Hedge methods have also been proposed to have improved power, and are less susceptible to violations of the GxE independence assumption. These include: (1) Two-step procedures that filter on marginal effects, gene environment correlations in the full sample population, or other tests [Dai, et al. 2012a; Gauderman, et al. 2013; Kooperberg and Leblanc 2008; Murcray, et al. 2011; Murcray, et al. 2009]; and (2) Data adaptive methods, such as empirical Bayes, Bayes-model averaging, or frequentist model averaging [Li and Conti 2009; Mukherjee, et al. 2012; Mukherjee and Chatterjee 2008]. Several papers have compared these two types of hedge methods for the genome-wide discovery setting [Cornelis, et al. 2012; Gauderman, et al. 2013; Mukherjee, et al. 2012; Murcray, et al. 2011; Thomas, et al. 2012]. These methods all performed relatively equivalently and were shown to generally have more power compared to standard unconditional logistic regression and better control of type I error when compared to case-only approaches [Mukherjee, et al. 2012]. Standard unconditional logistic regression methods maintained proper type 1 error control.
