 

M1 M2 M3 M4 M5

Figure 3 - Genetic Marker Loci and One QTL on an Autosomal Chromosome.
Five marker loci are equally spaced. A single QTL is at the midway between the

third and fourth marker loci.

33

The QTL and marker alleles were generated in linkage equilibrium across
base population animals and each offspring inherited a QTL-marker haplotype
subject to chance recombination between the loci.

Five marker loci on an autosomal chromosome with equally spaced
marker intervals were simulated. A biallelic QTL with equal frequencies (.5) was
located midway between the third and fourth marker loci. This location
assignment associated higher power of test than the midway location between
the fourth and fifth. The two QTL allelic effects were set to or and -or,

respectively, i.e., gene action was assumed to be additive. Therefore, the

variance due to QTL (2 03 ) was 203, and thus the value of or was set to be the

square root of of.

The magnitude of QTL effects was denoted by the ratio of QTL allelic

variance to the total additive genetic variance:

2
0'
v2- 0;.

 

The maximum value of v2 is .5, in which case, 02 e uals to zero.
U

Selection Schemes

The selection schemes refer to alternative strategies of sampling
phenotyped individuals for genotyping. The idea behind selective genotyping is

that scoring characters is often much less expensive than scoring genetic

34

markers. Hence, those may be merit in choosing a subset of phenotyped
individuals for genotyping.

Most of the studies assumed that genotype individual are randomly
chosen from population. However, the situation is not true in commercial
populations. Usually, such populations have been selected for many generations
toward a desired breeding goal. The selection could be truncated in most cases
or stabilizing occasionally. The typical selection scheme for detection of QTL is
disruptive, in which the uppermost and lower most fractions of scored individuals
are genotyped.

The 100 sons of each sire in granddaughter design in this study are
selected on their daughter yield deviations (DYD) (Van Raden and Wiggans,
1991) according to the following alternative selection schemes:

1. Random selection: Sons are randomly chosen from the population;

2. Disruptive selection: Those sons with their DYD away from the population
mean by one standard deviation (SD) in either direction are chosen;

3. Truncation selection: Sons with DYD greater than the mean are chosen.

4. Stabilizing selection: Sons with DYD within the range of one positive and

negative SD from the mean are chosen.

Statistical Analysis

A reduced animal model including QTL effects linked to markers and

polygenic effects was used to analyze the simulated data:

35

y=XB+ZTuu+ZDTvv+e
where y was an Nx1 vector of DYD evaluations on sons with N being the number
of DYDs, which was equal to the number of sons; [5 was a vector of fixed effects;
X was the design matrix relating B to y; u was an pr1 vector of the polygenic
effects unlinked to the markers with Np being the number of sires; Z was the
incidence matrix relating elements in y to sons; Tu was a transformation matrix
relating sons to sires for polygenic effect; v was a 2pr1 vector of the QTL allelic
effects linked to the markers; D was the incidence matrix relating each animal to
its two QTL alleles; TV was the transformation matrix relating sons to sires for
QTL effect; and e was the vector of residual effects. The random effects were

assumed to follow a normal distribution with a (co)variance structure of

Variance v = 0 A 02 0

 

 

 

 

where Au was the numerator additive polygenic relationship matrix (Henderson,
1975); of was the additive polygenic variance; A, was the relationship matrix of
QTL effects linked to markers; of was the QTL allelic variance; and

R = 103+ Auof + Ava":
where I is an identity matrix of MN, Au is the correlation matrix of Mendelian

polygenic effects and Av is the correlation matrix of Mendelian QTL effects. The

theory of building Av and Av was presented by Wang et al. (1995).

36

An algorithm of REML by Grignola et al. (1996a) was adapted for the estimation
of variance components in this study. The analysis was conducted at a number

of successive positions along a chromosome. Then, the likelihood was
maximized with respect to of, of, of and r (recombination rate) at each

position. The estimated location of QTL was determined by the largest likelihood

value over the grid of possible QTL locations.

Experimental Design

The objective of this study is to investigate the effects of different selection
schemes, lengths of marker interval (r), magnitudes of QTL effects (v2),
heritability levels (hz), and number of daughters (nd) in a granddaughter design
on the estimates of QTL location and variance components of QTL, polygenic
and residual effects. A sub-population is defined by each combination of r (10 or
30 cM), v2 (.125 or .25), h2 (.05 or .4), nd (50 or 100) and selection schemes
(random, disruptive, truncation and stabilizing), for a total of 64 combinations.

For each sub-population, fifty replicates are generated and are analyzed
separately. Phenotypic- variance (of + of) is set to 10,000. Bias of estimation is

defined as the difference between true parameter value and its estimate. The
bias rate is defined as the ratio of bias to the true value of parameter. The
median bias rate from 50 replicates is used as the measurement for each sub-
population. Power on each combination is estimated by the proportion of

significant results over the total replicates.

Chapter 3

BIASES IN GENETIC PARAMETER ESTIMATION FROM A POLYGENE AND
QTL MIXED MODEL UNDER GRANDDAUGHTER DESIGN

Abstract

Statistical biases in the estimation of quantitative trait loci (QTL) location,
additive genetic variance due to markers linked to QTL, additive genetic variance
due to polygene unlinked to markers, and residual variance were evaluated in a
simulation study. Genetic populations with a granddaughter design were
simulated under a mixed model with polygenic effects and QTL effects for
various combinations of selection schemes, marker intervals, numbers of
daughters, magnitudes of QTL effects and heritability levels with respect to a
single trait. For each of the combinations, 50 replicate populations were
generated and analyzed separately by a restricted maximum likelihood algorithm.

Estimates of QTL location, variance due to segregating QTL, and
polygenic and residual effects were all unbiased in unselected populations.
Estimates of QTL location were unbiased not only in unselected populations, but
also in populations under various selection schemes. However, estimates of

variances due to QTL, polygenic and residual effects were biased significantly in

37

38

populations under nonrandom selection schemes. The magnitudes of the biases
were dependent on marker intervals, numbers of daughters, magnitudes of QTL

effect and heritability levels.

Introduction

The advantages of marker assisted selection (MAS) in breeding schemes
have been discussed rigorously recently (Lande and Thompson, 1990; Gimelfarb
and Lande, 1995; Hospital and Moreau, 1997; Spelman and Garrick, 1997).
However, when genetic and economic responses to MAS were estimated in
these studies, genetic parameters were assumed known without error.
Simulation studies (Zhang and Smith, 1993 and Wang et al., 1995) have
demonstrated that the overestimation of variance due to QTL effects would
decrease estimated long term genetic gain under MAS. For an error of only 15
cM in the location of QTL, estimated genetic superiority of MAS would be
reduced by 80% after the first generation of selection (Spelman and Van
Arendonk, 1997).

The major difference between the classical polygenic animal model and a
mixture model of marker-associated QTL effects and polygenic effects is the
additional (co)variance matrix of QTL effects linked to markers. The algorithm to
construct this matrix for a single marker was presented by Fernando and
Grossman (1989), with multiple marker extensions provided by Goddard (1991).
Wang et al. (1995) proposed an algorithm for the case of incomplete information

of the origin of marker genes, which was required in the algorithm by Fernando.

39

Variances of single marker associated QTL effects, additive polygenic
effects, and residual effects can be estimated by restricted maximum likelihood
(REML) as shown by Van Arendonk et al. (1994). Grignola et al. (1996a)
expanded their method for multiple marker associations.

The accuracy and precision of the estimates using REML have been
evaluated in many studies for the polygenic model in unselected and selected
populations (Henderson, 1975a; Banks etal., 1985; Gianola etal., 1986;
Beaumont, 1991; Sorensen and Kennedy, 1984). Grignola et al. (1996b)
evaluated the accuracy and precision of the estimates using REML in a single
marked QTL and polygenic mixed model in unselected populations.

In most livestock species, data for linkage analysis must be obtained from
existing commercial populations. Usually, such populations had been selected for
many generations toward a desired breeding goal (Vukasinovic et al., 1998). One
sampling strategy that has been advocated to increase power of marker
associated QTL effect detection, given limited resources, is to select animals with
extreme phenotypic values for genotyping. However, the frequency of favorable
alleles and genetic variances are plausibly changed under selection (Mackinnon
and Georges, 1992; Keightley and Bulfield, 1993), thereby potentially causing
large biases in estimates of QTL effects.

Granddaughter design (GDD) proposed by Weller in 1990, together with
daughter design, are two typical designs proposed for the purpose of identifying
genetic markers associated with QTL. A GDD requires the genotyping of

grandsires and their sons at polymorphic loci and the recording of phenotypes on

40

granddaughters. One practical advantage of GDD over daughter design is that it
may be easier to collect blood or semen samples from sires and sons which are
normally concentrated in Al centers than from their daughters which are normally
scattered over many commercial farms.

The objective of this study was to investigate the effects of different
selection schemes, lengths of marker interval, magnitudes of QTL effects,
heritability levels, and number of daughters in a granddaughter design on the
estimation of QTL location and variance components of QTL, polygenic and

residual effects.

Materials and Methods

Data was generated given specified underlying parameters for a marked
QTL and polygene mixed model. The estimates of the parameters were obtained
by analyzing the simulated data using an algorithm of REML. The design of the
breeding populations simulated, the genetic model for simulation, and the

statistical model for analysis are described below.

Population

Each simulated GDD population consisted of 20 sires from 9 different
grandsires as in the study by Grignola et al (1996b). Each sire had 100 sons for
a total of 2000 sons. The number of granddaughters per son was set to either 50

or 100. The 100 sons of each sire were selected based on their daughter yield

41

deviations (DYD) (Van Raden and Vlflggans, 1991) according to the following

alternative selection schemes:

1. Random selection: Sons were randomly chosen from the population;

2. Disruptive selection: Those sons with their DYD away from the population
mean by one standard deviation (SD) in either direction were chosen;

3. Truncation selection: Sons with DYD greater than the mean were chosen.

4. Stabilizing selection: Sons with DYD within the range of one positive and

negative SD from the mean were chosen.

Genetic Model

A model with a mixture of polygenic effects and the allelic effect due to the
QTL linked to marker was defined for the purpose of evaluation of the sons
based on the phenotypic observations on granddaughters. The evaluation

criterion was called daughter yield deviation, or DYD:

N
DYDi=§1—2(v:j+vfj)+0.5u,+8,

d j=l

where DYD, is the evaluation on son i; N, is the number of daughters per son; v;
was the additive genetic effect of one allele at QTL linked to genetic markers of
granddaughter j of son i; viz]. was the additive genetic effect of the other allele at
QTL of granddaughter j of son i; ui was the additive genetic effect of the
polygene unlinked to genetic markers of son i; and 81 was the random residual

effect corresponding to DVDs. The variances of vi]. and v3. were both assumed

42

to be 03 , which was the additive genetic variance due to the QTL allele linked to
genetic markers. Therefore, the total additive genetic variance was of =2 of + of .
The ui was generated from a normal distribution with zero mean and a

(co)variance matrix of la: with a: being the additive genetic variance due to

polygenic effects unlinked to genetic markers. The residual effect (8;) was

generated from normal distribution with mean of zero and variance of

El—(OJS of + of), where of is the environmental variance of the
d

granddaughters’ yield. All covariances between v3, v3, ui , and si were assumed

0,2

to be zero. The heritability of the yield trait was defined as h2 = 2 ° 2
o" + a",

 

The QTL and marker alleles were generated in linkage equilibrium across
base population animals and each offspring inherited a QTL-marker haplotype
subject to chance recombination between the loci.

Five marker loci on an autosomal chromosome with equally spaced
marker intervals were simulated. A biallelic QTL with equal frequencies (.5) was
located midway between the third and fourth marker loci. This location
assignment associated higher power of test than the midway location between

the fourth and fifth. The two QTL allelic effects were set to or and -or,
respectively, i.e., gene action was assumed to be additive. Therefore, the

variance due to QTL (2 of ) was 203, and thus the value of or was set to be the

square root of of.

43

The magnitude of QTL effects was denoted by the ratio of QTL allelic

variance to the total additive genetic variance:

2

0'
v2- 0;.

 

The maximum value of v2 is .5, in which case, 0': equals to zero.

Statistical Model

A reduced animal model including QTL effects linked to markers and
polygenic effects was used to analyze the simulated data:

y=Xl3 +ZTuu+ZDTvv+e
where y was an Nx1 vector of DYD evaluations on sons with N being the number

of DYDs, which was equal to the number of sons; B was a vector of fixed effects;
X was the design matrix relating [3 to y; u was an pr1 vector of the polygenic
effects unlinked to the markers with Np being the number of sires; 2 was the
incidence matrix relating elements in y to sons; Tu was a transformation matrix
relating sons to sires for polygenic effect; v was a 2pr1 vector of the QTL allelic
effects linked to the markers; D was the incidence matrix relating each animal to
its two QTL alleles; Tv was the transformation matrix relating sons to sires for
QTL effect; and e was the vector of residual effects. The random effects were

assumed to follow a normal distribution with a (co)variance structure of

Variance v = 0 Avg: 0
e 0 0 R

 

 

 

 

— -i L- .I

where A, was the numerator additive polygenic relationship matrix (Henderson,
1975); of was the additive polygenic variance; A" was the relationship matrix of
QTL effects linked to markers; of was the QTL allelic variance; and

R =10:2 + Ana: + Ava:
where I is an identity matrix of MN, Au is the correlation matrix of Mendelian
polygenic effects and Av is the correlation matrix of Mendelian QTL effects. The
theory of building A, and A, was presented by Wang et al. (1995).

An algorithm of REML by Grignola et al. (1996a) was adapted for the

estimation of variance components in this study. The analysis was conducted at

a number of successive positions along a chromosome. Then, the likelihood was
maximized with respect to of, of, of and r (recombination rate) at each

position. The estimated location of QTL was determined by the largest likelihood

value over the grid of possible QTL locations.

Design

A sub-population was defined by each combination of marker density (10
or 30 cM), v2 (.125 or .25), h2 (.05 or .4), number of daughters (50 or 100) and
selection schemes (random, disruptive, truncation and stabilizing), for a total of

64 combinations. For each sub-population, fifty replicates were generated and

45

were analyzed separately. Phenotypic variance (of +03) was set to 10,000.

Bias of estimation was defined as the difference between true parameter value
and its estimate. The bias rate was defined as the ratio of bias to the true value
of parameter. The median bias rate from 50 replicates was used to assess the

relative degree of bias for each parameter within each sub-population.

Results and Discussions

QTL Location

Mackinnon and Weller (1995) showed that maximum likelihood estimation
of recombination rate was inaccurate compared to estimation of other
parameters. Accuracy of estimation can be improved by using interval mapping
(Knott and Hally, 1992). Grignola et al. (1996b) showed that the estimates of QTL
location were unbiased using REML within an interval mapping framework.
Spelman and Van Arendonk (1997) showed that genetic gain by marker assisted
selection with 5 cM error was significantly less than that achieved when the QTL
position was estimated correctly. In the framework of our study, estimates of the
location of QTL were unbiased in all sub-populations, and selection schemes did
not affect the estimate of QTL location. The median biases in all sub-populations
ranged from -1.3 to 1.1 cM and were not significantly different from zero

(P>0.05).

46

Figure 4 - Estimates of Polygenic, QTL and Residual Variances

The estimates were from populations with various combinations of underlying
heritability (hz) levels, magnitudes of QTL effect (v2), numbers of daughters (Nd),
and lengths of maker interval (in cM) when undergone alternative selection
schemes. L denotes the populations with .05 for hz, .125 for v2, and 50 for Nd; H

denotes the populations with .4 for hz, .25 for v2, and 100 for Na.

47

 

 

QTL allele variance

Polygenic variance

Residual variance

Selection Scheme

+Random +Disruptive ‘0—Truncation +Stabilizing

    

 

 

5
4
3
2
1
o
-1
12345678910111213141516
12345678910111213141516
12345678910111213141516
h“ LHLHLHLHLHLHLHLH
v’ LLHHLLHHLLHHLLHH
tr,| LLLLHHHHLLLLHHHH
cM 10101010101010103030303030303030

Sub Population

 

Figure 4 - Estimates of Polygenic, QTL and Residual Variances

 

48

Random Selection Scheme

Unbiased estimates of of, of and of were obtained under the random

selection scheme. These results were consistent with those of Grignola et al.
(1996b), who also found that estimates of the variance components were

unbiased for different underlying marker intervals, number of daughters, v2 and

h2 levels under random selection. The bias rates of the estimates of 02 of and

v 9

of in each sub-population were not significantly different from zero (P>0.05).

Disruptive Selection Scheme

Disruptive selection is usually used in selective genotyping for QTL
mapping where individuals with observations of extreme values were chosen for

genotyping. Under this selection scheme, those individuals with deviant

polygenic or QTL effects, or both, were chosen. We found that estimates of of

and of tended to be biased upwards for sub-populations based on this selection
protocol (Figure 4). The magnitude of bias depended on number of daughters
per son (Nd), and the levels of underlying v2 and hz. The bias rates of of and of

decreased significantly with more daughters and higher v2 and h2 values (Table 1

and 2). An increase in the number of daughters from 50 to 100 led to a significant
reduction of bias in the estimates of a: (p<0.01), but not for of (p>0.05).

Residual variance estimates could be reduced due to a decrease in

heterozygosity of QTL. For the mating of sire (00) with dam (qq) for example,

49

Table 1 - Median* and SE of Bias Rates (%) in the Estimation of QTL Allelic
Variance. The estimates were from populations subjected to disruptive selection
where the populations differed in different levels of heritability (hz) and

proportions of additive genetic variance due to QTL effect (v2).

 

 

Marker interval (cM)

 

 

 

 

v2 hz 1o 30
No. of daughters No. of daughters
50 100 50 100
.125 .05 412' i 20 405 ”‘° 1 21 441‘ i 28 365 ”d“ s 29
.4 234""c s 20 284“” a 16 311Moe i 18 293“ i 19
.25 .05 418' i 23 396“ a. 17 423' i 20 364'”c i 20
4 234“ s 8 19' i- 7 237" i 13 198' a 11

 

*Medians with the same superscripts were not different significantly (P>0.05).

50

Table 2 - Median* and SE of Bias Rates (%) in the Estimation of Polygenic
Variance. The estimates were from populations subjected to disruptive selection
where the populations differed in different levels of heritability (hz) and

proportions of additive genetic variance due to QTL effect (v2).

 

 

 

Marker interval (cM)

 

 

 

 

v2 hz 1o 30
Number of daufighters Number of daughters
50 100 50 100
.125 .05 376' i 23 286'“ s 13 379' :t 24 302“ :t 16
.4 93° a 6 61 °° s 4 9° : 6 64° a 5
.25 .05 379° 1: 35 162° 4. 2 432' s 3 204° i 21
.4 9° :1: 6 -7° : 6 16“ i 1 -28° :1: 9

 

*Medians with the same superscripts were not different significantly (P>0.05).

51

Table 3 - Median* and SE of Bias Rates (%) in the Estimation of Residual
Variances. The estimates were from populations subjected to disruptive selection
where the populations differed in different levels of heritability (hz) and

proportions of additive genetic variance due to QTL effect (v2).

 

 

Marker interval (cM)

 

 

 

 

V2 h2 1O 30
Number of daughters Number of daughters
50 100 50 100
.125 .05 379' a 15 304° 8 9 385' i 18 308° i 9
.4 146° a 3 119°' 3 3 146° 3 3 122° 3. 3
.25 .05 402' a 14 291° i 8 402' i 13 310° : 8
.4 125° 3 3 94° : 3 135°° : 4 100° : 3

 

*Medians with the same superscripts were not different significantly (P>0.05).

52

the probability of tracing “Q” of offspring back to sire is 100% if no recombination
exists between QTL and markers. However, the probability reduces to 50% if the
genotypes of sire and dam are both “Qq”. As the proportion of homozygotes with
high or low genotypic values was increased under disruptive selection, the
certainty of knowing the identity by descent of markers in parent to an offspring
was increased. Johnson (1992) and Wright (1991) showed that estimates of

residual variance were greater with relationship coefficients set not equal to zero

than set to zero. Estimates of of in this study showed an upward bias under

disruptive selection. The bias rates of or: estimates were smaller with greater

number of daughters, v2 and h2 values (Table 3).

Truncation Selection Scheme

Individuals selected under truncation selection scheme were those with

large genetic effects. Mackinnon and Georges (1992) showed that selection may

lead to underestimation of QTL effects. This study found that estimates of of

and a: were generally biased downwards (p<0.01) in populations that underwent

truncation selection, but they were not influenced significantly by daughter

numbers, marker interval length, and v2 and h2 levels.
The situation of estimation biases in of under truncation selection is

opposite to that under disruptive selection. The increased uncertainty of tracing
QTL from parents to progeny lead to a decrease in the relationship coefficients

on QTL effects, and thus an underestimation of residual variance. This was

53
because the homozygotes with large effects became a larger portion of the
population under the truncation selection. Estimates of of were generally biased

downwards (p<0.01) in populations that undenlvent truncation selection, but also
they were not influenced significantly by daughter numbers, marker interval

length, and v2 and h2 levels.

Stabilizing Selection Scheme

The individuals with either large or small QTL or polygene effects are
excluded under stabilizing selection, in which individuals within the range of a
defined SD from the mean were chosen. Polygene and QTL variances from
populations that underwent stabilizing selection appeared to be underestimated
in this study.

The individuals favored under stabilizing selection scheme were
heterozygotes at QTL. The certainty of knowing the parent-offspring relationship
with respect to a genetic marker was reduced from matings between
heterozygous individuals. The result was similar to matings between
homozygotes with large effects under the truncation selection scheme. This may
help explain why no difference was found (p>0.05) between populations

underwent stabilizing selection and those undergoing truncation selection in

2

V I

estimates of 0' of and of.

54

Conclusions and Implications

By using the REML method, QTL locations can be estimated without
significant bias in populations under a wide array of sampling strategies.
However, there were significant biases in estimates of variances due to QTL,
polygene and residual effects from populations subjected to selection schemes
except from those subject to only random selection. Variances due to QTL,
polygene and residual effects were overestimated from populations subjected to
disruptive selection, but were underestimated from populations subjected to
truncation and stabilizing selection schemes.

The biases in variance estimates for QTL, polygene and residual effects
declined with more daughters per sire, higher levels of underlying heritability and
greater proportion of additive genetic variance that was due to QTL effects.

This study confirmed the results by Grignola et al. (1996b) that QTL
location, variances of QTL, polygenic and residual effects could be unbiasedly
estimated from populations that did not undergo selection. Furthermore,
estimates of QTL location from populations undergone disruptive, truncation or
stabilizing selection schemes were also unbiased. From populations undergone
the same selection schemes, however, biases were evident in estimates of
variances of QTL, polygenic and residual effects. Since biases in variance
components can potentially reduce the advantage of marker assisted selection,

which is the ultimate application of QTL discovery, estimation biases for

55

variances of QTL, polygenic and residual effects in various circumstances need

to be monitored.

Chapter 4

POWER OF DETECTING MARKER ASSOCIATED QTL EFFECTS
IN GRANDDAUGHTER DESIGN

Abstract

Power was examined by simulation for the test of marker linked QTL effect
using restricted maximum likelihood. Granddaughter design populations were
simulated from a mixed model with polygenic and QTL effects for each
combinations of following characteristics: selection scheme, marker interval,
number of daughters, magnitude of QTL effect and heritability of trait. Sons in the
design were selected by four alternative selection schemes of random,
disruptive, truncation and stabilized. Fifty replicates were generated for each
population and were analyzed using restricted maximum likelihood. Results
indicated that selection scheme has significant influence on the power of testing
linkage between genetic markers and QTL using restricted maximum likelihood.
Disruptive selection generated higher power than random selection, whereas
truncation and stabilizing selection had less power than random selection. Power
of test using restricted maximum likelihood also depended on number of
daughters of each son, marker interval, magnitude of QTL effect and heritability.

Power of test was higher with more daughters per sire, smaller marker interval,

56

57

larger magnitude of QTL effect, and higher heritability. The magnitude of the
difference due to changing a factor was larger when the power was less

saturated by other factors.

Introduction

Several studies have shown that individual loci affecting quantitative traits
can be detected via linkage to genetic markers (Soller and Beckman, 1990;
Weller and Wyler, 1992). Notable examples were the QTL gene for ovulation rate
in swine linked to a marker on chromosome 8 with an additive effect of 3 ova
(Rathje et al., 1997), the QTL gene of fat percentage linked to markers clustered
on chromosome 4 of pig (Andersson et al., 1994) and the QTL with a significant
effect on protein yield in dairy cattle linked to beta-lactoglobulin (Bovenhuis and
Weller, 1994)

Identification of quantitative trait loci involves many animals to be
genotyped and performance tested. Consequently, experimental designs need to
be optimized to minimize the costs of data collection and genotyping for an
appropriate power (van der Beek et al., 1995).

Most successful QTL mapping efforts described to date have exploited F2
or backcrosses obtained from parental populations divergent for the traits of
interest (Soller et al., 1976; Paterson et al., 1988). This method results in more
power by introducing linkage disequilibrium and can be analyzed by standard
software. Linkage analysis for outcross data structure is more complicated and

more complex designs and analyses are needed.

58

ANOVA is the traditional method, which was performed by comparing
marker genotype effects (Weller et al., 1990a; Mackinnon and Georges, 1992).
The shortage of ANOVA is that no information on QTL location can be provided.
Weller (1986) developed a maximum likelihood method to detect marker
associated QTL effect. Location of QTL was estimated by maximization of
likelihood with respect to recombination rate and other parameters (e.g., mean,
additive QTL effect, and within QTL genotype residual variance). Haley and
Knott (1992) and Martinez and Curnow (1992) independently introduced a
regression method. Regression is performed on the probability of an individual
having a QTL genotype, given the genotype for the flanking markers. The QTL
location is estimated by the recombination rate with minimum residual sum
square. Generally, regression generated similar result as maximum likelihood.

Method of ANOVA, regression and maximum likelihood method were
developed mainly for linecross populations. They can not fully account for the
more complex data structures in outcross populations, such as data on several
families with relationships across families, unknown linkage phases in parents,
unknown number of QTL alleles in the population.

For polygenic effects without marker information, BLUP had proved to be
a very flexible method. It can handle data with many nongenetic effects, eg.
season, with arbitrary pedigree structure, and with nonrandom mating. Several
contributions have been made, which lead to the method of using BLUP for the

identification of linkage between markers and QTL.

59

Fernando and Grossman (1989) showed that information on a single
marker can be used in BLUP by fitting the additive effect for alleles at QTL linked
to genetic markers and additive polygenic effects for alleles at the remaining
quantitative trait loci. Goddard (1991) extended the method to include
information from more than one marker. Wang et al. (1995) presented an
algorithm for building the relationship matrix with the genetic markers that the
information on the origin of the markers is not required. Meuwissen and Goddard
(1996) presented a method in which the covariance matrix of effects at the
marked QTL was approximated. This approximation reduces the computational
requirements.

Restricted maximum likelihood method based on BLUP was presented by
Van Arendonk et al. (1994) for a single marker. Grignola et al., 1996a, 1996b)
extended this method to multiple markers. Identification of markers associated
QTL effect involves maximizing and comparing the likelihood of the data under
different genetic models to ascertain the most likely genetic structure. The
restricted maximum likelihood of error contrasts under a polygenic model were
compared with that under the combined model containing a major gene and
polygenic component. A significant improvement in the likelihood obtained by
incorporating a major gene in the model provides evidence for a linkage between
QTL and genetic markers.

Different statistical methods generated different power of test. The power
of test also depends on other factors (e.g. QTL effect, marker interval, sample

size etc.). The power of test has been examined for the statistical method of

60

ANOVA (Weller et al., 1990b), maximum likelihood (Knott and Haley, 1992b;
Le-Roy and Elsen, 1995; Carbonell et al., 1993; Jensen, 1989; Lander and
Botstein, 1989; Knapp and Bridges, 1990; Elsen et al., 1997) and regression
(Moreno-Gonzalez, 1992; Jansen, 1994b; Hyne and Kearsey, 1995; Rebai et al.,
1995). Knowledge of power using restricted maximum likelihood remains
unknown.

The objective of this study was to examine the power of test using REML
in circumstances of combination of different selection scheme, marker interval,
magnitude of QTL effect, heritability and number of daughters in granddaughter

designs.

Materials and Methods

Data was generated given specified underlying parameters for a marked
QTL and polygene mixed model. The estimates of the parameters were obtained
by analyzing the simulated data using an algorithm of REML. The design of the
breeding populations simulated, the genetic model for simulation, and the

statistical model for analysis are described below.

Population

The granddaughter design (GDD) was proposed for the purpose of
identifying genetic markers associated with QTL (Weller, 1990). A GDD requires
the genotyping of sires and their sons at polymorphic loci and the recording of

phenotypes on granddaughters.

61

In this study, the base population originated from 9 unrelated grandsires
that produced 20 sires, just as Grignola et al (1996b). Each sire had100 sons for
a total of 2000 sons. Each son produced 50 or 100 daughters (granddaughters of
sires). The dams that produced the 20 sires, 2000 sons and the daughters were
assumed to be unrelated. The 100 sons of each sire were selected based on
their daughter yield deviations (DYD) (Van Raden and Wiggans, 1991)
according to the following alternative selection schemes:

1. Random selection: Sons were randomly chosen from the population;

2. Disruptive selection: Those sons with their DYD away from the population
mean by one standard deviation (SD) in either direction were chosen;

3. Truncation selection: Sons with DYD greater than the mean were chosen.

4. Stabilizing selection: Sons with DYD within the range of one positive and

negative SD from the mean were chosen.

Genetic Model

A model with a mixture of polygenic effects and the allelic effect due to the
QTL linked to marker was defined for the purpose of evaluation of the sons
based on the phenotypic observations on granddaughters. The evaluation

criterion was called daughter yield deviation, or DYD:

N
DYD, =§1—§:(v' + v3.) + 0.5u, + a,

ii
if i=1

where DYD; is the evaluation on son i; Nd is the number of daughters per son; v}j

was the additive genetic effect of one allele at QTL linked to genetic markers of

62
granddaughter j of son i; v3. was the additive genetic effect of the other allele at

QTL of granddaughter j of son i; ui was the additive genetic effect of the

polygene unlinked to genetic markers of son i. The variances of v; and v; were

both assumed to be 03 , which was the additive genetic variance due to the QTL
allele linked to genetic markers. Therefore, the total additive genetic variance

was of =2 03 + 0': . The ui was generated from a normal distribution with zero

mean and a (co)variance matrix of la: with of being the additive genetic

variance due to polygenic effects unlinked to genetic markers; and e; was the

residual effect corresponding to DYD;., which was inducted by Mendelian random
effect, unknown additive genetic effect of dam and random environmental effect.

The residual effect was generated from normal distribution with mean of zero and

variance of bI—(OJS of + of), where of is the environmental variance of the
d

granddaughters’ yield. All covariances between v3, v3, ui , and 8; were assumed

0,2

to be zero. The heritability of the yield trait was defined as h2 = —2—°——2—.
o" + 0.

The QTL and marker alleles were generated in linkage equilibrium across
base population animals and each offspring inherited a QTL-marker haplotype
subject to chance recombination between the loci.

Five marker loci on an autosomal chromosome with equally spaced
marker intervals were simulated. A biallelic QTL with equal frequencies (.5) was

located midway between the third and fourth marker loci. This location

63

assignment associated higher power of test than the midway location between

the fourth and fifth. The two QTL allelic effects were set to or and -or,
respectively, i.e., gene action was assumed to be additive. Therefore, the

variance due to QTL (2 of ) was 2012, and thus the value of a was set to be the

square root of of.

The magnitude of QTL effects was denoted by the ratio of QTL allelic

variance to the total additive genetic variance:

2
v2= “v

0,2

 

The maximum value of v2 is .5, in which case, of equals to zero.

Statistical Model

A reduced animal model including QTL effects linked to markers and
polygenic effects was used to analyze the simulated data:

y =XB +ZTuu+ZDTvv+e
where y was an Nx1 vector of DYD evaluations on sons with N being the number
of DYDs, which was equal to the number of sons; [3 was a vector of fixed effects;
X was the design matrix relating B to y; u was an pr1 vector of the polygenic
effects unlinked to the markers with Np being the number of sires; Z was the
incidence matrix relating elements in y to sons; Tu was a transformation matrix
relating sons to sires for polygenic effect; v was a 2pr1 vector of the QTL allelic

effects linked to the markers; D was the incidence matrix relating each animal to

64

its two QTL alleles; Tv was the transformation matrix relating sons to sires for
QTL effect; and e was the vector of residual effects. The random effects were

assumed to follow a normal distribution with a (co)variance structure of

p c-

u “A 8° 0 0
Variancev = 0 A02 0

ej o 0 RJ

 

 

 

 

where Au was the numerator additive polygenic relationship matrix (Henderson,
1975); of was the additive polygenic variance; A" was the relationship matrix of
QTL effects linked to markers; of was the QTL allelic variance; and

R = 103+ Audi + AV0'3
where I is an identity matrix of MN, Au is the correlation matrix of Mendelian
polygenic effects and Av is the correlation matrix of Mendelian QTL effects. The
theory of building A, and A" was presented by Wang et al. (1995).

An algorithm of REML by Grignola et al. (1996a) was adapted for the

estimation of variance components in this study. The analysis was conducted at

a number of successive positions along a chromosome. Then, the likelihood was
maximized with respect to of, of, of and r (recombination rate) at each

position. The estimated location of QTL was determined by the largest likelihood

value over the grid of possible QTL locations.

65

Design

A sub-population was defined by each combination of marker density (10
or 30 cM), v2 (.125 or .25), h2 (.05 or .4), number of daughters (50 or 100) and
selection schemes (random, disruptive, truncation and stabilizing), for a total of

64 combinations. For each sub-population, fifty replicates were generated and
were analyzed separately. Phenotypic variance (0': + of) was set to 10,000.

Power on each combination was estimated by the proportion of significant results

over 50 replicates.

Results and Discussions

Profile of Test Statistics

Twice the log likelihood ratio was used as statistics for test of linkage
between QTL and genetic markers. This statistic has a Chi-square distribution
with one degree of freedom. There are 50 replicates for each sub population.
Figure 5 shows the profile of test statistics in the sub population with truncation
selection scheme, 30 cM marker interval, 50 daughters, v2=0.125 and h2=0.05.
The significant threshold for test statistics is 3.841 from Chi-square distribution
with one degree of freedom. There were 17 replicates with test statistics beyond

the threshold. Power was estimated at 0.34.

66

 

 

 

15
.3 o
3810- ' ° '
:8 - .
gé 5.. o . . . . 0 o o
l-o Q C G
5 oo. oo ... g 00' .0
-' o o. o
0 ' . .:."—r°-—r-°'.l——r—°1——l‘—°1—¢. ' '

 

O 5101520253035404550

Replicates

Figure 5 - Profiles of Test Statistics. The statistics were the twice log likelihood
ratio from 50 replicates in sub-population with truncation selection scheme, 30
centi Morgan marker interval, 50 daughters, v2=0.125 and h2=0.05. Each point

represents an observation of the test statistic.

67

random selection, whereas truncation and stabilizing selection have less power
than random selection.

Power was saturated in the sub-populations under both the disruptive and
random selection schemes. They reached the maximum except for those
populations with larger marker interval (30 cM), fewer daughters (50), lower QTL
allele variance ratio (0.125) and lower heritability (0.05) with random selection.

Disruptive selection increased the proportion of homozygote individuals
with either large or with small effect on QTL. QTL variance was increased. The
certainty of knowing the identity of markers in offspring to a parent increased.
Therefore disruptive selection had more power of testing the QTL effect linked to
markers. Significant increases in power for disruptive selection compared to
random selection was founded in the situation where power was not saturated in
random selection.

Mackinnon and Georges (1992) found that selection by truncation for the
trait of interest significantly reduced the difference between marker genotype
means and thus reduced the power to detect linked quantitative trait loci by
ANOVA method. Preferred individuals under truncation selection were the
homozygote with large QTL effects. The proportions of homozygote with small

effects and heterozygotes were decreased. This lead to the reduction of QTL

68

Table 4 - Power* from Populations Subjected to Alternative Selection Schemes

with Different Levels of Heritability (hz) and Magnitudes of QTL Effect (v2).

 

 

 

 

Marker Number of Selection Magnitude Of QTL Effect (v2)
Interval Daughters Scheme 0125 0-25
(CM) h2 h2

0.05 0.4 0.05 0.04

10 Random 100° 100° 100° 100°

Disruptive 1.00a 1 .00a 1 .00“ 1.00‘

Truncation 0.56ij 0.98°° 0.94°°°°' 0.98°°

Stabilizing 0.62hi 0.96°°° 0.92°°°° 100°

Random 100° 100° 100° 100°

Disruptive 1 .00“ 1 .00a 1.00a 1.00a

Truncation 0.76’9" 100° 100° 0.98°°

Stabilizing 080°“g 0.98°° 100° 100°

30 Random 0.84def 100° 100° 100°

Disruptive 100° 100° 100° 100°

Truncation 0.34k 0.76fgh 0.66ghi 0.98°°

Stabilizing 0.40ik 0.70'9hi 0.70““hi 100°

Random 098°b 0.98°°b 100° 100°

Disruptive 100° 100° 100° 100°

Truncation 0.58hij 0.90°°°|° 080°”9 100°

Stabilizing 0.56ij 0.88°°°° 0.92°°°° 100°

 

*Powers with the same superscripts were not different significantly (P>0.05).

69

 

 

 

 

h

o

% —°—Random

0' +Truncation
0.4‘
0.2 IIITIVIUIIIIIIII

12345678910111213141516
h’LaLaLaLaLaLnLaLa
VZLLHHLLHHLLHHLLHH
NdLLLLHHHHLLLLHHHH
on 10 10 10 10 10 10 10 10 30 30 30 30 30 30 30 30
Sub Population

Figure 6 - Comparison of Powers between Random and Truncation Selection
Schemes. The comparison are across populations with various combinations of
underlying heritability (11°) levels, magnitudes of QTL effect (v2), numbers of
daughters (Nd), and lengths of maker interval (in cM). L denotes the populations

with .05 for hz, .125 for v2, and 50 for Nd; H denotes the populations with .4 for h2,
.25 for v2, and 100 for Na.

70

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8 “'1—-
'- 0.6 L.#— ——« - e F — ._
g a b b b b
o . 4H --- e. .— _——_.
a 04
0.2
0
h’ 0.05 0.4 0.05 0.05 0.05
v2 0.125 0.125 0.25 0.125 0.125
N, 50 50 50 100 50
6M 30 30 30 30 10
Sub Population

Figure 7 - Marginal Effects on Power of Test with Random Selection. The effects
are from heritability (hz), magnitude of QTL effect (v2), number of daughters (Nd)
and marker interval (cM). The bars with different symbols (a and b) are different

at 0.05 level. Error bars indicate 95% confidence intervals.

71

 

0.75

 

0.5

 

Power

\E/ Number of Daughters -o—50 +100
0.25

 

 

 

 

o I T I T I I

v2 0.125 0.125 0.25 0.25 0.125 0.125 0.25 0.25
h2 0.05 0.05 0.05 0.05 0.4 - 0.4 0.4 0.4
CM 10 30 10 30 10 30 10 30

Sub Population

Figure 8 - Effect of Number of Daughters on Power of Test Under Truncation

Selection Scheme.

72

 

 

  

 

Power

Heritability -0—0.05 +0.4
0.25-————n—— A -.

 

 

 

 

0 I I fir I I I

v2 0.125 0.125 0.125 0.125 0.25 0.25 0.25 0.25
Nd 50 50 1 00 1 00 50 50 100 1 00
CM 10 30 10 30 1O 30 10 30

Sub Population

Figure 9 - Effect of Heritability on Power of Test Under Truncation Selection

Scheme.

73

 

 

 

 

Power
(3
01
l
l

Marker Interval -0— 30 +10

 

 

 

 

0 I I I T I I I

v2 0.125 0.125 0.25 0.25 0.125 0.125 0.25 0.25
N, 50 100 50 100 50 100 50 100
112 0.05 0.05 0.05 0.05 0.4 0.4 0.4 0.4

Sub Population

Figure 10 - Effects of Marker Interval on Power of Test under Truncation

Selection Scheme.

74

 

 

0.75 e. ; _- _-._ 7— _

 

 

Power

0.25 -~—— —-—

0 5 _ D.
VMagnitude of QTL Effect +0425 +025

 

 

 

0 I I I I I

 

 

h2 0.05 0.05 0.05 0.05 0.4 0.4
N, 50 50 100 100 50 50
0M 10 30 10 30 10 30

Sub Population

5

0.4 0.4
100 100
10 30

Figure 11- Effects of Magnitude of QTL Effect on Power of Test Under Truncation

Selection Scheme.

75

Selection Schemes

Selection schemes have great impact on power of test of linkage between
QTL and genetic markers. Disruptive selection generated higher power than
variance. However, the certainty of knowing that a genetic marker in offspring
was identical to a parent was decreased. Therefore truncation selection leads to
reduction of power. Significant reductions of truncation selection to random
selection were found in most sub-populations in this study (Figure 6).

The behavior of stabilizing selection was similar to truncation selection,
whereas the reason for the reduction of power by stabilizing selection was
different from truncation. Heterozygotes were the favorable individual under
stabilizing selection. The proportions were reduced for the homozygotes with
large or small genotype effects. Stabilizing selection reduced QTL variance. No
difference between truncation and stabilizing selection was found in this study

(p>0.05).
Other Factors

Power in sub-populations under disruptive selection was saturated.
Exploring the impact of number of daughters, marker interval, QTL effect and
heritability was based on other selection schemes. Further, the analysis focused
on random and truncation selection schemes since no difference was found
between truncation and stabilizing selection.

The result of this study showed the power of test depended on number of

daughters, marker interval, magnitude of QTL effect and heritability. Power was

76

increased with more daughters, closer marker interval, larger QTL effect and
higher heritability.

Under random selection, the power was 0.84 with h°=0.05, v2=0.125, 50
daughters and 30 cM marker interval. Significant increases of power (p<0.05)
were found by changing the level of anyone of these factors; higher hz, higher v2,
more daughters, or closer marker interval (Figure 7).

Under truncation selection, higher heritability, larger magnitude of QTL
effect, more daughters on shorter marker interval generally generated higher
power (Figure 8-11). Reverse Situations were found in two sub populations °
(Figure 9 and Figure 11). However, the differences reversed were not significant
(p>0.05, see Table 4).

The difference of power between two levels of a factor was larger when
power was less saturated by other factors.

The results indicated that when v2 was 0.125, a 10 cM marker interval
increased power by 65% when compared to a 30 cM marker interval with 50
daughters and heritability of 0.05 under truncation selection. Whereas, when v2
was 0.25, a 10 cM marker interval increased power by 42% when compared to a
30 cM marker interval with the same situation.

When marker interval was 30 cM, a v2=0.125 increased power by 94%
when compared to when v2 was 0.25 and heritability was 0.05 with 50 daughters
under truncation selection. Whereas, when marker interval was 10 cM, a
v2=0.125 increased power by 68% when compared to a v2 of 0.25 with same

sfluafion.

77

Conclusions and Implications

Selection scheme has significant influence on the power of testing linkage
between genetic marker and QTL using restricted maximum likelihood. Disruptive
selection generated higher power than random selection, whereas truncation and
stabilizing selection had less power than random selection.

Power of test using restricted maximum likelihood also depended on
number of daughters per sire, marker interval, magnitude of QTL effect and
heritability. Power of test was higher with more daughters per sire, smaller
marker interval, larger magnitude of QTL effect, and higher heritability.

The magnitude of the difference due to changing a factor was larger when

the power was less saturated by other factors.

GENERAL CONCLUSION

This study confirmed that estimates of QTL location, polygene variance,
QTL allele variance and residual variance were unbiased in unselected
populations. The unbiaseness was released to the number of daughters, marker
densities, magnitude of QTL effect and heritability of trait.

Estimates of QTL location were unbiased even in selected populations.
However, estimates of QTL variance, polygenic variance and residual variance
were biased in all selection schemes except random selection.

Variances of polygenic, QTL and residual effects were overestimated with
disruptive selection and underestimated with truncation and stabilizing. The
biases with disruptive selection reduced with more daughters, larger magnitude
of QTL effect and higher heritability of traits.

The selection schemes has significant influence on the power of testing
linkage between genetic markers and QTL using residual maximum likelihood.
Disruptive selection generated higher power than random selection, whereas
truncation and stabilizing selection had less power than random selection.

Power of test increased with more daughters of each sire, smaller marker
intervals, larger magnitude of QTL effect and higher heritability.

The magnitude of the difference due to changing a factor was larger when

power was less saturated by other factors.

78

APPENDICES

APPENDIX A

APPENDIX A
Expectation of Heritability Weighted DYD

Let Y1. yn be the n daughters' yields of a son and p be the population

mean, then daughters' yield deviation (DYD) can be expressed as following:

DYD f [(yt-u)+ + [(Yn-lull

l (0.55 + 0.5d1 + m1 + e1 + + 0.5s + 0.5dn + m..+ en)

’1

=0.5s + l (0.5d1 + + 0.5dn + m1 + + rnn + e1+ + en)
’1

where s is additive genetic effect of sire of daughter, d, is additive genetic effect
of dam of daughter l, m is Mendlian random effect of daughter i, e, is

environmental effects.

Var<DYD>= 0.3+ 1 (03+ 0.5a:+a.°)
n
=0.2so§+ l (0.2soj+ 0.5o§+of)
n

=0.2soj+ l (0.7585+of)
n

The variance of DYD was decomposed into two parts. The fist part

(0.25 of) is the variance of additive genetic of son. The other part is variance of

residual effect. Therefore, the heritability of DYD is the proportion of .25 a": over

IVar(DYD) if DYD is analyzed with weight of one.

79

80

2

2
= 0.750ll +0:c 0.250:

0.25no:

 

l (0.75o§+of)
n

Let

_ 0.750: + of
0.25mi:

 

If DYDs are analyzed in REML with weight of i, the expectation of residual
W

effect on DYD is 0.25 02. Therefore, the expectation of herityability of DYD is 0.5.

Here, w can be proved as:

 

w: 1 — Re liability

 

 

Reliability
where Reliability = n
n + K
_ 2
K = 4 hzh

The derivation is following:

Under sire model, the heritability is four times of the sire variance over the
total variance. The sire variance equal to 25% of additive genetic variance.
The total variance is sum of sire variance and residual variance. The
residual variance equal to 75% of additive genetic variance plus

environmental variance. Therefore,

2_ 402

S

- of + (0.750: + oi)

81

 

 

 

 

 

k = h2
0.756: + of
Re liability =
n + K
._ 63
2 0.750”: +6:
+
n

_ 1 — Re liability
Re liability

 

_ 0.756: +0:

2
nos

 

 

_ 0.750: +0:
0.25no:

Hence, there is an option of treating heritability of DYD as known (0.5) in REML

analysis when phenotypic observation is DYD.

APPENDIX B

APPENDIX 8

Input and Output of Grignola’s Program

Input

1. Pedigree of Sires and sons

2. Genotypes of sires and sons

3. DYDs of sons

4. Heritability of yield
Output:

1. Estimate of QTL location

2. Estimate of heritability of DYD

3. v2

4. Residual variance of DYD (dime)

Estimates of polygenic, QTL and residual variance on yield can be directly

calculated from the output of Grignola’s program.
Let h° be the heritability of yield and him, be the estimate of heritability of
DYD, then variance of polygenic(a§ ). QTL (63) allele and residual effect (oi) on

yield are calculated as following:

2 2
2 __ 4hDYDGDYDc

° 1—h§,,,,,
0,2:vzoz

, (1-h2)6§
e=‘——h§"__

82

 

 

APPENDIX C

APPENDIX C

Flow Chart of Pedigree
Function: Generating pedigree of sons from pedigree of sires.
Input: Pedigree of sire and ancestors of sires
Output: Pedigree of sons

 

Input pedigree of sire and ancestors
of sires: indiv(I), sire(I), I=1 to 29

 

l

I=10

 

 

 

 

Yes

 

 

 

 

 

/ Output pedigree /

 

 

 

 

 

 

 

   

J =Number

 

I=I+ l

 

 

of sons?

   

 

 

indiv(29+(I- 10)* 100+J)=29+(I-10)* 100+]
sire(29+(I- 10)* 100+J)=I

 

 

l

J=J+1

 

 

 

 

 

 

 

83

APPENDIX D

APPENDIX D

Flow Chart of Subroutine of crossover

Function: Generating a gamete from a individual.
Input: Number of gene loci, recombination rates, genotype of individual
Output: Gene type of gamete

 

Input number of loci recombination
rate genotype of individual

I

Generate random value x=1 with probability
of recombination rate and x=0 with probability
of l-recombination rate

 

 

 

 

 

No Yes

 

 

 

 

 

No genes exchange Exchange genes

 

 

 

 

 

I |

 

Generate random value y=1 with probability
of 0.5 and y=O with probability of 0.5

 

 

 

Yes

 

  

v

Chose gene set 1

 

 

Chose gene set 2

 

 

 

 

 

 

 

 

 

 

/ Output gamete gene type /

84

APPENDIX E

APPENDIX E

Flow Chart of subroutine of randasign

Function: Generating a gamete from a probability distribution.
Input: Allele probability
Output: Gene type of gamete

 

Input number of alleles (n) and
probability of alleles:

p(I), I=1 to n

I

Generate random value x=I with probability of p(I)

/ Output gamete gene type /

 

 

 

 

 

85

APPENDIX F

APPENDIX F

Flow Chart of subroutine of Phenotype

Function: Generating DYD
Input: Pedigree, genotype, and variance component

Output: DYD

Input pedigree, genotype, and
variance component

I

 

 

 

 

I=1

 

 

 

 

 

 

   

I=I+1 I>number of sons?

 

 
   

 

Generating polygenic effect and

 

 

residual effect for son I

 

 

     
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Average QTL effect I>number of
for son I granddaughters‘7
i No
Sum Averaged QTL Accumulate QTL effect
effect, polygenic and I
residual effect of son I J: 1+1

 

 

 

 

 

86

APPENDIX G

APPENDIX G

FORTRAN (90) Code of Simulation Program

c*********************************************************************

00000

Simulation program
Writen by Zhiwu Zhang
Last update: April 12, 1998

ci********************************************************************

PROGRAM simu

USE MSIMSL
C .................
c Declare variables
C ..................

common intvl,markfr,qtlfr,rintvl,rmarkfr,rqtlfr
PARAMETER (nmkloci=5,

* $ * i * t i t i *

INTEGER seed,

* t * t i i t * 4 *

nmkal=5,

nloci=nmkloci+l,

nqtla1=5,

ndaughter=1,

qtlposi=4,

ngsire=100,

nsire=10,

nson=1,

nparent=ngsire+ngsire*nsire, lsires+ancestors
nfparent=ngsire*nsire) lsires

idparent(nloci,2),

idgam(nloci),
indiv(nparent+nfparent*nson),
sire(nparent+nfparent*nson),
idmale(nparent+nfparent*nson,nloci),
idfemale(nparent+nfparent*nson,nloci),
irmk(nmkloci),

irqt1(1),

ntemp(nmkloci),

nqtlaraylnqtlal)

REAL recomb(nloci-l),

* i i i i *

probmk(nmkal),
probqtl(nqtlal),
dyd,

qtlefect,
eqt1a1(nqtlal),
weight,

87

88

nrmdev(1),

varu,

varv,

vare,
apoly(nparent+nfparent*nson),
epoly(nparent+nfparent*nson),
temp,

stdp,alpha,h2,dydh2,v2

*fi-II-fl'I-I-fifl-

seed=0

stdp=100

h2=0.4

v2=0.125

data recomb/.3,.3,.15,.15,.3/
data probmk/.2,.2,.2,.2,.2/
data probqtl/.50,.50,.0,.0,.0/
sampling=1.0

vara=h2*stdp*stdp

varv=v2*vara
alpha=sqrtlvarv/(probqtl(l)*probqtl(2)))
eqtlal=0

eqtlal(1)=0.5*alpha

eqtla1(2)=-.5*alpha

varu=vara-2*varv

vare=(1-h2)*stdp*stdp
dydh2=vara/(vara+(vare+0.75*vara)/ndaughter)

weight is function of h2 and number of daughter
temp=(4-h2)/h2

temp=ndaughter/(ndaughter+temp)

weight=temp/(1-temp)
selection=sampling*sqrt(.25*vara+(vare+0.75*vara)/ndaughter)

open(10,file='mremlpq',status='unknown') l parameter file of
mremlpq

open(11,file='pedmkqtl.dat',status='unknown') 1 output of
marker and atl

open(12,file='dydzw.dat',status='unknown') 1 output of dyd
open(9,fi1e='simu1ate.par',status='unknown') l parameter file
write(9,*) 'Number of marker loci ',nmkloci

write(9,*) 'Number of marker alleles ',nmkal

write(9,*) 'Number of QTL allels ',nqt1al

write(9,*) 'NUmber of daughters ',ndaughter

write(9,*) 'Number of sons per sire ',nson
write(9,*) 'Order of QTL among markers',qt1posi

99991

1020
1030
1040

89

write(9,*) 'Seed for random number
write(9,*) "

write(9,*) '

var(e)'

write(9,99991) varv,varu,vare
temp=0

do i=1,qt1posi-2
temp=temp+recomb(i)

end do
temp=temp+recomb(qtlposi-l)
write(9,*) '

DQTL'

write(9,99991) v2,h2,dydh2,temp

write(9,*) "

write(9,*) 'Recobination rate'
write(9,99991) recomb
write(9,*) 'QTL allele Frequ'
write(9,9999ll probqtl
write(9,*) 'QTL effect '
write(9,99991) eqtlal
write(9,*) 'Marker Frequency'
write(9,99991) probmk
write(9,*) "

write(9,*) 'Weight

close (9)
FORMAT (10X,7F10.2)

write(10,1020) .125, .50, 1000.0,
write(10,1020) .499, .99, 9000.0,
write(10,1020) .001, .01, 0.1,
write(10,1040) 0
write(10,1020) 0
write(10,1020) .0, .3, .6, .9, 1.2
write(10,1040) 5,5,5,5,5
do i=1,nparent

do j=1,5

',seed

var(v) var(u)

v2 h2 dydh2

',weight

write(10,1030) i,j,0.2,0.2,0.2,0.2,0.2

end do
end do
write(10,1020) 0.01, 1.19
write(10,1020) 0.01,0.01
FORMAT (1X,5F10.3)
FORMAT (1X,2I4,5F5.2)
FORMAT (1X,SI4)

end of output paramaters

90

write(*,*) 'It is generating data, please wait.’

Generating pedigree
by Zhiwu Zhang, August 30, 1997

00000

pedigree of grandsire
do i=1,ngsire
indiv(i)=i

sire(i)=0

end do

c pedigree of sire
do i=1,ngsire
do j=1,nsire
indiv(ngsire+(i—1)*nsire+j)=ngsire+(i-1)*nsire+j
sire(ngsire+(i-1)*nsire+j)=i
end do
end do

c pedigree of son
do i=1,ngsire*nsire
do j=1,nson
indiv(ngsire+ngsire*nsire+(i-l)*nson+j)=
* ngsire+ngsire*nsire+(i—l)*nson+j
sire(ngsire+ngsire*nsire+(i-l)*nson+j)=
* ngsire+i
end do
end do

c seed 0 for ramdom, non zero for fixed initial
CALL RNSET (SEED)

99998 FORMAT (2I6,6I3)

do i=1,nparent+nfparent*nson

C .....................................................
C generating marker and qtl information

o by Zhiwu Zhang, August 20, 1997

C _____________________________________________________
777 if (sire(i).eq.0) then

c unknown sire

c male side

call ransign(nmkal,probmk,nmkloci,irmk)
do j=1,qtlposi-1
idmale(i,j)=irmk(j)

91

end do

do j=qtlposi+1,nloci
idmale(i,j)=irmk(j-1)

end do

call ransign(nqtlal,probqtl,1,irqtl)
idmale(i,qtlposi)=irqt1(1)

c femal side
call ransign(nmkal,probmk,nmkloci,irmk)
do j=1,qtlposi-1
idfemale(i,j)=irmk(j)
end do
do j=qtlposi+1,nloci
idfemale(i,j)=irmk(j-1)
end do
call ransign(nqtla1,probqt1,1,irqtl)
idfemale(i,qtlposi)=irqtl(1)

c known sire

else

c male side
do j=1,nloci
idparent(j,1)=idma1e(sire(i),j)
idparent(j,2)=idfemale(sire(i),j)
end do
call crosover(nloci,recomb,idparent,idgam)
do j=1,nloci
idmale(i,j)=idgam(j)
end do

c femal side
call ransign(nmkal,probmk,nmkloci,irmk)
do j=1,qtlposi-1
idfemale(i,j)=irmk(j)
end do
do j=qtlposi+1,nloci
idfemale(i,j)=irmk(j-1)
end do
call ransign(nqtlal,probqtl,1,irqtl)
idfema1e(i,qtlposi)=irqtl(1)

end if
C .....................................................
c polygene effect
c by Zhiwu Zhang, September 10, 1997
C .....................................................
CALL RNNOR ( 1, nrmdev)
if (sire(i).eq.0) then
c unknown sire
apoly(i)=sqrt(varu)*nrmdev(1)
else
apoly(i)=sqrt(0.75*varu)*nrmdev(1)
* +0.5*apoly(sire(i))

end if

0000

if (i .gt. nparent) then
CALL RNNOR (l, nrmdev)
epoly(i)=sqrt((0.75*varu+vare)/ndaughter)*nrmdev(1)

QTL effect
by Zhiwu Zhang, September 9, 1997

nqtlaray=0
preparation for male side
do k=1,nloci
idparent(k,1)=idmale(i,k)
idparent(k,2)=idfema1e(i,k)
end do

do j=1,ndaughter
female side
call ransign(nqt1a1,probqtl,1,irqtl)
nqtlaray(irqt1(1))=nqtlaray(irqtl(l))+1
male side
call crosover(nloci,recomb,idparent,idgam)
nqtlaray(idgam(qtlposi))=nqtlaray(idgam(qtlposi))+1
end do
qtlefect=0
do j=1,nqtlal
qtlefect=qtlefect+nqtlaray(j)*eqtlal(j)
end do
qtlefect=qtlefect/ndaughter

dyd=qtlefect+0.5*apoly(i)+epoly(i)

1 Random sampling

keep nothing

2 Extrem sampling

if (abs(dyd).1t. selection) goto 777
3 over average sampling

if (dyd .lt. 0.0) goto 777

4 middle sampling

if (abs(dyd).gt. selection) goto 777
5 high selection sampling

if (dyd .lt. selection) goto 777

output DYD information
write(12,99997) indiv(i),sire(i),1,dyd,weight

end if

93

do j=1,qtlposi-1
ntemp(j)=idma1e(i,j)*10+idfemale(i,j)
end do

do j=qtlposi+1,nloci
ntemp(j-1)=idmale(i,j)*10+idfemale(i,j)
end do
C output pedigree and marker information
write(11,99998) indiv(i),sire(i),0,
* (ntemp(j),j=l,nmkloci)

end do 1 do i=1,nparent+nfparent*nson

99997 FORMAT (1I6,116,1I2,2F20.10)

end
C .....................................................
c subroutine for cross over
c by Zhiwu Zhang, August 28, 1997
C .....................................................
subroutine crosover(nloci,recomb,idparent,idgam)
c nloci number of marker or qtl loci
c recomb recombination rate
c idparent Identification of alleles on parent
c idgam Identification of alleles on gamate
c for checking
common intvl,rintvl
integer intv1(5),
* rintvl(5)
c end of checking
PARAMETER (nr=1)
INTEGER ir(nr),idparent(nloci,2),
* idgam(nloci),temp
REAL recomb(nloci-l),prob(2)
c cross or not

do i=1,nloci-1
prob(1)=recomb(i)
prob(2)=1-recomb(i)

call ransign(2,prob,nr,ir)
call ransign(2,prob,nr,ir)

c end

0000

0000

94

if (ir(1) .eq. 1) then
cross over

for checking
intvl(i)=intvl(i)+1
of checking

do j=i+1,nloci
temp=idparent(j,1)
idparent(j,1)=idparent(j,2)
idparent(j,2)=temp

end do

end if

end do

randomly chose one of two gamates
prob(1)=.S

prob(2)=.5

call ransign(2,prob,nr,ir)
if (ir(1) .eq. 1) then
option 1: chose first

do i=1,nloci
idgam(i)=idparent(i,1)

end do

else

option 2: non cross

do i=1,nloci
idgam(i)=idparent(i,2)

end do

end if

end

subroutine to sign alleles randomly

by Zhiwu Zhang, August 25, 1997

subroutine ransign(nmal,probs,nr,ir)

nmal number of marker alleles

probs frequency of marker allleles

nr total number to generate

ir array to store random nuber generated

INTEGER IMIN, IOPT, IWK(nma1), NOUT,ir(nr)
REAL WK(nmal),probs(nma1)

CALL UMACH (2, NOUT)

IMIN 1

IOPT = 0

nmass=nmal

CALL RNSET (SEED)

CALL RNGDA (NR, IOPT, IMIN, NMASS, PROBS, IWK, WK, IR)
END

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