A Bayesian Hierarchically Structured Prior for Gene-Based Association Testing with Multiple Traits in Genome-Wide Association Studies

Abstract

While genome-wide association studies (GWAS) often collect data on multiple correlated traits for complex diseases, conventional gene-based analysis is usually univariate, and therefore, treating traits as uncorrelated. Multivariate analysis of multiple correlated traits can potentially increase the power to detect genes that affect some or all of these traits. In this study, we propose the multivariate hierarchically structured variable selection (HSVS-M) model, a flexible Bayesian model that tests the association of a gene with multiple correlated traits. With only summary statistics, HSVS-M can account for the correlations among genetic variants and among traits simultaneously and can also estimate the various directions and magnitudes of associations between a gene and multiple traits. Simulation studies show that HSVS-M substantially outperforms competing methods in various scenarios, particularly when variants in a gene are associated with a trait in similar directions and magnitudes. We applied HSVS-M to the summary statistics of a meta-analysis GWAS on four lipid traits from the Global Lipids Genetics Consortium and identified 15 genes that have also been confirmed as risk factors in previous studies.

1. Introduction

Genome-wide association studies (GWASs) have been widely used to detect associations between genetic variants and complex diseases. One of the popular paradigms of analysis in GWASs is the gene-based association test. Compared to the conventional approach that tests single-nucleotide polymorphisms (SNPs) individually, gene-based tests are often more powerful due to the reduced burden of multiple testing and the information borrowed from SNPs in linkage disequilibrium (LD). However, current gene-based tests are mostly limited to univariate analysis (Pan et al., 2014, 2015; Wu et al., 2011; Yang et al., 2018, 2020, 2021) that tests the association of a gene with only one trait at a time. Multiple correlated traits are often measured and studied together in the context of complex diseases. For example, total cholesterol, high-density lipoprotein cholesterol, low-density lipoprotein cholesterol, and triglycerides jointly contribute to the risk factors for coronary artery disease (Teslovich et al., 2010). There can be genes that affect some or all of these traits, and as a result, analyzing multiple correlated traits simultaneously can potentially increase the power to detect associations of genes with a disease. Such joint analysis also has the potential to reveal pleiotropic genes involved in the biological mechanism of the disease.

Recognizing the increasing availability of multiple traits in GWASs, gene-based multivariate association methods have been proposed to jointly analyze multiple traits. For example, MSATS uses an adaptive procedure to combine the variance component test and the burden test based on a multivariate normal regression model for summary statistics (Guo and Wu, 2019). MTaSPUsSet uses an adaptive combination procedure based on a weighted sum of powered score statistics for each individual variant and trait (Kwak and Pan, 2017). USAT is a unified score-based method that adaptively combines the sum of squared score test (Pan, 2009) and the multivariate analysis of variance to adjust for different patterns of association between genetic variants and traits (Ray et al., 2016). MGAS combines top association signals from conventional univariate SNP-based tests into a multivariate gene-based p-value based on an extended Simes procedure (Van der Sluis et al., 2015). Some of these methods such as USAT require individual-level genotype data that are not always available. Moreoever, few of these methods can effectively estimate the directions and magnitudes of associations between a gene and different traits, which are often of primary interest in multivariate GWASs (Stephens, 2013).

In this study, we propose the multivariate hierarchically structured variable selection (HSVS-M) model for testing the association of a gene with multiple traits. The HSVS framework was previously considered for univariate gene- and pathway-based GWASs using summary statistics (Yang et al., 2018, 2020) and genotype data (Yang et al., 2021). We extend the HSVS framework in this study to multivariate gene-based GWASs that require only summary statistics, which are typically meant for common (minor allele frequency (MAF)≥5%) and low-frequency (1%≤MAF<5%) variants. Specifically, we propose in HSVS-M a discrete mixture Bayesian prior composed of a point mass at zero that supports the null hypothesis and a multivariate scale-mixing normal distribution that models the effects of SNPs in a gene under the alternative hypothesis while accounting for the linkage disequilibrium among the SNPs and the correlation among the traits. The posterior inference gives the posterior probability of the null hypothesis, which is closely related to Bayes factors and can be interpreted as a Bayesian analog to a p-value for hypothesis testing (Yang et al., 2020). We also introduce in HSVS-M a novel trait-adaptive weight that flexibly models the various associations between a gene and multiple traits. Because a gene can be associated with each trait differently, ignoring such differences and treating traits equally can weaken the power. The trait-adaptive weight allows HSVS-M to effectively estimate the directions and magnitudes of associations between a gene and each trait and thus increases the power of HSVS-M. Therefore, HSVS-M tackles not only whether but also how a gene is associated with multiple traits, which has not been effectively addressed by most competing methods.

The rest of the paper is organized as follows. We propose the HSVS-M prior in Section 2. We demonstrate through simulations that HSVS-M with its trait-adaptive weights is often more powerful than the competing methods in Section 3.1. We apply HSVS-M to a meta-analysis GWAS for lipid traits and use HSVS-M to identify genes that are associated with lipid traits in Section 3.2. Finally, Section 4 gives a brief discussion.

2. Methods

2.1. HSVS-M for gene-based multivariate GWAS

Consider a gene of p SNPs and k traits for each patient. Let Z_i· = (z_i1, …, z_ik)′ be the ith SNP’s summary statistics for the k traits, θ_i be the effect of the ith SNP’s strongest association with the k traits, w = (w₁, …,w_k)′ be the gene-specific trait-adaptive weight, R be the LD matrix for p SNPs, and Σ be the correlation matrix for k traits. We consider the model

$$\mathit{Z}~N\left(\mathit{W}\mathit{\theta },{\sigma }^2{\mathbf{\Sigma }}_{\mathit{Z}}\right),$$

in which $\mathit{Z}={\left({\mathit{Z}}_{\mathbf{1}}^{\prime },\dots ,{\mathit{Z}}_{\mathit{p}.}^{\prime }\right)}^{\prime }$, θ = (θ₁, …, θ_p)′, W = I_p ⊗ w, and Σ_Z = R ⊗ Σ; θ and w are parameters to be inferred whereas R and Σ can be empirically estimated. Here, w is a vector with each element defined in [−1, 1] measuring the direction and magnitude of the gene’s association with each trait relative to the trait that has the strongest association with the gene. Because different traits can have different associations with the gene and treating traits equally can weaken the signal and power, we use w to account for the difference in traits and thus boost the power by adaptively weighing the association between the gene and each trait. We use the same w for p SNPs assuming that their relative strength and direction of association with the k traits are same across SNPs in the same gene, which also allows HSVS-M to conduct posterior inference on the association between the gene and each of the k traits. Σ can be empirically estimated by calculating the covariance matrix for (Z_·1, …,Z_·k), in which Z_·j = (z_1j, …, z_pj)′ is the jth trait’s summary statistics for p SNPs. R can be obtained from the LD information in established databases such as the 1000 Genomes Project (1000 Genomes Project Consortium et al., 2015). Therefore, the model accounts for both the LD among SNPs and correlation among traits in the covariance.

Our interest is to test the null hypothesis H₀ : θ = 0; that is, no SNPs in the gene are associated with the k traits. We propose to test the hypothesis using HSVS-M, a discrete mixture prior for the gene selection with a Bayesian fused lasso hierarchy modeling the effect of individual SNPs in the gene. The HSVS-M prior is structured as

$$\mathit{\theta }\mid \gamma ,{\sigma }^2,{\mathit{\tau }}^{\mathbf{2}},{\mathit{\psi }}^{\mathbf{2}}~(1-\gamma )I(\mathit{\theta }=\mathbf{0})+\gamma N\left(\mathbf{0},{\sigma }^2{\mathbf{\Sigma }}_{\mathit{\theta }}\right),$$

in which we use γ, a binary indicator, on θ for the hypothesis testing. The value of γ indicates whether we reject the null hypothesis and P(γ = 0|·) is considered as a Bayesian analog to a p value and closely related to the Bayes factor (Yang et al., 2020).

Under the alternative hypothesis, we assume that θ follows a multivariate normal distribution N(0, σ²Σ_θ). The (i, j)th element of ${\mathit{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}$ gives the conditional dependence between SNPs i and j. This motivates the specification of the (i, j)th element of ${\mathit{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}$

$${\left[{\mathit{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}\right]}_{ij}=\{\begin{array}{cc}\frac{1}{{\tau }_i^2}+\frac{1}{{\psi }_{i-1}^2}+\frac{1}{{\psi }_i^2},& i=j=1,\dots ,p,\frac{1}{{\psi }_0^2}=\frac{1}{{\psi }_p^2}=0,\\ -\frac{1}{{\psi }_i^2},& j=i+1,i=1,\dots ,p-1,\\ -\frac{1}{{\psi }_{i-1}^2},& j=i-1,i=2,\dots ,p,\\ 0,& \text{otherwise}.\end{array}$$

Here, we use the Bayesian fused lasso hierarchy (Casella et al., 2010) on θ to induce correlation in the effect of neighboring SNPs and thus to account for their LD, which allows HSVS-M to borrow information from SNPs in LD to boost the power. This is achieved by further specifying the hyperpriors for ${\tau }_i^2$ and ${\psi }_i^2$ in ${\mathit{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}$ as

$${\tau }_i^2|{\lambda }_1~\mathrm{Exp}\left(\frac{{\lambda }_1^2}{2}\right),{\psi }_i^2|{\lambda }_2~\mathrm{Exp}\left(\frac{{\lambda }_2^2}{2}\right).$$

The exponential hyperpriors for ${\tau }_i^2$ and ${\psi }_i^2$ lead to exponential-scale mixture normal priors, which are equivalent to a Bayesian lasso (Park and Casella, 2008) and a Bayesian fused lasso, respectively. To see this, we collapse the hierarchical priors and obtain the marginal prior

$$\pi (\mathit{\theta }\mid \cdot )\propto \exp \left(-\frac{{\lambda }_1}{\sigma }\sum _{i=1}^p\left|{\theta }_i\right|-\frac{{\lambda }_2}{\sigma }\sum _{i=1}^{p-1}\left|{\theta }_{i+1}-{\theta }_i\right|\right).$$

The marginal prior indicates that under the alternative hypothesis, HSVS-M induces two shrinkage effects through the L1-norm regularization: one independently shrinks |θ_i|’s, the individual SNP effects, to control the type I error rates, whereas the other independently shrinks |θ_i+1−θ_i|’s, the difference between the SNP effects, to borrow strength from neighboring SNPs in LD.

To further boost the power, HSVS-M incorporates a trait-adaptive weight w to account for the various directions and magnitudes of the associations between a gene and k traits. We consider a flat prior for w

$$\begin{gathered} w_j~\mathrm{Uniform}\left(b_{j1},b_{j2}\right),j=1,\dots ,k, \\ \left(b_{j1},b_{j2}\right)=\{\begin{array}{ll}(0,1),\hfill & \text{if}{\overline{\mathit{Z}}}_{\cdot \mathit{j}}\ge t,\hfill \\ (-1,0),\hfill & \text{if}{\overline{\mathit{Z}}}_{\cdot \mathit{j}}\le -t,\hfill \\ (-1,1),\hfill & \text{otherwise,}\hfill \end{array} \end{gathered}$$ (1)

in which t > 0 is a decision threshold that determines the values of b_j1 and b_j2, the lower and upper bounds for w_j, respectively. We will discuss the choice of t in the following section. We assume in this study that w_j is the same across the p SNPs, i.e., the relative strength and direction of association with the k traits are same across SNPs in the same gene. We constrain the range of |w_j| to [0, 1] so that θ_i measures the maximum effect of the ith SNP on the k traits. To avoid identifiability issues, we use a data-adaptive procedure to empirically determine the direction of w_j based on ${\overline{\mathit{Z}}}_{\cdot \mathit{j}}$ except when ${\overline{\mathit{Z}}}_{\cdot \mathit{j}}$ is close to zero, as shown in Equation 1. The flat prior leads to a truncated normal posterior distribution for w_j with a weighted mean of Z_·j based on the SNP-level effect θ, which allows HSVS-M to borrow information from all SNPs while adaptively updating w_j. In addition, HSVS-M can also use the posterior samples for w_j to conduct posterior inference on the direction and magnitude of the association between the gene and the jth trait.

For the remaining priors and hyperpriors, we specify the prior for γ as Bernoulli(p_γ), and the hyperprior for p_γ as Beta(a, b) with constant parameters a and b. We also specify the prior for ${\tau }_i^2$ as $\mathrm{Exp}\left(\frac{{\lambda }_1^2}{2}\right)$ and ${\psi }_i^2$ as $\mathrm{Exp}\left(\frac{{\lambda }_2^2}{2}\right)$, and the hyperprior for ${\lambda }_1^2$ as Gamma(r₁, δ₁) and ${\lambda }_2^2$ as Gamma(r₂, δ₂) with constant shape parameters r₁ and r₂ and constant rate parameters δ₁ and δ₂. For σ², we use the improper prior σ² ∝ 1/σ². These choices of priors and hyperpriors lead to closed-form full conditional posterior distributions, as shown in Appendix A, allowing an efficient Gibbs sampler.

The full HSVS-M model is formulated as follows:

$$\begin{gathered} \text{Likelihood}:\mathit{Z}\sim N\text{(}\mathit{W}\mathit{\theta }\text{,}{\sigma }^2{\mathit{\sum }}_{\mathit{Z}}\text{)} \\ \text{Priors:}\mathit{\theta }\mid \gamma ,{\sigma }^2,{\mathit{\tau }}^{\mathbf{2}},{\mathit{\psi }}^{\mathbf{2}}~(1-\gamma )\mathit{I}(\mathit{\theta }=\mathbf{0})+\gamma N\left(\mathbf{0},{\sigma }^2{\mathbf{\Sigma }}_{\theta }\right),\text{in}\text{which} \\ {\left[{\mathbf{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}\right]}_{ij}=\{\begin{array}{cc}\frac{1}{{\tau }_i^2}+\frac{1}{{\psi }_{i-1}^2}+\frac{1}{{\psi }_i^2},& i=j=1,\dots ,p,\frac{1}{{\psi }_0^2}=\frac{1}{{\psi }_p^2}=0,\\ -\frac{1}{{\psi }_i^2},& j=i+1,i=1,\dots ,p-1,\\ -\frac{1}{{\psi }_{i-1}^2},& j=i-1,i=2,\dots ,p,\\ 0,& \text{otherwise}.\end{array} \\ {w}_j~\text{Uniform}\left(b_{j1},b_{j2}\right),j=1,\dots ,k,\text{in}\text{which} \\ \left(b_{j1},b_{j2}\right)=\{\begin{array}{ll}(0,1),\hfill & \text{if}{\overline{Z}}_{\cdot \mathit{j}}\ge t\hfill \\ (-1,0),\hfill & \text{if}{\overline{Z}}_{\cdot \mathit{j}}\le -t,\hfill \\ (-1,1),\hfill & \text{otherwise}.\hfill \end{array} \\ {\sigma }^2\propto 1/{\sigma }^2 \\ \text{Hyperpriors:}\gamma \mid p_{\gamma }~\text{Bernoulli}\left(p_{\gamma }\right),p_{\gamma }~\text{Beta}(a,b) \\ {\tau }_i^2\mid {\lambda }_1~\mathrm{Exp}\left(\frac{{\lambda }_1^2}{2}\right),{\lambda }_1^2~\text{Gamma}\left(r_1,{\delta }_1\right) \\ {\psi }_i^2\mid {\lambda }_2~\mathrm{Exp}\left(\frac{{\lambda }_2^2}{2}\right),{\lambda }_2^2~\text{Gamma}\left(r_2,{\delta }_2\right) \end{gathered}$$

2.2. Choice of hyperparameters

For the beta hyperprior on p_γ, we set (a, b) = (0.1, 0.1) to induce a non-informative prior for γ. Different parameterizations of (a, b) can be used for more informative choices; for example, a sparse hyperprior parameterization can be used with an empirical Bayes estimate of b (Yang et al., 2018). For the gamma hyperpriors on ${\lambda }_1^2$ and ${\lambda }_2^2$, we set (r₁, δ₁) = (r₂, δ₂) = (0.1, 0.1) to again induce non-informative priors for ${\tau }_i^2$ and ${\psi }_i^2$. For the uniform prior on w_j, we set t = 0.5 to achieve a balance between the power and the accuracy of weight estimates; overall, HSVS-M is robust to the choice of t and the effects of different t’s on HSVS-M’s power and weight estimation will be evaluated in Section 3.1.1.

3. Results

3.1. Simulation studies

3.1.1. Power of HSVS-M and competing methods

We conducted simulation studies to compare the power of HSVS-M with MSATS and MTaSPUsSet. Both methods are among the latest state-of-the-art multi-SNP multi-trait methods based on summary statistics and are accessible through R packages. We additionally included MTaSPUs (Kim et al., 2015), a single-SNP multi-trait method, to compare with the multi-SNP multi-trait methods. Because MTaSPUs is a SNP-based method, we used the aggregated Cauchy association test (Liu et al., 2019) to combine SNP-based p-values into a gene-based p-value. We also included HSVS-M with different t’s as a prior sensitivity test because t, as a decision threshold, is more subject to arbitrary choices compared to other (hyper)parameters, for which we used non-informative priors. To mimic the real data from the meta-analysis GWAS for four lipid traits (Teslovich et al., 2010) in the simulations, we considered four traits and set Σ, the trait-level correlation, as the empirical correlation matrix for the four lipid traits in the real data. Particularly, total cholesterol has a high correlation with low-density lipoprotein cholesterol (i.e., Σ₁₃ = 0.8787) and a moderate correlation with triglycerides (i.e, Σ₁₄ = 0.3842). We considered 20 SNPs per gene and set R, the SNP-level correlation, as the LD matrix of a random gene that has 20 SNPs in the real data. In line with the simulation method in MSATS (Guo and Wu, 2019), we simulated GWAS summary statistics Z from N(S ⊗P,R⊗Σ), in which S is the SNP-level effect vector of length 20 and P is the trait-level weight vector of length four that indicates the directions and magnitudes of the associations between the gene and the four traits. In all simulations, we let S = c × (2, 1.5, 1, 0.5, 0, …, 0)′ with four causal SNPs and c being the effect size scale parameter, and let the jth element of P, P_j ∈ {−1, 0, 1}, which indicates negative association, no association, and positive association, respectively, between the gene and the jth trait. We evaluated a method’s power in scenarios with various types of associations between a gene and four traits using different P’s, as shown in Table 1.

Table 1.

Simulation scenarios with various types of associations between a gene and four traits using different P’s

P	# of + trait1	# of − trait2	# of noise trait
(1 1 1 1)	4	0	0
(0 1 1 1)	3	0	1
(0 1 1 0)	2	0	2
(0 1 0 0)	1	0	3
(−1 1 −1 1)	2	2	0
(0 1 1 −1)	2	1	1
(0 −1 1 0)	1	1	2
(0 −1 0 0)	0	1	3

^1:traits that are positively associated with a gene

^2:traits that are negatively associated with a gene

To demonstrate the power of HSVS-M in estimating the trait-adaptive weight w, we also included HSVS-MC, a competing method in which empirical estimates of the weights ${\widehat{w}}_j={\overline{\mathit{Z}}}_{\cdot \mathit{j}}/\underset{1\le j\le k}{\max }\left(\left|{\overline{\mathit{Z}}}_{\cdot \mathit{j}}\right|\right)$ are used for the HSVS-M. We increased the effect size scale parameter c so that we generated a power curve for each method. When c = 0, we simulated 1,000 null replicated datasets and obtained a calibrated significance threshold α that preserves the type I error rate at 0.05 for each method. When c ≠ 0, a method’s power was estimated over 1,000 replicates as the proportion of P(γ = 0|·) < α for HSVS-M and HSVS-MC with 500 burn-in and 1,000 sampling Markov chain Monte Carlo (MCMC) iterations or p values < α for MSATS, MTaSPUsSet, and MTaSPUs.

As shown in Figure 1, HSVS-M outperformed the competing methods in most scenarios and was followed by MSATS, HSVS-MC, MTaSPUs, and MTaSPUsSet. HSVS-M was substantially more powerful than HSVS-MC, its constant-weight counterpart, which demonstrates the power of the trait-adaptive weight. HSVS-M was also robust to the presence of traits that have a negative or no association with the gene. We note that the power of HSVS-M, HSVS-MC, and MSATS was higher when P = (0, 1, 1, 1)′ and P = (0, 1, 1, 0)′ than that when P = (1, 1, 1, 1)′, which seems counterintuitive. A probable explanation is that the first trait has a high and moderate correlation with the third and the fourth trait, respectively (i.e., Σ₁₃ = 0.8787 and Σ₁₄ = 0.3842). Therefore, collinearity is likely to occur and thus reduce the power when the first trait is associated with the gene in the same manner as the third and the fourth trait, as seen in P = (1, 1, 1, 1)′. HSVS-M’s power is not sensitive to the choice of t in most scenarios in the sense that the difference in power among different t’s is negligibly small. In few scenarios, HSVS-M’s power increases very slightly as t increases.

Figure 1. Power comparison of HSVS-M, HSVS-MC, MSATS, MTaSPUsSet, and MTaSPUs.

In each panel, the horizontal axis indicates the effect size scale parameter c in the SNP-level effect vector S = c × (2, 1.5, 1, 0.5, 0, …, 0)′. P is the trait-level weight vector that indicates the directions and magnitudes of the associations between the gene and the four traits. Power was estimated as the proportion of P (γ = 0|·) < α for HSVS-M and HSVS-MC or p values < α for MSATS, MTaSPUsSet, and MTaSPUs among 1000 replicated datasets, with α being the calibrated significance threshold that preserves the type I error rate at 0.05. HSVS-M with different t’s are included with t being the decision threshold in the prior for the gene-specific trait-adaptive weight w.

The power of HSVS-M arises from the trait-adaptive weights w that HSVS-M uses to borrow strength from multiple correlated traits. To see this, we show in Figure 2 HSVS-M’s estimated trait-adaptive weights for the various associations between genes and traits corresponding to each simulation scenario. The estimated weights were calculated as the median of the posterior median of w among all replicated datasets. As the effect size increases, HSVS-M is more likely to produce accurate weight estimates for non-null traits and distinguish traits that are in different associations with the gene. This not only increases the power of HSVS-M but also allows it to identify the direction and magnitude of the association between a gene and a trait, which is a desirable feature that effectively addresses the question of how a gene is associated with each trait in multivariate GWASs. The accuracy of HSVS-M’s weight estimates for non-null traits increases slightly when t decreases in scenarios of small or medium effect sizes. This is because a larger t would lead more w_j’s to have a prior distribution of Uniform(−1, 1), whereas a smaller t is more likely to capture the direction of non-null traits when the effect size is not large.

Figure 2. HSVS-M’s estimated trait-adaptive weights for various types of associations between a gene and four traits.

In each panel, the horizontal axis indicates the effect size scale parameter c in the SNP-level effect vector S = c×(2, 1.5, 1, 0.5, 0, …, 0)′. P is the trait-level weight vector that indicates the directions and magnitudes of the associations between the gene and the four traits. Estimated weights were calculated as the median of the posterior median of w among 1,000 replicated datasets. HSVS-M with different t’s are included with t being the decision threshold in the prior for the gene-specific trait-adaptive weight w.

Although we could not include every possible scenario, we have strived to cover a broad range of scenarios by using various combinations of positive, negative, and null associations in P’s. For skipped scenarios such as P=(−1, −1, −1, −1), the results for power were almost identical to that of P=(1, 1, 1, 1) and the results for HSVS-M’s weight estimates were that of P=(1, 1, 1, 1) with a negative sign, which are analogous to P=(0 1 0 0) and P=(0 −1 0 0).

We further implemented additional simulations, in which we assume different P’s for different SNPs, to investigate the robustness of HSVS-M when the model assumptions are violated. Specifically, instead of assuming the same P for all SNPs, we sampled P_ik from N(μ_k, 1) for the ith SNP and kth trait. As shown in Figure S1 in the Supplementary Material, HSVS-M was more powerful than MTaSPUsSet and MTaSPUs, though less powerful than MSATS when the effect size is large. This is not surprising because MSATS is based on the variance-component test, which gains more power when the directions and magnitudes of associations between SNPs and traits vary to a greater extent.

3.1.2. Type I error rates of HSVS-M and competing methods

HSVS-M uses a calibrated significance threshold to preserve the type I error rate. We randomly chose 1,000 genes from the real data and simulated 1,000 replicated datasets of null Z from N(0,R⊗Σ) with respective LD matrices. We then obtained the calibrated significance threshold α that gives a false positive rate of 0.05 using the null Z. To evaluate the preservation of the type I error rates for each method, we applied α to the 1,000 replicated datasets of null Z that we generated in the simulations based on a randomly selected gene and calculated the type I error rate for each method.

In Table 2, we show the type I error rates of HSVS-M and the competing methods using each method’s calibrated significance threshold. The results indicate that all methods control the type I error rate at around 0.05.

Table 2.

Type I error rates of HSVS-M, HSVS-MC, MSATS, MTaSPUsSet, and MTaSPUs using calibrated significance thresholds

HSVS-M	HSVS-MC	MSATS	MTaSPUsSet	MTaSPUs
0.051	0.054	0.049	0.050	0.046

Type I error rates were calculated as the proportion of P(γ = 0|·) < α for HSVS-M and HSVS-MC or p values < α for MSATS, MTaSPUsSet, and MTaSPUs under the null hypothesis, with α being each method’s calibrated significance threshold.

3.2. Application to GWAS summary statistics for lipid traits

We applied HSVS-M to the meta-analysis summary statistics of GWASs for total cholesterol (TC), high-density lipoprotein cholesterol (HDL-C), low-density lipoprotein cholesterol (LDL-C), and triglycerides (TG) (Teslovich et al., 2010) from the Global Lipids Genetics Consortium with the aim of identifying genes that are associated with lipid traits. The dataset contains the Z-statistics of 2,647,705 SNPs for the four traits from a meta-analysis of 46 lipid GWASs with a total of more than 100,000 participants. We also obtained the coding region of genes and the chromosomal position and the minor allele frequency (MAF) of SNPs from Ensembl, a BioMart database (Durinck et al., 2005, 2009). In line with other studies (Guo and Wu, 2019), we filtered out SNPs that have a p value < 5×10⁻⁸ for any trait so that we demonstrate the power of HSVS-M as a multivariate gene-based approach in spite of the lack of univariate genome-wide significant individual SNPs. We further filtered out SNPs that have a MAF < 0.05 and group SNPs within 20kb of a gene coding region into a gene. Our final dataset contains 20,541 genes that harbor 990,860 unique SNPs.

HSVS-M identified 249 significant genes, among which 15 genes have been confirmed as risk factors in previous studies of lipid traits. Specifically, fatty acid desaturase 3 (FADS3) has been implicated for HDL-C and TG in a meta-analysis of GWASs for lipids in Caucasians (Kathiresan et al., 2009) and increased risk for familial combined hyperlipidemia that is characterized by familial segregation of elevated fasting plasma TG and TC (Plaisier et al., 2009). Diacylglycerol O-acyltransferase 2 (DGAT2) is the major enzyme of TG biosynthesis in eukaryotes (Stone et al., 2009) and has been found to facilitate the expansion of lipid droplets (Xu et al., 2012) that play a central role in most cardiovascular diseases. Monoacylglycerol acyltransferase-2 (MGAT2) has been found to interact with DGAT2 to promote TG biosynthesis(Jin et al., 2014). Solute carrier family 10 member 1 (SLC10A1) has been demonstrated to effectively transport eprotirome, a thyroid hormone analog that lowers cholesterol in humans (Kersseboom et al., 2017). Acyl-CoA thioesterase 4 (ACOT4) acts as an auxiliary enzyme in human peroxisomes, which have an established role in lipid metabolism (Hunt et al., 2012, 2014). Integrin subunit beta 3 (ITGB3) has been found to be associated with TG in the non-Hispanic black population (Chang et al., 2010). Cytochrome P450 1B1 (CYP1B1) has been shown to significantly lower HDL-C in incinerator workers (Hu et al., 2008). Forkhead box a2 (FOXA2) has been found to increase HDL-C levels in mice by regulating apolipoprotein M (Wolfrum et al., 2008). Non-coding RNA activated by DNA damage (NORAD) has been confirmed as an upstream mechanism to regulate transthyretin while patients with a higher level of transthyretin tend to have higher cholesterol (Zuo et al., 2020). Studies have shown that ADP ribosylation factor related protein 1 (ARFRP1) is essential for lipid droplet growth (Hommel et al., 2010) and the deletion of ARFRP1 results in impaired lipidation of very low density lipoproteins, which in turn leads to reduced plasma TG levels in the fasted state (Hesse et al., 2014). A GWAS meta-analysis of 188,000 individuals has also shown that ubiquitin conjugating enzyme e2 l3 (UBE2L3) is in close proximity with miR-130b and miR-301b, the two miRNAs that control the expression of key proteins involved in the LDL receptor, and plectin (PLEC) is in close proximity with miR-661, a miRNA in genome-wide significant association with TC and LDL-C (Wagschal et al., 2015). MicroRNA 128–2 (MIR128–2) has been shown to be a regulator of cholesterol homeostasis in that cholesterol efflux is attenuated by overexpression of MIR128–2 (Adlakha et al., 2013). Growth differentiation factor 9 (GDF9) has been found to facilitate cholesterol biosynthesis by promoting cholesterol biosynthetic enzymes (Su et al., 2008). R-spondin 3 (RSPO3) has been identified as harboring a SNP in genome-wide significant association with HDL-C (Ligthart et al., 2016). Table 3 shows the P(γ = 0|·) for HSVS-M and the p values for MSATS and MTaSPUsSet for these 15 risk factor genes.

Table 3.

Analysis of 15 genes that have been identified as risk factors in previous studies of lipid traits and confirmed by HSVS-M

Gene	HSVS-M	MSATS	MTaSPUsSet
FADS3	<1.00e-06*†	6.21e-14*	<1.00e-06*
UBE2L3	<1.00e-06*	1.10e-07*	2.10e-04
DGAT2	<1.00e-06*	2.27e-06*	5.99e-03
RSPO3	<1.00e-06*	3.06e-06	5.00e-06
ITGB3	<1.00e-06*	2.63e-05	1.20e-03
PLEC	<1.00e-06*	3.23e-05	4.00e-05
NORAD	<1.00e-06*	8.60e-05	2.00e-04
ARFRP1	<1.00e-06*	4.37e-03	1.70e-02
MIR128-2	<1.00e-06*	5.95e-03	1.42e-01
ACOT4	<1.00e-06*	1.66e-02	5.26e-01
CYP1B1	<1.00e-06*	1.79e-02	8.63e-01
FOXA2	<1.00e-06*	2.65e-02	1.50e-01
GDF9	<1.00e-06*	4.20e-02	1.57e-01
MGAT2	<1.00e-06*	1.23e-01	3.09e-01
SLC10A1	<1.00e-06*	6.78e-01	5.36e-01

^*Identified by the corresponding methods after the Bonferroni corrections

^†P(γ = 0|·) <1.00e-06 indicates P(γ = 0|·) = 0 with 10⁶ MCMC samples

In comparison, MSATS and MTaSPUsSet identified 249 and 163 significant genes, respectively, after controlling each method’s family-wise type I error rate at 0.05 with the Bonferroni correction. However, MSATS identified only 3 of these 15 risk factor genes and MTaSPUsSet identified one of them. We note that although both HSVS-M and MSATS identified 249 genes, there were only 12 genes identified by both methods. We consider this as a result of the difference in the power of methods. For HSVS-M, because we use a gene-specific weight under the model assumption that SNPs in a gene are associated with a trait in similar directions and magnitudes, HSVS-M would gain more power when this assumption holds. On the other hand, MSATS is based on the variance-component test, which gains more power when the directions and magnitudes of associations between SNPs and traits vary to a greater extent.

To demonstrate HSVS-M’s trait-adaptive weights, we also show as an example in Table 4 the estimated weights of the associations between the traits and PLEC, a gene identified by HSVS-M but not the competing methods. The estimated weights were calculated as the posterior median of w. PLEC has strong negative associations with TC and LDL-C, a weak negative association with HDL-C, and a weak positive association with TG, which are in line with the finding that PLEC is in close proximity with a miRNA significantly associated with TC and LDL-C (Wagschal et al., 2015).

Table 4.

Estimated weights of the associations between PLEC and the lipid traits using HSVS-M

Trait	Weight1 [95% CI2]
TC	−0.93 [−0.99, −0.77]
HDL-C	−0.20 [−0.33, −0.07]
LDL-C	−0.98 [−0.99, −0.84]
TG	0.16 [0.05, 0.31]

¹Estimated weights were calculated as the posterior median of w with a range of [−1, 1]

²CI: credible interval

4. Discussion

In this study, we propose HSVS-M, a gene-based multi-trait method that uses summary statistics and a gene-specific trait-adaptive weight both to test the association of variants in a gene with multiple traits and to estimate the strength of association of the gene with each trait. HSVS-M also provides an integrated framework both to test and to interpret associations, an important feature that addresses how traits are associated with a gene, which is often a question of primary interest in multi-trait GWASs (Stephens, 2013) but has not been tackled by most existing gene-based multivariate association tests.

As an integral part of HSVS-M, the gene-specific trait-adaptive weight w allows HSVS-M to produce accurate weight estimates and identify traits that are in different associations with the gene. In this study, we use a uniform distribution as the prior for w_j. Other more informative choices of priors can be considered given different assumptions about the association between the gene and the traits. For example, a normal prior $N\left({\mu }_{w_j},{\sigma }_{w_j}^2\right)$ can be used if one has the prior biological evidence that the weight for the average strength of association between the gene and the jth trait would be around ${\mu }_{w_j}$.

We demonstrate the power of HSVS-M in various simulated scenarios and applications to the real data. Specifically, HSVS-M is powerful when multiple traits are correlated and SNPs in a gene are associated with a trait in similar directions and magnitudes. HSVS-M’s superior power is mainly due to the gene-specific trait-adaptive weight w that HSVS-M uses to borrow strength from the traits. A SNP-specific weight w_i = (w_i1, …,w_ik)′ can also be a viable choice of weight, which would presumably give HSVS-M more latitude in modeling the association between SNPs and traits in case that the gene-level weight assumption is violated. However, a SNP-specific weight does not directly yield a gene-level weight estimate, which would make it less straightforward to draw inference about the strength of association between a gene and a trait.

The runtime of HSVS-M to complete 1,000 MCMC posterior samples for a simulated replicate of 20 variants and 4 traits is 0.30 seconds on a 2.3 GHz Intel Core i5 processor, whereas the runtime is 0.02 and 0.95 seconds for MSATS and MTaSPUsSet, respectively. In large-scale analyses where more MCMC samples might be desired due to the Bonferroni correction, we suggest a multi-phase analysis in which researchers apply HSVS-M to a gene with 1,000 MCMC samples as a first attempt to evaluate its significance and only keep promising genes in the analysis. Researchers can then apply HSVS-M to these few promising genes with more MCMC samples. To further achieve scalability, we have the option to implement parallel computing for HSVS-M, which, as shown in our osteosarcoma trio data analyses in the work of Yang et al. (2018) can potentially improve the computational efficiency by 25.7% for our method.

Appendix A

Posterior inference for HSVS-M via MCMC

We present the full conditional distributions for the HSVS-M prior as follows. GIG is the generalized inverse Gaussian distribution and IG is the inverse gamma distribution.

• $\mathit{\theta }\mid \cdot ~(1-\gamma )\mathit{I}(\mathit{\theta }=\mathbf{0})+\gamma N\left({\tilde{\sigma }}^2\tilde{\mu },{\sigma }^2{\tilde{\sigma }}^2\right),$

  where $\tilde{\mu }={\mathit{W}}^T{\mathbf{\Sigma }}_{\mathit{Z}}^{-\mathbf{1}}\mathit{Z},$ and ${\tilde{\sigma }}^2={\left\{{\mathit{W}}^T{\mathbf{\Sigma }}_{\mathit{Z}}^{-\mathbf{1}}\mathit{W}+{\mathbf{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}\right\}}^{-1},$

• $\gamma \mid \cdot ~\mathrm{Bernoulli}\left({\tilde{p}}_{\gamma }\right),$

  where ${\tilde{p}}_{\gamma }=1-\left(1-p_{\gamma }\right)/\left\{1-p_{\gamma }+p_{\gamma }\cdot {\left|{\mathbf{\Sigma }}_{\theta }\right|}^{-0.5}{\left|{\tilde{\sigma }}^2\right|}^{0.5}\exp \left(\frac{{\tilde{\mu }}^T{\tilde{\sigma }}^2\tilde{\mu }}{2{\sigma }^2}\right)\right\},$

• $w\mid \cdot ~(1-\gamma )U(b_1,b_2)+\gamma N(\frac{{\sum }_{i=1}^p{\theta }_i{\mathit{Z}}_i}{{\sum }_{i=1}^p{\theta }_i^2},\frac{{\sigma }^2}{{\sum }_{i=1}^p{\theta }_i^2}\mathbf{\Sigma }),$

• ${\sigma }^2\mid \cdot ~\mathrm{IG}\left(\frac{kp+\gamma p}{2},\frac{1}{2}{(\mathit{Z}-\mathit{W}\mathit{\theta })}^T{\mathbf{\Sigma }}_{\mathit{Z}}^{-\mathbf{1}}(\mathit{Z}-\mathit{W}\mathit{\theta })+\frac{1}{2}{\mathit{\theta }}^T{\mathbf{\Sigma }}_{\mathit{\theta }}^{-\mathbf{1}}\mathit{\theta }\right),$

• $p_{\gamma }|\cdot \sim \text{Beta}\left(\gamma +a,1-\gamma +b\right),$

• ${\tau }_i^2\mid \cdot ~(1-\gamma )\mathrm{Exp}\left(\frac{{\lambda }_1^2}{2}\right)+\gamma \mathrm{GIG}\left(\frac{1}{2},{\lambda }_1^2,\frac{{\theta }_i^2}{{\sigma }^2}\right),i=1,\dots ,p,$

• ${\psi }_i^2\mid \cdot ~(1-\gamma )\mathrm{Exp}\left(\frac{{\lambda }_2^2}{2}\right)+\gamma \mathrm{GIG}\left(\frac{1}{2},{\lambda }_2^2,\frac{{\left({\theta }_i-{\theta }_{i+1}\right)}^2}{{\sigma }^2}\right),i=1,\dots ,p-1,$

• ${\lambda }_1^2\mid \cdot ~\mathrm{Gamma}\left(r_1+p,{\delta }_1+\frac{1}{2}{\sum }_{i=1}^p{\tau }_i^2\right),$

• ${\lambda }_2^2\mid \cdot ~\mathrm{Gamma}\left(r_2+p-1,{\delta }_2+\frac{1}{2}{\sum }_{i=1}^{p-1}{\psi }_i^2\right),$

Data Availability Statement

The summary statistics for the four lipid traits from the Global Lipids Genetics Consortium are publicly available at http://csg.sph.umich.edu/willer/public/lipids2010/. The R package for HSVS-M is publicly available at https://yiyangphd.github.io/software/.