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

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.

Without genotyping error, the observed genetic marker genotype is a deterministic function $u(g)$ of the underlying phase-known genotype. In particular $u$ can forget chromosome phase and parental origin. ThereforeSeveral different phased haplotype pairs can receive factor 1 when the observed multilocus genotype is unphased. With laboratory error, missing alleles or uncertain calls, replace the indicator by an error or observation model. A wholly unobserved genetic marker contributes factor 1. Thus a 0-or-1 factor reflects perfect observation of a deterministic genetic marker state, not a general property of every genetic measurement.