Increased accuracy of artificial selection by using the realized relationship matrix.

Abstract

Dense marker genotypes allow the construction of the realized relationship matrix between individuals, with elements the realized proportion of the genome that is identical by descent (IBD) between pairs of individuals. In this paper, we demonstrate that by replacing the average relationship matrix derived from pedigree with the realized relationship matrix in best linear unbiased prediction (BLUP) of breeding values, the accuracy of the breeding values can be substantially increased, especially for individuals with no phenotype of their own. We further demonstrate that this method of predicting breeding values is exactly equivalent to the genomic selection methodology where the effects of quantitative trait loci (QTLs) contributing to variation in the trait are assumed to be normally distributed. The accuracy of breeding values predicted using the realized relationship matrix in the BLUP equations can be deterministically predicted for known family relationships, for example half sibs. The deterministic method uses the effective number of independently segregating loci controlling the phenotype that depends on the type of family relationship and the length of the genome. The accuracy of predicted breeding values depends on this number of effective loci, the family relationship and the number of phenotypic records. The deterministic prediction demonstrates that the accuracy of breeding values can approach unity if enough relatives are genotyped and phenotyped. For example, when 1000 full sibs per family were genotyped and phenotyped, and the heritability of the trait was 0.5, the reliability of predicted genomic breeding values (GEBVs) for individuals in the same full sib family without phenotypes was 0.82. These results were verified by simulation. A deterministic prediction was also derived for random mating populations, where the effective population size is the key parameter determining the effective number of independently segregating loci. If the effective population size is large, a very large number of individuals must be genotyped and phenotyped in order to accurately predict breeding values for unphenotyped individuals from the same population. If the heritability of the trait is 0.3, and N(e)=100, approximately 12474 individuals with genotypes and phenotypes are required in order to predict GEBVs of un-phenotyped individuals in the same population with an accuracy of 0.7 [corrected].

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Increased accuracy of artificial selection by using
the realized relationship matrix
B. J. HAYES1*, P. M. VISSCHER2AND M. E. GODDARD1,3
1Biosciences Research Division, Department of Primary Industries Victoria, 1 Park Drive, Bundoora 3083, Australia.
2Queensland Institute of Medical Research, Brisbane, Australia.
3Faculty of Land and Food Resources, University of Melbourne, Parkville 3010, Australia.
(Received 23 July 2008 and in revised form 4 December 2008)
Summary
Dense marker genotypes allow the construction of the realized relationship matrix between
individuals, with elements the realized proportion of the genome that is identical by descent (IBD)
between pairs of individuals. In this paper, we demonstrate that by replacing the average
relationship matrix derived from pedigree with the realized relationship matrix in best linear
unbiased prediction (BLUP) of breeding values, the accuracy of the breeding values can be
substantially increased, especially for individuals with no phenotype of their own. We further
demonstrate that this method of predicting breeding values is exactly equivalent to the genomic
selection methodology where the effects of quantitative trait loci (QTLs) contributing to variation in
the trait are assumed to be normally distributed. The accuracy of breeding values predicted using the
realized relationship matrix in the BLUP equations can be deterministically predicted for known
family relationships, for example half sibs. The deterministic method uses the effective number of
independently segregating loci controlling the phenotype that depends on the type of family
relationship and the length of the genome. The accuracy of predicted breeding values depends on
this number of effective loci, the family relationship and the number of phenotypic records. The
deterministic prediction demonstrates that the accuracy of breeding values can approach unity if
enough relatives are genotyped and phenotyped. For example, when 1000 full sibs per family were
genotyped and phenotyped, and the heritability of the trait was 0.5, the reliability of predicted
genomic breeding values (GEBVs) for individuals in the same full sib family without phenotypes was
0.82. These results were verified by simulation. A deterministic prediction was also derived for
random mating populations, where the effective population size is the key parameter determining the
effective number of independently segregating loci. If the effective population size is large, a very
large number of individuals must be genotyped and phenotyped in order to accurately predict
breeding values for unphenotyped individuals from the same population. If the heritability of the
trait is 0.3, and Ne=1000, approximately 5750 individuals with genotypes and phenotypes are
required in order to predict GEBVs of un-phenotyped individuals in the same population with an
accuracy of 0.7.
1. Introduction
In best linear unbiased prediction (BLUP) of breeding
values, information from performance of relatives is
incorporated through the use of a relationship matrix.
Elements of this matrix are derived as the predicted
proportion of the genome that is identical by descent
(IBD) among two individuals given their pedigree
relationship. However, Mendelian sampling during
gamete formation results in variation in the realized
proportion of the genome, which is IBD between
pairs of individuals with the same predicted relation-
ship coefficients (Franklin, 1977; Hill, 1993; Guo,
1996). For example, between full-sib individuals the
* Corresponding author. Tel: +61 (0)3 9479 5439. Fax: +61 (0)3
9479 3113. e-mail: ben.hayes@dpi.vic.gov.au
Genet. Res., Camb. (2009), 91, pp. 47–60.
doi:10.1017/S0016672308009981
f 2009 Cambridge University Press
Printed in the United Kingdom
47

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predicted proportion of the genome that is IBD is
0.5, while its standard deviation is 0.04 for a species
with 30 chromosomes each of 1 M in length (Guo,
1996).
DNA marker information can be used to calculate
the realized relationship matrix with elements the ac-
tual proportion of the genome that is IBD between
two individuals, with a high degree of precision, pro-
vided that a sufficient number of markers are used.
Nejati-Javaremi et al. (1997) demonstrated with
simulation that if the loci contributing to trait vari-
ation were known, and the alleles at these loci were
used to derive the realized relationship matrix, the
accuracy of breeding values calculated using this ma-
trix could be higher than that calculated using the
predicted relationship matrix. In practice, all the loci
contributing to trait variation are unlikely to have
been identified. Villanueva et al. (2005) demonstrated
by simulation that using the realized relationship
matrix derived from markers rather than the pre-
dicted relationship matrix in the calculation of esti-
mated breeding values (EBVs) could lead to higher
accuracies of selection. They proposed that marker
information used in this way could offer benefits in
selection programmes when no quantum trait locus
(QTL) has been mapped or when the underlying
genetic model can be considered the infinitesimal
model, where no individual QTL has a moderate to
large effect on the trait. For some traits such as height
in humans, this is indeed the case, with the largest
reported QTLs explaining only a small fraction of the
genetic variance (e.g. Sanna et al., 2008; Visscher,
2008).
While Villanueva et al. (2005) considered estimat-
ing realized relationships conditional on a known
pedigree (exploiting linkage information) realized re-
lationship coefficients can also be estimated for ‘un-
related’ individuals within a population. This requires
sufficient marker density to identify chromosome
segments in two individuals that are descended from
the same common, but unknown ancestor.
An alternative method by which DNA marker data
can be used to estimate breeding values is genomic
selection (Meuwissen et al., 2001). In this method, the
markers are used to track QTLs whose effects are es-
timated and summed to predict the breeding value of
each individual. However, if there are many QTLs
whose effects are normally distributed with constant
variance, then genomic selection can be equivalent
to the use of the realized relationship matrix (e.g.
Fernando, 1998; Habier et al., 2007; Van Raden,
2007 and Goddard, 2008).
Currently, there is no analytical method available
to predict the accuracy of EBVs calculated using the
genomic relationship matrix considering information
from relatives. Analytical expressions would be de-
sirable to guide the design of experiments aiming to
achieve a given accuracy of genomic breeding values
(GEBVs).Ourobjectivewastoderivesuchexpressions
for the accuracy of GEBV considering information
from relatives. We also modify the expression of
Goddard (2008) for the accuracy of GEBV in random
mating populations to improve the predictions. Our
starting point for all derivations was the equivalent
genomic selection model. We then verified the ana-
lytical predictions using two simulation approaches.
First, we derive a prediction of the accuracy based on
the prediction error variance (PEV) where the realized
relationship matrix is determined by a large number
of informative markers. Secondly, we derive accuracy
from simulations with both markers and QTLs seg-
regating as the correlation between true and predicted
breeding values. We then investigate the sensitivity of
the results to the number of markers used, the number
of QTLs and effective population size.
2. Methods
(i) An equivalent model for genomic selection
This material is also contained in Goddard (2008) but
is included here for completeness. Consider a model
of the true breeding value of the ith individual (gi)
based on a large number of QTLs of small effect. To
simplify our analytical derivation, we will define a
parameter q as the number of independent chromo-
some segments. This model can be pictured as divid-
ing the chromosomes into segments that effectively
segregate independently and defining the effect of the
segment as the sum of the effects of the QTL carried
on that segment. The assumption here is that there are
at least as many QTLs as there are effective chromo-
some segments. Alternatively if QTLs are unlinked,
then q is the number of unlinked QTLs. Then
gi= g
q
j=1
Wijuj,
where ujis the allele substitution effect at the jth QTL
and is normally distributed uyN(0, su
the variance of the effect of QTL alleles sampled ran-
domly from the population, and Wijis 0, 1 or 2 if
individual i carries 0, 1 or 2 copies of the second allele
at the jth QTL. In practice, it is convenient to subtract
the mean value of w from each element so that
Wij=0x2pjor 1x2pjor 2x2pj, where pj=the allele
frequency of the second allele at locus j. This
corresponds to the genomic selection model that
Meuwissen et al. (2001) called the BLUP model.
A simple version of genomic selection is to define the
Wijbased on markers instead of the QTL. Then the
best estimates of the ujand hence gican be obtained
by BLUP.
In matrix form g=Wu and V(g)=WWksu
W is a design matrix allocating QTL allele effects to
2), where su
2is
2, where
B. J. Hayes et al.48

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individuals. g is also normally distributed since it is
the sum of many normally distributed effects.
Now a vector of phenotypic records y can be mod-
elled as either
y=Xb+ZWu+e (1)
or
y=Xb+Zg+e,(2)
where X is a design matrix, b is a vector of fixed effects
and Z is matrix allocating records to individuals. The
two models are equivalent provided V(g)=Gsg
WWksu
trix calculated from the markers, and sg
variance. Elements of G are Gik, the proportion of the
genome that is IBD between individuals i and k.
Relationships, like inbreeding coefficients, are always
relative to a base population. By subtracting the mean
allele frequencies from the elements of W, the re-
lationships Gikare relative to the current population.
Consequently, they average approximately zero and
some are negative. This means that the genetic vari-
ance sg
population. The two models give the same estimates
of g=Wu. That is, a genomic selection model (1) with
normally distributed QTL effects is equivalent to a
conventional individual model (2) with the relation-
ship matrix among the individuals estimated from the
markers. Note that it assumes that the genotypes are
known without error.
2=
2, where G=WWksu
2/sg
2is the relationship ma-
2is the genetic
2is also the genetic variance in the current
(ii) Derivation of accuracies for breeding values
predicted with the equivalent model
When the marker data have been collected on a
sample of individuals, we can use either (1) or
(2) to calculate GEBVs for those individuals and their
reliabilities. However, it would be useful to predict in
general, before collecting the marker data, the accu-
racy that this form of genomic selection would
achieve. It is difficult to derive a formula for reliability
based on (2) because G is a complicated matrix. It is
perhaps easier to work with (1) but this is still difficult
because the design matrix, ZkWkWZ, the inverse of
which occurs in the system of equations required to
predict the QTL effects, is complex and likely to have
singularities. This complexity comes about because wij
for closely linked markers are correlated. Therefore,
we will approximate (1) by a model in which there are
q independent chromosome segments as described
above. In what follows, we first derive the number of
independent chromosome segments in different family
relationships or a random mating population, and
then use these numbers in the derivation of accuracy
of GEBV.
(a) Effective number of independent chromosome
segments within families
We will determine the effective number of chromo-
some segments by considering the variation in re-
lationship between pairs of individuals with the same
pedigree. For instance, based on pedigree all full sibs
have a relationship of 0.5 but in reality this relation-
ship varies from about 0.4 to 0.6 (Hill, 1993; Visscher
et al., 2006). This variation in relationship comes
about because sibs inherit large segments of chromo-
somes from their parents. The more the independent
chromosomes segments make up the genome the
more closely all full sibs would come to sharing ex-
actly 50% of their genome.
Formulae for the variation in realized relationship
between the different types of relatives have been
published by Hill (1993) and Guo (1996) and we will
use their formulae.
Consider a single locus and calculate the relation-
ship between relatives i and j, i.e. Gij. For full-sibs
25% of the time Gij=1, 50% of the time Gij=0.5 and
25% of the time Gij=0. So the variance is 1/8. If there
are q independent chromosome segments, then the
variance of Gij=1/(8q). However, the variance in Gij
can also be calculated for a genome consisting of
chromosomes of known length in Morgan. Hill (1993)
and Guo (1996) present formulae for this. For in-
stance, using their formula, if the genome consists of a
single chromosome 35 M long, then the variance in
relationship between pairs of full-sibs is 0.00177.
Equating this to the variance of Gij in our model
with q independent chromosome segments, 1/(8q)=
0.00177. So the effective number of loci is q=1/
(8*0.00177)=70.6, close to the reported empirical
value of 82 for human full sibs (Visscher et al., 2006).
So if two gametes produced by the same sire (corre-
sponding to two sibs) are considered, then a 35 M
chromosome will experience approximately 70 cross-
overs(35foreachgamete).Therefore,thetwogametes
can be considered as composed of 70 segments and for
each segment the probability that the two gametes are
identical is 0.5. Although we have assumed a single
chromosome here, Hill (1993) showed that the vari-
ation in relationship is not particularly affected by
assumptions on the number of chromosomes, pro-
vided the total length of the genome was kept con-
stant.
With the same assumptions as above, for half sibs,
the V(Gij) for a single locus is 1/16 and the variance
of relationship for a genome with one chromosome
35 M long is 0.00088 and so again qy70. For double
cousins, V(Gij) for a single locus is 3/32 and the vari-
ance in relationship from the formula of Guo (1996) is
0.00107, so q=88.
The number of effective loci is similar to the recom-
bination index for humans, assumed by Rasmusson
Selection using the realized relationship matrix 49

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(1993) to be the number of independently segregating
units in the genome.
(b) Effective number of chromosome segments in
a random mating population
To derive the effective number of loci in a random
mating population, consider two gametes taken at
random from the population. The position at one end
of a chromosome in both gametes can be traced back
until they coalesce. Positions close to this first point
will coalesce in the same ancestor but, as one moves
along the chromosome, a recombination will be
reached such that the next position coalesces in a dif-
ferent ancestor. Thus the two gametes can be seen to
be composed of a series of short chromosome seg-
ments that coalesce. The average length of these seg-
ments is 1/(4Ne), where Neis the effective population
size (Stam, 1980). Therefore, the two gametes of
length L Morgan are divided into 4NeL segments.
However, some segments are larger by chance than
others so that if the effective number of segments is
calculated from the variation in relationship between
individuals in the population it is approximately
2NeL/log(4NeL) per chromosome (Goddard, 2008).
However, this approximation does not consider
the fact that the small segments may still contain as
many QTL mutations as the larger segments since
they have on average a longer time to trace back to
the same common ancestor and hence a longer time
for mutations to accumulate. Therefore, the most
appropriate value for the number of effective seg-
ments might be in between 4NeL and 2NeL/log(4NeL)
per chromosome. As an approximation, we will as-
sume that the effective number of loci is 2NeL and
then test the validity of this assumption with simu-
lated data.
The variation in relationship between two gametes
arises for two reasons. First, some pairs of gametes
are more closely related by pedigree than others. For
instance, some pairs of gametes may share a common
parent or grandparent, whereas other pairs do not.
Secondly, even considering pairs of gametes that have
the same pedigree relationship, they may share more
or less alleles than the average expected for that
relationship, due to Mendelian sampling. The first
source of variation in relationship is used by a con-
ventional individual model BLUP to estimate the
breeding values of individuals including those with no
phenotypic record. The second source of variation in
realized relationship is the source of the increase in
reliability of GEBVs. For a pair of gametes with
constant pedigree relationship, the ancestor in which
one chromosome segment coalesces is independent of
the ancestor in which another chromosome segment
coalesces. Consequently, the variation in relationship
due to this source would be zero if there were an
infinite number of unlinked loci. Even though
the number of positions in the genome may be very
large, linkage causes variation in relationship by gen-
erating chromosome segments that coalesce. Since
each segment coalesces independently conditional on
the pedigree, it again seems appropriate to estimate
the effective number of loci as in between the number
of segments (4NeL) and the number of segments
weighted by length (2NeL/log(4NeL) per chromo-
some), e.g. as 2NeL.
(c) Accuracy of genomic EBVs with information
from relatives
In what follows we will assume that fixed effects can
be adequately estimated and that the data have been
corrected for them, so y=Zg+e, where y is corrected
for fixed effects.
Even without any genetic markers, the breeding
value of an individual can be estimated from pedigree
and phenotypic records. We will focus on predicting
the increase in reliability due to markers of the GEBV
of an individual that has marker data but no pheno-
typic record and no offspring because that is the most
important use of genomic selection. That is, we will
calculate the increase in accuracy of GEBV above that
obtainable simply from the pedigree and records on
ancestors and collateral relatives. In this scenario, the
breeding value of the ith individual (gi) can be ex-
pressed as the mean of individuals with the same
pedigree as individual i( f) and a deviation from that
mean caused by the actual genes the individual in-
herited:
gi=f+ g
q
j=1
usij+ g
q
i=1
umij,
where f=family mean breeding value, usij=paternal
allele effect inherited by the ith individual at the jth
independent chromosome segment as a deviation
from family mean, umij=maternal allele effect in-
herited by the ith individual at the jth independent
chromosome segment as a deviation from the family
mean and summation is over all independent chromo-
some segments.
The variance of the breeding values is then
V(g)=V( f )+ g
q
j=1
V(usj)+ g
q
j=1
V(umj):
If we analyse the data y with the model and esti-
mate f, usjand umj:
gˆ =fˆ+ g
q
j=1
uˆsj+ g
q
j=1
uˆmj:
The components of this equation are independent,
as the effects of the sire and dam alleles are expressed
B. J. Hayes et al. 50

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as deviations from the family mean. Therefore
V(gˆ )=V(fˆ)+ g
q
j=1
V(uˆsj)+ g
q
j=1
V(uˆmj):
(3)
The reliability of GEBVs is V(g ˆ )/V(g) and the ac-
curacy is the square root of this reliability. The re-
liability can be calculated using eqn (3) and compared
with that obtained using only the family mean to
quantify the increase in reliability due to the marker
data.
With N progeny per family the reliability from
phenotypic and pedigree information alone is
V(fˆ)
V(f )=
N
N+lf,
where
lf=(V(y)xV( f ))=V( f ):
Now we calculate the increase in reliability with
genomic information. Assuming that there are n pa-
ternal alleles per family that are equally represented in
the data so that there are N/n individuals carrying
each paternal allele. As explained in the appendix
ls=2q/h2. Then
V(uˆs)=s2
appendix)
u(1x1=n)N=(N+nls)(derived in the
g
j=1
q
V(u ˆsj)=0?5V(g)(1x1=n)N=(N+nls)because
V(g)=2qs2
u
gq
As an example consider the case of selecting among
a population of individuals consisting of full sib fam-
ilies. In this case V( f )=0.5 V(g), lf=(V(y)xV( f ))/
V(f )=(1x0.5h2)/(0.5h2)
above, for full-sibs, if the genome consists of a single
chromosome 35-Morgan long, the variance in re-
lationship among pairs of full sibs is 0.00179 corre-
sponding to q=70 effective chromosome segments.
Consequently, su
Within a family of full-sibs there are two paternal
alleles and two maternal alleles, so n=2,
j=1V(uˆdj) is calculated in a similar manner.
andV( fˆ)=0?5V(g)N
N+lf:
As
2=V(g)/(2*70), ls=2q/h2=140/h2.
g
j=1
q
V(uˆsj)=0?5V(g)(1=2)N=(N+2ls)
and gq
h2=0.5, then
j=1V(uˆdj) is the same. If V(g)=1, N=99 and
V(fˆ)=0?5*99=(99+3)=0?486:
So from eqn (3)
V(gˆ )=0?486+0?5*(1=2)*99=(99+2*280)
+0?5*(1=2)*99=(99+2*280)=0?561:
As V(g)=1, the reliability is 0.561.
(d) Accuracy of GEBVs in a random breeding
population
We also want to predict the increase in reliability of an
EBV using the realized relationship matrix compared
with the pedigree relationship matrix in a random
breeding population. As before, assume the breeding
value is the sum of many QTLs each of which is in
perfect Linkage disequilibrium (LD) with a marker.
That is, the breeding value of individual i is gi=
gq
press wijas a deviation from the mean, e.g. wij=
xijx2pj,wherexijis0,1or2representinghomozygote,
heterozygote and other homozygote and pjis the allele
frequency at independent chromosome segment j and
as before ujis the allele substitution effect at the jth
independent chromosome segment assuming there
are only two alleles per independent chromosome
segment, and q is the number of independently segre-
gating chromosome segments, which for a randomly
mating population is derived above. The phenotypes
are modelled as in (1).
This derivation of accuracy of breeding value is
similar to those for full-sib families but differs in an
important way. In the full-sib case, each parent is as-
sumed to have two different alleles at each indepen-
dent chromosome segment, so the number of alleles at
one independent chromosome segment is four times
the number of families. Consequently, the genetic
variance at one independent chromosome segment in
the parental generation is 2su
a random mating population, there are assumed to be
only two alleles per independent chromosome seg-
ment and the variance contributed by the jth inde-
pendent chromosome segment is V(wj) su
su
where V(w) is the average value of V(wj) over all in-
dependent chromosome segment. In the full-sib case,
we were estimating the effect of markers within a
family and so only the number of individuals within
the family could be used to estimate u. On the other
hand, the effective number of loci within a full-sib
family is small because large segments of chromosome
segregate within a family. By contrast in a random
mating population we are estimating the effect of u
across the population, so all individuals with pheno-
types (N) can be used but the effective number of loci
is large because there must be a marker close enough
to any QTL to be in high LD with it.
There is assumed to be no LD between the QTLs
so the BLUP equations for estimating u are approxi-
mately block diagonal with the jth independent
chromosome segment having a block of equations
j=1wijujas before, except here it is convenient to ex-
2. However, in the case of
2=2pj(1xpj)
2=qV(w)su
2and the total genetic variance is sg
2,
[WkjWj+l] u ˆj=Wkjy
Selection using the realized relationship matrix51