Abstract

Genetic susceptibility and environmental exposures play a synergistic role in the aetiology of many diseases. We consider a case-control study of a rare disease in relation to a categorical exposure and a genetic factor under the assumption that the genotype and the exposure occur independently in the population under study. Using a logistic model for risk, we describe maximum likelihood methods based on log-linear models that explicitly impose the independence assumption, something the usual logistic regression analyses cannot do. The estimator of the genotype-exposure interaction effect depends only on data from cases. Estimators for genotype and for exposure effects depend also no data from controls, but only through their respective marginal totals. All three estimators have smaller variance than they would were independence not enforced. These results have important implications for design: (i) Case-only studies can efficiently estimate gene-by-environment interactions. (ii) Studies where controls are genotyped anonymously can estimate genotype, exposure, and interaction effects as efficiently as designs where genotype and exposure data are linked. This feature addresses a growing concern of human subjects review boards. (iii) Exposure and interaction effects, but not genotype effects, can be estimated from studies where genetic information is only collected from cases (although one can recover the genotype effect if external gene prevalence data exist). Such designs have the compensatory benefit that the response rate (hence, validity) is higher when controls are not subjected to intrusive tissue sampling. However, the independence assumption can be checked only with linked genotype and exposure data for some controls. We illustrate the methods by applying them to recent study of cleft palate in relation to maternal cigarette smoking and to a variant of the transforming growth factor alpha gene in the child.