A maximum likelihood method for studying gene–environment interactions under conditional independence of genotype and exposure

Given the biomedical interest in gene–environment interactions along with the difficulties inherent in gathering genetic data from controls, epidemiologists need methodologies that can increase precision of estimating interactions while minimizing the genotyping of controls. To achieve this purpose, many epidemiologists suggested that one can use case‐only design. In this paper, we present a maximum likelihood method for making inference about gene–environment interactions using case‐only data. The probability of disease development is described by a logistic risk model. Thus the interactions are model parameters measuring the departure of joint effects of exposure and genotype from multiplicative odds ratios. We extend the typical inference method derived under the assumption of independence between genotype and exposure to that under a more general assumption of conditional independence. Our maximum likelihood method can be applied to analyse both categorical and continuous environmental factors, and generalized to make inference about gene–gene–environment interactions. Moreover, the application of this method can be reduced to simply fitting a multinomial logistic model when we have case‐only data. As a consequence, the maximum likelihood estimates of interactions and likelihood ratio tests for hypotheses concerning interactions can be easily computed. The methodology is illustrated through an example based on a study about the joint effects of XRCC1 polymorphisms and smoking on bladder cancer. We also give two simulation studies to show that the proposed method is reliable in finite sample situation. Copyright © 2006 John Wiley & Sons, Ltd.