= Multiplicative penetrance preserves Hardy-Weinberg equilibrium
{title2=$\pi_i^*=\pi_i\psi_i/\sum_u\pi_u\psi_u$}
For ordered alleles $I,J$ independently drawn with probabilities $\pi_i$, suppose disease <penetrance> is $c\psi_i\psi_j$. Conditional on disease, the allele draws remain independent with probabilities $\pi_i^*=\pi_i\psi_i/\sum_u\pi_u\psi_u$. This follows by normalizing the product $c\pi_i\pi_j\psi_i\psi_j$. Unordered homozygote probabilities are $(\pi_i^*)^2$ and heterozygote probabilities are $2\pi_i^*\pi_j^*$. Unaffected controls do not generally share this factorization: their genotype weights contain $1-c\psi_i\psi_j$.
// Target: biology.bigb
Back to article page
