Re: “Genetic Association and Gene-Environment Interaction: A New Method for Overcoming the Lack of Exposure Information in Controls”

We read with interest the recent paper by Kazma et al. (1), in which the authors propose a multinomial analysis to jointly test for genetic (G) and gene-environment interaction (G × E) effects using a reference control group for whom one has genotypes but no other data. The proposed logistic regression test makes clever use of a method developed for phenotypic subcategories, by treating exposed and unexposed cases as if they had separate phenotypes. The authors show that the resulting joint hypothesis test is powerful, sometimes even more powerful than a full case-control analysis. We wish to offer some clarifications and cautions related to that proposed analysis. The proposed likelihood ratio joint test statistic can alternatively (because the likelihood factors) be calculated by summing 2 chi-squared likelihood ratio test statistics, one arising from the usual case-only analysis assessing G × E and the other from a case-control analysis of genotype alone assessing the marginal effect of G. With this design, the former statistic captures all available information about violation of a multiplicative interaction null, while the latter captures all available information about genotype effects. Because the case-only analysis conditions on genotype, these 2 statistics are independent, and (under the joint null and required assumptions) their sum is itself a chi-squared statistic, with degrees of freedom equal to the sum of their respective degrees of freedom. We verified using toy data that the proposed multinomial test statistic for the composite null hypothesis is identical to the sum of the 2 standard test statistics, one for G × E and one for G. Partitioning the evidence in this way, rather than using the proposed multinomial approach, is conceptually simple and allows one to see how much of an apparent effect is attributable to genotype effects versus gene-by-environment interaction. The fact that the proposed test breaks into these components also provides insight into its properties. Why is its type I error inflated when G and E are not independent in the source population? We know the case-only method is invalid without that independence, and the proposed test inherits that flaw from its case-only component. Why is the proposed test's power sometimes better than that of a corresponding case-control analysis using full information? The case-only component imposes the G-E independence assumption, whereas the case-control analysis does not. Consequently, the case-only approach has power for G × E equivalent to that of the corresponding case-control study with infinitely many controls. Why is the proposed test's power only slightly reduced either compared with a test of G alone, when the actual effect only involves G, or compared with a case-only test of G × E, when the actual effect only involves interaction? Under these component-specific alternatives, the noncentrality parameters for the proposed test and corresponding component test are exactly the same, so the power difference depends entirely on different degrees of freedom: 2 for the joint test versus 1 for either component test. We feel that readers should be cautioned on several issues. First, the proposed test carries some of the potential benefits but also the potential invalidity of a case-only analysis of G × E. Second, in practice, the type I error rate is unknowable for the proposed procedure, although the authors exploited simulators’ omniscience when they compared power. Finally, a control group may be convenient but not appropriate for the cases at hand. Selection of suitable controls remains a difficult challenge in epidemiology, and the attractiveness of a one-size-fits-all control group in genetic studies can easily seduce investigators into affordable but biased analyses.