Fast eQTL Analysis for Twin Studies

Abstract

Twin data are commonly used for studying complex psychiatric disorders, and mixed effects models are one of the most popular tools for modeling dependence structures between twin pairs. However, for eQTL (expression quantitative trait loci) data where associations between thousands of transcripts and millions of single nucleotide polymorphisms need to be tested, mixed effects models are computationally inefficient and often impractical. In this paper, we propose a fast eQTL analysis approach for twin eQTL data where we randomly split twin pairs into two groups, so that within each group the samples are unrelated, and we then apply a multiple linear regression analysis separately to each group. A score statistic that automatically adjusts the (hidden) correlation between the two groups is constructed for combining the results from the two groups. The proposed method has well-controlled type I error. Compared to mixed effects models, the proposed method has similar power but drastically improved computational efficiency. We demonstrate the computational advantage of the proposed method via extensive simulations. The proposed method is also applied to a large twin eQTL data from the Netherlands Twin Register.

Introduction

Genome-wide association studies (GWAS) have been widely used over the last decade for identifying genetic variants associated with a diversity of complex human diseases, such as type 2 diabetes, breast cancer and psychiatric disorders [Garcia-Closas et al., 2013; Hanson et al., 2014; Winham et al., 2013]. These studies have identified a large number of disease associated SNPs (single nucleotide polymorphisms). However, the majority of the SNPs detected by GWAS individually explain a very small fraction of the total heritability associated with these traits, with no immediately clear functional or regulatory roles [McCarthy and Hirschhorn, 2008]. Gene expression, as an intermediate molecular phenotype, may provide additional insight into the regulatory roles of SNPs implicated by GWAS. With widely available high throughput technologies, gene expression and genetic variant data can be collected simultaneously on disease-relevant tissues from the same individuals, and expression quantitative trait loci (eQTL) analysis can be performed to assess which genomic regions and genetics variants lead to gene expression [Gilad et al., 2008]. eQTL analyses have been used to identify eQTL hotspots (genomic regions affecting multiple transcripts), construct causal networks, elucidate subclasses of clinical phenotypes, and determine lists of candidate genes for clinical trials [see the reviews of Kendziorski and Wang, 2006 and Wright et al., 2012]. Recent research has also shown that SNPs detected by GWAS are significantly more likely to be eQTL, which can be used to boost the discovery of trait-associated SNPs, and improve understanding of the biology underlying complex traits [Hsu et al., 2010; Nicolae et al., 2010; Schadt et al., 2008; Zhang et al., 2012; Zhong et al., 2010].

Various eQTL analytical approaches have been established where an association test is performed between one transcript and one SNP at a time by linear regression analysis, analysis of variance [Shabalin, 2012], generalized linear regression [Hernandez et al., 2012], or mixed effects models [Kang et al., 2008] depending on the type of eQTL data. More complicated analytical procedures, such as Bayesian regression [Bottolo et al., 2011; Chipman et al., 2011; Stegle et al., 2010] and partial least square regression [Chun et al., 2009] have also been applied to eQTL data. In addition, several methods have been proposed for detecting associations between a group of SNPs and the gene expression of each transcript [Hoggart et al., 2008; Michaelson et al., 2009; Zeng, 1994]. User friendly software such as Genevar [Yang et al., 2010] and eQTL viewer [Zou et al., 2007] have been developed for eQTL data analysis, output visualization and result interpretation. The high dimensionality of eQTL data makes modern eQTL analysis computationally intensive, given the fact that associations between several million SNPs and tens of thousands of transcripts need to be tested. To mitigate this heavy computational burden, analysis may be restricted to a small number of SNP-transcript pairs [Ghazalpour et al., 2008]. Alternatively, Shabalin [2012] has developed a fast eQTL analytical tool which is thousands of times faster than any existing QTL/eQTL software. Matrix eQTL is extremely computationally efficient because it expresses the association test between a SNP and the transcript pair as a function of the correlation between pair, which can be realized by a quick matrix operation.

For complex psychiatric disorders, such as schizophrenia and major depressive disorder, twin studies have received attention for establishing the general extent to which genes and environment are etiologically important [Boomsma et al., 2002; Chou et al., 2009; Neale and Cardon, 1992; Park et al., 2012; Silventoinen et al., 2003; Vaccarino et al., 2008]. Typical twin data include both monozygotic twins (MZ) and dizygotic twins (DZ), plus unpaired individual twins (singletons). MZ twins are assumed to be genetically identical, while DZ twins share 50% of their genes on average. Assuming that MZ and DZ twins share the same environment, a higher phenotypic similarity between MZ twins compared to DZ twins indicates that the phenotype is genetically controlled. Unlike data with independent samples, twin data require more careful statistical modeling since ignoring genetic relatedness and shared environment among twin pairs may lead to high false and/or low true positive findings. Several statistical approaches are available for twin data. One of the most common approaches is structural equation modeling (SEM) Neale et al. [1989]. Several software programs to perform SEM are available, such as Mx [Neale et al., 1999], Mplus [Muthen and Muthen, 1998], LISREL [Jorsekog and Sorborn, 1986] and OpenMx [Boker et al., 2011, 2012]. Another popular alternative for twin and family data is the mixed-effects model where random effects are used to properly account for the correlations among subjects [Carlin et al., 2005; Kuna et al., 2012; Wang et al., 2011]. Mixed model have a well-established theory which is familiar to statisticians. Moreover, it is conveniently implemented in most statistical software and can flexibly adjust other non-genetic and genetic covariates [Feng et al., 2009; Rabe-Hesketh et al., 2008].

Though powerful, the mixed effects models are computationally intensive and impractical for modern twin eQTL analysis. Moreover, the ultra fast tool Matrix eQTL is not readily applicable to twin data since it does not model the dependence structure between twin pairs. To overcome these computational challenges, we propose a novel fast twin eQTL analysis approach. In this approach, we first randomly split the twin pairs into two groups such that within each group, the samples are unrelated. We then run a separate analysis for each group using any statistical procedure valid for independent data, such as multiple linear regression or analysis of variance. When combining the results from the two groups, we find traditional meta-analysis procedures, such as Fisher’s test, is no longer applicable since the two sets of results are not independent due to the correlation between twin pairs. Naively combining the two sets of p-values without consideration of the dependence structure of the data would lead to inflated false positive findings. In this paper, we propose a novel score test which automatically adjusts the (hidden) correlation structures between twin pairs, and therefore controls the type I error accurately. To demonstrate the computational advantages and evaluate the performance of the proposed method, we conduct extensive simulations under various settings to mimic real world twin data.

Our simulation results establish that our proposed approach controls type I error rates well, with negligible power loss compared to the mixed effects model, which is the gold standard for analyzing twin data. Furthermore, the computational efficiency of the proposed method is dramatically improved. The proposed method is more than a thousand times faster than mixed effects models. The fast performance of the proposed method is achieved by computing the most computationally intensive part in the score test by matrix operations, similar to what has been done in matrix eQTL. The utility of the proposed method is further illustrated by analyzing a twin eQTL data where the twin samples (~ 4,000) are from the Netherlands twin registry.

Materials and Methods

The first step of the proposed approach is to randomly split each twin pair into two groups such that all samples within each group are unrelated. Thus a statistical procedure valid for independent data can be directly applied to each group for testing SNP-transcript pair associations. For simplicity and because of its popularity for eQTL data, multiple linear regression is considered. Specifically, for each group, given a transcript and a SNP, we fit the following linear regression model:

$$y_i=\beta g_i+{\mathit{x}}_i\mathit{\gamma }+{\epsilon }_i,i=1,\cdots ,n,$$

where for the ith individual, y_i is the gene expression, g_i is the SNP genotype which is coded 0, 1 or 2 according to the number of minor alleles in the genotype, β is the fixed effect of the SNP; and γ = (γ₀, γ₁, …, γ_q)′ is a vector of parameters corresponding to the vector of non-genetic covariates plus the intercept x_i = (1, x_i1, …, x_iq). Rewriting the above model in matrix form, we get

$$Y=\beta G+X\mathit{\gamma }+\epsilon ,$$

where Y = (y₁, y₂, …, y_n)′, G = (g₁, g₂, …, g_n)′ and $X=({\mathit{x}}_1^{\prime },\cdots ,{\mathit{x}}_n^{\prime })\prime$. The log-likelihood function is therefore

$$l(\theta )=\sum _{i=1}^n{l}_i(\theta )$$

where θ = (β, γ, σ²) and $l_i=-\frac{1}{2}\text{log}{\sigma }^2-\frac{{(y_i-\beta g_i-{\mathit{x}}_{\mathbf{i}}\mathit{\gamma })}^{\mathbf{2}}}{2{\sigma }^2}$. For the eQTL analysis, the null hypothesis is H₀ : β = 0 where the vector of the nuisance parameters is η = (γ, σ²). Following Zou et al. [2004], we get the score function of the ith individual as

$$U_i=U_{\beta ,i}(0,\mathit{\eta })-{\mathrm{\Sigma }}_{\beta ,\mathit{\eta }}(0,\mathit{\eta }){\mathrm{\Sigma }}_{\mathit{\eta },\mathit{\eta }}{(0,\mathit{\eta })}^{-1}{U}_{\mathit{\eta },i}(0,\mathit{\eta }),$$

where U_β,i(β, η) and U_η,i(β, η) are defined as

$$U_{\beta ,i}(\beta ,\mathit{\eta })=\frac{\partial l_i(\beta ,\mathit{\eta })}{\partial \beta }=\frac{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma })g_i}{{\sigma }^2},$$

$$U_{\mathit{\eta },i}(\beta ,\mathit{\eta })=\frac{\partial l_i(\beta ,\mathit{\eta })}{\partial \mathit{\eta }}=\left(\begin{array}{c}\hfill \frac{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma }){\mathit{x}}_i^{\prime }}{{\sigma }^2}\hfill \\ \hfill \frac{{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma })}^2}{2{\sigma }^4}-\frac{1}{2{\sigma }^2}\hfill \end{array}\right),\text{respectively},$$

and Σ_β,η(β, η) is the limit of $n^{-1}{\sum }_{i=1}^n\frac{{\partial }^2{l}_i}{\partial \beta \partial \mathit{\eta }}$, and Σ_η,η(β, η) is the limit of $n^{-1}{\sum }_{i=1}^n\frac{{\partial }^2{l}_i}{\partial \mathit{\eta }\partial \mathit{\eta }}$. The two Hessian matrices are

$$\frac{{\partial }^{2}l(\beta ,\mathit{\eta })}{\partial \beta \partial \mathit{\eta }}={\sum }_{i=1}^n(-\frac{g_i{z}_i}{{\sigma }^2},-\frac{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma })g_i}{{\sigma }^4})\text{and}$$

$$\frac{{\partial }^{2}l(\beta ,\mathit{\eta })}{\partial \mathit{\eta }\partial \mathit{\eta }}=n^{-1}{\sum }_{i=1}^n\left(\begin{array}{cc}\hfill -\frac{{\mathit{x}}_i^{\prime }{\mathit{x}}_i}{{\sigma }^2}\hfill & \hfill -\frac{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma }){\mathit{x}}_i^{\prime }}{{\sigma }^4}\hfill \\ \hfill -\frac{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma }){\mathit{x}}_i^{\prime }}{{\sigma }^4}\hfill & \hfill \frac{1}{2{\sigma }^4}-\frac{{(y_i-\beta g_i-{\mathit{x}}_i\mathit{\gamma })}^2}{{\sigma }^6}\hfill \end{array}\right).$$

Let the restricted MLE η̂ be the solution of Σ_i=1 U_η,i(β, η) = 0, where β is set to 0 in this equation. Specifically, we have

$$\hat{\mathit{\gamma }}={(X\prime X)}^{-1}X\prime Y,$$

$${\hat{\sigma }}^2={\sum }_{i=1}^n{(y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})}^2/n={\Vert Y-X\hat{\mathit{\gamma }}\Vert }^2/n.$$

Note that η̂ is estimated under H₀ and thus does not depend on the genotypes of any given SNP. Therefore it does not change from SNP to SNP, and needs only to be estimated once for each transcript. Also note that all of the off-diagonal elements in ${\frac{{\partial }^{2}l(\beta ,\mathit{\eta })}{\partial \mathit{\eta }\partial \mathit{\eta }}|}_{\beta =0,\mathit{\eta }=\hat{\mathit{\eta }}}$ equal to zero. Replacing all unknown parameters by their sample estimators in the score function, we get

$${Û}_i=U_{\beta ,i}(0,\hat{\mathit{\eta }})-{\mathrm{\Sigma }}_{\beta ,\mathit{\eta }}(0,\hat{\mathit{\eta }}){\mathrm{\Sigma }}_{\mathit{\eta },\mathit{\eta }}{(0,\hat{\mathit{\eta }})}^{-1}{U}_{\mathit{\eta },i}(0,\hat{\mathit{\eta }})=\frac{(y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})g_i}{\widehat{{\sigma }^2}}-\sum g_i{\mathit{x}}_i{(\sum {\mathit{x}}_i^{\prime }{\mathit{x}}_i)}^{-1}\frac{(y_i-{\mathit{x}}_i\hat{\mathit{\gamma }}){\mathit{x}}_i^{\prime }}{\widehat{{\sigma }^2}}+\frac{\sum (y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})g_i}{n\widehat{{\sigma }^2}}-\frac{\sum (y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})g_i}{n\widehat{{\sigma }^4}}{(y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})}^2.$$

For computational efficiency, we express the score function Û_i in terms of a matrix operation below. Denote

$$M_x=X{(X\prime X)}^{-1}X\prime ,$$

$$J_n=(1,\cdots ,1)\prime \text{and is of length}n,$$

$$a={\sum }_{i=1}^n{g}_i{\mathit{x}}_i=G\prime X,$$

$$A={\sum }_{i=1}^{n_k}{\mathit{x}}_i^{\prime }{\mathit{x}}_i=X\prime X,$$

$$c=\frac{\sum (y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})g_i}{n\widehat{{\sigma }^2}}=\frac{1}{n\widehat{{\sigma }^2}}(G\prime Y-G\prime X{(X\prime X)}^{-1}X\prime Y)=\frac{1}{n\widehat{{\sigma }^2}}G\prime (I-M_x)Y,$$

we have

$${Û}_i=\frac{y_i-{\mathit{x}}_i\hat{\mathit{\gamma }}}{\widehat{{\sigma }^2}}[(g_i-a{A}^{-1}{\mathit{x}}_i^{\prime })-c(y_i-{\mathit{x}}_i\hat{\mathit{\gamma }})]+c.$$

Let Û be the vector of the score function of all individuals. That is,

$$Û=\left(\begin{array}{c}\hfill {Û}_1\hfill \\ \hfill {Û}_2\hfill \\ \hfill ⋮\hfill \\ \hfill {Û}_n\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \frac{y_1-{\mathit{x}}_1\hat{\mathit{\gamma }}}{\widehat{{\sigma }^2}}\hfill \\ \hfill \frac{y_2-{\mathit{x}}_2\hat{\mathit{\gamma }}}{\widehat{{\sigma }^2}}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{y_n-{\mathit{x}}_n\hat{\mathit{\gamma }}}{\widehat{{\sigma }^2}}\hfill \end{array}\right)\times \left[\right(\begin{array}{c}\hfill g_1-{\mathit{x}}_1{A\prime }^{-1}a\prime \hfill \\ \hfill g_2-{\mathit{x}}_2{A\prime }^{-1}a\prime \hfill \\ \hfill ⋮\hfill \\ \hfill g_n-{\mathit{x}}_n{A\prime }^{-1}a\prime \hfill \end{array})-c(\begin{array}{c}\hfill y_1-{\mathit{x}}_1\hat{\mathit{\gamma }}\hfill \\ \hfill y_2-{\mathit{x}}_2\hat{\mathit{\gamma }}\hfill \\ \hfill ⋮\hfill \\ \hfill y_n-{\mathit{x}}_n\hat{\mathit{\gamma }}\hfill \end{array}\left)\right]+\left(\begin{array}{c}\hfill c\hfill \\ \hfill c\hfill \\ \hfill ⋮\hfill \\ \hfill c\hfill \end{array}\right)=\frac{(I-M_x)Y}{\widehat{{\sigma }^2}}\times [(G-X{A\prime }^{-1}a\prime )-c(I-M_x)Y)]+J_{n}c.$$

Substituting a, A and c into the above equation, we get

$$Û=\frac{(I-M_x)Y}{\widehat{{\sigma }^2}}\times [(I-M_x)G-\frac{1}{n\widehat{{\sigma }^2}}(I-M_x)YY\prime (I-M_x)G]+\frac{1}{n\widehat{{\sigma }^2}}{J}_{n}Y\prime (I-M_x)G=\{\frac{(I-M_x)Y}{\widehat{{\sigma }^2}}\times [(I-M_x)-\frac{1}{n\widehat{{\sigma }^2}}(I-M_x)YY\prime (I-M_x)]+\frac{1}{n\widehat{{\sigma }^2}}{J}_{n}Y\prime (I-M_x)\}G.$$ (*)

When the covariates X and G are independent of each other, the elements inside {} can be simplified further to

$$\frac{(I-M_x)Y}{\widehat{{\sigma }^2}}\times [(I-\frac{1}{n\widehat{{\sigma }^2}}(I-M_x)YY\prime ]+\frac{1}{n\widehat{{\sigma }^2}}{J}_{n}Y\prime .$$

Note that the elements inside {} only depend on Y and the covariates X, and thus are the same across all SNPs. This motivates us to derive the above matrix operation to calculate the score vectors across a large number of SNPs for a given transcript simultaneously. Specifically, we replace the vector G in equation (*) by a matrix H = (G₁, …, G_m), where G_j is the G vector corresponding to SNP j (j = 1, …, m). The score matrix Û then is n × m, where the jth column represents the score vector at SNP j. Remember that twin pairs are randomly split into two groups. Notationally let’s add superscripts to Û above and denote Û⁽¹⁾ and Û⁽²⁾ as the score matrices of group 1 and group 2, respectively. Also let n_k be the number of individuals in the kth group (k = 1, 2) and the score function of the ith subject for the jth SNP in the kth group be $U_{i,j}^{(k)}$ (i = 1, …, n_k), or specifically we now have

$${Û}^{(k)}=\left(\begin{array}{cccc}\hfill {Û}_{1,1}^{(k)}\hfill & \hfill {Û}_{1,2}^{(k)}\hfill & \hfill \cdots \hfill & \hfill {Û}_{1,m}^{(k)}\hfill \\ \hfill {Û}_{2,1}^{(k)}\hfill & \hfill {Û}_{2,2}^{(k)}\hfill & \hfill \cdots \hfill & \hfill {Û}_{2,m}^{(k)}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \cdots \hfill & \hfill ⋮\hfill \\ \hfill {Û}_{n_k,1}^{(k)}\hfill & \hfill {Û}_{n_k,2}^{(k)}\hfill & \hfill \cdots \hfill & \hfill {Û}_{n_k,m}^{(k)}\hfill \end{array}\right).$$

The linear regression results for the jth SNP from the two groups may be naively combined as

$$W_{\text{naive}}(j)=\frac{{(\hat{\beta }(1)+\hat{\beta }(2))}^2}{{(SE(\hat{\beta }(1)))}^2+{(SE(\hat{\beta }(2)))}^2}$$

and assumed to follow ${\chi }_1^2$ asymptotically under H₀. Here, β̂(k) is the estimate of the SNP effect β with its standard error SE(β̂(k)) from the linear regression analysis of the kth group (k = 1, 2). This test is only valid when the results from the two groups are independent of each other, which is not likely to be true for twin data. To account for the dependence structure of the two groups, we derive a new score statistic to automatically adjust the correlation between the two groups:

$$W_{\text{proposed}}(j)=\frac{{({\sum }_{i=1}^{n_1}{Û}_{i,j}^{(1)}+{\sum }_{i=1}^{n_2}{Û}_{i,j}^{(2)})}^2}{{\sum }_{i=1}^{n_1}{({Û}_{i,j}^{(1)})}^2+{\sum }_{i=1}^{n_2}{({Û}_{i,j}^{(2)})}^2+2{\sum }_{i=1}^{n_{\mathit{\text{twin}}}}{Û}_{i,j}^{(1)}\times {Û}_{i,j}^{(2)}},$$

where n_twin is the total number of twin pairs, and groups 1 and 2 individuals are arranged in such way that the first n_twin samples in groups 1 and 2 are paired twins, and the remaining samples are singletons. The proposed test statistic W_proposed(j) follows ${\chi }_1^2$ asymptotically under H₀.

Results

Simulation studies

The proposed method is applied to simulated twin data to evaluate its performance. Each dataset includes 900 or 1800 individuals consisting of MZ twins, DZ twins and singletons in the ratio 2 : 2 : 1. Two continuous covariates (x₁, x₂) with effects γ = (0.3, 0.1) are correlated to the response variable y, where x₁ and x₂ are generated from N(3, 1) and N(5, 2), respectively. The response variable y is generated from the following model,

$$y_i=\mu +g_i\beta +{\mathit{x}}_i\mathit{\gamma }+a_i+c_i+d_i+e_i,i=1,\cdots ,n,$$

where y_i is the gene expression for the ith individual, μ is the grand mean, g_i is the SNP genotype, and x_i is a vector of non-genetic covariates. The SNP genotype g_i and the vector of covariates x_i are generated independently. The random terms a_i, d_i, c_i, e_i are the additive, dominant, common environment effects and random error, respectively, which are mutually independent and normally distributed with mean zero and variance ${\sigma }_a^2,{\sigma }_c^2,{\sigma }_d^2$ and ${\sigma }_e^2$, respectively. For subjects i and j who are a twin pair, we have $\mathit{\text{cov}}(a_i,a_j)={\sigma }_a^2$ and $\mathit{\text{cov}}(d_i,d_j)={\sigma }_d^2$ if they are MZ pair; $\mathit{\text{cov}}(a_i,a_j)={\sigma }_a^2/2$ and $\mathit{\text{cov}}(d_i,d_j)={\sigma }_d^2/4$ if they are DZ pair, while $\mathit{\text{cov}}(c_i,c_j)={\sigma }_c^2$ for all twin pairs. According to Neale et al. [1989], the above model is referred to as the ACE or ACDE model depending ${\sigma }_d^2=0$ or not. For each simulation set up, 1000 datasets are generated. We set ${\sigma }_e^2=1,{\sigma }_a^2=0.75$ and ${\sigma }_c^2=0.75$, resulting in the additive heritability $h_a^2$ of 0.462 and the variance explained by the shared environmental c², of 0.231 under the ACE model. For the ACDE model, we set ${\sigma }_d^2$ to $2{\sigma }_a^2$, leading to $h_a^2=0.316$ and c² = 0.158. Since there are 1000 association tests for each simulated dataset, 1000 datasets give a total of 1000 * 1000 = 1 million tests, which are used for the type I error evaluation. The type I error of the proposed method is compared with three other methods: 1) a multiple linear regression model on the full twin data, where the dependence between the twin pairs is ignored; 2) the naive approach; and 3) the mixed effects model: y_ij = g_ijβ + x_ijγ + a_ij + d_ij + c_ij + ε_ij, where i and j are family id and individual index respectively, x_ij is the vector of non-genetic covariates plus intercept, g_ij is the SNP genotype, and the definition of a_ij, c_ij, d_ij and their covariance structures are described in the above ACE and ACDE models.

Results from Tables 1 and 2 demonstrate that the type I error rates of both the proposed method and the mixed effects model are well controlled. In contrast, if the linear regression model (lm) is directly applied to the full twin data, the type I error rates have been dramatically inflated. For example, under the settings of n = 1800 and the targeted type I error rate α = 0.05, the type I errors for the data from the ACE model and ACDE model are 0.098 and 0.102, respectively. The type I error inflation in the naive method is also clear.

Table 1.

Type I error comparison for data from the ACE model

	n = 900				n = 1800
α	0.05	0.01	0.001	0.0001	0.05	0.01	0.001	0.0001
proposed	0.0522	0.0108	0.0011	0.0001	0.0507	0.0100	0.001	0.0001
naive	0.1000	0.0310	0.060	0.0012	0.0989	0.0301	0.0055	0.0010
mixed	0.0509	0.0103	0.0011	0.0001	0.0502	0.0100	0.0010	0.0001
lm	0.0988	0.0301	0.0057	0.0011	0.0984	0.0298	0.0053	0.0010

proposed: proposed score test statistic

naive: naive test statistic

mixed: mixed effects model

lm: multiple linear regression

Table 2.

Type I error comparison for data from the ACDE model

	n = 900				n = 1800
α	0.05	0.01	0.001	0.0001	0.05	0.01	0.001	0.0001
proposed	0.0524	0.0107	0.0011	0.0001	0.0510	0.0104	0.0010	0.0001
naive	0.1027	0.0321	0.0064	0.0012	0.1025	0.0320	0.0062	0.0013
mixed	0.0509	0.0103	0.0010	0.0001	0.0502	0.0101	0.0010	0.0001
lm	0.1024	0.0317	0.0061	0.0012	0.1020	0.0317	0.0061	0.0012

proposed: proposed score test statistic

naive: naive test statistic

mixed: mixed effects model

lm: multiple linear regression

For power comparisons, we generate 1000 datasets under the alternative hypothesis, where we set β to 0.32, 0.37, 0.45, and 0.50 in the ACE model and β to 0.40, 0.45, 0.50, and 0.55 in the ACDE model. For both of the models, the non-genetic effects γ and the variance components ${\sigma }_a^2,{\sigma }_e^2,{\sigma }_c^2$ and ${\sigma }_d^2$ are kept the same as in the above type I error investigation. The results in Tables 3 and 4 show that the proposed method has negligible power loss compared to the gold standard mixed effects models regardless of the sample size. All simulations and data analyses are conducted in R (a programming language, R Core Team [2014]).

Table 3.

Power comparison for data from the ACE model

		β = 0.32			β = 0.37			β = 0.45			β = 0.50
	Method	proposed	naive	mixed	pro	naive	mixed	pro	naive	mixed	pro	naive	mixed
n = 900	α = 0.05	0.733	0.817	0.761	0.834	0.886	0.865	0.933	0.966	0.951	0.970	0.985	0.986
	0.01	0.503	0.662	0.557	0.664	0.790	0.710	0.830	0.903	0.882	0.899	0.951	0.933
	0.001	0.225	0.416	0.286	0.363	0.584	0.434	0.629	0.796	0.696	0.745	0.873	0.811
	0.0001	0.103	0.240	0.130	0.166	0.372	0.224	0.368	0.633	0.466	0.530	0.753	0.630
n = 1800	α = 0.05	0.945	0.971	0.966	0.984	0.991	0.996	1.000	1.000	0.999	1.000	1.000	1.000
	0.01	0.831	0.919	0.889	0.940	0.973	0.963	0.989	0.998	0.999	0.999	1.000	0.999
	0.001	0.592	0.781	0.690	0.784	0.912	0.867	0.953	0.988	0.983	0.988	0.998	0.999
	0.0001	0.344	0.600	0.459	0.567	0.796	0.693	0.864	0.953	0.928	0.951	0.989	0.983

proposed: proposed score test statistic

naive: naive test statistic

mixed: mixed effects model

lm: multiple linear regression

Table 4.

Power comparison for data from the ACDE model

		β = 0.40			β = 0.45			β = 0.50			β = 0.55
	Method	proposed	naive	mixed	pro	naive	mixed	pro	naive	mixed	pro	naive	mixed
n = 900	α = 0.05	0.740	0.849	0.786	0.842	0.906	0.873	0.904	0.950	0.932	0.949	0.974	0.963
	0.01	0.501	0.670	0.561	0.633	0.792	0.688	0.755	0.870	0.804	0.844	0.926	0.890
	0.001	0.231	0.426	0.268	0.330	0.569	0.395	0.459	0.695	0.538	0.597	0.805	0.672
	0.0001	0.094	0.242	0.122	0.151	0.342	0.201	0.237	0.489	0.299	0.334	0.627	0.426
n = 1800	α = 0.05	0.964	0.982	0.980	0.986	0.993	0.995	0.995	0.999	0.999	0.999	1.000	0.999
	0.01	0.859	0.942	0.913	0.948	0.980	0.971	0.981	0.991	0.990	0.994	0.999	0.998
	0.001	0.664	0.831	0.745	0.800	0.918	0.869	0.895	0.974	0.948	0.969	0.988	0.985
	0.0001	0.431	0.676	0.511	0.605	0.813	0.703	0.756	0.907	0.844	0.865	0.972	0.927

proposed: proposed score test statistic

naive: naive test statistic

mixed: mixed effects

lm: multiple linear regression

Computational Efficiency

Our proposed method combines the advantages of multiple linear regression and the matrix operation to achieve fast computational performance. The proposed method took 361 seconds with one 2.93GHz Intel processor to perform 1 million association tests whereas the mixed effects model took 21, 000 seconds using a cluster of 100 Intel processors to run the same number of tests. The R function that implements the proposed method can be found at www.bios.unc.edu/~feizou/software.

Netherlands Twin Registry (NTR) eQTL Study

A total of 2, 752 individuals from the Netherlands Twin Registry (NTR) had their SNP genotypes and gene expression data measured [Wright et al., 2014] on Affymetrix 6.0 SNP arrays and U219 expression arrays. The goals of this project were to quantify the heritability of each transcript, to build a comprehensive list of eQTL in peripheral blood, and to assess the biomedical relevance of these eQTLs. After quality control [see Wright et al., 2014, for details], 686, 895 SNPs and 47, 585 transcripts on 2, 494 individuals were used for the eQTL analysis. The 2, 494 individuals include 638 MZ pairs, 529 DZ pairs and 160 singletons. A total of 96 covariates including the plate number, hybridization well position, age at blood sampling, sex, time intervals from sampling to RNA extraction and hybridization, and total white and red cell counts, 5 principal components (PCs) from the gene expression data, and 3 PCs derived from the SNP data are included as covariates in the eQTL analysis [see Wright et al., 2014].

In contrast to controlling the family-wise error, controlling the false discovery rate (FDR) is less stringent and handles the dependence between test statistics reasonably well [Benjamini and Yekutieli, 2001]. Instead of saving the p-values for all the SNP-transcript pairs, Shabalin [2012] only saves the p-values that pass a pre-specified significance threshold, on which the Benjamini and Hochberg procedure is applied. Following the procedure of Shabalin [2012], we identified 184, 474 and 127, 891 SNP-transcript pairs at FDR = 0.01 and FDR = 0.001, respectively, with the proposed method. In contrast, the naive method identified 225, 419 and 136, 509 significant SNP-transcript pairs at FDR = 0.01 and FDR = 0.001, which are significantly more than what have been detected by the proposed method. These additional findings are likely to be false positives, given the fact that the type I error inflation of the naive method. Based on the results from the proposed method, SNP-transcripts pairs with p-values passing certain thresholds are shown in the heat map (upper panel of Figure 1). Significant SNP-transcript pairs along the 45 degree diagonal line in the heat map are cis-eQTLs, whereas those off-diagonal are trans-eQTLs. The lower panel of Figure 1 clearly suggests that there are two master regulator regions on chromosomes 3 and 17 that affect a large number of transcripts, which deserve further investigation.

Figure 1.

eQTL hotspot for NTR twin data

Upper panel: heat map of the SNP-transcript pairs with p-values< 10⁻¹⁵. Green, blue, red and dark dots indicate the pairs whose −log10(p) falls into [15, 25), [25, 50), [50, 100) and [100, ∞), respectively. The colored lines on the 4 edges are used to differentiate the 22 autosomes. X-axis: SNP locations; Y-axis: transcript locations.

Lower panel: the number of transcripts (y-axis) that are significantly associated with each SNP (P < 10⁻¹⁵). X-axis: SNP locations.

The top 20 SNP-transcript pairs detected by the proposed approach are summarized in Table 5. Note that the p-values of these pairs from the naive method and Matrix eQTL method are too small to be reported precisely and thus are reported as 0 in the software R.

Table 5.

Top 20 SNP-transcript pairs identified from the proposed method

	SNP	chr(SNP)	start(SNP)	end(SNP)	GENE	chr(GENE)	start(GENE)	end(GENE)	log10(pval)	q value
1	rs3909451	chr5	96295120	96295121	11745601_a_at	chr5	96215266	96225192	228.4077	1.278352e-218
2	rs27300	chr5	96363406	96363407	11745601_a_at	chr5	96215266	96225192	227.2055	8.544353e-218
3	rs12229020	chr12	10117690	10117691	11729479_a_at	chr12	10124007	10138056	227.1056	8.544353e-218
4	rs7313235	chr12	10132274	10132275	11729479_a_at	chr12	10124007	10138056	226.8456	1.166032e-217
5	rs2235918	chr1	17422605	17422606	11727597_at	chr1	17393255	17445948	226.6566	1.441486e-217
6	rs2076607	chr1	17422659	17422660	11727597_at	chr1	17393255	17445948	225.8381	7.908774e-217
7	rs9272346	chr6	32604371	32604372	11750528_x_at	chr6	32410960	32411702	225.5650	1.271430e-216
8	rs4808485	chr19	16439497	16439498	11764269_at	chr19	16438314	16438642	224.0849	3.360241e-215
9	rs27290	chr5	96350093	96350094	11745601_a_at	chr5	96215266	96225192	221.9954	3.670607e-213
10	rs4698634	chr4	17630191	17630192	11736024_at	chr4	17616279	17627249	221.6546	7.240198e-213
11	rs27300	chr5	96363406	96363407	11747137_x_at	chr5	96215265	96253609	221.4142	1.145032e-212
12	rs3909451	chr5	96295120	96295121	11747137_x_at	chr5	96215265	96253609	221.2025	1.708595e-212
13	rs2549794	chr5	96244540	96244541	11747137_x_at	chr5	96215265	96253609	221.0543	2.218743e-212
14	rs12229020	chr12	10117690	10117691	11753241_s_at	chr12	10124195	10137625	220.9615	2.550835e-212
15	rs7673500	chr4	17621378	17621379	11736024_at	chr4	17616279	17627249	220.8615	2.939277e-212
16	rs12229020	chr12	10117690	10117691	11762101_at	chr12	10124033	10136838	220.8420	2.939277e-212
17	rs2549794	chr5	96244540	96244541	11745601_a_at	chr5	96215266	96225192	220.7022	3.816818e-212
18	rs2076608	chr1	17422301	17422302	11727597_at	chr1	17393255	17445948	220.1213	1.373340e-211
19	rs2235914	chr1	17424844	17424845	11727597_at	chr1	17393255	17445948	220.0841	1.417414e-211
20	rs9902260	chr17	62399869	62399870	11757545_x_at	chr17	62399863	62400230	219.5656	4.443247e-211

Discussion

eQTL analysis usually involves thousands of transcripts and millions of SNPs assayed in thousands of individuals. The challenges for eQTL analysis for twin data are the heavy computational burden and the dependency of twins within pairs. The mixed effects model is the gold standard for analyzing twin data. However, it is extremely time-consuming for modern eQTL data. Although matrix eQTL is an ultra fast tool, it is not applicable to twin data for eQTL analysis, since the covariance matrix V is assumed to be known. This assumption is not true for twin data, where the covariance matrix can be decomposed into several parts. For example, $V={\sigma }_a^2(A+\frac{{\sigma }_e^2}{{\sigma }_a^2})$ where ${\sigma }_a^2$ and ${\sigma }_e^2$ are unknown. If the heritability is much larger than random error, covariance for the additive genetic effect A can be used to approximate the covariance V, otherwise dropping the second term in the parentheses may result in an invalid inference. Alternatively, the heritability for each transcript needs to be estimated from the mixed effects model before the matrix eQTL is applied, and the computation advantage will be lost dramatically. In contrast to the mixed effects model, we randomly split the twin pairs into two groups and then applied multiple linear regression in each group to calculate the score vector for examining the association of each SNP-transcript pair. The simplicity of linear regression models mitigates the heavy computational burden dramatically. Simulation studies demonstrate the computational advantages of the proposed methods. Specifically, the proposed method is > 6000 times faster than the mixed effect model. The proposed method achieves the fast performance for two reasons: the simplicity of the linear regression model and large matrix operations (especially the matrix multiplication), expressing the most computationally intensive part in an efficient way.

To combine the analytical results from two groups, we propose a novel score statistic which automatically adjusts the correlation between the two groups in a natural way. Results from simulations show that the proposed method controls type I error rates much better at the target levels than do the linear regression model and the naive method. Moreover, this proposed method is competitive and does not lose much power compared to the mixed effects model but is much more computationally efficient. The slight power loss is likely due to the following two reasons: a) the fixed non-genetic covariate effects need to be estimated twice in our analysis while in the mixed effects model, they only need to be estimated once; and b) the shared environment effects are specifically modeled by the mixed effects model but our method lumps that effect into the random error term and inexplicitly models it. Score statistics have been well studied [Cox and Hinkley, 1974, Section 9.3(iii)] and widely used in practice. For example, the similar idea based on score statistic has been applied to meta-analysis of GWAS with overlapping samples, where a robust estimator for variance-covariance matrix was proposed [Lin and Sullivan, 2009]. Although we have focused our attention on twin studies, the proposed method could be applied and extended to multivariate phenotypes, where multiple correlated phenotypes are measured for each subject.

In simulation studies, we have also tried another test statistic,

$$T_2={\left(\begin{array}{c}\hfill {\sum }_{i=1}^{n_1}{Û}_{i,j}^{(1)}\hfill \\ \hfill {\sum }_{i=1}^{n_2}{Û}_{i,j}^{(2)}\hfill \end{array}\right)}^T\left(\begin{array}{cc}\hfill {\sum }_{i=1}^{n_1}{({Û}_{i,j}^{(1)})}^2\hfill & \hfill {\sum }_{i=1}^{n_{\mathit{\text{twin}}}}{Û}_{i,j}^{(1)}\times {Û}_{i,j}^{(2)}\hfill \\ \hfill {\sum }_{i=1}^{n_{\mathit{\text{twin}}}}{Û}_{i,j}^{(1)}\times {Û}_{i,j}^{(2)}\hfill & \hfill {\sum }_{i=1}^{n_2}{({Û}_{i,j}^{(2)})}^2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\sum }_{i=1}^{n_1}{Û}_{i,j}^{(1)}\hfill \\ \hfill {\sum }_{i=1}^{n_2}{Û}_{i,j}^{(2)}\hfill \end{array}\right)~{\chi }_2^2\text{under}{H}_0.$$

However, it is less efficient than the proposed one. This is probably due to the fact that our proposed score test follows ${\chi }_1^2$ under H₀ while T₂ follows ${\chi }_2^2$ under H₀. In addition, we examined the robustness of the proposed method under two model misspecification scenarios where (1) if the IBD value for DZ pairs is not fixed at 0.5, but are randomly generated from N(0.50, 0.04) to reflect the IBD variation among DZ samples that are commonly observed; (2) the random error e_i is non-normally distributed with a skewed distribution or a distribution with heavier tails (see Table 6). Expect these changes, the simulation set ups are kept the same as those in Table 1 with sample size n = 900. Clearly, both the proposed method and the computationally intensive mixed effects model are very robust to the model misspecifications, further suggesting broad applications of the proposed method.

Table 6.

Type I error for data from misspecified ACE model with n=900

	ρdz ~ N		E ~ Gamma1		E ~ Gamma2		E ~ t1		E ~ t2
	pro	mix	pro	mix	pro	mix	pro	mix	pro	mix
α = 0.05	0.0522	0.0510	0.0524	0.0502	0.0524	0.0508	0.0525	0.0519	0.0545	0.0491
α = 0.01	0.0108	0.0103	0.0110	0.0111	0.0108	0.0101	0.0110	0.0113	0.0116	0.0118
α = 0.001	0.0011	0.0011	0.0011	0.0011	0.0011	0.0010	0.0012	0.0011	0.0013	0.0018
α = 0.0001	0.0001	0.0001	0.0001	0.0004	0.0001	0.0001	0.0001	0.0002	0.0001	0.0001

pro: proposed score test statistic

mix: mixed effects

E: random error

ρ_dz: IBD value for DZ pairs

N: Normal(0.501,0.0389)

Gamma¹: Gamma(2, $\sqrt{2}$) and Gamma²: Gamma(6, $\sqrt{6}$)

t¹: t_df=4 and t² t_df=3

Finally, it is worth mentioning that the twin samples are randomly split, therefore the analysis result is not fixed and varies from one split to another split. As an example, we split 10, 000 simulated data from the ACE model twice and the scatter plot of the proposed test statistics from the two splits is given in Figure 2. The two results are not identical but highly consistent with R² > 0.99. To reduce the splitting variation, we can repeat splitting of the samples multiple times and combine the analysis results from the multiple splits. But how to combine the highly correlated results deserves further investigation.

Figure 2.

Scatter plot of the proposed test statistics from 10, 000 simulated data with each data randomly split twice. The data are generated from the ACE model and sample size is 1800.