A linear mixed model framework for gene-based gene-environment interaction tests in twin studies

Abstract

Interaction between genes and environments (GxE) can be well investigated in families due to the shared genetic and environment among family members. However, majority of the current tests of GxE interaction between a set of variants and an environment are only suitable for studies with unrelated subjects. In this paper, we extend several GxE interaction tests to linear mixed model framework to study interaction between a set of correlated environments and a candidate gene in families. The correlated environments can either be modeled separately or jointly in one model. We demonstrate theoretically that the tests developed by modeling correlated environments separately are valid and present a computationally fast alternative to detect GxE interaction in families. For either strategy, we also propose treating the genetic main effects as a random effect to reduce the number of main-effect parameters and thus improve the power to detect interactions. Additionally, we propose a generalization of a test of interaction that adaptively sums the interactions using a sequential algorithm. This generalized set of tests, referred to as the Seq-SPU family of tests, can be expressed as a weighted version of the sum of power score tests (SPU). We find that the adaptive version of our test, Seq-aSPU, can outperform aSPU in cases where the interactions effects are in opposite directions. We applied these methods to the Minnesota Center for Twin and Family Research dataset and found one significant gene in interaction with four psychosocial environmental factors affecting the alcohol consumption among the twins.

1 Introduction

Over the past two decades, researchers have concentrated on studying the genetic contribution to complex diseases using genome-wide association studies (GWAS). However, the single nucleotide polymorphisms (SNPs) identified in GWAS only explain a small proportion of the disease heritability. This may be because the genetic risk of the SNPs is modified by environmental factors. Understanding the interplay between genes and environments can further help us to understand complex diseases. Many examples of gene-environment (GxE) interaction have been found for a variety of diseases [Hunter, 2005]. Identification of these interactions is important for understanding of the underlying disease etiology and developing disease prevention and intervention strategies. Here, we aim to identify GxE interactions in a family study by testing interactions between a group of SNPs from a candidate gene and a set of environmental factors.

One strategy to identify GxE interaction between a candidate gene and a set of correlated environmental factors is to test the interaction of each SNP G, and each environmental factor E separately and subsequently apply a multiple testing correction. A severe limitation of this approach is that the type I error rate can be inflated if any SNP correlated with G or an environment correlated with E and associated with the trait is excluded from the model [Lin et al., 2013]. Separate tests can also produce inflated type I error in the presence of G-E dependence. Another limitation is that single marker tests do not incorporate the possible joint effects among the SNPs and the environments. By not taking advantage of possible joint effects, single marker tests of interaction can lose power [Lin et al., 2013; Coombes et al., 2016]. Recently, several gene-based tests for interaction between a group of SNPs in a candidate gene and one environmental measure have been developed [Lin et al., 2013, 2015; Wang et al., 2015; Coombes et al., 2016]. Most of these tests can be implemented using the score vector of the pairwise GxE interactions and its covariance matrix corresponding to a regression set-up with G and E main effects and their pairwise interactions. We can perform a GxE interaction test between a candidate gene and a set of correlated environmental factors by separately testing for interaction between the set of SNPs and each environmental factor. However, similar to the arguments made above, including all of the environments and their interactions with the SNP-set in one model can increase power for detection of GxE interaction. Moreover, separate tests for interaction can produce inflated type I error due to a misspecified model for the same reason mentioned above.

In their current forms, the methods [Lin et al., 2013, 2015; Wang et al., 2015; Coombes et al., 2016] for interaction between a group of SNPs and an environmental measure are not applicable for correlated subjects such as families. Using these methods for GxE interaction without accounting for within family similarities can result in an inflated type I error [Chen et al., 2013]. It is therefore important to extend these methods for family data because family studies such as twin studies can provide more power for detection of GxE interaction. By matching on genotype in these studies, the proportion of genotype-concordant, exposure-discordant pairs may be much higher than in studies with unrelated subjects [Thomas, 2010; Yang and Khoury, 1997]. Additionally, with the increasing emphasis on finding GxE interactions, many family studies, such as the Framingham Heart Study [Dawber et al., 1951], the National Heart, Lung, and Blood Institute Family Heart Study [Higgins et al., 1996], and the STANISLAS cohort [Visvikis-Siest and Siest, 2008] would require tools to analyze GxE interactions within the family framework.

Our work is motivated by data originating from the Minnesota Center for Twin and Family Research (MCTFR) study which investigates psychological outcomes such as substance use disorders (SUDs) [Miller et al., 2012]. Addiction to several different substances including caffeine, nicotine, alcohol, cannabis, sedatives, stimulants, cocaine, and hallucinogens all appear to be moderately to strongly heritable, but genetic association studies with common and rare variants have explained little of the estimated heritability in this study [McGue et al., 2013; Vrieze et al., 2014]. Using biometric models, the MCTFR has shown how particular environmental factors relate to substance abuse risk and interact with genetic risk for the twin sample [Hicks et al., 2011; Samek et al., 2016]. Our methodological work in this paper aims to facilitate the detection of specific genes that are involved in this interaction process.

Finally, the methods for GxE interaction mentioned above have only focused on interactions of a set of SNPs from a candidate gene with a single environmental factor. However, in the MCTFR study, we focus on a group of four correlated environmental factors that may interact with genes to influence alcohol consumption.

In this article, we use the score vector for the GxE interactions and its covariance within the classic linear mixed model (LMM) framework of ACE and AE models [Falconer and Mackay, 1981] to extend and propose new tests of GxE interaction for family data. We either test one environment at a time or all at once in this model. Within both LMM approaches, we also implement a ridge penalty on the genetic main effect using a random effect. This reduces the number of parameters we need to estimate which in turn produces computationally efficient tests of GxE interaction. Using the resulting score vector of GxE interaction and its covariance, we extend the score test, gene-environment set association test (GESAT) [Lin et al., 2013], adaptive sum of powered score tests (aSPU) [Pan et al., 2014], and the interaction test using a sequential adaptive Sum (iSeq-aSum) [Coombes et al., 2016] from independent subjects to families. Additionally, we propose a generalization of iSeq-aSum using the family of powered score tests as first proposed by Pan et al. [2014]. In fact, the resulting family of tests, the sequential algorithm for the sum of powered score (Seq-SPU) tests, is equivalent to a weighted version of the SPU tests when weights are chosen using a sequential algorithm. Finally, we study the performance of the methods using the MCTFR dataset. We perform a gene-based GxE interaction analysis on the twin cohort of the MCTFR to study how genes from select candidate genes interact with a set of environmental factors to affect alcohol consumption.

2 Methods

To set up the GxE interaction model for family data, assume we have M independent families with m_i individuals where i = 1, ···, M. For the j^th individual from the i^th family, let Y_ij, G_ij = (G_ij1, ···, G_ijq)^T, E_ij = (E_ij1, ···, E_ijp)^T, X_ij = (X_ij1, ···, X_ijL)^T be the phenotype, the q minor allele counts for the common genetic variants from a candidate gene, which are standardized by their mean and standard deviation, the p environmental factors, and L covariates, respectively. Define ${\mathbf{S}}_{ij}={\mathbf{G}}_{ij}\otimes {\mathbf{E}}_{ij}={(G_{ij1}{\mathbf{E}}_{ij}^T,\cdots ,G_{\mathit{ijq}}{\mathbf{E}}_{ij}^T)}^T$ to be the pq pairwise GxE interactions for the ij^th individual where ⊗ is the Kronecker product.

An approach to modeling dependent observations is to model the dependency structure with random effects. While it is possible to use random effects for discrete outcomes, estimation is much more difficult [Pinheiro and Chao, 2006] and using quantitative outcomes allow us to extend classic models originally proposed in linear mixed models (LMMs) to test for GxE interactions. Therefore, we restrict our scope to continuous outcomes and focus on obtaining the score vector for the GxE interactions as well as its covariance matrix in the LMM setup. We consider testing for GxE interaction among the q SNPs and the p environments in either one model with all of the pq pairwise interaction terms as discussed in Section 2.1 or in p separate models each with q pairwise interaction terms included as discussed in Section 2.2.

2.1 Joint modeling of GxE interaction

To test for all pq interactions in one model, we use the following LMM:

$$Y_{ij}={\alpha }_0+{\mathit{X}}_{ij}^T{\mathit{\alpha }}_1+{\mathbf{E}}_{ij}^T{\mathit{\alpha }}_2+{\mathbf{G}}_{ij}^T{\mathit{\alpha }}_3+{\mathbf{S}}_{ij}^T\mathit{\beta }+a_{ij}+c_{ij}+e_{ij}$$ (1)

where α₀,α₁,α₂,α₃, β are the regression coefficients for the intercept, covariates, environmental factors, genetic variants, and GxE interactions, respectively. For Model 1, we are interested in testing the null hypothesis that there is no GxE interaction (H₀ : β = 0). We refer to this strategy that includes all environmental factors in the model as the Joint approach because it considers possible joint effects between environments and their interactions with the SNP-set.

To account for shared genetic effects within families, let a_i = (a_i1, ···, a_imi) ~ MVN(0, A_i) where ${\mathbf{A}}_i={\sigma }_A^2{\mathbf{K}}_i$ and K_i is equal to two times the kinship matrix for the i^th family, where [K_i]_r,s is the probability that a gene is identical-by-decent for the r^th and s^th member of the i^th family. For example, if we have a family consisting of either monozygotic (MZ) or dizygotic (DZ) twins, the off-diagonal elements of K_i are 1 or 1/2, respectively. To account for shared environmental effects within families, let c_i = (c_i1, ···, c_imi) be a random intercept for family defined as c_i ~ MVN(0, C_i) where ${\mathbf{C}}_i={\sigma }_C^2{\mathbf{J}}_{m_i}$ and J_mi is an m_i × m_i matrix of ones. While Model 1 includes some shared environments as fixed effects, there may still be unmeasured shared environments which can be accounted for by ${\sigma }_C^2$. Finally, we let e_i = (e_i1, ···, e_imi) ~ MVN(0, E) where $\mathbf{E}={\sigma }_E^2{\mathbf{I}}_{m_i}$. This term accounts for all other unshared effects. The covariance between different families is zero because different families are assumed to be independent. We refer to Model 1 as the ACE model because it splits the covariance within families into three parts: A = shared genetic effects, C = shared environmental effects, and E = unshared environmental effects [Falconer and Mackay, 1981]. Note that depending on our family structures, ${\sigma }_A^2$ and ${\sigma }_C^2$ may not be identifiable. However, for our simulations and real data application to the MCTFR which contains MZ and DZ twins, these parameters are identifiable. In our simulations, we investigate the effect of not estimating ${\sigma }_C^2$ in twins. We refer to this as the AE model. In our simulations, we explore the consequences of failing to adjust for shared environments in the AE model when our goal is test for GxE interaction.

A concern common in analyzing high dimensional genetic data is that Model 1 requires estimation of K + p + q + 1 main effects where q can be very large for some genes. This may cause estimation issues and impact our GxE interaction test. Lin et al. [2013] and Lin et al. [2015] have previously penalized genetic main effects using a ridge penalty to alleviate this issue. However, using a ridge penalty within an LMM framework can be computationally intensive. Instead, we re-cast the ridge penalization of the SNPs using a random effect by allowing ${\mathit{\alpha }}_3~\text{MVN}(\mathbf{0},{\sigma }_G^2{\mathbf{I}}_q)$ in Equation 1 [Hodges, 2013; Shen et al., 2013]. Under this setup, the traditional ridge penalization parameter λ is equivalent to ${\sigma }_E^2/{\sigma }_G^2$. Now, we only need to estimate one additional variance parameter. The formulation of α₃ implies that $q\times {\sigma }_G^2$ can be interpreted as the proportion of variation explained by the SNPs within a gene [Speed and Balding, 2014; Yang et al., 2011].

To test the null hypothesis H₀ : β = 0, we use the score vector of β which can be derived as U(β) = SΣ̂⁻¹ (Y −1α̂₀−Xα̂₁–Eα̂₂) where S = (S₁₁ ··· S_ij ··· S_MmM)^T, Y = (Y₁₁, ···, Y_MmM)^T, X = (X₁₁ ··· X_MmM)^T, and E = (E₁₁ ··· E_MmM)^T. The estimated covariance is $\widehat{\mathbf{\sum }}={\widehat{\sigma }}_G^2\mathbf{G}{\mathbf{G}}^T+{\widehat{\sigma }}_A^2\mathbf{K}+{\widehat{\sigma }}_C^2\mathbf{C}+{\widehat{\sigma }}_E^2\mathbf{I}$ where G = (G₁₁ ··· G_MmM)^T and K and C are block diagonal with K_i and J_i on the diagonals, respectively. We use the R package regress to obtain our estimates of α̂ = (α̂₀, α̂₁, α̂₂) and Σ̂ under the H₀ : β = 0. The Fisher Information of θ = (α, β) is I(θ) = (X̃|S)^T Σ̂⁻¹ (X̃|S) where X̃ = [1|X|E]. I(θ) can be partitioned into matrices I_XX, I_XS, I_SX, and I_SS according to the dimensions of α and β. Thus, the covariance of the score vector of β is $\mathbf{V}={\mathbf{I}}_{SS}-{\mathbf{I}}_{SX}{\mathbf{I}}_{XX}^{-1}{\mathbf{I}}_{XS}$.

We use the LMM score vector U = U(β) with its covariance matrix V to extend current score tests of GxE interaction to families: the score test, aSPU [Pan et al., 2014], GESAT [Lin et al., 2013], and iSeq-aSum [Coombes et al., 2016]. In similar fashion to Pan et al. [2014], we develop a larger family of tests called Seq-aSPU which can incorporate iSeq-aSum. These methods are summarized in Table 1.

Table 1.

A summary of methods used and their dependency on the score vector U and its covariance matrix V (Section 2.1). Note that all these tests require V to derive their null distribution.

Method	Dependency on U, V to compute test statistic	Description	Resampling-based
Score Test	U,V	Requires both U and V to construct test statistic	No
iSeq-aSum	U, d	computes dT U using a sequential search of the allocation vector (d)	Yes
aSPU	U, γ	adaptively sums U for different power of γ	Yes
Seq-aSPU	U, d, γ	Combines the strategies of iSeq-aSum and aSPU	Yes

2.1.1 Score Test

Given U and V, the score test can be calculated as T_sco = U^TV⁻¹U which has an asymptotic chi-square distribution with pq degrees-of-freedom (df) under H₀. However, if the number of variants q in a candidate gene is large, the score test can lose power to detect interaction due to its large df.

2.1.2 Adaptive Sum of Powered Score Tests

If we instead calculate our test statistic without using V, Pan [2009] showed that T_ssu = U^TU may more efficiently test for the combined effect of β because it uses fewer df. As an extension, Pan et al. [2014] proposed aSPU for testing for genetic main effects. This test was recently extended to tests of GxE interaction for independent subjects [Coombes et al., 2016]. Using the score vector U, we can construct the GxE interaction SPU test for families as T_SPU(γ) = 1^TU^γ where ${\mathbf{U}}^{\gamma }=(U_1^{\gamma },\cdots ,U_{pq}^{\gamma })$ for a set of integers γ ≥ 1. As γ increases, the larger components of U are weighted higher. The null distribution of these SPU test statistics may be difficult to derive. However, under H₀, the score vector U ~ 𝒩(0, V). Therefore, we can generate B copies of the null score vector by sampling from 𝒩(0, V) for which we calculate B copies of the SPU test $T_{\mathit{SPU}(\gamma )}^{(b)}$ where b = 1, ···, B. The p-value for a given γ is thus $P_{\mathit{SPU}(\gamma )}=({\sum }_{b=1}^{B}I(\mid T_{\mathit{SPU}(\gamma )}^{(b)}\mid >\mid T_{\mathit{SPU}(\gamma )}\mid )+1)/(B+1)$. Using the p-values for a set of γs, Pan et al. [2014] also proposes the aSPU test

$$T_{\mathit{aSPU}}=\underset{\gamma \in \mathrm{\Gamma }}{min}{P}_{\mathit{SPU}(\gamma )}.$$

where Γ = {1, 2, ···, 8,∞}. With γ = ∞, this is similar to the UminP test by using the maximum value from the score vector. Using the same B copies of the null score vector, we can calculate the aSPU test statistic for each null score vector $T_{\mathit{aSPU}}^{(b)}$ and find the proportion of null aSPU test statistics that are smaller than our observed aSPU test statistic. Thus, the p-value for the aSPU test is $P_{\mathit{aSPU}}=({\sum }_{b=1}^{B}I(T_{\mathit{aSPU}}^{(b)}>T_{\mathit{aSPU}})+1)/(B+1)$.

Here, the SSU test (SPU(2)) statistic is very similar to extending the gene-environment set association test (GESAT) statistic as proposed by Lin et al. [2013] to family data. However, rather than modeling the genetic main effect as a random effect, [Lin et al., 2013] proposed reducing the dimension of the genetic main effect using a ridge penalty which is tuned using a cross-validation technique. By reformulating the ridge penalty as a random effect, we can incorporate the estimation of the variance parameter into our test and avoid post-hoc cross-validation schemes.

A benefit of using our SPU test where γ = 2 is that p-values can be calculated using Davies method [Davies, 1980] rather than by sampling which can speed calculations. The SPU test of H₀ : β = 0 is equivalent to the score test of H₀ : τ = 0 when we define β ~ MVN(0, τI_pq). Now, rather than using a sampling method to calculate a p-value, we can use the characteristic function inversion method to calculate an asymptotic p-value [Lin et al., 2013]. Under the null hypothesis, the variance of the residuals is

$$var(\mathit{Y}-\stackrel{\sim }{\mathit{X}}\widehat{\mathit{\alpha }})=\widehat{\mathbf{\sum }}-\stackrel{\sim }{\mathit{X}}{({\stackrel{\sim }{\mathit{X}}}^T{\widehat{\mathbf{\sum }}}^{-1}\stackrel{\sim }{\mathit{X}})}^{-1}{\stackrel{\sim }{\mathit{X}}}^T={\mathbf{P}}_0.$$

Thus, ${\mathbf{U}}^T\mathbf{U}~{\sum }_{k=1}^q{\lambda }_k{\chi }_{1,k}^2$ where λ_k are the eigenvalues of the matrix S^T Σ̂⁻¹ P₀Σ̂⁻¹ S. The p-value is then computed analytically using the Davies method [Davies, 1980].

2.1.3 A Weighted SPU test

Another subset of the SPU test, SPU(1), is very similar to the Sum test. The Sum test uses a pooled regression estimate to model GxE interaction:

$$Y_{ij}={\alpha }_0+{\mathit{X}}_{ij}^T{\mathit{\alpha }}_1+{\mathbf{E}}_{ij}^T{\mathit{\alpha }}_2+{\mathbf{G}}_{ij}^T{\mathit{\alpha }}_3+{\beta }_c\sum _{k=1}^{pq}{d}_k{S}_{\mathit{ijk}}+a_{ij}+c_{ij}+e_{ij}$$ (2)

where d_k = 1 for all k and β_c is the pooled GxE interaction regression estimate. The null hypothesis we wish to test is H₀ : β_c = 0. A score vector of β_c can be derived as

$$\begin{gathered} U({\beta }_c)=\left(\sum _{k=1}^{pq}{d}_k{\mathbf{S}}_k\right){\widehat{\mathbf{\sum }}}^{-1}(\mathit{Y}-\mathbf{1}{\widehat{\alpha }}_0-\mathit{X}{\widehat{\mathit{\alpha }}}_1-\mathbf{E}{\widehat{\mathit{\alpha }}}_2) \\ =\sum _{k=1}^{pq}{d}_k\left({\mathbf{S}}_k{\widehat{\mathbf{\sum }}}^{-1}(\mathit{Y}-\mathbf{1}{\widehat{\alpha }}_0-\mathit{X}{\widehat{\mathit{\alpha }}}_1-\mathbf{E}{\widehat{\mathit{\alpha }}}_2)\right) \\ =\sum _{k=1}^{pq}{d}_k{U}_k={\mathit{d}}^T\mathbf{U} \end{gathered}$$

where d = (d₁, ···, d_pq)^T = 1 and S_k is the ${\sum }_{i=1}^M{m}_i\times 1$ vector for the k^th GxE interaction. Notice that the score vector for β_c is a linear combination of U. The score vector U ~ 𝒩(0, V) under the null hypothesis, thus, d^TU ~ 𝒩(0, d^TVd). Therefore, the score test statistic of β_c is

$$T(\mathit{d})={({\mathit{d}}^T\mathbf{U})}^2/({\mathit{d}}^T\mathbf{V}\mathit{d})$$ (3)

which has an asymptotic χ² distribution with one df under the null hypothesis.

The benefit of the Sum test is that it tests a single parameter β_c. Thus, it has low df and possibly increased power to detect interactions. However, a common issue with the Sum test is that it loses power if there are a combination of interactions with positive and negative effects. Instead of simply summing up the score vector using d = 1, Coombes et al. [2016] adaptively sums the GxE interactions with a chosen allocation vector d = (w₁s₁, ···, w_pqs_pq) where w_k is a chosen weight and s_k indicates the directionality of the effect for the k^th interaction. We generally set w_k = 1 and set s_k = 1 or −1 which indicates an interaction effect is positive or negative. The strategy of this test, which we refer to as iSeq-aSum, is to maximize the T(d) statistic by best combining positive and negative effects. Coombes et al. [2016] and Basu and Pan [2011] showed that using an adaptively pooled effect estimate can avoid the power loss associated with the Sum test when both positive and negative effects are present. However, if many interactions with no effect are pooled together with causal interactions, the regression estimate β_c will be pulled toward zero, which can result in a loss of power for both the Sum test and iSeq-aSum.

The SPU family of tests [Pan et al., 2014] can avoid losing power when there are many null interactions by increasing γ, so we use this strategy to propose a generalization of iSeq-aSum. To do this, we replace U in Equation 3 with U^γ to obtain the Seq-SPU test:

$$T({\mathit{d}}_{\gamma },\gamma )={({\mathit{d}}_{\gamma }^T{\mathbf{U}}^{\gamma })}^2/({\mathit{d}}_{\gamma }^T\mathbf{V}{\mathit{d}}_{\gamma })$$ (4)

where d_γ is allowed to vary for different γs. By taking the square root of T(d_γ, γ), it is easy to see that this test is equivalent to the weighted version of the SPU test [Kim et al., 2014] with weights equal to ${\mathit{d}}_{\gamma }{({\mathit{d}}_{\gamma }^T\mathbf{V}{\mathit{d}}_{\gamma })}^{-1/2}$. Notice that iSeq-aSum is equivalent to Seq-SPU(1). Also, it is clear that as γ increases, the denominator weight ${\mathit{d}}_{\gamma }^T\mathbf{V}{\mathit{d}}_{\gamma }$ will have less impact on the allocation. If there are many interactions with no effect, which causes Seq-SPU(1) to lose power, in concordance with the SPU tests, we increase γ to avoid power loss. To find the optimal d_γ for a given γ, we proceed through the following sequential algorithm:

1. Initialize d_γ = 1

2. for k in 1 : pq

   • Set d_γ,k = −1 or 1 corresponding to the allocation that maximizes T(d_γ, γ)

To compute p-values, we first generate B copies of the null score vector as before and find the optimal allocation ${\mathit{d}}_{\gamma }^{(b)}$ for a given γ. We then calculate $T{({\mathit{d}}_{\gamma }^{(b)},\gamma )}^{(b)}$ for each γ ∈ Γ and b = 1, ···, B. The Seq-SPU(γ) p-values are computed as $P(\gamma )=({\sum }_{b=1}^{B}I(\mid T{({\mathit{d}}_{\gamma }^{(b)},\gamma )}^{(b)}\mid >\mid T({\mathit{d}}_{\gamma },\gamma )\mid +1)/(B+1)$. The Seq-aSPU test statistic is calculated as min_γ∈Γ P(γ). The p-value for Seq-aSPU can be calculated as before for aSPU.

While we perform a search over the entire set of Γ = {1, ···, 8,∞}, we expect that only odd-valued γs will show power gain in comparison to their SPU counterparts because these γs are susceptible to power loss if there are a mix of positive and negative effects.

2.2 Univariate modeling of GxE interaction

An alternative to jointly testing for GxE interaction between the SNPs and all of the environments at once is to test for GxE interaction using each environment separately. This approach is especially useful if we have a large number of correlated environments. For the k^th environment, we use the following LMM:

$$Y_{ij}={\alpha }_0^{\ast }+{\mathit{X}}_{ij}^T{\mathit{\alpha }}_1^{\ast }+E_{\mathit{ijk}}{\alpha }_{2,k}^{\ast }+{\mathbf{G}}_{ij}^T{\mathit{\alpha }}_3^{\ast }+{\mathbf{S}}_{\mathit{ijk}}^T{\mathit{\beta }}_k^{\ast }+{\stackrel{\sim }{a}}_{ij}+{\stackrel{\sim }{c}}_{ij}+{\stackrel{\sim }{e}}_{ij}$$ (5)

where S_ijk = G_ij × E_ijk for the j^th individual for the i^th family. For Model 5, we are interested in testing the null hypothesis H₀ : β_k = 0 for all k = 1, ···, p. To obtain the score vector and covariance for each β_k, we follow the same steps as in Section 2.1. For each environment, we use the LMM score vector of its interaction with the SNPs and the covariance matrix to calculate p-values for the Score, SPU, and Seq-SPU tests as outlined in Section 2.1.1. We obtain the p-value for the test of H₀ : β_k = 0 for all k = 1, ···, p by applying a Bonferroni correction to the minimum p-value of the p tests. We refer to this testing procedure as the minP approach. Note that this approach could also be used for pq separate models with a single variant and a single environment in each model.

This univariate model, though computationally efficient, assumes a misspecified model for each genetic marker or environment. Similar to the argument provided in Lin et al. [2013], one can show that the asymptotic limits of the MLEs ( ${\widehat{\alpha }}_0^{\ast },{\widehat{\mathit{\alpha }}}_1^{\ast },{\widehat{\alpha }}_{2,k}^{\ast },{\widehat{\alpha }}_3^{\ast },{\widehat{\beta }}_k^{\ast }$) are, in general, not equal to the true values of (α₀,α₁, α_2,k, α₃, β_k) including when there is G-E dependence. Moreover, if there is G-E dependence or dependence among the multiple environmental or genetic factors, a test not incorporating all SNPs and environments into a model may have inflated type I error [Lin et al., 2013]. However, in Equation 5, we correct for the existence of such additional variants or environments through the random effects ã_ij and c̃_ij (Appendix 6.1). Hence even if the interpretability of ã_ij and c̃_ij will be different from the random effects in Equation 2.1, the minP approach will have correct type I error. We investigate this in Appendix 6.1 and in Section 3. As shown in Appendix 6.1, if the true disease model is a model in equation 1 where multiple SNPs and environments influence the disease, this minP approach will lose power to detect interactions because it will inflate the variance of the estimator of the G-E interaction. Nevertheless, this approach provides a computationally fast way to perform GxE interaction testing in families, especially when we test between a large set of variants and environmental factors.

We also demonstrate that for testing GxE interaction through the minP approach, it is generally sufficient to approximate the genetic and the environmental effect through a common random effect in twin studies. In other words, one could construct a valid minP test for G-E interaction with an ‘AE’ model instead of an ‘ACE’ model, where the random effect ‘A’ in the ‘AE’ model captures the effect of genetic and shared environmental effect in twins. As derived in Appendix 6.1 and illustrated in Simulation 1, we demonstrate that this additional random effect produces a valid test for interaction for the minP approach.

3 Results

We first compared through simulation studies the performance of different methods to test for GxE interaction between a SNP-set from a candidate gene and a set of environments for a twin dataset. We generated datasets with 400 MZ and 250 DZ twin pairs using the genotype and environmental data from the MCTFR study described in Section 4. This sample size is chosen to mimic the sample size of our stratified analysis in Section 4. To preserve the correlation structures for the SNPs and environments in the MCTFR, we jointly sampled the SNPs, environments, and sex from the twins with complete data in the MCTFR dataset. We selected a candidate gene for alcoholism [Olfson and Bierut, 2012], ADH1B, which has 11 SNPs in low LD genotyped for the Illumina Human660W-Quad Array chip. Approximately 46% of the twins are male. The four environment scores sampled for our simulations are approximately normal with mean zero and standard deviations ranging from 0.77 to 0.89. Higher scores for the environments indicate greater risk for developing alcoholism. The correlation between pairs of environments ranges from 0.25 to 0.41. The environments are also correlated within twin pairs with correlations ranging from 0.63 to 1. The genes and environments were correlated less than 0.03 which indicates that there is not gene-environment correlation.

We varied the number of causal SNPs Q in ADH1B and the proportion of total variance explained by these SNPs ( $R_G^2$). We let the causal SNPs interact P environments and varied the proportion of total variance explained by the interactions ( $R_S^2$). There were no other GxE interactions in our model. With Q SNPs interacting with P environments, we let

$$var(Y)=1=R_{\mathit{sex}}^2+R_E^2+R_G^2+R_S^2+{\sigma }_A^2+{\sigma }_C^2+{\sigma }_E^2$$ (6)

where $R_{\mathit{sex}}^2=0.008,R_E^2=0.35,{\sigma }_A^2=0.3,{\sigma }_C^2=0$, and ${\sigma }_E^2=1-(R_{\mathit{sex}}^2+R_E^2+R_G^2+R_{GE}^2+{\sigma }_A^2+{\sigma }_C^2)$ are the proportion of total variance explained by sex, four environments, genetic similarity, unmeasured shared environment, and error, respectively. For the i^th set of twins, we used the variance explained by each predictor, the male indicator X_i = (X_i1, X_i2), four environments E_i,k = (E_i1,k, E_i2,k) for k = 1, · · ·, 4, and the minor allele counts for the causal SNPs G_i,k = (G_i1,k, G_i2,k) for k = 1, · · ·, Q to simulate the phenotype Y_i = (Y_i1, Y_i2) as

$${\mathit{Y}}_i=b_1{\mathit{X}}_i+b_E\sum _{p=1}^4{\mathbf{E}}_{i,p}+b_G\sum _{q=1}^Q{\mathbf{G}}_{i,q}+b_S\sum _{p=1}^P\sum _{q=1}^Q{d}_q{\mathbf{G}}_{i,q}\otimes {\mathbf{E}}_{i,p}+{\mathit{a}}_i+{\mathit{c}}_i+{\mathit{e}}_i$$ (7)

where · is the dot product, $b_1=\sqrt{R_{\mathit{sex}}^2/(0.46(0.54))},{\mathit{a}}_i~\mathcal{N}(0,{\sigma }_A^2{\mathbf{K}}_i)$ where the off-diagonal elements of K_i are either 1 or 1/2 for MZ or DZ pair, respectively, and ${\mathit{e}}_i~\mathcal{N}(0,{\sigma }_E^2{\mathbf{I}}_2)$. We assumed that each causal SNP had the same effect b_G. We set

$$b_G=\sqrt{\frac{R_G^2}{var\left({\sum }_{k=1}^Q{G}_k\right)}}=\sqrt{\frac{R_G^2}{{\sum }_{k=1}^{Q}var(G_k)+{\sum }_{k\ne l}cov(G_k,G_l)}}$$ (8)

where G_k for k = 1, · · ·, Q are correlated binomial random variables representing each causal SNP. We estimated the variance and covariance terms in Equation 8 using only one twin from each pair from the MCTFR data. With this setup, the causal SNPs collectively explain $R_G^2$ of the total variance. We also determined the values of b_E and b_S in this way. For the causal interactions, we set (d₁, · · ·, d_Q) where d_q = 1 or −1 so that the q^th SNP interacts with the P environments either with a positive effect or negative effect.

We are interested in testing the null hypothesis that there are no GxE interactions. To estimate the type I error for testing GxE interaction, we generated 10,000 null datasets with $R_S^2=0$ which corresponds to setting b_S = 0. To estimate the power for each method, we generated 1,000 datasets with $R_S^2=0.02$. The power and type I error were calculated at an α = 0.05 or 0.01 level. To compute p-values for the SPU and Seq-SPU tests, we used B = 1000 to sample the null score vector. For each simulated dataset, we either tested for GxE interaction using the minP approach as described in Section 2.2 or the Joint approach as described in Section 2.1.

In Simulation 1, we assessed the performances of the ACE and AE models and compared the power to detect interaction for the minP and Joint approach. In Simulation 2, we assessed the impact of including a ridge penalty for the ACE model. Finally, we compared the performances of the SPU and Seq-SPU tests using the Score test as a control in Simulation 3.

3.1 Simulation 1

To compare the performances of the ACE and AE models for the minP and Joint approaches, we set Q = 2 and the fraction of total variance explained by the SNPs as $R_G^2=0.005$. We let the two SNPs interact in opposite directions, (d₁, d₂) = (1, −1), with either one or four environments.

In Model 7, we set ${\sigma }_C^2=0$ for unmeasured shared environments. However, the minP approach models each environment separately. Consequently, the environments not included in the test of interaction are “unmeasured” and resulted in ${\widehat{\sigma }}_C^2=0.13$ for the ACE model in Table 3. Meanwhile, the AE model approximately increased ${\widehat{\sigma }}_A^2$ by the same amount to account for the environments that were left out. The other variance components were similar among the ACE and AE models and both of these models maintained type I error and had similar power for either choice of P. Because we have used only twin data, specifying the A component as approximately A+C results in negligible difference between the ACE covariance structure and the AE covariance structure. When we used the Joint approach, there was no difference in type I error between the ACE and AE models because the true ${\sigma }_C^2$ was set to zero in Equation 7.

Table 3. Simulation 1.

Type I error and power comparison of AE and ACE models for using environments separately or all together. Analyses were simulated using the ADH1B gene with two causal SNPs interacting with P environments. The first SNP interacts with the P environments with a positive effect, while the second interacts with a negative effect. The interactions explain 2% of the total variance of the simulated phenotype. ${\widehat{\sigma }}_G^2$ is multiplied by 11 so that it can be interpreted as the proportion of variance explained by the 11 SNPs in the simulation.

	Test: Model	minP Approach		Joint Approach
		ACE	AE	ACE	AE
Mean (SD) of Variance Parameter Est.	$11\times {\widehat{\sigma }}_G^2=$	0.005 (0.006)	0.005 (0.006)	0.005 (0.006)	0.005 (0.006)
	${\widehat{\sigma }}_A^2=$	0.313 (0.095)	0.444 (0.03)	0.253 (0.061)	0.293 (0.027)
	${\widehat{\sigma }}_C^2=$	0.126 (0.083)	-	0.037 (0.049)	-
	${\widehat{\sigma }}_E^2=$	0.364 (0.028)	0.356 (0.026)	0.338 (0.025)	0.334 (0.024)
Type I error		ACE	AE	ACE	AE
α = 0.05	Score	0.0422	0.0402	0.0406	0.0420
	aSPU	0.0473	0.0452	0.0496	0.0524
	Seq-aSPU	0.0459	0.0451	0.0516	0.0519
α = 0.01	Score	0.0072	0.0071	0.0081	0.0074
	aSPU	0.0089	0.0081	0.0133	0.0120
	Seq-aSPU	0.0096	0.0081	0.0092	0.0122
Power with P=4		ACE	AE	ACE	AE
α = 0.05	Score	0.3890	0.3840	0.4770	0.4820
	aSPU	0.4620	0.4540	0.7930	0.7960
	Seq-aSPU	0.5030	0.5010	0.7730	0.7710
α = 0.01	Score	0.1550	0.1510	0.2470	0.2520
	aSPU	0.2660	0.2590	0.5840	0.5980
	Seq-aSPU	0.2910	0.2820	0.5410	0.5390
Power with P=1		ACE	AE	ACE	AE
α = 0.05	Score	0.5460	0.5350	0.5430	0.5460
	aSPU	0.6720	0.6660	0.8530	0.8570
	Seq-aSPU	0.7020	0.6950	0.8630	0.8630
α = 0.01	Score	0.3460	0.3420	0.3170	0.3130
	aSPU	0.5200	0.5010	0.6960	0.6980
	Seq-aSPU	0.5390	0.5300	0.7150	0.7100

We next compared power between the minP and Joint approaches in Table 3. When all four environments interacted with the two SNPs, the Joint approach was much more powerful than the minP approach for any method because the model for the minP approach excludes three environments that interact with the SNPs (Appendix 6.1). Interestingly, the aSPU and Seq-aSPU methods had more power with the Joint approach even when P = 1. Likewise, due to the large increase in df for the Score test (11 df → 44 df), the Score test’s power for the Joint approach was similar to the Score test’s power for the minP approach. For simplicity, the rest of our simulations will use the ACE model with the minP approach and P = 1. Alternative model choices did not affect our conclusions for Simulations 2 and 3.

3.2 Simulation 2

To assess the potential gain of using a ridge penalty for the ACE model, we either fitted the genetic main effects as fixed effects or as a random effect. We used Q = 2 with interactions in opposite directions. In Table 2, the number of SNPs in ADH1B (11) times ${\widehat{\sigma }}_G^2$ correctly estimated the specified proportion of variance explained by the causal SNPs $R_G^2$. The A, C, and E variance components in Table 2 were very similar for all models, and there was no evidence of inflated type I error for any method or model. Due to the reduction in parameters to estimate, the model using ridge penalization was usually more powerful than the model that fitted the main effects as fixed effects regardless of the genetic main effect size; although the power difference was very small.

Table 2. Simulation 2.

Type I error and power comparison of the ACE model with a ridge penalty implemented or not. Simulations used two causal interactions acting in opposite directions. The interactions explain 2% of the total variance of the simulated phenotype. ${\widehat{\sigma }}_G^2=$ is multiplied by 11 so that it can be interpreted as the proportion of variance explained by the 11 SNPs in the simulation.

Ridge Penalty:	$R_G^2=$	0.005		0.01		0.1
		Yes	No	Yes	No	Yes	No
Mean (SD) of Variance Parameter Est.	$11\times {\widehat{\sigma }}_s^2=$	0.005 (0.006)	-	0.011 (0.01)	-	0.16 (0.032)	-
	${\widehat{\sigma }}_A^2=$	0.313 (0.095)	0.313 (0.096)	0.312 (0.095)	0.313 (0.096)	0.313 (0.083)	0.313 (0.083)
	${\widehat{\sigma }}_C^2=$	0.126 (0.083)	0.125 (0.084)	0.127 (0.083)	0.126 (0.084)	0.129 (0.074)	0.129 (0.074)
	${\widehat{\sigma }}_E^2=$	0.364 (0.028)	0.364 (0.028)	0.36 (0.028)	0.36 (0.028)	0.272 (0.022)	0.272 (0.022)
Type I error		Yes	No	Yes	No	Yes	No
α = 0.05	Score	0.0420	0.0418	0.0416	0.0416	0.0441	0.0427
	aSPU	0.0457	0.0474	0.0449	0.0473	0.0465	0.0479
	Seq-aSPU	0.0438	0.0466	0.0439	0.0470	0.0468	0.0479
α = 0.01	Score	0.0072	0.0076	0.0074	0.0078	0.0077	0.0081
	aSPU	0.0088	0.0094	0.0088	0.0090	0.0089	0.0094
	Seq-aSPU	0.0096	0.0106	0.0091	0.0102	0.0108	0.0111
Power		Yes	No	Yes	No	Yes	No
α = 0.05	Score	0.5460	0.5190	0.5560	0.5220	0.6650	0.6520
	aSPU	0.6720	0.6690	0.6770	0.6740	0.7660	0.7630
	Seq-aSPU	0.7020	0.6970	0.7030	0.7000	0.7930	0.7870
α = 0.01	Score	0.3460	0.3340	0.3570	0.3430	0.4770	0.4600
	aSPU	0.5200	0.5200	0.5220	0.5100	0.6260	0.6190
	Seq-aSPU	0.5390	0.5260	0.5340	0.5340	0.6540	0.6370

3.3 Simulation 3

For our last simulation, we compared the performances of aSPU and Seq-aSPU by studying the power of each family of tests over the entire Γ = {1, · · ·, 8,∞} set. We set the fraction of total variance explained by the SNPs at $R_G^2=0.005$. We used the minP approach with the ridge-penalized ACE model. Using α = 0.05, Figure 1 shows the comparison between the powers of the Score, SPU, and Seq-SPU tests for two different scenarios: interactions with only one environment in the same or opposite directions.

Figure 1. Simulation 3.

Comparison of the SPU and Seq-SPU family of tests in two scenarios: (a,b) interactions in the same direction, or (c,d) interactions in opposite directions. There are either 2/11 interactions with effect (a,c) or 4/11 interactions with effect (b,d) that explain 2% of the total variation. For a given γ, significant differences in power between the SPU test and Seq-SPU test are denoted with a star. Significant differences are defined as 95% confidence interval (CI) of the power of $\text{Seq}-\text{SPU}(\mathit{power}\pm z_{0.025}\sqrt{(0.95)(0.05)/1000})$ not containing the power of its SPU counterpart.

For either scenario in Figure 1, we let the interactions of either two or four of the possible 11 SNPs (Q = 2 or 4) with the first environment explain 2% of the total variation in the simulated phenotype. When only two interactions had effect, higher-valued γs for either SPU or Seq-SPU had higher power because the nine interactions with no effect are weighted much smaller in comparison. Therefore, as the number of non-null interactions was increased to four, the lower-valued γs for both SPU and Seq-SPU increased in power because more interactions should be equally weighted. The biggest difference between aSPU and Seq-aSPU was seen in this case for either scenario because there are large differences between SPU and Seq-SPU with γ = 1 which corresponds to the Sum and iSeq-aSum tests, respectively.

In the first scenario in Figure 1a and 1b, we set the causal interactions to be in the same direction d = 1. Because the effects of the causal interactions are in the same direction, using the sequential algorithm of iSeq-aSum to determine directional effects is unneeded. Consequently, iSeq-aSum is less powerful than the Sum test and thus aSPU has higher power than Seq-aSPU. This difference is most evident when four out of 11 interactions had effect because the Sum test had much higher power than the higher df test, iSeq-aSum.

For the second scenario in Figure 1c and 1d, we set half of the interactions to be positive and the other half to be negative. As discussed in Section 2.1.3, this is a situation in which the even-valued γs in SPU are expected to perform much better than odd-valued γs. This pattern for the SPU tests held in our results regardless of the number of causal interactions and was especially true for γ = 1. Meanwhile, the odd-valued γs for Seq-SPU did not lose dramatic power due to the sequential algorithm summing the powered score statistics in the correct way. This was most evident when four out of 11 interactions had effect. In this case, all odd-valued γs for Seq-SPU did significantly better than their SPU counterparts. As a result, Seq-aSPU was much better than aSPU in this scenario.

4 Minnesota Center for Twin and Family Research

Finally, we applied the methods studied in Section 3 to the MCTFR dataset. The MCTFR follows MZ and DZ twins through adolescence into at least early adulthood to study psychological outcomes such as SUDs [Miller et al., 2012]. Previous studies of the MCTFR data have estimated the amount of alcohol consumption to be highly heritable with about 50% to 80% of the phenotype explained by genetic variation in the four-member sample consisting of parents and genetic offspring [McGue et al., 2013]. The development of this SUD reflects the influence of genes that are modulated by environmental factors such as the quality of the parent-child relationship, affiliation with deviant peers, personality characteristics, antisociality (e.g., conduct disorder) and life stress. The goal of this analysis is to perform gene-based tests of GxE interaction to study whether association between drinking score and candidate genes for alcoholism are modified at age 17 in the twin cohort by four different environmental factors.

The phenotype used in our models was a drinking index formed by a factor analysis of questions from an in-person interview and questionnaire. Our environmental factors of interest, ‘deviant peers’, ‘environmental assets’, ‘family conflict’, ‘family climate’ scores, were computed as factor scores from a range of questionnaires completed by the twins and their parents. The environmental factor scores are normally distributed and correlated with each other as described in Section 3.

We tested for GxE interaction between the SNPs in the candidate genes for alcoholism listed in Olfson and Bierut [2012] and the four environments. 50 of these 54 genes had genotyped SNPs in the MCTFR data. Covariates used in the model include age and the first four principal components from an Eigenstrat analysis [Price et al., 2006] of the SNP data were used as covariates to adjust for population stratification. Because Eigenstrat can be sensitive to pairs of close relatives, one relative was removed from each pair in the initial computation, but genotypes of the relatives were projected onto components from the unrelated set of subjects. More details can be found in Miller et al. [2012]. We used the Joint approach with the ridge penalized ACE model to test for GxE interaction in these genes because this model and approach performed best in our simulations. We limited our analyses to the seventeen-year-olds with non-zero drinking scores to avoid confounding the mechanisms that influence teenagers to start drinking with the mechanisms that influence how much a teenager drinks. Lastly, we analyzed males and females separately because two out of the four environments had a significant interaction with sex. This finding is consistent with previous findings that the disease etiology of alcoholism differs among sex [Prescott, 2002; Hardie et al., 2008]. We also found that the A,C, and E variance components shown in Table 4 varied greatly between males and females.

Table 4.

Real data analysis of MCTFR twin data stratified by sex. Genes with large differences between aSPU and Seq-aSPU are shown. P-values less than 0.05 are marked in bold.

	Males		Females
Gene	SLC6A2		ADH7		CNR1
Location	16q12.2		4q23		6q15
# SNPs	44		38		39
$\#\mathbf{SNPs}\times {\widehat{\mathit{\sigma }}}_{\mathit{G}}^{\mathbf{2}}=$	0.0003		0.0022		2.1e-09
${\widehat{\mathit{\sigma }}}_{\mathit{A}}^{\mathit{2}}=$	0.410		0.171		0.172
${\widehat{\mathit{\sigma }}}_{\mathit{C}}^{\mathbf{2}}=$	6.6e-06		0.085		0.088
${\widehat{\mathit{\sigma }}}_{\mathit{E}}^{\mathbf{2}}=$	0.293		0.273		0.274
γ =	SPU	Seq-SPU	SPU	Seq-SPU	SPU	Seq-SPU
1	0.92208	0.01299	0.00899	0.46653	0.10190	0.00009
2	0.17982	0.34466	0.45055	0.59241	0.04995	0.09855
3	0.64635	0.32368	0.16384	0.52647	0.07692	0.09337
4	0.32567	0.42358	0.54146	0.62837	0.12887	0.17647
5	0.73127	0.47353	0.33167	0.62438	0.13387	0.19106
6	0.44855	0.51249	0.61039	0.66533	0.21079	0.24257
7	0.80619	0.59441	0.45554	0.65834	0.20480	0.23206
8	0.53646	0.58741	0.65534	0.67433	0.26374	0.27757
∞	0.71728	0.74925	0.71129	0.71429	0.23676	0.23322
adaptive	0.38462	0.02697	0.02298	0.69131	0.13487	0.00031
Score	0.13947		0.82886		0.05701

To compute p-values for aSPU and Seq-aSPU, we initially sampled the null score vector with B = 1000. For smaller p-values, we increased B by a factor of ten until the final p-value was greater than 5/B. Only one p-value, Seq-aSPU’s p-value for CNR1 in the female analysis, met the α = 0.05/50 = 0.001 threshold to be identified as a significant GxE interaction. We also found that the largest estimated percent of variance in alcohol consumption explained by a gene was 0.47% for the GABRB1 gene in females and 0.46% for the HTR1B gene in males.

While most of the p-values are very similar for aSPU and Seq-aSPU, in Table 4 we investigated the genes, SLC6A2, CNR1, and ADH7, for which the two tests show large differences. In these three genes, the main difference between the SPU and Seq-SPU tests is for γ = 1. For SLC6A2 and CNR1, iSeq-aSum had a more significant p-value than the Sum test, which caused Seq-aSPU to have a much smaller p-value than aSPU. Conversely, the Sum test had a much smaller p-value than iSeq-aSum for ADH7 which caused aSPU to have a more significant p-value than Seq-aSPU. These findings are consistent with our simulations in Section 3.

5 Discussion

In this paper, we have developed tests of interaction between SNPs in a candidate gene and multiple environments for family data using an LMM framework. This framework allows us to incorporate covariance terms to adjust for genetic similarity and unmeasured shared environments in families using the ACE model. We incorporated the measured environments using the minP and Joint modeling approaches. We found that even if only one environment has interaction with the gene, the Joint approach can still have an advantage over the minP approach and we explore in Appendix 6.1 the reason behind the loss in power for the minP test.

One important finding of our simulation study is that even in presence of correlation among multiple environmental factors, the minP approach produced a valid test for GxE interaction. We also showed that in twin studies, the random effect to capture the shared environment or shared polygenic effect corrects for the effects of the unaccounted environmental factors (See Simulation 1 and Appendix 6.1). Hence, our proposed minP approach provides a fast, sensible, and valid way of testing for GxE interactions, especially in the presence of a large number of correlated genetic variants and environmental factors.

In this paper, we have proposed a ridge penalty re-expressed as a random effect to capture the genetic main effects, which can reduce the number of parameters we need to estimate. By expressing this penalty term as a random effect, we were able to easily incorporate the variance estimate for the penalization into our score test instead of performing computationally intensive cross-validation. We also avoided computational difficulties with parameter estimation in penalized mixed models.

Finally, we proposed a generalization of the iSeq-aSum test which is equivalent to a weighted version of the SPU test with weights calculated using the sequential algorithm of Basu and Pan [2011]. Because aSPU and Seq-aSPU adaptively choose the best γ over the set Γ, if a γ-value included in the search performs poorly, the adaptive test can potentially lose power. For instance, when there were a mix of positive and negative interactions, iSeq-aSum had much higher power than the Sum test; thus, Seq-aSPU was more powerful than aSPU. Thus, by improving the power of odd-valued γs when there are a combination of positive and negative effects, specifically γ = 1, Seq-aSPU can gain substantial power over aSPU. However, if the interactions were in the same direction, aSPU was more powerful than Seq-aSPU, because the Sum test was more powerful than iSeq-aSum. For our analysis of the MCTFR data, most of the p-values for aSPU and Seq-aSPU were very similar. However, the only significant GxE interaction was identified by Seq-aSPU and not aSPU because iSeq-aSum’s p-value was more significant compared to the Sum test which was not significant.

The models presented here do have some limitations. First, Seq-aSPU is more computationally intensive than aSPU because it performs a stepwise search for every γ. However, this search only provided a benefit for odd-valued γs in our simulations. Thus, a more computationally efficient approach would be to not perform this search for even γs and instead either use the equivalent SPU test or let a_γ = 1 for even γ. Second, using LMMs to compute the score vector U and its covariance V can be quite computationally intensive. For our simulations using the R package regress, this calculation accounted for the majority of our computation time (Table 5). While R packages such as lme4 and nlme can be quite fast, these packages are currently not able to implement kinship matrices or a ridge penalty expressed as a random effect. However, the recent package GMMAT [Chen et al., 2016] can potentially be implemented here to speed up our computation. Alternatively, it is possible to calculate the score vector for GxE interaction using generalized estimating equations (GEEs). However, implementing a ridge penalty for the genetic main effect in GEEs is not straightforward. Finally, while the framework proposed here is intended for testing for GxE interaction, the same framework can be used to incorporate GxG interactions in families. By using random effects in an LMM framework, we can account for family relatedness and reduce the number of genetic main effect parameters needed to be estimated.

Table 5.

Mean computation time for one iteration in our simulations. The computation time for either method is computed after U and V are estimated through our LMM fit. The computation time for the score test is negligible given U and V.

	minP approach (per model)	Joint Approach
Fitting the LMM	Time (s)	Time (s)
ACE w/ridge penalty	541.8	683.3
AE w/ridge penalty	430.9	492.1
ACE	191.3	-
Method	Time (s)	Time (s)
SPU	0.6	0.7
Seq-aSPU	3.4	24.4

6 Appendix

6.1 Bias Caused by Omission of Relevant Variables and validity of minP test

In this subsection, we evaluate the impact of not including correlated environmental variables (confounders) in the regression model and the validity of the computationally fast minP test in Section 2.2 in families. First, we derive the bias and standard error inflation in the estimation of GxE interaction parameter when we have a sample of unrelated individuals. For the convenience of derivation, we will assume that the phenotypes are standardized such that the mean is equal to zero and the variance is equal to one. Suppose that a correctly specified regression model would be

$$\mathit{Y}={\beta }_0+\mathbf{G}{\beta }_1+{\mathbf{E}}_1{\beta }_2+{\mathbf{E}}_2{\beta }_2^{\ast }+{\mathbf{S}}_1{\beta }_3+\mathit{\epsilon },$$

where $\mathit{\epsilon }~N(0,{\sigma }_e^2\mathbf{I})$, S₁ = G* E₁, E₁ and E₂ have 1 and (P − 1) columns, respectively, and they E₁ and E₂ are correlated. In our minP approach (Section 2.2), we adjust for one environmental factor at a time. If we regress Y on G, E₁ and S₁ without including E₂, then the estimator of β₃ will be biased due to the exclusion of E₂. For simplicity, let us assume that we have already regressed out the main effects of G and E₁, i.e β₁ = β₂ = 0 in Equation 6.1. Then the parameter estimate for β₃ will be

$$b_3={({\mathbf{S}}_1^T{\mathbf{S}}_1)}^{-1}{\mathbf{S}}_1^T\mathit{Y}={\beta }_0+{({\mathbf{S}}_1^T{\mathbf{S}}_1)}^{-1}{\mathbf{S}}_1^T{\mathbf{E}}_2{\beta }_2^{\ast }+{({\mathbf{S}}_1^T{\mathbf{S}}_1)}^{-1}{\mathbf{S}}_1^T\mathit{\epsilon }.$$

Taking the expectation, we see that unless ${\mathbf{S}}_1^T{\mathbf{E}}_2=0$ or ${\beta }_2^{\ast }=0$, b₃ is biased. The bias can be quantified as

$$\mathbb{E}[b_3\mid {\mathbf{S}}_1,{\mathbf{E}}_2]={\beta }_3+{\mathbf{P}}_{3.2}{\beta }_2^{\ast },\text{where}{\mathbf{P}}_{3.2}={({\mathbf{S}}_1^T{\mathbf{S}}_1)}^{-1}{\mathbf{S}}_1^T{\mathbf{E}}_2,$$

Each column of the P_3.2 matrix is the column of slopes in the least squares regression of the corresponding column of E₂ on the columns of S₁ Green [2002]. The variance of the GxE interaction parameter estimate b₃ will be $\mathit{Var}[b_3\mid {\mathbf{S}}_1]={\sigma }_e^2{({\mathbf{S}}_1^T{\mathbf{S}}_1)}^{-1}$.

If we had computed the correct regression by including E₂, then the parameter estimate b_3.2 of the GxE interaction parameter β₃ in Equation 6.1 would have been unbiased and would have had a covariance matrix equal to the upper left block of ${\sigma }_e^2{({\mathbf{S}}_1^T{\mathbf{S}}_1)}^{-1}$. The variance of b_3.2 will be given by

$$\begin{gathered} \mathit{Var}[b_{3.2}\mid {\mathbf{S}}_1,{\mathbf{E}}_2]={\sigma }_e^2{({\mathbf{S}}_1^T{\mathit{M}}_2{\mathbf{S}}_1)}^{-1},\text{where}{\mathit{M}}_2=\mathbf{I}-{\mathbf{E}}_2{({\mathbf{E}}_2^T{\mathbf{E}}_2)}^{-1}{\mathbf{E}}_2^T, \\ ={\sigma }_e^2{[{\mathbf{S}}_1^T{\mathbf{S}}_1-{\mathbf{S}}_1^T{\mathbf{E}}_2{({\mathbf{E}}_2^T{\mathbf{E}}_2)}^{-1}{\mathbf{E}}_2^T{S}_1]}^{-1}. \end{gathered}$$ (9)

We can compare the covariance matrices of b₃ and b_3.2 more easily by comparing their inverses, i.e $\mathit{Var}{[b_3\mid {\mathbf{S}}_1]}^{-1}-\mathit{Var}{[b_{3.2}\mid {\mathbf{S}}_1,{\mathbf{E}}_2]}^{-1}=(1/{\sigma }_e^2){\mathbf{S}}_1^T{\mathbf{E}}_2{({\mathbf{E}}_2^T{\mathbf{E}}_2)}^{-1}{\mathbf{E}}_2^T{\mathbf{S}}_1$, which is nonnegative definite. Hence, although b₃ is biased, its variance is not larger than that of b_3.2 (since the inverse of its variance is at least as large). Hence omitting the correlated environmental factors could produce an inflated test statistic depending on the degree and direction of the bias and the variance reduction. The extent of reduction in variance of b₃ will depend on the proportion of variance explained by the expected value of ${\mathbf{E}}_2^T{\mathbf{E}}_2$ and the degree of dependence between E₂ and S₁. Hence, under this setup, our proposed minP test (Section 2.2) will be invalid.

On the other hand, our regression model in Equation 1 in families can be written as a combination of fixed and random effects. For the sake of illustrating our point, we will consider that our model is comprised of the interaction term and the unaccounted confounder E₂, i.e

$$\mathit{Y}={\mathbf{E}}_2{\beta }_2+{\mathbf{S}}_1{\mathit{\beta }}_3+\mathit{a}+\mathit{c}+\mathit{e}.$$ (10)

The random effects a, c, e represents the polygenic, shared and non-shared environmental effects in a family, respectively (Equation 1). If E₂ is not accounted for in the model, so the analysis model will have the following components:

$$\mathit{Y}={\mathbf{S}}_1{\beta }_3+\mathit{a}+\mathit{c}+\mathit{e}.$$ (11)

Let us assume that the random effect is only comprised of the shared environment and the non-shared environment. One could think of this random effect c as the unmeasured set of environmental confounders that contribute to the shared environment among family members. If we had observed all these unmeasured shared environmental factors, E₂, the random environment c can be written as c = E₂β₂, then $\mathit{c}~\text{MVN}\left(0,{\sigma }_c^2{\mathbf{E}}_2{\mathbf{E}}_2^T\right)$. With Equation 11, the estimator of β₃ will be ${\widehat{\beta }}_3={({\mathbf{S}}_1^T{\mathbf{V}}^{-1}{\mathbf{S}}_1)}^{-1}{\mathbf{S}}_1^T{\mathbf{V}}^{-1}\mathit{Y}$. We assume $\mathbf{V}={\sigma }_c^2{\mathbf{E}}_2{\mathbf{E}}_2^T+{\sigma }_e^2\mathbf{I}$ and thus ${\widehat{\beta }}_3={({\mathbf{S}}_1^T{\mathbf{V}}^{-1}{\mathbf{S}}_1)}^{-1}\left({\mathbf{S}}_1^T{\mathbf{V}}^{-1}{\mathbf{S}}_1{\beta }_3+{\mathbf{S}}_1^T{\mathbf{V}}^{-1}\mathit{a}+{\mathbf{S}}_1^T{\mathbf{V}}^{-1}\mathit{c}+{\mathbf{S}}_1^T{\mathbf{V}}^{-1}\mathit{e}\right)$. Hence $\mathbb{E}({\widehat{\beta }}_3)={({\mathbf{S}}_1^T{\mathbf{V}}^{-1}{\mathbf{S}}_1)}^{-1}\left({\mathbf{S}}_1^T{\mathbf{V}}^{-1}{\mathbf{S}}_1{\beta }_3\right)={\beta }_3$, since 𝔼(a) = 𝔼(c) = 𝔼(e) = 0. Hence, the random effect model in Equation 11 will generate an unbiased estimate β̂₃ of β₃. The variance of the interaction parameter estimate would be

$$\begin{gathered} \mathit{Var}({\widehat{\beta }}_3\mid {\mathbf{S}}_1,\mathit{c},{\mathbf{E}}_2)={({\mathbf{S}}_1^T{\mathbf{V}}^{-1}{\mathbf{S}}_1)}^{-1}={\left({\mathbf{S}}_1^T{\left({\sigma }_c^2{\mathbf{E}}_2{\mathbf{E}}_2^T+{\sigma }_e^2\mathbf{I}\right)}^{-1}{\mathbf{S}}_1\right)}^{-1} \\ ={\sigma }_e^2{\left[{\mathbf{S}}_1^T{\left(\mathbf{I}+\frac{{\sigma }_c^2}{{\sigma }_e^2}{\mathbf{E}}_2{\mathbf{E}}_2^T\right)}^{-1}{\mathbf{S}}_1\right]}^{-1}. \end{gathered}$$

Hence the estimator β̂₃ will have a bigger variance or lower precision than the estimator b_3.2. However, the precision difference will be significantly reduced as compared to Equation 9. The precision difference will depend on the extent to which the environment is shared among individuals (i.e how big ${\sigma }_c^2$ is). However, the estimator β̂₃ from the mixed effect model in Equation 11 will perform better than b₃, since it will produce an unbiased estimator of β₃. Unfortunately, E₂ is usually unknown, and, in general, we instead consider the random effect $\mathit{c}~\text{MVN}(0,{\sigma }_c^2\mathbf{J}{\mathbf{J}}^T)$ where J is the unit matrix. By replacing the variance ${\sigma }_c^2{\mathbf{E}}_2{\mathbf{E}}_2^T$ with the variance ${\sigma }_c^2\mathbf{J}{\mathbf{J}}^T$ we will further increase the variance of our estimator to $\mathit{Var}[{\widehat{\beta }}_3\mid {\mathbf{S}}_1,c]={\sigma }_e^2{\left[{\mathbf{S}}_1^T{\left(\mathbf{I}+\frac{{\sigma }_c^2}{{\sigma }_c^2}\mathbf{J}{\mathbf{J}}^T\right)}^{-1}{\mathbf{S}}_1\right]}^{-1}$.

When we add a random effect to capture additional genetic variation and environmental variation, the residual confounding effect in the mean will get absorbed in the random effect, so the minP test mentioned in Section 2.2 will produce a valid test of GxE interaction. In fact, we got correct type I errors for the minP approach in all of our comparisons in Simulation 1.

One could potentially show that we do not in fact need separate random effects to capture the unmeasured confounded effects for both gene and environment in twin studies provided both the effects are linear in unmeasured confounders. As we did in the previous paragraph, we can consider one random effect a in Equation 11 instead of c, where a ~ MVN(0,A) where $\mathbf{A}={\sigma }_A^2\mathbf{K}$. Hence the performance of the new estimator ${\widehat{\beta }}_3^{\ast }$ will depend on the correlation between the measured and omitted environmental factors can be captured through the polygenic effect a. In twin studies, the correlation between the polygenic effects for MZs will be 1 and DZ twins will be $\frac{1}{2}$. Due to the high genetic correlation among twins, the impact of the shared environment will be absorbed in the misspecified AE model as shown below. In our simulation studies, we noticed correct type I error, but loss in power for the minP approach due to larger variance of the estimator for GxE interaction. However, the power loss was less severe for the AE model as compared to the ACE model due to lower inflation in the variance of β̂₃. In general, the random effect variance, ${\sigma }_a^2$, would be inflated to capture the unmeasured confounders. It may be possible to quantify the relationship between the random-effect parameters between the models as we did in the next section.

6.2 Impact on parameter estimates for assuming AE model when ACE is correct

The log-likelihood based on n_MZ MZ twin pairs and n_DZ DZ twin pairs is

$$l(\theta )=-(n_{MZ}+n_{DZ})log(2\pi )-\frac{n_{MZ}}{2}log\mid {\mathbf{\sum }}_{MZ}\mid -{\mathbf{\sum }}_{MZ}\left(\frac{1}{2}\sum _{\text{MZ}\text{paris}}(Y_i-\mu ){\mathbf{\sum }}_{MZ}^{-1}(Y_i-\mu )\right)-\frac{n_{DZ}}{2}log\mid {\mathbf{\sum }}_{DZ}\mid -{\mathbf{\sum }}_{DZ\mathit{pairs}}\left(\frac{1}{2}\sum _{\text{DZ}\text{paris}}(Z_i-\mu ){\mathbf{\sum }}_{DZ}^{-1}(Z_i-\mu )\right)$$

where μ is the mean vector and the Σ_MZ, Σ_DZ are the covariance matrices for the MZ and DZ sib-pairs. Under the true ACE model, we have

$${\mathbf{\sum }}_{MZ}=\left(\begin{array}{cc}{\sigma }_a^2+{\sigma }_c^2+{\sigma }_e^2& {\sigma }_a^2+{\sigma }_c^2\\ {\sigma }_a^2+{\sigma }_c^2& {\sigma }_a^2+{\sigma }_c^2+{\sigma }_e^2\end{array}\right),{\mathbf{\sum }}_{DZ}=\left(\begin{array}{cc}{\sigma }_a^2+{\sigma }_c^2+{\sigma }_e^2& \frac{1}{2}{\sigma }_a^2+{\sigma }_c^2\\ \frac{1}{2}{\sigma }_a^2+{\sigma }_c^2& {\sigma }_a^2+{\sigma }_c^2+{\sigma }_e^2\end{array}\right)$$

Under the assumed AE model, we have

$${\mathbf{\sum }}_{MZ}=\left(\begin{array}{cc}{\lambda }_a^2+{\lambda }_e^2& {\lambda }_a^2\\ {\lambda }_a^2& {\lambda }_a^2+{\lambda }_e^2\end{array}\right),{\mathbf{\sum }}_{DZ}=\left(\begin{array}{cc}{\lambda }_a^2+{\lambda }_e^2& \frac{1}{2}{\lambda }_a^2\\ \frac{1}{2}{\lambda }_a^2& {\lambda }_a^2+{\lambda }_c^2\end{array},\right).$$

where the assumed additive genetic variance, common environmental variance and random environmental variance are ${\sigma }_a^2,{\sigma }_c^2,{\sigma }_e^2$, respectively, in the ACE model. and the additive genetic variance and random environmental variance are ${\lambda }_a^2,{\lambda }_e^2$, respectively, in the AE model. Meanwhile, Y_i follows a bivariate normal distribution with mean 0 and covariance matrix Σ_MZ and Z_i follows a bivariate normal distribution with mean 0 and covariance matrix Σ_DZ. Note that the maximum likelihood estimations of are unbiased estimators of the solutions of equations $\mathbb{E}[\frac{\delta l(\theta )}{\delta {\lambda }_a^2}]=0$ and $\mathbb{E}[\frac{\delta l(\theta )}{\delta {\lambda }_e^2}]=0$. By some matrix operations, one can show that the solutions to the above equations would derive the relationship between the parameters in ACE and AE model. If the sample only consists of MZ twins, then one can show that ${\lambda }_a^2={\sigma }_a^2+{\sigma }_c^2$ and ${\lambda }_e^2={\sigma }_e^2$. If the sample only consists of DZ twins, one can show that ${\lambda }_a^2={\sigma }_a^2+2{\sigma }_c^2$ and ${\lambda }_e^2={\sigma }_e^2-{\sigma }_e^2$. However when the sample contains both MZ and DZ twins, one can show that the relationship will be non-linear and as shown in the previous section and in Simulation 1, the test statistic derived under the misspecified AE model will still produce a valid test for GxE interaction.