Case-control studies of gene-environment interaction: Bayesian design and analysis

Abstract

With increasing frequency, epidemiologic studies are addressing hypotheses regarding gene-environment interaction. In many well-studied candidate genes and for standard dietary and behavioral epidemiologic exposures, there is often substantial prior information available that may be used to analyze current data as well as for designing a new study. In this article, first, we propose a proper full Bayesian approach for analyzing studies of gene-environment interaction. The Bayesian approach provides a natural way to incorporate uncertainties around the assumption of gene-environment independence, often used in such an analysis. We then consider Bayesian sample size determination criteria for both estimation and hypothesis testing regarding the multiplicative gene-environment interaction parameter. We illustrate our proposed methods using data from a large ongoing case-control study of colorectal cancer investigating the interaction of N-acetyl transferase type 2 (NAT2) with smoking and red meat consumption. We use the existing data to elicit a design prior and show how to use this information in allocating cases and controls in planning a future study that investigates the same interaction parameters. The Bayesian design and analysis strategies are compared with their corresponding frequentist counterparts.

Figure 1

Posterior density corresponding to NAT2*RMGF (top) and NAT2*SMOKE (bottom) interaction log odds-ratio under three different levels of uncertainty around gene-environment independence. In each case, the Dirichlet prior parameter on p₁ is kept fixed at α₁ = (5, 5, 5, 5) and the prior parameter on p₀, namely, β₀ is varied at (5,5,5,5), (20,20,20,20) and (80,80,80,80) corresponding to the three posteriors denoted by s₀ set at 20, 80 and 320 in the above figure. The arrows on the horizontal axis mark the values of the case-control (CC), the case-only (CO) and the empirical Bayes (EB) estimate. The corresponding numerical results are collected in Table 2.

Figure 2

A graphical display of LPC and ALC criterion, plotted against each candidate n for estimating NAT2*RMGF and NAT2*SMOKE interaction log odds-ratio parameter. Optimal allocation of cases and controls for each candidate n is as determined in Web Table 5 by using equation (5). In each case we consider a 95% HPD credible interval. The Dirichlet prior parameters for each strength are chosen as given in Web Table 2, based on data from MECC study. We chose the threshold values as l = 0.8 and δ = 0.05, where the horizontal lines on each graph are drawn. The n at which the graph first exceeds the horizontal line is the desired optimal sample size. Corresponding numerical results are collected in Table 4.