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 ${\psi }_A{\psi }_C$, thenThe 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