Objective: Disease risk elevation due to an environmental factor only for individuals with a susceptible genotype is a typical example of gene-environment interaction. In order to identify risk factors interacting with susceptible genotypes in case-control studies, presumptions on minimal size of cases with the susceptible genotype (S (min)) and odds ratio (OR) among the susceptible individuals (OR(susceptible)) are useful.
Model: Proportion of exposed cases (P(1)) and OR for whole cases (OR(whole)) statistically detectable in a case-control study can be calculated in a conventional method. P(1) was assumed to be a weighted sum of the exposed among cases with the genotype (P(x)) and cases without the genotype (equal to proportion of the exposed among controls, P(0)), i.e., S P(x) + (1 - S) P0, where S is the size (proportion) of cases with the genotype. For each calculated P(1), S became the minimum (S(min)) in case of P(x) = 1. OR(susceptible) was calculated by {P(x) (1 - P(0))} / {(1 - P(x)) P(0)}.
Results: S(min) and OR(susceptible) were listed for the combinations of the above components. For example, a detectable P(1) was 0.638 for P(0)=0.5 in a case-control study with 200 cases (N(1)) and 200 controls (N(0)), when a error of a two-sided test was 0.05 with an 80% of power. In case of P(1)=0.638, OR(whole) was 1.77, producing S(min) = 0.277 for infinite OR(susceptible). It indicates that an environmental factor cannot be detected in case that a high-risk genotype frequency is less than 0.277.
Interpretation: If the size of cases with a susceptible genotype is expected to be less than S(min), case-control studies are unlikely to detect a significant OR of the environmental factor.