A unified powerful set-based test for sequencing data analysis of GxE interactions

Summary

The development of next-generation sequencing technologies has allowed researchers to study comprehensively the contribution of genetic variation particularly rare variants to complex diseases. To date many sequencing analyses of rare variants have focused on marginal genetic effects and have not explored the potential role environmental factors play in modifying genetic risk. Analysis of gene–environment interaction (GxE) for rare variants poses considerable challenges because of variant rarity and paucity of subjects who carry the variants while being exposed. To tackle this challenge, we propose a hierarchical model to jointly assess the GxE effects of a set of rare variants for example, in a gene or regulatory region, leveraging the information across the variants. Under this model, GxE is modeled by two components. The first component incorporates variant functional information as weights to calculate the weighted burden of variant alleles across variants, and then assess their GxE interaction with the environmental factor. Since this information is a priori known, this component is fixed effects in the model. The second component involves residual GxE effects that have not been accounted for by the fixed effects. In this component, the residual GxE effects are postulated to follow an unspecified distribution with mean 0 and variance . We develop a novel testing procedure by deriving two independent score statistics for the fixed effects and the variance component separately. We propose two data-adaptive combination approaches for combining these two score statistics and establish the asymptotic distributions. An extensive simulation study shows that the proposed approaches maintain the correct type I error and the power is comparable to or better than existing methods under a wide range of scenarios. Finally we illustrate the proposed methods by a exome-wide GxE analysis with NSAIDs use in colorectal cancer.

1. Introduction

Both genetic and environmental factors contribute to the development of complex diseases. Understanding the interplay between genes and environment is of great interest in genetic epidemiology, as it may help researchers elucidate the underlying biology and devise effective clinical prevention and intervention strategies. Many methods have been proposed to enhance power for detecting genome-wide gene-environment interaction (GxE) effects for individual common variants (Thomas, 2010; Hsu and others, 2012 and references therein). However, these methods do not provide adequate power for GxE testing on rare variants. In some instances, because few individuals carry variant alleles and have environmental exposure, the asymptotic-based inference becomes unreliable. To improve both type I error and power, variants are aggregated by a priori defined sets (e.g., genes and functional classes). Instead of testing GxE variant-by-variant, the GxE testing will be performed set-by-set. The idea is then to combine several signals in the set that would otherwise be difficult to detect individually.

Limited work has been done on testing for the interaction between a set of rare genetic variants and an environmental factor. A natural approach is to test for the overall association of the set with the outcome by standard -degrees of freedom likelihood ratio or score tests. However, when the genetic variants are rare, such tests may not keep the correct type I error and the power is also limited. To improve type I error and power, Lin and others (2013) used the kernel machine regression framework by imposing the GxE effects to follow an arbitrary distribution with mean 0 and variance and test for heterogeneity of GxE effects with an extension that allows for correlated GxE effects (Lin and others, 2016). Tzeng and others (2011) developed a set-based GxE test for continuous traits based on similarity matrices, similar to the variance component test as in Lin and others (2013). We also proposed a set-based GxE test where the variant alleles are summed over the set of genetic variants with weight informed by the screening statistics (burden) (Jiao and others, 2013), and recently extended it to allow for heterogeneous GxE effects (Jiao and others, 2015). However, our works treated genetic main effects as fixed, and the type I error may be inflated when the variants are rare. Generally speaking, the variance component-based test is powerful when both positive and negative directions of GxE effects exist, whereas the burden test is powerful when the variants in the set have the same direction of GxE. Since the pattern of the GxE effects for a set of variants is typically complex and unknown, it is important to devise a unified approach that combines test statistics that capture a particular feature of GxE effects, where a feature can be a priori defined variant functions or some generic distributional assumption of GxE effects. By this the combined test statistic may be powerful under a wide range of scenarios for GxE.

In this article we introduce a unified hierarchical regression framework for modeling GxE effects that account for a priori information about variant characteristics such as functional features and data-driven screening statistics as fixed effects, and heterogeneous GxE effects as random effects. We show that all previously proposed tests can be derived under this framework by constraining certain parameters to 0. Under this regression framework we develop a novel and rigorous approach to deriving independent score statistics for fixed effects and the variance component. The approach is broadly applicable to any mixed effects model where the hypothesis of interest is to test both the fixed and random effects equal to 0. Our proposed score statistics have two advantages: (1) both score statistics have trackable asymptotic distributions, and (2) the score statistics corresponding to the fixed effects and the random effects are asymptotically independent. The independence of score statistics is very desirable, because it not only allows study of the properties of various combinations of independent statistics with trackable asymptotic distributions but also facilitates the search of optimum within a particular class of combinations of the two score statistics. Towards this end we proposed two data-adaptive approaches to optimally combine two independent score statistics. Our framework and tests provide, nearly uniformly, more powerful approaches to identifying GxE of rare variants.

The rest of the article is organized as follows. In Section 2 we describe the hierarchical model for GxE interaction effects, derive the independent score statistics for fixed and random effects, respectively. The two data-driven combination approaches for combining the score statistics are also presented. The results from an extensive simulation study are presented in Section 3. The proposed methods are applied to a large exome-wide study of GxE analysis with NSAIDs use in colorectal cancer (CRC) in Section 4, and finally, the article is concluded with some remarks.

2. Methods

2.1. Notation and model

Consider an outcome , which can be binary or continuous. Let be a set of variants where is a function of the th variant genotype (e.g., the number of copies of the variant allele), and an environmental factor. The model for GxE is

$$\begin{array}{c}g\{\text{E}(D|\mathbf{G},E)\}={\beta }_0+{\beta }^{E}E+\sum _{p=1}^P{\beta }_p^G{G}_p+\sum _{p=1}^P{\beta }_p^{GE}{G}_{p}E,\end{array}$$ (2.1)

where is a link function, which depends on the type of outcome. For binary outcome, a commonly used link function is logit. For continuous outcome, linear or log transformation may be used. In this model, are the intercept, the main effects of E and G, and the interaction effects, respectively. The main parameters of interest in this article are the interaction effects between the genetic variants and , .

A direct approach for drawing statistical inference on ’s is based on the likelihood ratio test treating ’s as fixed effects. However, this approach may yield an inflated type I error and lose power when the variants are rare and the number of the variants is large. To overcome these issues, we propose a hierarchical model to reduce the dimension of parameters while leveraging information across variants. Specifically, let denote known attributes associated with the th variant. These attributes can be functional annotations such as whether a variant is missense, nonsense, or other characteristics. It can be data-driven weights such as those based on minor allele frequency or screening statistics of marginal association of with disease risk and correlation between and that were shown to be informative for GxE yet independent of GxE interaction tests (Hsu and others, 2012). Under the hierarchical modeling, we model the genetic main and interaction effects as a function of these variant attributes as following

$$\begin{array}{c}{\beta }_p^G={\mathbf{Z}}_p^T{\mathit{\gamma }}^G+{\delta }_p^G,{\beta }_p^{GE}={\mathbf{Z}}_p^T{\mathit{\gamma }}^{GE}+{\delta }_p^{GE},\end{array}$$ (2.2)

where and are row vectors of regression coefficients associated with the R attributes for the main and interaction effects, respectively, and and are the respective variant-specific main and interaction effects that cannot be explained by . To further leverage the information across variants allowing for robust statistical inference for rare variants and improving power, we assume these residual variant-specific effects, and , follow arbitrary distributions with mean 0 and variance and , respectively. Plugging model (2.2) into model (2.1), we obtain

$$\begin{array}{ccc}\hfill g\{\text{E}(D|\mathbf{G},E)\}& \hspace{0.17em}=\hfill & {\beta }_0+{\beta }^{E}E+({\mathit{\gamma }}^G{)}^T(\sum _{p=1}^P{\mathbf{Z}}_p{G}_p)+\sum _{p=1}^P{\delta }_p^G{G}_p\hfill \\ & & \hspace{0.17em}+({\mathit{\gamma }}^{GE}{)}^T(\sum _{p=1}^P{\mathbf{Z}}_p{G}_p)E+\sum _{p=1}^P{\delta }_p^{GE}(G_p\cdot E).\hfill \end{array}$$ (2.3)

It becomes clear that characterizes the interaction effects of genetic risk (or burden) scores weighted by one of the variant attributes, whereas is the variance of the residual GxE effects. For example, suppose , a scalar, for all variants, then is the number of variant alleles that a subject carries. For another example, let if the th variant is missense and 0 otherwise, for . Then is the total number of missense variant alleles that a subject carries, and characterizes the “dosage” interaction effect of (missense) variant alleles with . To guard against the possibility that such “dosage” interaction effect may not fully account for the interaction effect, the remaining variant-specific interaction effects are captured by , which are assumed to follow an arbitrary distribution with mean 0 and variance . Testing the interaction effect between and a set of variants therefore amounts to test and , and the number of tested parameters is , which is typically smaller than if we were to test variant GxE.

Model (2.3) encompasses scenarios that give rise to previously proposed tests. If we set , testing GxE is to test , which is the variance component test proposed by Lin and others (2013). If is a screening statistics-based weight with as fixed effects, the score statistic for GxE derived under this model is same as our early works on screening informed GxE tests assuming (Jiao and others, 2013) and (Jiao and others, 2015). Further if is 1 for all variants, then our model is same as the random effects model in Lin and others (2016), where the interaction effects have an exchangeable correlation.

Model (2.3) provides a basis for a general framework for modeling the interaction effect of variants with . It can straightforwardly include adjustment of other covariates, for example, principal components to account for population substructure. It can be further generalized to allow for non-linear genetic effects by replacing and by and , respectively, where is an unknown function belonging to the functional space implicitly specified by some positive definite kernel function (Cristianini and Shawe-Taylor, 2000). Common examples of kernel functions include the polynomial kernel, identity-by-state, and the Gaussian Kernel (Schaid, 2010). Without loss of generality, we focus on the generalized linear regression model (2.3) for the development of proposed GxE test statistics.

2.2. Proposed score statistics

Computation remains one of the foremost considerations in large-scale genome-wide discovery because of the large sample size and the large number of variants being tested. It is important that test statistics are quick to compute, and their p values can be obtained based on asymptotic distributions. Maximum likelihood or Bayesian approaches based on the mixed effects model (2.2) involve -dimensional integration, for which the required computation can be intensive for genome-wide discovery. We therefore propose to use score statistics to test GxE interaction effects because they only depend on the null and are easy to compute with trackable asymptotic distributions.

Consider the data consist of subjects such that , and . These subjects can be sampled randomly from the population or under a retrospective sampling scheme such that cases (diseased) and controls (non-diseased) are randomly selected from their respective subpopulations. We establish the asymptotic distributions under both scenarios and the proof are provided in Section A in the supplementary material available at Biostatistics online. We denote to be the (weighted) burden scores, where is a vector of , for . Further, we denote the score statistics for by .

The usual approach for deriving score statistics is to take the partial derivatives of the log-likelihood function with respect to the parameters of interest under the null hypothesis. Here, we propose a novel modification to the derivation of score statistics. Specifically, we derive the score statistic for corresponding to the burden scores under and as usual. However, for the score statistic , we propose to derive it under without constraining , instead of and . This seemingly simple modification to has an important property: the resulting score is independent of asymptotically. In fact, it is asymptotically orthogonal to the projection of the score statistics for onto the space of under and .

To explicitly express the two score statistics, additional notation is introduced as follows. Define to be the fitted value of under , and to be the fitted value of under without constraining . Then the score statistics for and can be written as

$$\begin{array}{ccc}{U}_{{\gamma }^{GE}}& \hspace{0.17em}=& \hspace{0.17em}(\mathbf{D}-\tilde{\mathit{\mu }}{)}^T(\mathbb{B}\cdot \mathbf{E}){\mathbb{V}}^{-1}(\mathbb{B}\cdot \mathbf{E}{)}^T(\mathbf{D}-\tilde{\mathit{\mu }}),\end{array}$$ (2.4)

$$\begin{array}{ccc}{U}_{{\tau }^2}& \hspace{0.17em}=& \hspace{0.17em}(\mathbf{D}-\hat{\mathit{\mu }}{)}^T(\mathbb{G}\cdot \mathbf{E})(\mathbb{G}\cdot \mathbf{E}{)}^T(\mathbf{D}-\hat{\mathit{\mu }}),\end{array}$$ (2.5)

where , , and is a covariance matrix of and is equal to , where with is a vector of 1, is an identity matrix, and with . We term the score statistics for testing both the fixed and random interaction effects as Mixed effects Score Tests for interaction (MiSTi).

As the fitted value and involve the random main effects for the variants, maximum posterior estimate (MPE) (also known as best linear unbiased prediction, BLUP, in the context of linear mixed effects model) can be used; however, it requires calculation of the posterior distribution of the random effects given observed data and estimation of variance , for which the computation can be intensive especially under the generalized linear model because -dimensional integration is required and there is no closed form for the integration. Instead, we propose to use ridge regression estimators (Hastie and others, 2005) for estimating the random effects. Note that for linear models, the ridge regression estimator is equivalent to the BLUP for a given penalty (de Vlaming and Groenen, 2015). Furthermore, the ridge regression estimator is -consistent if the penalty grows at the rate of (Knight and Fu, 2000). Following these results, it is easy to show that as , follows a distribution with degrees of freedom and follows a weighted sum of i.i.d. distributions with weights being eigen-values of , where with , the design matrix of proposed logistic model (2.3) under .

2.3. Combinations of score statistics

The asymptotic independence of and offers many possibilities for combining these two statistics. For example, since each score statistic has an asymptotic distribution, we can obtain the p value based on each score statistic and combine these p values using for example, the commonly used Fisher’s or Tippett’s combinations. However, unlike in the conventional meta-analysis where the combined components are from different studies but test the same parameter, and do not test the same parameters. The score statistic tests the association of weighted burden scores when , whereas the score statistic tests . Hence, a usual weighting for combining test statistics in the meta-analysis, which often involves sample sizes of individual studies, does not apply here. In the following we propose two data-driven combination approaches: grid-search optimal linear combination and adaptive weighted linear combination.

2.3.1. Gird-search-based optimal linear combination

The perhaps most straightforward approach for combining the two score statistics is to take the weighted sum of and as , where controls the relative contribution of the burden score versus the variance components. A natural approach to choosing the optimal value of is by minimizing the p values as , where is the p value based on for a given . We call the corresponding test statistic, , oMiSTi with “o” referring to optimal. Now let be the observed minimal p value obtained from data we can show that under the null

$$\begin{array}{cc}Pr\hfill & (p_{{\rho }^{\ast }}\le p_{{\rho }^{\ast }}^{obs})=1-\text{Pr}(p_{\rho }\ge p_{{\rho }^{\ast }}^{obs},\mathrm{\forall }\rho \in [0,1])\hfill \\ & \hfill \hspace{0.17em}=1-\text{E}\{\text{I}(U_{{\gamma }^{GE}}<q_{U_{{\gamma }^{GE}}}(1-p_{{\rho }^{\ast }}^{obs}))\text{Pr}(U_{{\tau }^2}<\underset{\rho \ne 1}{min}\frac{1}{1-\rho }[q_{T_{\rho }}(1-p_{{\rho }^{\ast }}^{obs})-\rho U_{{\gamma }^{GE}}]\mid U_{{\gamma }^{GE}})\},\end{array}$$ (2.6)

where stands for the 100-th quantile of the random variable , and the expectation is with respect to . We refer to Section B in the supplemental material available at Biostatistics online for a detailed derivation of (2.6). It is clear that the independence between and facilitates an easy evaluation of the above conditional probability of given . We employ a numerical method proposed by Liu and others (2009) to approximate the distribution of that is a weighted sum of i.i.d. random variables by a skewed distribution with skewness and degrees of freedom obtained by matching the fourth moment while minimizing the third moment between the two distributions. This numerical method has been applied in SKAT-related methods and shown to perform well in the rare variants association analysis with finite sample sizes Lee and others, 2012. The expectation can be obtained by numerical integration on , which is very fast for univariate integrals.

In practice, evaluating the conditional probability in (2.6) for all can be computationally intensive. To ease this, we consider a grid-search method to evaluate the conditional probability on a set of pre-specified grid points , and the search of optimal is restricted on this given set of grid points. Based on our numerical experiences, a set of 20 grid points usually achieves good performance at a reasonable computational cost.

2.3.2. Adaptive weighted linear combination

Since each of the score statistics and has an asymptotic distribution, it is natural to first calculate the p value based on each score statistic and then combine the two independent p values. Fisher’s combination (Fisher and others, 1970) is a very popular approach for combining independent p values. This can be represented by , where and . It is expected that Fisher’s combination is very powerful when both the burden and variance components are non-null, but could potentially lose power when only one is non-null. To allow for flexibility of the combined test to accommodate the evidence of association mainly from either the burden or variance component, we propose an adaptive weighted linear combination with weights determined by and , respectively. Specifically, the adaptive weighted linear combination can be represented as

$$T_a=Z_{\gamma }^2+Z_{{\tau }^2}^2,$$ (2.7)

where the subscript refers to “adaptive”. We term this combination as aMiSTi. Note that is equivalent to the square of , where and . Interestingly, the weights and are equivalent to the sine and cosine functions of the angle between the direction of the observed 2D test statistics and the x-axis. We note that a similar idea of adaptive weighting has been proposed in set-based association testing for main effects (see e.g., Cai and others, 2012). Compared to the grid-search weighted combination, the adaptive weighting has the advantage that it does not require a prior decision on the number and placement of grid points. The nice property of and , and the independence between the two components facilitate an easy way to calculate the p value for through numerical integration at low computational cost.

3. Simulation

We conducted an extensive simulation study to evaluate the performance of our proposed combinations (oMiSTi) and (aMiSTi), Fisher’s combination (fMiSTi) and an existing approach, iSKAT (Lin and others, 2016). To help understand the performance of various combination methods, we also included tests based on the burden component only () and the variance component only (). We evaluated the performance of all tests for both continuous and binary outcomes. Here we are focused on results for binary outcome because in our real data application the phenotype is binary and it is also generally more challenging than continuous outcome. The results for continuous outcome are provided in Section D, supplementary materials available at Biostatistics online. Briefly, all tests maintain correct type I error and MiSTi’s are generally more powerful than or as powerful as iSKAT. When the signals mainly come from either the burden or the variance component, oMiSTi and aMiSTi are slightly more powerful than fMiSTi. On the other hand, when the signals come from both components, fMiSTi is slightly more powerful, probably due to the cost of estimating the weight from the data for oMiSTi and aMiSTi.

Specifically, we generated the binary outcome according to logistic regression model

$$\text{logit}\{Pr(D=1\mid \mathbf{G},E)\}=-3+0.5E+\stackrel{P}{\sum _{p=1}}{\beta }_p^G{G}_p+\stackrel{P}{\sum _{p=1}}{\beta }_p^{GE}{G}_{p}E.$$ (3.1)

We generated from independently from . We generated under two different scenarios. The first scenario was to generate = or independent SNPs with minor allele frequencies (MAF) equally spaced from to . The purpose of this simulation was to study how all methods perform under various alternatives without being confounded by complicated genetic structures. To save space, the results are presented in Section D.3, supplementary material available at Biostatistics online. Here we only provide a brief summary. All tests maintain correct type I error across all settings considered under this scenario. The various combinations of MiSTi’s have comparable power, and they all have greater power than iSKAT. The pattern and strength of the genetic main effects do not affect the general pattern of the powers across all methods.

The second scenario is to mimic a more realistic genetic structure by generating based sequencing data from the Dallas Heart Study (DHS, DHS, Victor and others, 2004). Specifically, haplotypes were inferred based on the sequencing data on a candidate gene ANGPTL5 for 3409 subjects and randomly paired to achieve a desirable sample size. There are a total genetic variants in ANGPTL5. Of these, variants have MAF , and are functional variants. Unless otherwise stated, a total of 10 000 simulated data sets were generated each with 5000 cases and 5000 controls to mimic our real data example. The type I error and power were evaluated at three significance levels , , and .

Type I error We considered two settings for the genetic main effects: (1) Null, , (2) Sparse, 20% of and 0 otherwise. The set of variants is defined in two different ways: functional variants only and all variants in the gene. When all variants are tested for GxE, we also assessed the type I error with and without including a function indicator, which is 1 if the variant is functional and 0 otherwise. Ridge regression was used in fitting the null models and the penalty was selected by generalized cross validation following the suggestion in Lin and others (2013). Overall, the type I error of all MiSTi combinations are well within the 95% confidence intervals of the true type I error (Table 1). iSKAT also has correct type I error; however, the type I error is somewhat inflated when . We also examined the type I error of the proposed methods at the exome-wide significance level. A large scale simulation with 1 000 000 datasets was conducted. Figure 1 shows the estimated type I error of MiSTi’s at various significance levels ranging from to the exome-wide significance level . All MiSTi’s appear to keep a correct type I error all the way to the exome-wide significance level.

Fig. 1.

Type I error rate of three different combination approaches for DHS genetic structure on the functional variants under genetic main effects as setting 2n. Sample size is 10 000 and the number of simulation runs is 10 00 000.

Table 1.

Type I error of four combinations oMiSTi, aMiSTi, fMiSTi, and iSKAT as well as tests for burden () and variance component () at significance level , and for binary outcome. A total of variants are in the gene-set, and are functional. For , the type I error for MiSTi’s with and without incorporating the functional indicator, denoted by FA and No FA, respectively, are also shown.

		Null genetic main effects			Sparse genetic main effects
			No FA	FA		No FA	FA
0.05
	oMiSTi	0.0519	0.0519	0.0499	0.0515	0.0483	0.0484
	aMiSTi	0.0522	0.0516	0.0487	0.0512	0.0490	0.0476
	fMiSTi	0.0554	0.0529	0.0505	0.0517	0.0512	0.0498
	iSKAT	0.0568	0.0555	—	0.0548	0.0589	—
		0.0496	0.0519	0.0498	0.0518	0.0521	0.0510
		0.0546	0.0519	0.0480	0.0530	0.0469	0.0467
0.01
	oMiSTi	0.0107	0.0120	0.0121	0.0107	0.0110	0.0108
	aMiSTi	0.0107	0.0116	0.0118	0.0105	0.0109	0.0104
	fMiSTi	0.0116	0.0116	0.0115	0.0104	0.0107	0.0093
	iSKAT	0.0125	0.0134	—	0.0118	0.0143	—
		0.0093	0.0121	0.0117	0.0091	0.0111	0.0104
		0.0121	0.0107	0.0103	0.0113	0.0100	0.0095
0.005
	oMiSTi	0.0051	0.0060	0.0065	0.0060	0.0053	0.005
	aMiSTi	0.0053	0.0057	0.0066	0.0062	0.0052	0.0048
	fMiSTi	0.0054	0.0059	0.0062	0.0055	0.0054	0.0045
	iSKAT	0.0065	0.0079	—	0.0070	0.0084	—
		0.0046	0.0066	0.0064	0.0044	0.0056	0.0055
		0.0060	0.0060	0.0058	0.0063	0.0054	0.0051

Power To compare power among different tests, we randomly selected two functional variants to have GxE. We considered two scenarios, the GxE effects of the two variants are in same direction (GxE same direction) and opposite direction (GxE opposite direction). The results of power comparison are shown in Table 2. All MiSTi tests are much more powerful than iSKAT with MiSTis having comparable power with each other, and oMiSTi and aMiSTi being more powerful than fMiSTi when the signal is from only the variance component as in the GxE opposite direction model. When all variants are included in the set, all tests lose power; however, iSKAT still has lower power than MiSTi tests in most of the cases. Furthermore, incorporating an indicator for whether or not a variant is functional can improve power for MiSTi tests considerably when the functional indicator is informative, as in the case of the GxE same direction model. Importantly, we note that the power of the burden component is much greater after incorporating the functional indicator, suggesting that the interaction signals are mainly from the functional variants. This demonstrates that using the mixed effects model leveraging functional annotation, if known and informative, cannot only improve power for detecting the overall association, but also help identify sources of the signals that may inform the follow-up studies.

Table 2.

Power of various methods without and with functional annotations: , , , , , and for dichotomous outcomes for 27 functional variants (FV Only) and 100 variants (All) from the DHS without or with functional annotations (No FA and FA, respectively) under scenarios with no main effects and two settings of interaction effects on two variants with same and opposite directions respectively (GxE Same Direction and GxE Opposite Direction). The significance levels are 0.05, 0.01, and 0.005, respectively

		GxE same direction			GxE opposite direction
			No FA	FA		No FA	FA
0.05
	oMiSTi	0.798	0.151	0.370	0.726	0.063	0.077
	aMiSTi	0.813	0.159	0.365	0.727	0.069	0.072
	fMiSTi	0.842	0.160	0.396	0.703	0.079	0.078
	iSKAT	0.554	0.086	—	0.633	0.064	—
		0.439	0.066	0.373	0.062	0.053	0.056
		0.745	0.193	0.133	0.810	0.083	0.093
0.01
	oMiSTi	0.576	0.054	0.181	0.480	0.010	0.016
	aMiSTi	0.590	0.053	0.172	0.484	0.011	0.017
	fMiSTi	0.647	0.056	0.186	0.454	0.013	0.017
	iSKAT	0.241	0.022	—	0.316	0.014	—
		0.230	0.015	0.201	0.013	0.013	0.011
		0.498	0.068	0.024	0.581	0.013	0.016
0.005
	oMiSTi	0.486	0.031	0.124	0.394	0.007	0.009
	aMiSTi	0.490	0.032	0.122	0.397	0.007	0.009
	fMiSTi	0.542	0.031	0.131	0.354	0.006	0.009
	iSKAT	0.166	0.011	—	0.222	0.007	—
		0.164	0.007	0.146	0.006	0.005	0.006
		0.408	0.042	0.013	0.480	0.006	0.010

4. Application to genome-wide exome chip GxE analysis

CRC is a commonly diagnosed cancer, and has a sizable genetic component and well-established environmental risk factors. Identifying GxE is key to understand the interplay between genes and environment and their role in the development of CRC. We applied our proposed MiSTi’s and iSKAT to the case–control data from the Genetics and Epidemiology of Colorectal Cancer Consortium (GECCO) (Peters and others, 2013). In particular, this data set includes exome-wide genotyping on 17 864 white study participants (9135 CRC cases and 8729 controls) from 11 studies. The description of these studies is provided in supplementary Table S6 available at Biostatistics online. To illustrate, we assessed association of exome-wide interaction with nonsteroidal anti-inflammatory drugs (NSAIDs) use (nonuser, regular user) with CRC risk. A logistic regression model was used, adjusting for study, age, sex, and three major principal components to account for population substructure. NSAIDS use is associated with decreased risk of CRC with odds ratio (OR) 0.71 (95% confidence interval: 0.66–0.76).

We aggregated rare variants (MAF ) by gene and assessed the interaction effect of each gene with NSAIDs use on CRC risk. Genes with two or fewer rare variants were excluded, resulting a total of 7600 genes. The number of variants ranges from 3 to 357, with an average of 4.8. Since the colorectal tissue specific functional annotation database is still under development in GECCO, detailed annotation is not yet available for analysis. To illustrate, we simply set for all variants, creating a burden score (B) that is the sum of variant alleles in a gene. The MiSTi’s test both the BxE and the variance component for GxE equal to 0.

Two genes are identified at the exome-wide significance level : PTCHD3 at 10p12.1 and TELO2 at 16p13.3 (Table 3). The interaction of PTCHD3xNSAIDs is only detected by fMiSTi, and the pvalues of oMiSTi and aMiSTi are close to the threshold ( and , respectively). However, the p value for iSKAT is 0.0189, which is highly non-significant. Gene TELO2 is identified by both fMiSTi and iSKAT at the exome-wide significance level, though the p values of oMiSTi and aMiSTi are close to the threshold. No other genes have reached exome-wide significance by any test.

Table 3.

P-values of MiSTi’s, iSKAT, burden and variance component for the interaction of PTCHD3 and TELO2 with NSAIDs use from the analysis of GECCO exome chip data

Gene	oMiSTi	aMiSTi	fMiSTi	iSKAT
PTCHD3
TELO2

Gene PTCHD3 has been previously reported by Jiao and others (2013). There are eight rare variants in PTCHD3 with MAF ranging from to (see supplementary Table S7 available at Biostatistics online). A ridge regression model on the eight variants in PTCHD3 and MSAIDs with interactions shows that the NSAIDs use has no protective effect if a subject carries any minor allele in PTCHD3 (see supplementary Table S8 available at Biostatistics online). For TELO2, there are seven rare variants with MAF ranging from to . A ridge regression with GxE shows that NSAIDs use has a stronger protective effect if the subject does not carry any minor allele in any of the seven variants (OR = 0.69 = than subjects who carry at least one minor allele (). For both genes, the interaction appears to be driven by a few variants as shown in single variant GxE test, and there is substantial variability on the individual variant GxE effects, where both positive and negative effects exist. The variance component test would be more powerful than the burden test under this scenario. However, our approach suggests that despite highly heterogeneous GxE effects, there appears to be a concerted GxE as shown in the burden GxE. A search of literature suggests that TELO2 encodes a protein that is a regulator of the DNA damage response and associated with telomere maintenance (Takai and others, 2007), which could potentially be modified by NSAIDs use. Though these findings are very preliminary and need to be replicated in an independent data set, it is clear that our proposed tests are able to detect as many, if not more, genes than the existing test.

5. Discussion

In this article, we proposed a mixed effects model for assessing the association of GxE for a set of genetic variants and a novel approach for constructing score test statistics for testing both the fixed effect and the variance component equal to 0. Our novel construction ensures that the score statistics corresponding to fixed and random effects are asymptotically independent, which enables one to combine these two score statistics efficiently with p values that can be easily computed. We also proposed two data-adaptive combinations: linear combinations based on grid-search and adaptive weighted. Extensive simulation shows that these two and Fisher’s combination all have comparable power and they are more powerful than existing tests under a wide range of scenarios. This is particularly appealing for genome-wide search of GxE, as all possible interaction models could exist.

Model (2.2) has fixed and random effects for both the main genetic effects and GxE. As the focus here is on testing GxE, there is no need to model the main effects of as in (2.2). Instead we can directly estimate by ridge regression. This is because tests for GxE are valid, as long as the main effects of and are adequately modeled. In fact, we show that the fitted values from the two ridge regression modeling on the genetic main effects are asymptotically equivalent if the penalty term is (see Section C supplementary material available at Biostatistics online). However, if the interest is in GxE effects, it is important that the main effects and GxE have the same models to ensure the hierarchical structure and interpretable estimates.

Sample size determination for set-based GxE is important in the study design phase. The power of a set-based GxE test depends on the size of the set and the number of causal variants with GxE in the set, as well as the effect sizes and MAFs of the causal variants, and the underlying linkage disequilibrium structure among the variants. One needs to balance out between not missing causal variants that would require a larger set and not including too many neutral variants that would require a smaller set. We note that when a set includes many neutral variants, if the functional annotation is somewhat informative, our proposed mixed effects score tests can improve power significantly. A closed form of power calculation would be desirable for determining the sample size required for particular scenarios; however, it is difficult to derive such a formula because the power depends on many factors. It would be more realistic to calculate power based on simulations for any particular scenarios that investigators deem to be reasonable. As our score statistics have asymptotic distributions, it is computationally feasible to conduct such a simulation-based study to assess the power with given sample size. As a back-of-the-envelope calculation, we may calculate power by a “composite” single variant GxE, where the MAF of the composite variant is the sum of the MAFs of rare variants in the set and the effect size for the composite variant would be such that the explained variation, that is, is equivalent to the total sum of variation explained by the hypothesized causal variants. By doing this transformation, we can then easily calculate the power for single variant GxE using the popular power calculator Quanto (http://biostats.usc.edu/Quanto.html). For example, under the first scenario in Table 2, the back-of-the-envelop power calculations give the power estimates 0.863, 0.689, and 0.603, respectively, when and , which are largely consistent with the power as shown for MiSTi’s.

Supplementary material

Supplementary material is available online at http://biostatistics.oxfordjournals.org.

Funding

The work is supported by the grants from the National Institute of Health (R01 CA189532, R01 CA195789, R21 ES022332, P01 CA53996, R21 CA195789). The GECCO Coordinating Center is funded by the grants from the National Institute of Health (R01 CA189532, R21 ES022332, P01 CA53996, U01 CA137088, R01 CA59045, U01 CA164930).