Penetrance

Penetrance is the probability of an expressed phenotype conditional on a specified genotype. It can depend on age, sex and environment. A fully penetrant recessive model assigns disease probability one to the recessive homozygote and zero to other genotypes; incomplete penetrance or phenocopies replace these indicators by intermediate probabilities.

For a binary disease trait $Y$ with genotype $G$, the genetic variance of penetrance is the between-genotype term in the law of total variance. The averaged within-genotype term is $\mathbb{E}[q_G(1-q_G)]$. Under multiplicative diallelic penetrance, the genetic term is $p^2({\beta }^2-{\alpha }^4)$. This decomposition attributes variance to the modeled locus, not automatically to all hereditary causes.

With Hardy-Weinberg equilibrium, let $G$ count copies of the risk allele. The penetrance $p{\theta }^G$ must lie in $[0,1]$ for every possible genotype. Writing $\alpha =1-\varphi +\varphi \theta$ and $\beta =1-\varphi +\varphi {\theta }^2$ gives $\mathbb{E}(q_G)=p{\alpha }^2$ and $\mathbb{E}(q_G^2)=p^2{\beta }^2$.