10 Resemblance between relatives

The resemblance between relatives is one of the basic genetic phenomena displayed by metric characters, and the degree of resemblence is a property of the character.

  • casual components
  • observational components

The resemblance between related individuals, can be looked at

  • similarity of individuals in the same group
  • difference between individuals in different groups

The covariance of related individuals is a new property of the population.

Both genetic and environmental sources of variance contribute to the covaraince of relatives, the covariance of phenotypic values being the sum of the genetic and environmental covariances.

10.1 Genetic covariance

Assumpation: HWE, random mating, no epistatic interaction.

10.1.1 Offspring and one parent

Way 1:

The covariance is that of an individual's genotypic value with half its breeding value.

covOP=cov(G,12A)=cov(A+D,12A)=cov(A,12A)+cov(D,12A)=12VA

Note: cov(D,A) = 0

The genetic covariance of offspring and one parent is therefore half the additive genetic variance of the parents.

Way 2:

For one locus with two alleles, we can get the table below.

Table 9.1

Parents' Gentoype Genotype Frequence Genotypic value Breeding value Offspring mean genotypic value
A1A1 p2 2q(αqd) 2qα qα
A1A2 2pq (qp)α+2pqd (qp)α 12(qp)α
A2A2 q2 2p(α+pd) 2pα pα

Note: the genotypic value in the table is the deviations from the population mean.

covOP=[2q(αqd)]×[qα]×[p2]+[(qp)α+2pqd]×[12(qp)α]×[2pq]+[2p(α+pd)]×[pα]×[q2]=pqα2=12VA

Summing over all loci we again reach the conclusion that the covariance of offspring and one parent is equal to half the additive variance.

The regression of offspring on one parent is got by dividing the covariance by the variance of the parents, which is the phenotypic variance of the population.

bOP=covOPVP=12VAVP

10.1.2 Offspring and mid-parent

Way 1:

covOˉP=cov(12(G1+G2),12A)=12(covOP1+covOP2)=12VA If G1 and G2 have the same variance, then covOP1 = covOP2/

Thus, provided the two sexes have equal variances, the covariance of offspring and mid-parent is the same as that of offspring with one parent.

Way 2:

Table 9.2

Genotype of parents Frequencies of mating Mid-parent value Progeny genotypic value for A1A1 (a) Progeny genotypic value for A1A2 (d) Progeny genotypic value for A2A2 (-a) Mean value of progeny
A1A1 A1A1 p4 a 1 - - a
A1A1 A1A2 4p3q 12(a+d) 12 12 - 12(a+d)
A1A1 A2A2 2p2q2 0 - 1 - d
A1A2 A1A2 4p2q2 d 14 12 14 12d
A1A2 A2A2 4pq3 12(a+d) - 12 12 12(a+d)
A2A2 A2A2 p4 -a - - 1 -a

covOˉP=MPM2

where MP is the mean product, and M2 is the square of population mean.

covOˉP=a2×p4+[12(a+d)]2×4p3q+0+12d2×4p2q2+[12(a+d)]2×4pq3+a2×q4[a(pq)+2dpq]2

covOˉP=pqα2=12VA

Although covOP is same with covOˉP, the regression (degree of resemblance) is not the same.

bOˉP=covOˉPσ2OˉP

σ2ˉP=var(12(P1+P2))=14×(var(P1)+var(P2))=12VP

So

bOˉP=VAVP

The regression of offspring on parents is a useful measure of the degree of resemblance.

10.1.3 Half sibs

Way 1:

Half sibs are individuals that have one parent in common and the other parent differnt. The mean genotypic value of the group of half sibs is by definition half the breeding value of the common parent. The covariance is the variance of the true means of the half-sib groups, and is therefore the variance of half the breeding values of the common parents:

cov(HS)=V12A=14VA

Way 2:

From table 9.1, we can get

cov(HS)=[qα]2×p2+[12(qp)α]2×2pq+[pα]2×q2=12pqα2=14VA

The degree of resemblance between sibs is expressed as the intraclass correlation, which is the between-group varaince, i.e., the covariance as a proportion of the total varaince. So the correlation of half sibs is

t=14VAVP

10.1.4 Full sibs

The covariance of full sibs is less simple than those of the relationships so far considered, because the domiance variance contributes to it.

Way 1:

Consider the covariance due to the additive variance first.

Full sibs have both parents in common, so the covariance is the variance of 12(A1+A2), which is 12VA if the additive variance is the same in the two sexes.

Now consider the contribution of domiance. It should be 14VD.

Cov(FS)=12VA+14VD

Way 2:

From Table 9.2,

cov(FS)=a2×p4+[12(a+d)]2×4p3q+d2×2p2q2+14d2×4p2q2+[12(a+d)]2×4pq3+a2×q4[a(pq)+2dpq]2

cov(FS)=covOˉP+d2p2q=12VA+14VD

The correlation of full sibs is:

t=12VA+14VDVP

In principle the difference between the covariances of full sibs and of half sibs provides a way of estimating the domiance variance, since cov(FS)2cov(HS)=14VD. In practice, however, this can be done only if there is no environmental contributions to the phenotypic covariances.

10.1.5 Twins

For dizygotic (fraternal) twins are related as full sibs and their genetic covariance is that of full sibs.

For monozygotic (identical) twins have identical genotypes, so there is no genetic variance within pairs and the whole of the genetic vaiance appears in the between-pair component. The genetic covariance is therefore

cov(MZ)=VG

10.1.6 General

The generalized covariance for any sort of relationship is

cov=rVA+uVD

The coefficient r of the additive varaince is sometimes called the coefficient of relationship, or the theoretical correlation.

Table 9.3 Coefficients of the variance components in the covariances of relative.

Relationship Coefficient r (of VA) Coefficient u (of VD)
MZ twins 1 1
First-degree Offspring:parent 1/2 0
Full sib 1/2 1/4
Second-degree Half sib 1/4 0
Offspring:grandparent - -
Uncle(aunt):nephew(niece) - -
Double first cousin - -
Third-degree Offspring:great-grandparent 1/8 0
Single first cousin - -

10.1.7 Epistatic interaction

10.2 Environmental covariance

10.3 Phenotypic resemblance