Testing for Gene--Environment and Gene--Gene Interactions Under Monotonicity Constraints

Recent genome-wide association studies (GWASs) designed to detect the main effects of genetic markers have had considerable success with many findings validated by replication studies. However, relatively few findings of gene--gene or gene--environment interactions have been successfully reproduced. Besides the main issues associated with insufficient sample size in current studies, a complication is that interactions that rank high based on <italic>p</italic>-values often correspond to extreme forms of joint effects that are biologically less plausible. To reduce false positives and to increase power, we develop various gene--environment/gene--gene tests based on biologically more plausible constraints using bivariate isotonic regressions for case--control data. We extend our method to exploit gene--environment or gene--gene independence information, integrating the approach proposed by Chatterjee and Carroll. We propose appropriate nonparametric and parametric permutation procedures for evaluating the significance of the tests. Simulations show that our method gains power over traditional unconstrained methods by reducing the sizes of alternative parameter spaces. We apply our method to several real-data examples, including an analysis of bladder cancer data to detect interactions between the <italic>NAT2</italic> gene and smoking. We also show that the proposed method is computationally feasible for large-scale problems by applying it to the National Cancer Institute (NCI) lung cancer GWAS data.