Case-Control Studies of Gene-Environment Interaction: Bayesian Design and Analysis

Summary

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 which may be used to analyze current data as well as for designing a new study. In this paper, 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 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 which investigates the same interaction parameters. The Bayesian design and analysis strategies are compared with their corresponding frequentist counterparts.

1 Introduction

Case-control studies are popular epidemiological tools for assessing association between a disease and a candidate gene. Most human diseases have a multifactorial etiology, involving complex interplay of multiple genetic and environmental factors. Thus, while searching for the disease causing variant(s), one cannot ignore the environmental risk factors modifying the disease risk. Similarly, one has to understand the genetic architecture of a disease while evaluating risk due to standard environmental exposures like environmental toxins, dietary exposures and physical activity levels. The National Institute of Health has recently undertaken the Genes, Environment and Health Initiative (GEI: http://www.gei.nih.gov) to integrate genomic sciences with the world of assessing environmental exposures in researching disease etiology. With these new initiatives, the emphasis on searching for gene-environment interaction (GxE) effects is becoming more common and fundamental in determining the genetic and environmental roots of complex diseases.

An important issue in estimating GxE interaction is the use of an assumption that there is no association between the genetic factor and the environmental factor (gene-environment independence assumption- Piegorsch et al., 1994). Under this assumption, the multiplicative interaction odds-ratio parameter can be estimated by data on cases only. Exploiting this assumption by means of retrospective likelihoods leads to an enormous gain in efficiency for estimating the GxE interaction parameter in a general regression model (Chatterjee and Carroll, 2005). However, the methods using gene-environment independence assumption incur bias and result in inflated Type I error rates under departures from this assumption (Albert et al., 2001; Mukherjee et al., 2008). There has been recent proposals on relaxing the gene-environment independence assumption in an empirical Bayes fashion (Mukherjee and Chatterjee, 2008) or through Bayesian model averaging (Li and Conti, 2009). Several simulation studies in the above papers illustrate that under violation of the independence assumption, one can trade off between bias and efficiency in a data adaptive way by using such composite estimators. In the current paper, we propose a full Bayes (FB) analysis and design strategy to incorporate prior belief on the assumption of gene-environment independence. Unlike the empirical Bayes (EB) or Bayesian model averaging (BMA) approach, the FB approach retains the advantage of proper Bayesian inference based on the exact posterior, without relying on the validity of the large sample Wald tests based on the asymptotic distribution of the EB or the BMA estimator. We derive a closed form expression for the posterior distribution of the interaction odds-ratio in terms of standard Beta random variables and thus there is no additional computational burden associated with the FB approach. Based on the new FB analysis strategy devised in the first half of the paper, we proceed to evaluate sample size considerations from a proper Bayesian perspective.

There has been considerable amount of work on the sample size determination (SSD) problem for unmatched case-control studies of GxE interaction (Luan et al., 2001; Garcia-Closas and Lubin, 1999; Foppa and Spiegelman, 1997, Goldstein et al., 1997; Hwang et al., 1994;). Yang et al. (1997) discuss the SSD problem for case-only designs. Gauderman (2002) considered the SSD problem under several study designs and offered a popular software, called QUANTO, widely used by many investigators (http://hydra.usc.edu/gxe). Mukherjee et al. (2008) evaluate the newly proposed shrinkage estimators of GxE interaction in terms of power and Type I error in unmatched case-control studies. Another genre of closely related literature where sample sizes are calculated for indirect use of GxE interaction in association studies for detecting genetic main effects (Kraft et al., 2007; Hein et al., 2007) has emerged, and the proposed Bayesian SSD methods could easily be adapted to the latter context.

In all of the above papers, the design approach followed is purely frequentist, where, in the planning stages of a study which aims at testing significance of the interaction odds-ratio ψ (say), certain characteristics of the sampling distribution of the test statistic (type I error, power, length of confidence interval) are controlled. Since the criteria typically involve unknown parameter values, for example, prevalence of G and E, odds ratio (OR) corresponding to main effects of G and E, OR between G and E in the control population, and a plausible value for the interaction parameter ψ, initial guesses for the true parameters are needed for evaluation and implementation of these procedures. In this sense the SSD problem has a natural Bayesian flavor even in a frequentist setting as it requires some form of prior information. The resulting sample sizes are only locally optimal choices and depend on these initial guesses which are often overly optimistic or potentially incorrect. Bayesian methods do not suffer from this limitation and one can allow different levels of uncertainty/information on the design parameters via prior distributions.

Bayesian SSD methods control certain aspects of the posterior distribution, such as precision of posterior estimates, length of Bayesian credible intervals and alike. These “characteristics” of the posterior distributions are functionals of data, and thus are random quantities under a pre-experimental setting. The probability distribution governing these random functionals is the prior predictive distribution of data. The prior predictive distribution is the marginal distribution of data integrated with respect to the prior distribution. Thus the resultant sample sizes which control for these criteria depend on the entire prior distribution, not just fixed point guesses for the parameters. Another less critical limitation of many of the frequentist sample size methods is their dependence on asymptotic distribution of the maximum likelihood estimate (MLE), whereas Bayesian methods rely on exact pre-posterior computations.

The literature on Bayesian SSD has been flourishing lately. An ensemble of papers on Bayesian SSD had appeared in The Statistician from 1995 to 1997 (Lindley, 1997; Bernardo 1997; Adcock, 1997; Joseph and Wolfson (1997); Joseph et al. (1997); Pham-Gia (1997); to name a few). Wang and Gelfand (2002) consider Monte Carlo methods for the Bayesian SSD problem whereas Walker (2003) presents a Bayesian nonparametric alternative to the SSD problem. More recently, De Santis and his colleagues have made a series of important new contributions to the field (De Santis (2007); De Santis (2006)), with the first Bayesian SSD paper on case-control studies by De Santis et al. in 2004. M'Lan et al. (2006) propose Monte Carlo methods for Bayesian SSD and propose several criteria related to interval estimation of the odds ratio parameter in a case-control study. In a section of their paper, M'Lan et al. (2006) consider the Bayesian SSD problem for GxE studies, which, to the best of our knowledge is so far the only published work on Bayesian SSD for GxE problems. M'Lan et al consider an array of Bayesian SSD criteria, study them analytically and propose elegant computational tools to identify the optimal SSD. Though M'Lan et al do consider Bayesian SSD for case-only and case-control analysis of GxE studies, the paper does not address the particular issue of incorporating prior belief on gene-environment independence. M'Lan et al. (2006) consider Bayesian SSD criteria related to interval estimation whereas we consider criteria related to testing as well. The current paper attempts to present a more complete and comprehensive account of Bayesian analysis and SSD for GxE problems, elaborates on a real application, thus adding to the timely first presentation in M'Lan et al (2006).

The rest of the article is organized as follows. Section 2 presents a description of the FB approach for GxE studies. We use data on NAT2, smoking and red meat consumption from the molecular epidemiology of colorectal cancer (MECC) study to illustrate the analytical methods. We address the problem of relaxing gene-environment independence in Section 2.1 and compare the case-control, case-only, EB, and the newly proposed FB estimator in terms of bias and mean squared error (MSE). Section 3 describes the different Bayesian SSD criteria we consider. Section 4 contains description of the general method to determine optimal allocation of cases and controls and the computational algorithm. Section 5 uses the analysis results obtained in Section 2 to plan a new study and elicit design priors based on the current results. This section then provides sample size choices under different criteria that we consider. Sensitivity of the allocation rule with respect to prior strength is assessed. We evaluate the Bayesian SSD criteria under corresponding frequentist case-control designs that are routinely used. Section 6 presents concluding discussion while some algebraic and numerical details are relegated to the Web Appendix.

2 Bayesian Analysis of Gene-Environment Interaction

Bayesian analysis of case-control data has recently received much attention (Müller and Roeder, 1997; Gustafson et al., 2002; Sinha et al., 2004, 2005; Mukherjee et al, 2007;). Lindley (1964) remains a classic reference on Bayesian analysis of contingency tables. Latorre (1982, 1984) consider deriving analytical expression for posterior inference regarding cross-ratio estimation in 2 × 2 and 2 × 4 tables. We propose the following simple Bayesian analysis of a 2 × 4 table for the situation with a binary G and a binary E and the a representation of the posterior distribution of the interaction odds-ratio parameter. The results can be directly extended to categorical G and E.

We consider the set-up of an unmatched case-control study with a binary genetic factor G and a binary environmental exposure E. Let E = 1 (E = 0) denote an exposed (unexposed) individual and G = 1 (G = 0) denote whether an individual is a carrier (non-carrier) of the susceptible genotype. Let D denote disease status, where D = 1 (D = 0) stands for an affected (unaffected) individual. Let n₀ and n₁ be the number of selected controls and cases, respectively. The data can be represented in the form of a 2 × 4 table as displayed in Table 1.

Table 1.

Data structure for a unmatched case-control study with a binary genetic factor and a binary environmental exposure.

	G = 0		G = 1
	E = 0	E = 1	E = 0	E =1	total
D = 0	r 000	r 001	r 010	r 011	n 0
D = 1	r 100	r 101	r 110	r 111	n 1

Let r₀ = (r₀₀₀, r₀₀₁, r₀₁₀, r₀₁₁) and r₁ = (r₁₀₀, r₁₀₁, r₁₁₀, r₁₁₁) denote the vector of observed cell frequencies in the controls and cases respectively. The population parameters, namely, the cell probabilities corresponding to a particular G-E configuration in the underlying control and case populations are denoted as p₀ = (p₀₀₀, p₀₀₁, p₀₁₀, p₀₁₁ = 1 − p₀₀₀ − p₀₀₁ − p₀₁₀) and p₁ = (p₁₀₀, p₁₀₁, p₁₁₀, p₁₁₁ = 1 − p₁₀₀ − p₁₀₁ − p₁₁₀), respectively. The observed vectors of cell counts can be viewed as realizations from two independent multinomial distributions, namely, r₀|p₀ ~ Multinomial(n₀, p₀) and r₁|p₁ ~ Multinomial(n₁, p₁). Let OR₁₀ = p₀₀₀p₁₀₁/p₀₀₁p₁₀₀ denote the odds-ratio associated with E for nonsusceptible subjects (G = 0), OR₀₁ = p₀₀₀p₁₁₀/p₀₁₀p₁₀₀ denote the odds-ratio associated with G for unexposed subjects (E = 0) and OR₁₁ = p₀₀₀p₁₁₁/p₀₁₁p₁₀₀ denote the odds-ratio associated with G = 1 and E = 1 compared to the baseline category G = 0 and E = 0. Therefore, ψ = OR₁₁/(OR₁₀OR₀₁) = (p₀₀₁p₀₁₀p₁₀₀p₁₁₁) / (p₀₀₀p₀₁₁p₁₀₁p₁₁₀) is the multiplicative interaction parameter of interest. Also note that

$$\psi =\frac{p_{100}{p}_{111}}{p_{101}{p}_{110}}∕\frac{p_{000}{p}_{011}}{p_{001}{p}_{010}}={\psi }_1∕{\psi }_0=\frac{\mathrm{GE}\text{odds-ratio in cases}}{\mathrm{GE}\text{odds- ratio in controls}}.$$ (1)

In (1), ψ₀ is the gene-environment association parameter which is assumed to be unity under G-E independence assumption. The likelihood of the parameters given the data is then a product of two independent multinomial likelihoods with probability vectors p_d, d = 0, 1. We assume independent Dirichlet prior distribution on p_d, namely, π(p_d) ~ Dirichlet(α_d = (α_d00, α_d01, α_d10, α_d11)), d = 0, 1. One can derive the posterior distribution of p_d again as a Dirichlet distribution via this multinomial-Dirichlet conjugate analysis.

$$\begin{array}{cc}\hfill p({\mathit{r}}_d\mid {\mathit{p}}_d)& \sim \text{Multinomial}(n_d,{\mathit{p}}_d);\pi \left({\mathit{p}}_d\right)\sim \text{Dirichlet}\left({\alpha }_d\right);\hfill \\ \hfill \Rightarrow \pi ({\mathit{p}}_d\mid {\mathit{r}}_d)& \sim \text{Dirichlet}(r_{d00}+{\alpha }_{d00},r_{d01}+{\alpha }_{d01},r_{d10}+{\alpha }_{d10},r_{d11}+{\alpha }_{d11}).\hfill \end{array}$$ (2)

Result 1: Let us consider the following transformation from p_d → (X_d, Y_d, Z_d), d = 0, 1.

$$Z_d=p_{d00},Y_d=\frac{p_{d01}}{1-p_{d00}},X_d=\frac{p_{d10}}{1-p_{d00}-p_{d01}}.$$

Then, one can show that the posterior distributions

$$\begin{array}{cc}\hfill Z_d\mid {\mathit{r}}_d& \sim \text{Beta}(r_{d00}+{\alpha }_{d00},r_{d01}+{\alpha }_{d01}+{\alpha }_{d10}+r_{d11}+{\alpha }_{d11}),\hfill \\ \hfill Y_d\mid {\mathit{r}}_d& \sim \text{Beta}(r_{d01}+{\alpha }_{d01},r_{d10}+{\alpha }_{d10}+r_{d11}+{\alpha }_{d11}),\hfill \\ \hfill X_d\mid {\mathit{r}}_d& \sim \text{Beta}(r_{d10}+{\alpha }_{d10},r_{d11}+{\alpha }_{d11}).\hfill \end{array}$$

And (X_d, Y_d, Z_d|r_d) are mutually independent random variables.

Proof: Follows by elementary Jacobian calculation corresponding to the transformation. For a similar argument for cross-ratio estimation in a 2 × 2 table, see Lecoutre and Charron (2000). Note that, we can express the GE odds-ratio parameter in cases and controls as, (for d = 0, 1,)

$${\psi }_d=\frac{Z_d(1-X_d)}{X_d{Y}_d(1-Z_d)}.$$ (3)

Finally, the interaction log odds-ratio can be expressed as β = log(ψ) = log(ψ₁) − log(ψ₀).

We assume that the stochastic model parameters in cases and controls, namely, p₁ and p₀ are independent, so π(p₀, p₁) = π(p₀)π(p₁). Thus in the proposed FB analysis with the above prior-likelihood-posterior structure, it is extremely easy to generate the posterior distribution of β, namely π(β|r₀, r₁) in terms of (X_d, Y_d, Z_d|r_d) which are independent Beta random variables and one does not have to resort to any numerical integration techniques. This, we believe is a very attractive feature of the FB approach as it provides the flexibility of obtaining the full posterior distribution of the interaction parameter using extremely inexpensive computation. Latorre (1984) derived expression for the posterior distribution of ψ using sum of four infinite series, but we work with exact simulation from the posterior distribution of ψ instead of using these expressions.

Remark 1: Asymptotic Normality of the Posterior: The asymptotic normality of the posterior distribution follows by approximating the joint posterior distribution of p₀ and p₁ by Laplace approximation around the posterior mode (or the MLE) (Ghosh et al., 2006, p. 62). Multivariate delta theorem can then be used to derive the asymptotic approximation for the distribution of log(ψ₀) and log(ψ₁). By independence of the data and prior parameters in cases and controls, the asymptotic posterior distribution for log(β) is derived as,

$$\text{log}\left(\beta \right)\mid {\mathit{r}}_0,{\mathit{r}}_1\stackrel{d}{\to }N({\mu }_1-{\mu }_0,{\sigma }_0^2+{\sigma }_1^2),$$

where, for d, g, e = 0, 1,

$${\mu }_d=\text{log}\left(\frac{{\tilde{p}}_{d00}{\tilde{p}}_{d11}}{{\tilde{p}}_{d01}{\tilde{p}}_{d10}}\right),{\sigma }_d^2=\sum _{g=0}^1\sum _{e=0}^1(1∕{\tilde{p}}_{dge}),{\tilde{p}}_{dge}=\frac{r_{dge}+{\alpha }_{dge}}{n_d+{\sum }_{g=0}^1{\sum }_{e=0}^1{\alpha }_{dge}}.$$

For approximation around the MLE one can simply replace p̃_dge by p̂_dge = r_dge/n_d. Again, though sample size calculations in terms of a normal variate my be convenient, we do not use this approximation in our subsequent calculations as generation from the exact posterior distribution itself is straightforward.

Remark 2: Prior Structure: Instead of the cell probability vectors (p₀, p₁), from an epidemiologic perspective, it may be more natural to have prior guesses on an alternative parametrization, say for example in terms of (P_G, P_E, OR₀₁, OR₁₀, ψ₀, ψ). We have formulated the problem in terms of the multinomial-Dirichlet structure for computational convenience. However, one can use the following relationships to translate the prior guesses back and forth between the two parameterizations. Directly imposing normal priors on the log-odds ratio scale will distort the conjugacy of the analysis which we would like to retain.

The control probability vector p₀ is related to (P_G, P_E, ψ₀), by the following equation

$${\psi }_0=\frac{p_{000}(p_{000}-(1-P_G-P_E))}{(1-P_G-p_{000})(1-P_E-p_{000})},p_{001}=1-P_G-p_{000},p_{010}=1-P_E-p_{000}.$$

The case-probability vector p₁ can be obtained from the disease risk parameters and p₀ as,

$$p_{100}=\frac{p_{000}}{p},p_{101}=\frac{p_{001}}{p}\cdot O{R}_{10},p_{110}=\frac{p_{010}}{p}\cdot O{R}_{01},p_{111}=\frac{p_{011}}{p}\cdot O{R}_{10}\cdot O{R}_{01}\cdot \psi ,$$

where p = p₀₀₀ + p₀₀₁ · OR₁₀ + p₀₁₀ · OR₀₁ + p₀₁₁ · OR₁₀ · OR₀₁ · ψ.

In the next section, we present FB inference regarding the interaction log odds-ratio parameter β, through our analysis of the MECC data. We specially focus on the issue of incorporating uncertainty around the gene-environment independence assumption through our prior specification on the G-E association parameter log(ψ₀).

2.1 Incorporating uncertainties around G-E independence

In the 2 × 4 Table 1, the unconstrained MLE of the interaction parameter β based on case-control data is given by,

$${\widehat{\beta }}_{CC}=\text{log}\left(\frac{r_{001}{r}_{010}{r}_{100}{r}_{111}}{r_{000}{r}_{011}{r}_{101}{r}_{110}}\right).$$

The G-E independence assumption is realistic for external exposures like environmental pollutants, toxins or radio-active substances, but is at best dubious for genes and behavioral exposures which may share a common metabolic pathway. Note that under the assumption of G-E independence, the denominator in (1), namely ψ₀ 1, and one can unbiasedly estimate β = log(ψ) by data on cases only, namely, by the case-only (CO) estimator of β,

$${\widehat{\beta }}_{CO}=\text{log}\left(\frac{r_{100}{r}_{111}}{r_{101}{r}_{110}}\right).$$

This case-only estimator is much more efficient than ${\widehat{\beta }}_{CC}$, but incurs bias when ψ₀ ≠ 1.

Mukherjee et al. (2008) present a comprehensive discussion of different estimators of GxE interaction parameter including an EB estimator which combines ${\widehat{\beta }}_{CC}$ and ${\widehat{\beta }}_{CO}$ in a data-adaptive way, depending on the strength of G-E association. The FB analysis proposed in this paper provides another option to incorporate uncertainty around G-E independence and still provides shrinkage when the assumption is tenable in the light of the data. We work with a completely general case-control likelihood which does not impose any assumption on G-E independence. The independence assumption (log(ψ₀) = 0) is reflected only through the prior structure. We assume a prior on the G-E association parameter log(ψ₀) that is centered around zero to reflect G-E independence but vary the strength/variance of the prior to allow uncertainty around the assumption of independence. The advantage of this approach over the EB approach is to retain the flexibility of a full Bayesian inference and obtain the entire posterior of data. Moreover, the FB approach is based on exact computations, whereas the variance expression and derived Wald tests for the EB approach depend on the accuracy of certain large sample approximations and asymptotic normality of the sampling distribution of the EB estimator. The limitation of the FB approach is sensitivity of the results to prior choices and how to select the prior strength/uncertainty.

We first define the strength of the Dirichlet prior on the control and case probability vector as $s_d={\sum }_{g=0}^1{\sum }_{e=0}^1{\alpha }_{dge}$ for d = 0, 1. The total prior strength is given by s = s₀ + s₁. One can choose s₀ and s₁ based on the level of uncertainty of the data coming from the control and case population respectively. For example, under the same level of uncertainty in both the case and control parameters, and a fairly noninformative prior choice which centers the prior on β roughly around zero, one may select α₀ = α₁ = (5, 5, 5, 5) implying a total prior strength of s = s₀ + s₁ = 20 + 20 = 40. Different choices of s₀ and s₁ induce different variances on the parameters log(ψ₀) and log(ψ₁), an expression is provided in the Web Appendix. For reflecting different degrees of belief on G-E independence assumption, we treat p₀ and p₁ asymmetrically in our prior specification. We choose a fairly non-informative prior on p₁, and we choose a prior on p₀ which leads to a prior on log(ψ₀) roughly centered at zero (or G-E independence). We then vary s₀, to induce different prior variances around the independence assumption.

We now describe FB analysis of the MECC study data with different choices of s₀. The MECC study is a population-based case-control study of patients who received a diagnosis of invasive colorectal cancer (CRC) in northern Israel between March 31, 1998 and March 31, 2004. Participants were interviewed to obtain demographic information, personal and family history of cancer, medical history, medication use, and health habits. They also completed a dietary questionnaire and a blood sample was collected. NAT2 genotyping was used to classify individuals as fast or slow acetylators for consistency with previously published literature (Roberts-Thomson et al., 1996). The acetylator phenotype has been hypothesized to modulate the relationship between red meat and risk of CRC, with increasing red meat consumption associated with increased risk of CRC among fast, but not slow acetylators in some, (Roberts-Thomson et al., 1996) but not all studies (Barrett et al., 2003). NAT2 has also been suggested as a potential modifier of the relationship between cigarette smoking and CRC as well as colorectal adenomas, although studies are not consistent (Barrett et al., 2003). We analyze the interaction between NAT2 and frequency of grilled red meat consumption (RMGF) as well as NAT2 and smoking status (SMOKE) with the existing study base of 1785 cases and 1971 controls. The observed cell counts are presented in Web Table 1 for NAT2, SMOKE, and RMGF, stratified by case-control status.

Table 2 presents the dataset from the MECC study analyzed with the three different methods (CC, CO, EB), and the proposed FB approach with varying levels of prior variance on log(ψ₀), whereas the prior parameter on p₁, namely, α₁ is fixed at a fairly non-informative choice (5,5,5,5). One can notice that for NAT2*RMGF, the CC estimator of β is 0.39, whereas the CO estimator is 0.11. The MLE of log(ψ₀) is -0.27 with s.e.=0.14. By varying the prior strength on p₀ from 20 to 320 (inducing a change in prior variance on log(ψ₀) from 0.9 to 0.05), one can trade off between the case-control and case-only analysis with FB estimates varying from 0.38 to 0.13 respectively. For NAT2*SMOKE, the CC estimator is 0.18 whereas the CO estimator is -0.12. In contrast to the situation with RMGF, for SMOKE, even with a very sharp prior around zero with strength 320, reflecting substantial prior belief in G-E independence, the FB inference tracks the case-control estimator (FB estimate with prior strength 320 is 0.11). This is because the data contain quite strong evidence that log(ψ₀) departs from 0, that NAT2 and SMOKE exhibit departures from independence (MLE of log(ψ₀) is -0.30, s.e.=0.09). Thus unlike the NAT2*RMGF data, the NAT2*SMOKE data resist the prior assumption of G-E independence and cannot be shrunk towards a case-only analysis even with a quite strong prior around this assumption. The EB estimator (Mukherjee and Chatterjee (2008)) in Table 2 also trades off between the CC and CO estimators depending on the strength of G-E independence reflected in the data. In Web Table 2, we present posterior estimates of log(ψ₁) and log(ψ₀) separately, to assess effects of the prior. Figure 1 illustrates the posterior of β under three different prior uncertainties on G-E independence with the CC,CO and EB estimates marked by arrows on the horizontal axis. One can notice the changes in posterior behavior corresponding to these two different exposures reflecting different levels of evidence in support of the independence assumption in the two datasets. The differences in the CC and CO estimates in both datasets reflect how drastically the assumption of G-E independence can affect inference regarding GxE interaction. A supplementary analysis that treats p₀ and p₁ symmetrically is presented in Web Table 3 and Web Figure 1.

Table 2.

The different estimates of log-interaction parameter β in the MECC data corresponding to NAT2*RMGF and NAT2*SMOKE interaction. CC, CO and EB refer to the case-control, case-only and empirical Bayes estimator of β while under the FB analysis, we use a fairly noninformative prior on the G-E log odds-ratio in cases, namely, log(ψ₁), and vary the strength of the prior s₀ on G-E log odds-ratio among controls, i.e., log(ψ₀), to reflect different levels of uncertainty on gene-environment independence. For FB analysis “Est” refers to the posterior mean whereas “PSD” is the posterior standard deviation.

		NAT2*RMGF		NAT2*SMOKE
Method		Est.	s.e.	Est.	s.e.
CC		0.394	0.194	0.177	0.137
CO		0.114	0.132	-0.121	0.099
EB		0.303	0.206	0.125	0.145

	s0	Est	PSD	Est	PSD
FB	20	0.377	0.189	0.175	0.137
	80	0.304	0.178	0.165	0.135
	320	0.130	0.174	0.112	0.132

†For FB, the case Dirichlet parameter α₁ is fixed as (5,5,5,5) with s₁ = 20 whereas the prior strength on controls namely s₀ of 20, 80, and 320 correspond to the control Dirichlet parameter α₀ being set at (5,5,5,5), (20,20,20,20), and (80,80,80,80) respectively.

‡Prior variance corresponding to log(ψ₁) is 0.90 whereas prior variance on log(ψ₀) under s₀=20, 80, and 320 are 0.90, 0.20, and 0.05 respectively.

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.

Table 3 presents results of a small scale simulation study comparing the CC, CO, EB and the FB estimate with varying prior strength on log(ψ₀). We consider four simulation settings,(i) when independence assumption holds with ψ₀ = 1; (ii) under modest departure from the independence assumption, with positive G-E association, ψ₀ = 1.25. Settings (iii) and (iv) are based on the observed NAT2*RMGF and NAT2*SMOKE configurations which exhibit negative association between G and E. Under the independence assumption in (i) FB 80 and FB 320 have significant MSE advantage over the classical CC estimator for modest sample sizes, the gain is comparable to the CO estimator. Under violation of the independence assumption, FB 20 and FB 80 provide a better control on the bias when compared to the CO estimator. FB 80 often exhibits reduced bias compared to the EB estimator with similar or less MSE values. The trade-off between bias and MSE of the FB approach depends on the strength of the prior. Though the FB prior strength regarding G-E association can be chosen in a data adaptive way, resembling the EB approach, or be elicited from historical data, using a generic prior of moderate strength like FB 80 appears to be a fairly robust choice across all sample sizes and G-E association scenarios for this example.

Table 3.

The estimated bias and MSE corresponding to the different estimation procedures under four simulation settings. The four settings correspond to the following true values for the parameters (P_G, P_E, OR₀₁, OR₁₀, ψ₀, ψ): (i) (0.3, 0.5, 1, 1, 1, 1.3); (ii) (0.3, 0.5, 1, 1, 1.25, 1.3) (iii) (0.36, 0.13, 1.02, 1.06, 0.76, 1.48) and (iv) (0.36, 0.44, 1.02, 0.87, 0.74, 1.19). Setting (i) reflects G-E independence (ψ₀ = 1) and setting (ii) reflects moderate but positive G-E dependence (ψ₀ = 1.25). Whereas Settings (iii) and (iv) exactly replicate the observed data in MECC study for NAT2*RMGF and NAT2*SMOKE interaction, where negative association is observed between G and E. CC, CO and EB refer to the case-control, case-only and empirical Bayes estimator of β while under the FB analysis, we use a fairly noninformative prior on the G-E log odds-ratio in cases, namely, log(ψ₁), and vary the strength of the prior on G-E log odds-ratio among controls, namely, log(ψ₀), to reflect different levels of uncertainty on gene-environment independence. Results based on 5000 simulated datasets under each setting.

n1	n0	Method	Simulated 1		Simulated 2		NAT2*RMGF		NAT2*SMOKE
			bias	MSE	bias	MSE	bias	MSE	bias	MSE
200	200	CC	0.007	0.200	0.001	0.200	0.023	0.400	0.002	0.181
		CO	0.006	0.098	0.228	0.148	-0.291	0.256	-0.304	0.181
		EB	0.008	0.131	0.101	0.152	-0.112	0.288	-0.126	0.156
		FB 20	0.013	0.176	0.075	0.167	-0.076	0.226	-0.015	0.146
		FB 80	0.018	0.122	0.087	0.147	-0.101	0.249	-0.098	0.124
		FB 320	0.016	0.104	0.156	0.141	-0.247	0.253	-0.202	0.125
500	500	CC	-0.007	0.076	0.003	0.077	0.004	0.153	0.003	0.071
		CO	0.000	0.037	0.227	0.090	-0.285	0.146	-0.299	0.124
		EB	-0.004	0.049	0.092	0.070	-0.112	0.129	-0.100	0.074
		FB 20	0.013	0.053	0.034	0.075	-0.064	0.121	-0.004	0.065
		FB 80	0.012	0.048	0.060	0.069	-0.111	0.114	-0.050	0.060
		FB 320	0.008	0.040	0.108	0.064	-0.204	0.136	-0.127	0.075
1000	1000	CC	0.004	0.037	0.000	0.038	0.010	0.073	-0.001	0.035
		CO	0.001	0.019	0.225	0.069	-0.279	0.110	-0.297	0.105
		EB	0.002	0.025	0.077	0.040	-0.091	0.072	-0.078	0.042
		FB 20	0.014	0.027	0.016	0.037	-0.027	0.064	-0.004	0.034
		FB 80	0.013	0.024	0.032	0.035	-0.089	0.074	-0.030	0.032
		FB 320	0.011	0.018	0.076	0.034	-0.185	0.107	-0.101	0.036
2000	2000	CC	0.000	0.019	-0.002	0.019	0.001	0.036	0.001	0.018
		CO	0.001	0.009	0.222	0.058	-0.282	0.096	-0.298	0.098
		EB	0.000	0.012	0.055	0.022	-0.077	0.042	-0.049	0.022
		FB 20	0.005	0.019	0.006	0.018	-0.018	0.034	-0.001	0.017
		FB 80	0.005	0.018	0.015	0.018	-0.061	0.038	-0.015	0.017
		FB 320	0.004	0.012	0.042	0.013	-0.141	0.091	-0.061	0.019

†For FB, the case Dirichlet parameter α₁ is fixed as (5,5,5,5) with s₁ = 20 and values of s₀ at 20, 80, and 320 correspond to the control Dirichlet parameter α₀ being set at (5,5,5,5), (20,20,20,20), and (80,80,80,80) respectively.

‡Under the above setting, prior variance corresponding to log(ψ₁) is 0.90 whereas prior variance on log(ψ₀) under s₀=20, 80, and 320 are 0.90, 0.20, and 0.05 respectively.

3 Sample Size Determination Criteria

The criteria for SSD in the Bayesian domain stem mainly from two different perspectives. One approach as introduced by Raiffa and Schlaifer (1961) and more recently by Bernardo (1997) and Lindley (1997) is to pose the problem in a pure utility based decision theoretic framework. The other approach, (which may not always reconcile with a maximizing expected utility principle) followed by Joseph et al. (1995), Pham-Gia (1997) and De Santis (2007), is to consider a functional of the posterior distribution, routinely used in Bayesian inference. Recall that at the design stage, before one observes data, functionals of the posterior distribution are random variables that are governed by the prior predictive density. For the 2 × 4 table described in Section 2, the prior predictive distribution of the random cell counts R_d under the data and prior structure of (2) is given by the following multinomial-Dirichlet distribution, for d = 0, 1,

$$\begin{array}{cc}\hfill \mathbb{P}({\mathit{R}}_d={\mathit{r}}_d)& =\int p({\mathit{R}}_d={\mathit{r}}_d\mid {\mathit{p}}_d)\pi \left({\mathit{p}}_d\right)d{\mathit{p}}_d\hfill \\ \hfill & =\left(\begin{array}{c}\hfill n_d\hfill \\ \hfill r_{d00},r_{d01},r_{d10},r_{d11}\hfill \end{array}\right)\frac{\Gamma ({\alpha }_{d00}+{\alpha }_{d01}+{\alpha }_{d10}+{\alpha }_{d11})}{\Gamma \left({\alpha }_{d00}\right)\Gamma \left({\alpha }_{d01}\right)\Gamma \left({\alpha }_{d10}\right)\Gamma \left({\alpha }_{d11}\right)}\times \int p_{d00}^{r_{d00}+{\alpha }_{d00}}{p}_{d01}^{r_{d01}+{\alpha }_{d01}}{p}_{d10}^{r_{d10}+{\alpha }_{d10}}{p}_{d11}^{r_{d11}+{\alpha }_{d11}}d{\mathit{p}}_d\hfill \\ \hfill & =\left(\begin{array}{c}\hfill n_d\hfill \\ \hfill r_{d00},r_{d01},r_{d10},r_{d11}\hfill \end{array}\right)\frac{\Gamma ({\alpha }_{d00}+{\alpha }_{d01}+{\alpha }_{d10}+{\alpha }_{d11})}{\Gamma \left({\alpha }_{d00}\right)\Gamma \left({\alpha }_{d01}\right)\Gamma \left({\alpha }_{d10}\right)\Gamma \left({\alpha }_{d11}\right)}\times \frac{\Gamma ({\alpha }_{d00}+r_{d00})\Gamma ({\alpha }_{d01}+r_{d01})\Gamma ({\alpha }_{d10}+r_{d10})\Gamma ({\alpha }_{d11}+r_{d11})}{\Gamma ({\alpha }_{d00}+r_{d00}+{\alpha }_{d01}++r_{d01}+{\alpha }_{d10}+r_{d10}+{\alpha }_{d11}+r_{d11})}.\hfill \end{array}$$ (4)

Since the data and prior parameters among cases and controls are assumed to be independent, the joint prior predictive distribution of R₀ and R₁ is simply the product of the prior predictive distributions of R₀ and R₁ .

We consider the following four Bayesian SSD criteria:

1. The average posterior variance (APV): For given γ > 0, we choose the smallest n satisfying

   $$\mathbb{E}\left\{Var[\beta \mid {\mathit{R}}_0,{\mathit{R}}_1]\right\}\le \gamma ,$$

   where V ar[β|R₀, R₁] represent the posterior variance of β given data. Based on the previous representation of β in terms of the difference of G-E log odds-ratio in cases and controls, this implies that $\mathbb{E}\left\{Var[\text{log}\left({\psi }_1\right)\mid {\mathit{R}}_1]\right\}+\mathbb{E}\left\{Var[\text{log}\left({\psi }_0\right)\mid {\mathit{R}}_0]\right\}\le \gamma$. We present a closed form expression for the approximate posterior variance in the Web Appendix.
2. The average length criterion (ALC): Let L_α (R₀, R₁) denote the random length of the 100(1-α)% level highest posterior density (HPD) interval for β. For a given l > 0, we find the smallest n satisfying

   $$\mathbb{E}\left\{L_{\alpha }({\mathit{R}}_1,{\mathit{R}}_0)\right\}\le l.$$
3. Length probability criterion (LPC): The ALC ensures that on an average one does not obtain large non-informative HPD intervals. However, it does not control for variability in the random length values L_α (R₁, R₀). A cautious investigator may not be satisfied with the average assurances provided by the ALC criterion. The LPC criterion provides a more conservative choice where we find the smallest n satisfying

   $$\mathbb{P}\{L_{\alpha }({\mathit{R}}_1,{\mathit{R}}_0)\ge l\}\le \delta ,$$

   for fixed l and δ ∈ (0, 1). Joseph and Bélisle (1997) derive this criterion as a special case of their proposed worse outcome criterion.
4. Criterion for hypothesis testing: Consider, for example, the problem of testing the hypothesis H₀ : β > 0 versus H_a : β ≤ 0. In a Bayesian paradigm one selects one of the two hypotheses, by comparing the posterior probability of the null and the alternative. We say that there is evidence in favor of the null (or alternative) hypothesis if π(H₀|r₀, r₁) ≥ ρ (or ≤ 1 − ρ)where ρ 2 (0.5, 1) is a chosen probability level. A state of indecision is reached when 1 − ρ ≤ π(H₀|R₀, R₁) ≤ ρ, when there is neither sufficient evidence to support H₀, or to refute it. The sample size criterion proposed by Verdinelli (1996) and also used in De Santis et al. (2004) is to choose the smallest n such that the probability of an inconclusive decision stays below a certain small threshold δ ∈ (0, 1), i.e.,

   $$TC1\stackrel{\text{def}}{=}\mathbb{P}\{1-\rho \le \pi ({\mathit{H}}_0\mid {\mathit{R}}_0,{\mathit{R}}_1)\le \rho \}\le \delta .$$

   In general, one can test any composite null and alternative hypothesis H₀ : β ∈ Φ₀ versus ${\mathit{H}}_a:\beta \in {\Phi }_1={\Phi }_0^c$ by comparing π(H_j|r₁, r₀) = ∫_Φj π(β|r₁, r₀)dβ, j = 0, 1, and employ the above SSD criterion. Along with the above testing criterion TC1, we also evaluate an analogue of frequentist power criterion which is simply an evaluation of $TC2\stackrel{\text{def}}{=}\mathbb{P}(\pi ({\mathit{H}}_a\mid {\mathit{R}}_0,{\mathit{R}}_1)\ge \rho )$ or the chance of selecting the alternative hypothesis with a pre-specified high posterior probability ρ. A discussion of various SSD criteria for hypothesis testing and model selection can be found in Weiss (1997) and Wang and Gelfand (2002).

Having introduced the ingredients of the SSD problem for GxE, we now proceed to describe the optimization scheme to allocate cases and controls and to determine the optimal total sample size n with respect to each specific criterion.

4 Computational Algorithm

We consider two approaches towards determining the optimal SSD.

1. The Direct allocation of optimal number of cases and controls and optimal sample size: For each possible sample size n, find the best pair of n₁ and n₀, by directly minimizing/optimizing any of the criteria under consideration, namely, (a)-(d). So, for each candidate sample size n, one has to evaluate the chosen criterion for (n − 1) configurations n₁ = 1, ··· , n − 1 and n₀ = n − n₁ and check whether the desired threshold is met. Then repeat this for all n over a grid and find the minimum n = n₁ + n₀ for which the pre-specified threshold levels are met.

2. The two-step allocation strategy: Another allocation strategy, akin to the one considered in De Santis et al. (2004) is to follow a two step procedure where at the first step, for each n, one allocates n₁ and n₀ such that the average posterior uncertainty in ψ_d, d = 0, 1 is the same in cases and controls. When prior information on cases is sharper than the one in controls, this requirement of equal posterior uncertainty intuitively will lead to allocating more controls than cases and vice versa. Thus, at the first step, one chooses the pair n₁ = (1, ··· , n − 1), n₀ = n − n₁, such that,

   $$\mid \mathbb{E}\left(\text{Var}({\psi }_1\mid {\mathit{R}}_1)\right)-\mathbb{E}\left(\text{Var}({\psi }_0\mid {\mathit{R}}_0)\right)\mid$$ (5)

   is closest to zero. The expression for V ar(ψ_d|r_d), d = 0, 1, is provided in the Web Appendix. One then evaluates any of the chosen criteria (a)-(d) for each n with this specific allocation of n₁ and n₀, and chooses the minimum n for which the threshold conditions are met. This second approach has the advantage that for each candidate n, the best split between cases and controls according to (5) remains the same for each criterion and only the second step of searching for the optimal n needs to be implemented separately for each criterion.

For a given n₁ and n₀, for evaluating the $\mathbb{P}$ and $\mathbb{E}$ with respect to the Multinomial-Dirichlet prior predictive distribution in (4), one first has to select the design prior parameters α_d, d = 0, 1. In the next section, we show how one can choose these prior parameters based on the observed data in Table 2. Given α_d and n_d one can, (I) generate r_d by following the Gibbs sampling algorithm described in Casella and George (1992) by sequentially generating a large number of observations (10000, say) from the full conditional distributions p(r_d|p_d) and π(p_d|r_d), as described in (2). As the length of the sequence increases, the samples from r_d converge stochastically to the marginal distribution $\mathbb{P}\left({\mathit{R}}_d\right)$ as given in (4). The convergence of the chain was assessed through the diagnostic proposed by Geweke (1992). (II) For each realization of r_d, generate 10000 random samples from the posterior distribution of (X_d, Y_d, Z_d|r_d), which are standard Beta distribution as described in Result 1. (III) Finally, in each of the above 10000 generations for a given r_d, compute ψ_d as a function of (X_d, Y_d, Z_d|r_d), d = 0, 1 as given in (3) and finally compute β in terms of ψ_d. This process generates 10000 observations from the posterior distribution of β for a given (r₁, r₀). One can now evaluate the length of the HPD interval or any of the other characteristics of the posterior distribution of β , as described in (a)-(d) of Section 3.

After repeating the above Steps (I)-(III) a large number of times, say 50000 times, one is able to obtain Monte Carlo estimates of the $\mathbb{P}$ and $\mathbb{E}$ involved in each of the criterion. One can then find the minimum n for which they meet the desired threshold requirement.

Remark 3: M'Lan et al. (2006) indicate that there may be substantial error in the Monte Carlo estimates for the criterion function, and propose a curve fitting technique to the criterion function over a grid of sample sizes instead of direct optimization. We recognize this possibility and use a large number of Monte Carlo samples to provide more stable estimates. The approach of M'Lan et al (2006) could be adapted as well for more efficient computation. However, unlike M'Lan et al, we simplify our search for optimal design by a somewhat different two-step allocation strategy (De Santis et al (2004)) with a closed form expression for the posterior variance being used to determine the best split for each given n at the first step. Consequently, the SSD obtained by direct optimization of criteria (a)-(d) may not necessarily reside in the subclass that minimize (5), and thus be different from the ones obtained by the two-step procedure. The use of the two-step strategy reduces our computational burden enormously by limiting the design search space at the second stage to only the optimal (n₀, n₁) that are identified at the first stage by minimizing (5) for each given n. This makes repeated evaluation of the SSD criteria at the second step feasible and one can take the average of the replicated values to reduce Monte Carlo errors further.

5 Illustrative Example

For a general discussion regarding the dual issue of choosing design priors versus analysis priors, we refer the reader to (Spiegelhalter et al. (2004), Wang and Gelfand (2002), De Santis, (2006)). For specifying the prior hyperparameters based on current MECC study data, we used the same technique applied in Bernardo et al. (1994) and De Santis et al. (2004). Based on observed data given in Section 2, the prior estimates of α_d were calculated by using the following equation.

$$\frac{{\alpha }_{dge}}{s_d}=\frac{r_{dge}}{n_d}(d=0,1;g=0,1;e=0,1),\frac{s_0}{s_1}=\frac{n_0}{n_1},\sum _{d=0}^1\sum _{g=0}^1\sum _{e=0}^1{\alpha }_{dge}=s.$$ (6)

The obtained design prior parameters are displayed in Web Table 4 for RMGF and SMOKE data. The variation in prior strength controls the prior variance, with the prior mean for β and all other parameters being fixed at the corresponding MLE obtained from the current MECC database. The details corresponding to each setting are provided in Web Table 4.

Based on the prior parameters as chosen above, the optimal pairs can be selected via either method (i) or (ii) previously explained. We present the best pairs for some candidate n = n₁ + n₀ obtained in the first step of the two-stage approach in Web Table 5 by equating the expected posterior uncertainty in cases and controls for both SMOKE and RMGF data respectively. One can notice that for each n the number of cases and controls needed is almost even for NAT2*SMOKE interaction, whereas more controls are needed for the NAT2*RMGF interaction. In fact, the large sample asymptotic variances corresponding to the MLEs of log(ψ₀) and log(ψ₁) are 0.0202 and 0.0175 for RMGF and 0.009 and 0.0098 for SMOKE data, intuitively justifying the reason for this type of allocation based on the information contained in the design prior. Note that the effect of prior on the optimal allocation rule is more pronounced for smaller n than larger n.

To get a sense of the nature of the different criteria, we report the criterion values for different candidate n in Table 4, with the optimal pairs as determined by Web Table 5. The three criteria LPC, ALC and APV focus on estimation accuracy. For the length based criteria we consider 95% HPD intervals. One can notice as expected that the LPC criterion provides a larger sample size than the ALC criterion as the former is controlling the probability distribution of the random interval lengths whereas the latter is simply controlling for the mean length. With increasing prior strength the value of each criterion decreases and the required n needed to attain a certain threshold becomes smaller. Comparing the results for RMGF with the results for SMOKE one can notice that everything else remaining same, much larger sample sizes are needed for RMGF, to reach the same level of precision when compared to SMOKE. Figure 3 presents a graphical display of the ALC and LPC criterion with l = 0.8, plotted against each candidate n which also illustrates the same features of the SSD summarized above.

Table 4.

The LPC, APC and APV criterion values corresponding to estimation of NAT2*RMGF and NAT2*SMOKE interaction for certain representative values of n. Optimal allocation of cases and controls for each candidate n is as determined in Web Table 5. For LPC and ALC we consider a 95% HPD credible interval. The Dirichlet prior parameters for each strength are chosen as given in Web Table 4, based on data from MECC study. We evaluated the LPC criterion by choosing the threshold length of the 95% HPD credible interval as l = 0.8 for illustration purposes. An investigator can find optimal sample sizes by selecting the smallest n for which the threshold that is desired for his/her criterion is first met or refine the search in that interval. For example, the sample sizes in bold face denote approximately optimal sample sizes if ALC was desired to be approximately 0.8 or lower.

Prior strength	NAT2*RMGF						NAT2*SMOKE
	n	n1	n0	LPC	ALC	APV	n	n1	n0	LPC	ALC	APV
200	500	202	298	1.000	1.870	0.232	500	256	244	1.000	1.259	0.104
	1000	412	588	1.000	1.428	0.136	1000	504	496	1.000	0.961	0.061
	2000	828	1172	0.999	1.060	0.075	1500	754	746	0.606	0.808	0.043
	3000	1262	1738	0.795	0.878	0.052	1700	848	852	0.061	0.764	0.038
	3500	1514	1986	0.521	0.816	0.045	2000	998	1002	0.010	0.711	0.033
	3700	1594	2106	0.416	0.795	0.042	2500	1250	1250	<0.001	0.642	0.027
	4000	1728	2272	0.258	0.760	0.039	3000	1500	1500	< 0.001	0.590	0.023
500	500	184	316	1.000	1.502	0.149	500	268	232	1.000	1.041	0.071
	1000	402	598	1.000	1.229	0.100	1000	516	484	1.000	0.851	0.048
	1500	620	880	1.000	1.062	0.075	1300	664	636	0.063	0.777	0.040
	2000	836	1164	1.000	0.953	0.060	1500	762	738	0.032	0.737	0.036
	2500	1054	1446	0.933	0.870	0.050	2000	1012	988	0.009	0.659	0.029
	3000	1266	1734	0.492	0.803	0.043	2500	1260	1240	<0.001	0.602	0.024
	3500	1480	2020	0.152	0.754	0.038	3000	1506	1494	<0.001	0.557	0.023
	3800	1606	2194	0.045	0.726	0.035	3500	1746	1754	<0.001	0.534	0.021
	4000	1704	2296	0.023	0.709	0.033	4000	2000	2000	<0.001	0.531	0.020
1000	500	150	350	1.000	1.214	0.097	500	288	212	1.000	0.847	0.047
	1000	370	630	1.000	1.051	0.073	700	386	314	0.357	0.796	0.042
	1500	586	914	1.000	0.940	0.058	780	421	349	0.051	0.780	0.040
	2000	806	1194	0.977	0.859	0.049	1000	536	464	0.013	0.734	0.035
	2500	1018	1482	0.423	0.795	0.042	1500	782	718	< 0.001	0.657	0.028
	2900	1198	1702	0.060	0.753	0.038	2000	1030	970	<0.001	0.599	0.024
	3000	1240	1760	0.032	0.743	0.036	2500	1278	1222	<0.001	0.555	0.020
	3500	1456	2044	0.002	0.701	0.032	3000	1526	1474	<0.001	0.519	0.018
	4000	1680	2320	<0.001	0.664	0.029	4000	2000	2000	<0.001	0.432	0.014

Web Table 6 presents the optimal pair (n₁, n₀) obtained by the direct approach (i) by evaluating the ALC criterion instead of the two-stage allocation approach. Though there are certain numerical differences, the main comparative features between the two datasets remain the same. The trend in differences in number of cases and controls is less obvious in Web Table 6 than the two-stage approach (Web Table 5) as the first-step criterion of equating posterior uncertainty in the two-step approach is not imposed in the direct optimization approach.

We will like to point out that the criteria considered in Table 4 only focus on variance but not the aspect of detecting given departures from a null hypothesis or the effect size. In Table 5, we present the results on the two testing criteria we discussed in (d) for testing H₀ : β ≤ 0 vs H_a : β > 0. Notice that with the test criterion TC1 which minimizes the probability of reaching the indifference zone, one can actually need more sample size with a more informative prior with higher strength (for SMOKE data in Table 9 with s = 200: n₁ = 1500, n₀=1500; with s = 500: n₁ = 2761, n₀ = 2739). This may seem quite surprising, but has also been observed in De Santis et al. (2004). The explanation lies in the fact that for the precision based criteria, larger number of observations are needed in absence of strong prior information to reach the target precision. However in the testing criterion the location and dispersion of the posterior, both play a role in determining the sample size. The sample observations also help to locate the posterior density away from the prior mean, and often in absence of strong prior information, few observations are enough to shift the posterior density to conclude in favor of one of the hypotheses. We also notice that for TC1, both RMGF and SMOKE require approximately similar sample sizes for weaker prior strength, RMGF needing much smaller sample sizes for higher prior strength. This is also expected as in our design prior, RMGF shows stronger departure from the null than the prior chosen for SMOKE, reflecting the findings of the current MECC study. In evaluating the analogue of the frequentist power criterion TC2, we evaluate the probability of concluding H_a with posterior odds larger than 3:1 in favor of H_a. This probability remains consistently higher for NAT2*RMGF interaction, than NAT2*SMOKE interaction in Table 5.

Table 5.

The two test criterion values (TC1 and TC2) corresponding to estimation of NAT2*RMGF and NAT2*SMOKE interaction. Optimal allocation of cases and controls for each candidate n is as determined in Web Table 5. The Dirichlet prior parameters for each strength are chosen as given in Web Table 4, based on data from MECC study. In both cases we consider H₀ : β ≤ 0 versus H_a : β > 0 The criterion $\mathrm{TC}1=\mathbb{P}\{0.4\le \pi ({\mathit{H}}_a\mid {\mathit{R}}_0,{\mathit{R}}_1)\le 0.6\}$ evaluates the probability of falling in an indecisive zone, whereas the criterion $\mathrm{TC}2=\mathbb{P}\{\pi ({\mathit{H}}_a\mid {\mathit{R}}_0,{\mathit{R}}_1)\ge 0.75\}$ evaluates the probability of concluding H_a with posterior odds larger than 3:1. One would like to determine sample sizes with small values of TC1 (say 0.02, for example) and higher values of TC2, the posterior probability of detecting a positive interaction effect.

Prior strength	NAT2*RMGF					NAT2*SMOKE
	n	n1	n0	TC1	TC2	n	n1	n0	TC1	TC2
200	500	202	298	0.105	0.568	500	256	244	0.123	0.468
	1000	412	588	0.079	0.601	1000	504	496	0.089	0.516
	1500	630	870	0.066	0.603	1500	754	746	0.073	0.522
	2000	828	1172	0.048	0.620	2000	998	1002	0.064	0.527
	2500	1046	1454	0.052	0.628	2500	1250	1250	0.054	0.550
	3000	1262	1738	0.049	0.636	3000	1500	1500	0.050	0.554
	4000	1683	2317	0.040	0.633	3500	1750	1750	0.048	0.547
	5000	2103	2897	0.040	0.635	4000	2000	2000	0.044	0.556
	6000	2524	3476	0.035	0.638	5000	2500	2500	0.036	0.577
	7000	2945	4055	0.029	0.655	6000	3000	3000	0.040	0.571
	8000	3365	4635	0.030	0.656	7000	3500	3500	0.032	0.587
	10000	4207	5793	0.026	0.665	10000	5000	5000	0.030	0.595
500	500	184	316	0.116	0.649	500	268	232	0.158	0.504
	1000	402	598	0.090	0.676	1000	516	484	0.127	0.541
	1500	620	880	0.078	0.697	1500	762	738	0.096	0.564
	2000	836	1164	0.069	0.695	2000	1012	988	0.084	0.583
	2400	1010	1390	0.065	0.700	2500	1260	1240	0.081	0.580
	3000	1266	1734	0.064	0.703	3000	1506	1494	0.074	0.601
	4000	1688	2312	0.053	0.710	4000	2008	1992	0.063	0.606
	5000	2110	2890	0.045	0.727	5000	2510	2490	0.064	0.615
	6000	2532	3468	0.043	0.718	5500	2761	2739	0.051	0.615
	7000	2954	4046	0.046	0.728	7700	3850	3850	0.048	0.624
	8000	3376	4624	0.037	0.727	8000	4000	4000	0.045	0.628
	9000	3798	5202	0.036	0.729	9000	4500	4500	0.041	0.633
	10000	4220	5780	0.034	0.738	10000	5000	5000	0.041	0.639

In Table 6, we evaluate the Bayesian SSD criteria under a case-control design which will typically be used to ensure 80% power with a type-1 error level of 5% in a classical frequentist setting for testing H₀ : β = 0 against H_a : β ≠ 0. We consider four parameter settings identical to Table 3. Table 6 gives one a sense of the thresholds for the Bayesian SSD criteria that are comparable with corresponding frequentist power criteria. There are certain numerical differences in the Bayesian criteria TC2 and ALC from simulation setting (i) (independence) to (ii) (dependence) under increasing strength in the design prior. In general, the criteria values remain fairly robust. Since the Bayesian and frequentist testing paradigms are very different, it is hard to directly translate the two sets of operating characteristics. We evaluate the frequentist type I error and power corresponding to the Bayesian decision rule employed in TC2 where the null hypotheses is rejected if π(H_a|r₀, r₁) ≥ 0.75. We include performances of the Wald tests based on the CO and EB estimators under this design. The unacceptable performance of the CO estimator under departures from G-E independence illustrates why the assumption should not be taken for granted at the design stage if there is reasonable doubt and the need for allowing uncertainty around this assumption. Especially under negative G-E dependence, (as also shown in Figure 1 of Li and Conti, 2009), CO even loses its power advantages along with unbelievably high type-I error rates. The EB procedure strikes a compromise and is more robust across G-E association scenarios.

Table 6.

Comparison of Bayes SSD criteria under frequentist design: The four Bayes SSD criteria estimates under two simulated data configurations, and data corresponding to inference regarding NAT2*RMGF and NAT2*SMOKE interaction are presented. The sample sizes are pre-determined to achieve a power of 0.8, with level of significance 5% to detect a significant interaction β under a classical case-control design. The two simulated datasets with 2000 cases and 2000 controls were created to yield the value of (P_G, P_E, OR₀₁, OR₁₀, ψ₀, ψ) as (i) (0.3, 0.5, 1, 1, 1, 1.3) and (ii) (0.3, 0.5, 1, 1, 1.25, 1.3), reflecting G-E independence and G-E dependence respectively. For NAT2*RMGF and NAT2*SMOKE interaction, the parameter guesses are fixed at the point estimates obtained in the MECC study, namely, at (0.36, 0.13, 1.02, 1.06, 0.76, 1.48) and (0.36, 0.44, 1.02, 0.87, 0.74, 1.19) respectively. The criterion considered for estimation are ALC and APV. For Bayesian hypotheses testing we consider the hypotheses H₀ : β ≤ 0 versus H_a : β > 0. The testing criterion $TC1=\mathbb{P}\{0.4\le \pi ({\mathit{H}}_a\mid {\mathit{R}}_0,{\mathit{R}}_1)\le 0.6\}$ evaluates the probability of falling in an indecisive zone, whereas the criterion $TC2=\mathbb{P}\{\pi ({\mathit{H}}_a\mid {\mathit{R}}_0,{\mathit{R}}_1)\ge 0.75\}$ evaluates the probability of concluding H_a with posterior odds larger than 3:1. In the last two columns we also add the performance of Wald tests based on the case-only estimator and the empirical Bayes estimator under each simulation setting, with Power (Type I error in parentheses) under the given sample sizes.

Data	Prior Strength	n1 (= n0)	TC1	TC2(Power*, Type 1 error*)	ALC	APV	Power (Type 1 error)
							CO	EB
Simulated 1	200	4300	0.030	0.659(0.965,0.242)	0.357	0.009
	500		0.038	0.752(0.974,0.241)	0.336	0.007	0.981	0.918
	1000		0.039	0.833(0.987,0.226)	0.316	0.006	(0.054)	(0.042)
Simulated 2	200	4300	0.030	0.641(0.963,0.250)	0.360	0.009
	500		0.038	0.721(0.974,0.238)	0.359	0.008	1.000	0.862
	1000		0.040	0.796(0.986,0.215)	0.338	0.007	(0.911)	(0.066)
RMGF	200	3600	0.032	0.662(0.989,0.251)	0.582	0.022
	500		0.034	0.733(0.995,0.234)	0.544	0.019	0.268	0.623
	1000		0.036	0.827(0.998,0.212)	0.521	0.018	(0.761)	(0.083)
SMOKE	200	8900	0.020	0.605(0.984,0.252)	0.249	0.004
	500		0.029	0.663(0.989,0.243)	0.244	0.004	0.811	0.705
	1000		0.034	0.701(0.994,0.242)	0.240	0.004	(1.000)	(0.051)

^*The power and type I error corresponding to TC2 are calculated based on the fraction of time the Bayesian decision rule π(H_a|r₀, r₁) ≥ 0.75 rejects the null hypothesis H₀ : β < 0, with r₀ and r₁ being generated from the multinomial distributions with the four parameter settings described above, under the alternative (ψ = 1.3) and the null (ψ = 1) respectively.

In terms of setting specific values for the different Bayesian SSD criteria, there are no established benchmarks like 80% power or 5% Type I error in the classical setting, and largely depend on the investigator and the study under consideration. In Table 7 we specify certain ad hoc thresholds for the ALC, LPC and TC2 criteria and study the changes in the SSD across prior choices for different simulation settings. One can notice the changes in the SSD requirements when estimation accuracy is the goal (ALC, LPC) in contrast to detecting an effect size (TC2) and also the reduced sample sizes with increase in prior strength. To reiterate that the requirements in Bayesian perspective could be quite different from that in a frequentist testing situation, we provide the frequentist power of (CC, CO, EB) methods under the Bayesian design.

Table 7.

The variation in sample sizes under three Bayesian SSD criteria:(i) ALC ≤ 0.6, (ii) LPC≤0.05, and (iii) mar. We evaluated the LPC criterion by choosing the threshold length of the 95% HPD credible interval as l = 0.6. CC, CO, and EB refer to the case-control, case-only and empirical Bayes estimator of β and we provide power of these procedures under the chosen Bayesian design. The two simulated datasets were created to yield the value of (P_G, P_E, OR₀₁, OR₁₀, ψ₀, ψ) as (i) (0.3, 0.5, 1, 1, 1, 1.3) and (ii) (0.3, 0.5, 1, 1, 1.25, 1.3), reflecting G-E independence and G-E dependence respectively. For NAT2*RMGF and NAT2*SMOKE interaction, the parameter guesses are fixed at the point estimates obtained in the MECC study, namely, at (0.36, 0.13, 1.02, 1.06, 0.76, 1.48) and (0.36, 0.44, 1.02, 0.87, 0.74, 1.19) respectively.

Data	prior strength	n1	n0	Power
ALC* ≤ 0.6				CC	CO†	EB
NAT2*RMGF	200	1935	2565	0.601	0.161	0.404
	500	1677	2223	0.536	0.132	0.335
NAT2*SMOKE	200	1050	1050	0.155	0.151	0.086
	500	900	900	0.141	0.148	0.076
Simulated Data 1	200	1316	1484	0.266	0.468	0.373
	500	1128	1272	0.234	0.419	0.336
Simulated Data 2	200	1610	1890	0.321	0.989	0.481
	500	1426	1674	0.290	0.977	0.457

LPC* ≤ 0.05				CC	CO	EB
NAT2*RMGF	200	4128	5472	0.901	0.270	0.788
	500	3225	4275	0.812	0.212	0.640
NAT2*SMOKE	200	1750	1750	0.229	0.229	0.122
	500	1400	1400	0.201	0.183	0.106
Simulated Data 1	200	2162	2438	0.415	0.682	0.573
	500	1739	1961	0.363	0.596	0.492
Simulated Data 2	200	2070	2430	0.401	0.997	0.530
	500	1656	1944	0.335	0.987	0.479

TC2** ≥ 0.7				CC	CO	EB
NAT2*RMGF	200	5160	6840	0.953	0.316	0.878
	500	1032	1368	0.346	0.096	0.201
NAT2*SMOKE	200	12500	12500	0.919	0.907	0.875
	500	11000	11000	0.883	0.862	0.825
Simulated 1	200	5405	6095	0.798	0.971	0.910
	500	2068	2332	0.400	0.661	0.547
Simulated 2	200	5750	6750	0.821	1.000	0.871
	500	2760	3240	0.506	1.000	0.622

^*ALC and LPC focus on estimation whereas

^**TC2 focus on detecting departures from a null hypothesis.

^†CO has elevated type-I errors except for Simulated 1 scenario (ψ₀ ≡ 1), exact type I error values not presented here.

Of possible interest is the design situation when one chooses all available number of cases and wants to select the optimal case:control ratio in terms of a given criterion. Web Table 7 provides evaluation of the ALC criterion at certain designated number of cases and varying case:control ratios of 1:1-1:3 from which an investigator can determine the optimal ratio for a chosen threshold. M'Lan et al (2006) point out that with a very diffuse design prior, the sample sizes provided by the ALC criteria are unstable, thus we have refrained from using a very diffuse Dirichlet prior as a design prior in the current study.

In conclusion, the results in terms of optimal allocation, as well as controlling the different aspects of the posterior illustrate the appealing data-adaptive feature of Bayesian SSD, by including prior information as well as uncertainty from an existing study into designing a new study. Also, by varying the prior strength one can calibrate ones faith in the findings of the current study and control the effective sample size. Bayesian computation becomes extremely fast due to inexpensive generation of the posterior samples. Software codes written in R for the Bayesian analysis and for determining the optimal SSD are available at http://www.sph.umich.edu/bhramar/public_html/research.

6 Discussion

The current paper presents a full Bayes approach to analyze and design studies of gene-environment interaction. The FB analysis of interactions presented in the paper is extremely straightforward, but have not previously been indicated in the literature. The paper illustrates how to incorporate prior uncertainties around the assumption of gene-environment independence in a FB framework as an alternative to previously proposed EB strategy. The FB formulation allows us to explore the SSD problem for GxE studies in a proper Bayesian framework. We illustrate through use of data from an ongoing case-control study, how to formulate a design prior in order to plan a future study. An ensemble of Bayesian SSD criteria are considered and evaluated. The methods are compared with other existing design and analytic choices, thus making the paper the first comprehensive work on proper full Bayesian analysis and design for GxE problems after the initial exposition in M'Lan et al (2006). In absence of existing data or historical information, choosing a design prior is a difficult task, just like fixing effect sizes agnostically in a traditional power calculation is, and the paper does not propose a remedy for that situation. In such situations the newly proposed robust Bayesian SSD criteria that protect against the choice of a single base prior (Brutti et al., 2008) may be gainfully employed.

The methods can be directly applied towards studies of gene-gene interactions where similar issues regarding gene-gene independence may arise. The model could be extended to accommodate a general categorical G, such as genotype data for single nucleotide polymorphisms (SNP) which is typically coded as 0, 1 or 2 by counting the number of variant alleles carried by an individual. A limitation of the proposed analysis is that the methods are restricted to binary/categorical G and E. However, it is often the case that approximate sample size and power calculations are carried out even for continuous exposures under such stratified setting in many study proposals. A hierarchical Bayesian analysis using the retrospective likelihood framework of Chatterjee and Carroll (2005) under a general regression set-up, is possible, but requires use of substantively greater computational time (Mukherjee et al, 2007) and a computationally more challenging SSD problem.

Though the paper captures important features of a GxE study design, it focusses on “one at a time” analysis of a single GxE parameter. In the modern era of genomewide association studies, for testing thousands of genes and their interactions simultaneously, an appropriate utility function that reflects the issue of error due to multiple testing should be considered in a proper Bayesian manner, a topic which remains of future research interest.

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.