Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 45 / 1 / b / i / Solution

For a binary disease phenotype, put $p_i(g)=P(X_i=\text{affected}\mid G_i=g)$. The penetrance function depends on the inheritance model, the disease alleles and genotype, and potentially age, sex, environment and other genetic modifiers. Phenocopies allow disease without the putative disease genotype, while incomplete penetrance allows a genetic carrier of that genotype to remain unaffected. Thus the phenotype factor is $p_i(g)$ for an affected individual and $1-p_i(g)$ for an unaffected one.

Under a fully penetrant model with no phenocopies the genotype determines disease status, so the factor is 1 for a compatible genotype and 0 for an incompatible genotype. For example, in fully penetrant recessive inheritance, $aa$ is affected and $AA,Aa$ are unaffected. Missing phenotype observations contribute the constant factor 1 rather than excluding genotypes.