Here's how we can calculate the covariance between half-siblings:
First, imagine selecting huge number of half-sibs pairs at random.
The phenotype of the first half-sib in the pair is a random variable
(call it
), as is the phenotype of the second (call it
).
The mean of
is just the mean phenotype in all the
progeny taken together,
. Similarly, the mean of
is
just
. Now with one locus, two alleles we have three possible
phenotypes:
(corresponding to the genotype
),
(corresponding to the genotype
), and
(corresponding to the genotype
). 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 of randomly chosen half-sibs based on the
frequenices in Table 1 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_{...
...lpha_2 - {\bar x})^2\right] \\
&=& \left({1 \over 4}\right)V_a
\end{eqnarray*}](img26.png)