= Transmitted and untransmitted alleles under multiplicative penetrance
Under random mating and <Hardy-Weinberg equilibrium>, label each parental allele by whether it is transmitted. Let $(A,B)$ and $(C,D)$ be the transmitted/untransmitted pairs of two parents. Before ascertainment all four draws are independent with population frequencies $\pi$. If offspring disease weights the transmitted pair by $\psi_A\psi_C$, then
$$
\mathbb P(A=a,B=b,C=c,D=d\mid\text{offspring disease})=\pi_a^*\pi_c^*\pi_b\pi_d.
$$
The transmitted pair is an affected-case genotype and the untransmitted pair a population <family pseudo-control genotype> under this homogeneous model. For unordered genotypes, compatible labelled transmissions must be summed. Conditioning on parental genotypes instead removes nuisance population frequencies and gives the usual <transmission disequilibrium test>.
// Target: statistical-modelling.bigb
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