Minimal Sizes of Cases with a Susceptible Genotype and Minimal Odds Ratios among Susceptible Individuals in Case-control Studies

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 (Smin) and ‍odds ratio (OR) among the susceptible individuals (ORsusceptible) are useful. ‍Model: Proportion of exposed cases (P1) and OR for whole cases (ORwhole) statistically detectable in a case-control ‍study can be calculated in a conventional method. P1 was assumed to be a weighted sum of the exposed among cases ‍with the genotype (Px) and cases without the genotype (equal to proportion of the exposed among controls, P0), i.e., ‍S Px + (1 - S) P0, where S is the size (proportion) of cases with the genotype. For each calculated P1, S became the ‍minimum (Smin) in case of Px = 1. ORsusceptible was calculated by {Px (1 - P0)} / {(1 - Px) P0}. ‍
Results: Smin and ORsusceptible were listed for the combinations of the above components. For example, a detectable ‍P1 was 0.638 for P0=0.5 in a case-control study with 200 cases (N1) and 200 controls (N0), when á error of a two-sided ‍test was 0.05 with an 80% of power. In case of P1=0.638, ORwhole was 1.77, producing Smin=0.277 for infinite ORsusceptible. ‍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 Smin, case-control studies ‍are unlikely to detect a significant OR of the environmental factor.