Family pseudo-control genotype

Combine the untransmitted maternal and paternal alleles into a hypothetical control genotype, paired with the child's transmitted genotype. Even when two heterozygous parents have a heterozygous child and parent-specific phase is ambiguous, the combined untransmitted genotype is unambiguously heterozygous. Such pseudo-controls are matched family controls, not independent random individuals; Hardy-Weinberg principle is therefore not guaranteed merely by this construction.

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