Covariance between half-siblings

Now we can calculate the covariance between half-siblings. To do so we imagine selecting huge number of half-sibs pairs at random. Now the phenotype of the first half-sib in the pair is a random variable (call it $S_1$), as is the second (call it $S_2$). The mean of $S_1$ is just the mean phenotype in all the progeny taken together, $\bar x$. Similarly, the mean of $S_2$ is just $\bar x$. Now with one locus, two alleles we have three possible phenotypes: $x_{11}$ (corresponding to the genotype $A_1A_1$), $x_{12}$ (corresponding to the genotype $A_1A_2$), and $x_{22}$ (corresponding to the genotype $A_2A_2$). So all we need to do to calculate the covariance between half-sibs is to write down all possible pairs of phenotypes and the frequency with which they will occur in our sample based on the table above and the frequency of maternal genotypes. It's actually a bit easier to keep track of it all if we write down the frequency of each maternal genotype and the frequency with which each possible phenotypic combination will occur in her progeny.

$$\begin{eqnarray*} \mbox{Cov}(S_1,S_2) &=& p^2[p^2(x_{11} - {\bar x})^2 + 2pq(x_... ...2\alpha_2 - {\bar x}]^2\} \\ &=& \left({1 \over 4}\right)V_a \end{eqnarray*}$$