The gory details

OK, so I've given you the basic idea. Where does it come from, and how does it work? Funny you should ask. The whole approach is based on calculations of the degree to which different relatives resemble one another. For these purposes we're going to continue our focus on phenotypes influenced by one locus with two alleles, and we'll do the calculations in detail only for half sib families. We start with something that may look vaguely familiar.⁴ Take a look at Table 1.

Table 1: Half-sib family structure in a population with genotypes in Hardy-Weinberg proportions.

Maternal		Offspring genotype
genotype	Frequency	$A_1A_1$	$A_1A_2$	$A_2A_2$
$A_1A_1$	$p^2$	$p$	q	0
$A_1A_2$	$2pq$	$\frac{p}{2}$	$\frac{1}{2}$	$\frac{q}{2}$
$A_2A_2$	$q^2$	0	p	q

Note also that the probabilities in Table 1 are appropriate only if the progeny are from half-sib families. If the progeny are from full-sib families, we must specify the frequency of each of the nine possible matings (keeping track of the genotype of both mother and father) and the offspring that each will produce.⁵