Response to Invited Commentary Mukherjee et al. Respond to ‘‘GE-Whiz! Ratcheting Up Gene-Environment Studies’’

We would like to thank Thomas et al. (1) for sharing their thoughtful perspectives on 2 articles (2, 3) about genome-wide gene-environment interaction analysis that are appearing in this issue of the American Journal of Epidemiology. In this rejoinder, we would like to respond to the points that were brought up with regard to the paper by Mukherjee et al. (2). Thomas et al. (1) raise the concern whether the increased power in empirical Bayes and model averaging methods could simply be due to more liberal control of type I error for these procedures. To see why this is not the case, it is useful to reevaluate what is the proper criterion for controlling false positives in such large-scale analysis. Suppose we are using a test-statistic T to test a set of null hypotheses H0k for k 1;...; K that, respectively, involve a set of nuisance parameters gk, k 1;...; K. For a genome-wide gene-environment interaction scan, for example, H0ks could be the null hypotheses for no multiplicative interactions for different single nucleotide polymorphisms, and gks could be the corresponding genotype-exposure correlation parameters in the underlying population. With this notation, the family-wise error rate (FWER) for the test-statistics T at a given critical level c can be written as FWER prH0f[ k ðTk> cÞg; where H0 is the global null hypothesis that all H0ks are true. Now applying Bonferroni’s inequality, it is easy to see tha