Sanja Franic, VU University Amsterdam 2010
Variance. Questions formulated within the context of genetic study of quantitative traits
pertain predominantly to variation, the basic idea being the decomposition of phenotypic
variance into components attributable to different causes. These variance components
correspond to components of value described in the last section, so that e.g. the genotypic
variance is the variance of the genotypic values. Assuming that the genotypic values and the
environmental deviations are not correlated and do not interact, the variance decomposition is:
VP = VG + VE
= VA + VD + VI + VE,
the more general expression being VP = VG + VE + 2covGE + VGE, where covGE is the
covariance between genotypic values and the environmental deviations, and VGE the variance
due to interaction between genotypes and the environment. The ratio VA/VP represents the
degree to which the variation in the phenotype is due to variation in the breeding values, and
is known as the heritability coefficient.
Covariance. The phenotypic covariance between individuals may generally be expressed in
terms of the aforementioned variance components. In terms of the covariation between
genotypic values, for instance, the resemblance between offspring and parent may be
represented as the covariance of the parents’ genotypic values of with the mean genotypic
values of their offspring. Since the mean value of the offspring is by definition half the
breeding value of the parent, the covariance to be deduced is that of the parent’s genotypic
value (G = A + D) with half of their breeding value (½A): covG_OP = [∑½A(A + D)]/N =
(½∑A2 + ½∑AD)/N = ½∑A2/N + ½∑AD/N = ½VA + ½covAD. The covariance between the
breeding values and dominance deviations (covAD) is zero, as can be verified by multiplying
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Sanja Franic, VU University Amsterdam 2010
each of the breeding values by the corresponding dominance deviation and frequency (given
in Table 2) and summing over the three genotypes: –4 p2q3αd + 4p2q2(q – p)αd + 4p3q2αd =
4p2q2αd(– q + q – p + p) = 0. Thus the genetic component of the parent-offspring covariance
equals covG_OP = ½VA.
In full siblings, the mean additive genotypic value of a group of siblings is the mean
breeding value of the two parents. Denoting the breeding values of the two parents A and A’,
the covariance between the additive genotypic values of offspring is covA_FS = ∑½(A + A’)½(
A + A’)/N = ∑¼(A + A’)2/N = ∑¼(A2 + 2AA’ + A’2)/N = ¼VA + ¼VA’ + ½covAA’. The
assumption of random mating implies that the covariance between the parents’ breeding
values (covAA’) is zero. Thus the covariance of the breeding values of full siblings reduces to
covA_FS = ¼VA + ¼VA’. If the additive genetic variance is equal in the two sexes, this
expression becomes covA_FS = ¼VA + ¼VA = ½VA. In addition, if parental genotypes at a
single locus are A1A2 and A3A4, the offspring may have one of the four possible genotypes:
A1A2, A1A4, A2A3, and A2A4. If the first sibling has any of these genotypes, the probability
that the second sibling has the same genotype is ¼. Thus, one quarter of full siblings have the
same genotype for this locus, and consequently the same dominance deviation. For these
pairs, the covariance due to dominance deviations is cov = ∑D2/N = VD. In other pairs the
covariance due to dominance deviations is zero. Thus, over all pairs of siblings, the
covariance due to dominance deviations is ¼VD. The total genotypic covariance between full
siblings is therefore covG_FS = ½VA + ¼VD.
The same expression holds true for dizygotic (fraternal, DZ) twins, whose degree of
genetic relatedness is the same as that of full siblings. Monozygotic (identical, MZ) twins
have identical genotypes and therefore share their entire genotypic variance, thus covG_MZ =
VA + VD.
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