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.
\[cov_{OP} = cov(G,\frac{1}{2}A) = cov(A+D,\frac{1}{2}A) = cov(A,\frac{1}{2}A) + cov(D,\frac{1}{2}A) = \frac{1}{2}V_A\]
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 |
|---|---|---|---|---|
| \(A_1A_1\) | \(p^2\) | \(2q(\alpha-qd)\) | \(2q\alpha\) | \(q\alpha\) |
| \(A_1A_2\) | \(2pq\) | \((q-p)\alpha+2pqd\) | \((q-p)\alpha\) | \(\frac{1}{2}(q-p)\alpha\) |
| \(A_2A_2\) | \(q^2\) | \(-2p(\alpha+pd)\) | \(-2p\alpha\) | \(-p\alpha\) |
Note: the genotypic value in the table is the deviations from the population mean.
\[cov_{OP} = [2q(\alpha-qd)] \times [q\alpha] \times [p^2] + [(q-p)\alpha+2pqd] \times [\frac{1}{2}(q-p)\alpha] \times[2pq] + [-2p(\alpha+pd)]\times [-p\alpha] \times [q^2] = pq\alpha^2 = \frac{1}{2}V_A\]
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.
\[b_{OP} = \frac{cov_{OP}}{V_P} = \frac{1}{2}\frac{V_A}{V_P}\]
10.1.2 Offspring and mid-parent
Way 1:
\[cov_{O\bar{P}} = cov(\frac{1}{2}(G_1+G_2),\frac{1}{2}A) = \frac{1}{2}(cov_{OP_1} + cov_{OP_2}) = \frac{1}{2}V_A\] If \(G_1\) and \(G_2\) have the same variance, then \(cov_{OP_1}\) = \(cov_{OP_2}\)/
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 \(A_1A_1\) (a) | Progeny genotypic value for \(A_1A_2\) (d) | Progeny genotypic value for \(A_2A_2\) (-a) | Mean value of progeny | |
|---|---|---|---|---|---|---|---|
| \(A_1A_1\) | \(A_1A_1\) | \(p^4\) | a | 1 | - | - | a |
| \(A_1A_1\) | \(A_1A_2\) | \(4p^3q\) | \(\frac{1}{2}(a+d)\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) | - | \(\frac{1}{2}(a+d)\) |
| \(A_1A_1\) | \(A_2A_2\) | \(2p^2q^2\) | 0 | - | 1 | - | d |
| \(A_1A_2\) | \(A_1A_2\) | \(4p^2q^2\) | d | \(\frac{1}{4}\) | \(\frac{1}{2}\) | \(\frac{1}{4}\) | \(\frac{1}{2}d\) |
| \(A_1A_2\) | \(A_2A_2\) | \(4pq^3\) | \(\frac{1}{2}(-a+d)\) | - | \(\frac{1}{2}\) | \(\frac{1}{2}\) | \(\frac{1}{2}(-a+d)\) |
| \(A_2A_2\) | \(A_2A_2\) | \(p^4\) | -a | - | - | 1 | -a |
\[cov_{O\bar{P}} = MP - M^2 \]
where MP is the mean product, and \(M^2\) is the square of population mean.
\[cov_{O\bar{P}} = a^2 \times p^4 + [\frac{1}{2}(a+d)]^2 \times 4p^3q + 0 + \frac{1}{2}d^2 \times 4p^2q^2 + [\frac{1}{2}(-a+d)]^2 \times 4pq^3 + a^2\times q^4 - [a(p-q)+2dpq]^2\]
\[cov_{O\bar{P}} = pq\alpha^2 = \frac{1}{2}V_A\]
Although \(cov_{OP}\) is same with \(cov_{O\bar{P}}\), the regression (degree of resemblance) is not the same.
\[b_{O\bar{P}} = \frac{cov_{O\bar{P}}}{\sigma^2_{O\bar{P}}}\]
\[\sigma^2_{\bar{P}} = var(\frac{1}{2}(P_1+P_2)) = \frac{1}{4}\times (var(P_1) + var(P_2)) = \frac{1}{2}V_P\]
So
\[b_{O\bar{P}} = \frac{V_A}{V_P}\]
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:
\[cov_{(HS)} = V_{\frac{1}{2}A} = \frac{1}{4}V_A\]
Way 2:
From table 9.1, we can get
\[cov_{(HS)} = [q\alpha]^2 \times p^2 + [\frac{1}{2}(q-p)\alpha]^2 \times 2pq + [-p\alpha]^2 \times q^2 = \frac{1}{2}pq\alpha^2 = \frac{1}{4}V_A\]
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
\[t= \frac{1}{4}\frac{V_A}{V_P}\]
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 \(\frac{1}{2}(A_1+A_2)\), which is \(\frac{1}{2}V_A\) if the additive variance is the same in the two sexes.
Now consider the contribution of domiance. It should be \(\frac{1}{4}V_D\).
\[Cov_{(FS)} = \frac{1}{2}V_A + \frac{1}{4}V_D\]
Way 2:
From Table 9.2,
\[cov_{(FS)} = a^2 \times p^4 + [\frac{1}{2}(a+d)]^2 \times 4p^3q + d^2\times 2p^2q^2 + \frac{1}{4}d^2 \times 4p^2q^2 + [\frac{1}{2}(-a+d)]^2 \times 4pq^3 + a^2\times q^4 - [a(p-q)+2dpq]^2\]
\[cov_{(FS)} = cov_{O\bar{P}} + d^2p^2q^ = \frac{1}{2}V_A + \frac{1}{4}V_D\]
The correlation of full sibs is:
\[t = \frac{\frac{1}{2}V_A + \frac{1}{4}V_D}{V_P}\]
In principle the difference between the covariances of full sibs and of half sibs provides a way of estimating the domiance variance, since \(cov_{(FS)}-2cov_{(HS)} = \frac{1}{4}V_D\). 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
\[cov_{(MZ)} = V_G\]
10.1.6 General
The generalized covariance for any sort of relationship is
\[cov = rV_A + uV_D\]
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 \(V_A\)) | Coefficient u (of \(V_D\)) | |
|---|---|---|---|
| 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 | - | - |