We can use similar arguments to compute the covariance of (4.4) at two different loci s and t on an unlinked chromosome. This will be very useful in the following chapter, where we consider the simultaneous use of many markers to map genes of unknown genomic location, which requires that we understand the joint distributions of (4.4) at different marker positions on the genome. Indeed, let x(s) and x(t) denote the indicators of heterozygosity at markers s and t, respectively. Then cov(x(s),x(t)) = E[x(s)x(t)] - E[x(s)]E[x(t)]
= Pr[x(s) = 1]E[x(t)|x(s) = 1] - 1/4 = (1/2)(1 - 9) - 1/4 = (1 - 29)/4 .
The crucial step in this chain of equalities is the observation that if we are given x(s) = 1, then x(t) = 1 if and only if there is no recombination between s and t, which has probability 1 - 9. Since var[x(s)] = var[x(t)] = 1/2, we have cor[x(s),x(t)] = 1 - 29.
Denote by Zs and Zt the statistic Z in (4.2) at the marker loci s and t, respectively. From (4.4) we see that Zt is a linear function of the x«(t), so by the preceding argument we find that the correlation function of Zs and Zt is the same as for x(s) and x(t). Hence, for any two loci on a chromosome that does not contain a QTL
where 9 is the recombination fraction between s and t. A similar, but more complicated, calculation shows that under local alternatives the same correlation function applies to linked chromosomes.
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