## Abstract

Two new gene-based association analysis methods, called *PrediXcan* and *TWAS* for GWAS individual-level and summary data, respectively, were recently proposed to integrate GWAS with eQTL data, alleviating two common problems in GWAS by boosting statistical power and facilitating biological interpretation of GWAS discoveries. Based on a novel reformulation of PrediXcan and TWAS, we propose a more powerful gene-based association test to integrate single set or multiple sets of eQTL data with GWAS individual-level data or summary statistics. The proposed test was applied to several GWAS datasets, including two lipid summary association datasets based on and samples, respectively, and uncovered more known or novel trait-associated genes, showcasing much improved performance of our proposed method. The software implementing the proposed method is freely available as an R package.

- aSPU test
- statistical power
- Sum test
- transcriptome-wide association study (TWAS)
- weighted association testing

IN spite of many successes, genome-wide association studies (GWAS) face two major challenges. The first is their limited statistical power, even with tens to hundreds of thousands of individuals in a typical GWAS or mega-GWAS, thus missing many associated genetic variants, mostly single nucleotide polymorphisms (SNPs), due to the polygenic effects and small effect sizes. The second is that, even for those few identified SNPs, since they often do not reside in protein-coding regions, it is difficult to interpret their function and thus biological mechanisms underlying complex traits. A new gene-based association test called *PrediXcan* was recently proposed to integrate GWAS individual-level data with an eQTL dataset, alleviating the above two problems in boosting statistical power of GWAS and facilitating biological interpretation of GWAS discoveries (Gamazon *et al.* 2015). It was extended to GWAS summary association data (Torres *et al.* 2017). A similar approach, called transcriptome-wide association study (TWAS), was proposed by another group for GWAS individual-level and summary data for one or more eQTL datasets (Gusev *et al.* 2016). These approaches are motivated by the key fact that many genetic variants influence complex traits through transcriptional regulation (He *et al.* 2013). Focusing on the genetic component of expression excludes environmental factors influencing gene expression and complex traits, and thus can increase statistical power. In addition, compared to standard GWAS, treating genes as analysis units reduces the number, and thus burden, of multiple tests, again leading to improved power. By applications to common diseases like type 2 diabetes (T2D) and complex traits like body mass index (BMI), lipids and height, the authors have convincingly shown the power of integrating GWAS and eQTL data, gaining biological insights into complex traits. There are more follow-up studies applying TWAS to other diseases. For example, Gusev *et al.* (2017) identified some new genes associated with schizophrenia; interestingly, they also confirmed a previous observation that, contrary to the usual GWAS practice, the nearest gene to a GWAS hit often is not the most likely susceptibility gene, highlighting the critical role of incorporating gene expression to unravel disease mechanisms that may not be achieved by GWAS alone. The current standard and popular view is that PrediXcan and TWAS work because of their predicting or imputing *cis* genetic component of expression for a larger set of individuals in GWAS, facilitating the following expression-trait association testing. Based on this view, some new methods have been proposed to improve over TWAS by addressing some existing weaknesses in gene expression prediction (Bhutani *et al.* 2017; Park *et al.* 2017). In spite of its intuition and usefulness, the current view on PrediXcan and TWAS may not have told the whole story. Here, we offer some new insights into PrediXcan and TWAS with a novel reformulation on their underlying association testing. Our key observation is that both PrediXcan and TWAS are based on the same weighted association test; the weights on a set of SNPs in a gene are the *cis*-effects of the SNPs on the gene’s expression level (derived from an eQTL dataset). In other words, PrediXcan and TWAS put a higher weight on an SNP (eSNP) that is more strongly associated with the gene’s expression level, in agreement with empirical evidence that eSNPs are more likely to be associated with complex traits and diseases (Nicolae *et al.* 2010). This new formulation also points out the connection to existing weighted association analysis (Roeder *et al.* 2006; Ho *et al.* 2014). More importantly, since the same association test in PrediXcan and TWAS suffers from power loss under some common situations, we develop an alternative and more powerful association test with broader applications. Since there is no uniformly most powerful gene-based association test, any single nonadaptive test will lose power in some situations; it is important to develop and utilize adaptive tests to yield consistently high power (Li and Tseng 2011; Lee *et al.* 2012; Pan *et al.* 2014). We propose using such an adaptive and powerful test under a general and rigorous framework of generalized linear models (GLMs), which can accommodate various types of quantitative, categorical and survival phenotypes and can adjust for covariates. It is applicable to both individual-level genotypic, phenotypic data and GWAS summary statistics. It is flexible to incorporate a single or multiple sets of weights derived from eQTL data or other data sources.

## Methods

### PrediXcan and TWAS

We briefly review PrediXcan and TWAS for GWAS individual-level data before giving our new formulation. One first builds a prediction model for a gene’s expression level, called “genetically regulated expression (GReX),” by using the genotypes around the gene based on an eQTL dataset. Next, for a GWAS dataset, one uses the prediction model to predict or “impute” the GReX of the gene using the SNPs around the gene for each subject in a main GWAS dataset. Specifically, for a given gene, suppose that in an eQTL dataset, and are the expression level of the gene, and the *p* SNP genotype scores (with additive coding) around the gene, respectively. A linear model is assumed: where is the *cis*-effect of SNP *j* on gene expression, and is the noise. Based on the eQTL dataset, one can use a method, *e.g.*, elastic net (Zou and Hastie 2005) or a Bayesian linear mixed model (Zhou *et al.* 2013) as used in PrediXcan and TWAS respectively, to obtain the estimates ’s. Now for a given GWAS dataset, for each subject *i* with the genotype scores for the gene, the predicted GReX is For a trait for subject *i* in the GWAS dataset, one simply applies a suitable GLM(1)to test for association between the trait and predicted/imputed expression with null hypothesis where is the canonical link function (*e.g.*, the logit and the identity functions for binary and quantitative traits, respectively), and is the mean of the trait. One of the asymptotically equivalent Wald, Score and likelihood ratio tests can be used.

### A novel reformulation and extensions

Here, we first point out that PrediXcan and TWAS can be regarded as a special case of general association testing with multiple SNPs in a GLM:(2)The goal is to test It can be verified that both PrediXcan and TWAS are a weighted Sum test in the above general model (Pan 2009), with weights on each SNP *j*; that is, PrediXcan and TWAS conduct the Sum test on with the genotype scores replaced by the weighted genotype scores in GLM (2). This new interpretation and formulation will facilitate our gaining insights into PrediXcan and TWAS, including their possible limitations, thus motivating some modifications for improvement. It offers a direct and intuitive justification for PrediXcan and TWAS: the two methods perform well due to their overweighting on expression-associated SNPs (eSNPs), as supported by empirical evidence that eSNPs are more likely to be associated with complex traits and disease (Nicolae *et al.* 2010). Obviously, it also suggests their extensions to other endophenotypes, and to incorporate prior knowledge and other data sources related to the GWAS trait of interest, such as previous linkage scans (Roeder *et al.* 2006) and imaging endophenotypes (Xu *et al.* 2017), although we do not pursue that here. More importantly, since the Sum test can be derived under the oversimplifying working assumption of in (1) and (2) (*i.e.*, all weighted SNPs have an equal effect size and the same effect direction, which is in general incorrect), we can see possible limitations of the Sum test and thus of PrediXcan and TWAS. As discussed in Pan (2009), Pan *et al.* (2014) and others (Wu *et al.* 2011), the Sum test may lose power if the effect directions of the (weighted) SNPs are different, or the effect sizes are sparse (*i.e.*, with many 0's). Accordingly, one may apply other tests, *e.g.*, the sum of squared score (SSU) test that is equivalent to a variance-component score test as used in kernel machine regression (also known as SKAT in rare variant analysis) with a linear kernel and a nonparametric MANOVA (also called genomic distance-based regression) with the Euclidean distance metric (Wessel and Schork 2006), which may yield higher power under many situations (Schaid 2010a,b; Pan 2011).

### New method: aSPU

A class of the so-called sum of powered score (SPU) tests cover both the Sum and SSU tests as special cases. Specifically, we denote the unweighted and weighted score vectors for *β* in (1) aswhere is the fitted mean of under (with ) in (1), and are the weights of the SNPs derived from an eQTL dataset. The effects of the weights can be regarded as replacing the unweighted genotype scores by the weighted genotype scores in GLM (2). The Sum (*i.e.*, PrediXcan and TWAS) and SSU tests based on the weighted genotypes are:More generally, for an integer an SPU(*γ*) test is defined asIt is clear SPU(1) = Sum and SPU(2) = SSU. Furthermore, for an even integer we have The is closely related to the UminP test (but ignoring possibly varying variances of ’s); often, they performed similarly (Pan 2009).

Since there is no uniformly most powerful test, for a given situation, any nonadaptive test may or may not be powerful. By using various values of *γ*, we yield a class of SPU tests, one of which is expected to be more powerful in any given situation. For example, the Sum = SPU(1) test treats each SNP equally *a priori*, yielding high power if all the SNPs are associated with the trait with similar effect sizes and the same association direction. On the other hand, when only a smaller subset of SNPs are associated with the trait, or their association directions are different, the SSU = SPU(2) test is often more powerful. As *γ* increases, SPU(*γ*) relies more on the SNPs that are more strongly associated with the trait, and is thus more powerful for more sparse association signals (*i.e.*, fewer associated SNPs). In the end, as *γ* approaches (as an even integer), it only considers the most significant SNP.

Since the optimal value of *γ* is unknown and data-dependent, we propose using an adaptive SPU (aSPU) test to data-adaptively approximate the most powerful SPU test among a set of versatile SPU(*γ*) tests with various values of *γ*, thus maintaining high power in a wide range of scenarios. Empirically, we have found that using often performs well and thus adopt it; the aSPU test is defined as(3)where is the *P*-value of the test.

#### P-value calculations:

Although asymptotic *P*-values for the SPU(1) = Sum and SPU(2) = SSU tests can be calculated (Pan 2009) [with possible small-sample adjustments (Lee *et al.* 2012; Chen *et al.* 2015; Wang 2016)], in general, we can use *one layer of Monte Carlo simulations* to estimate the *P*-values for all the SPU and aSPU tests *simultaneously* (Pan *et al.* 2014). Specifically, we simulate null score vectors for from its asymptotic null distribution, a multivariate normal with mean 0 and covariance matrix *V*; there is a closed form solution for *V* (Pan *et al.* 2014). Then, the null statistics are calculated from the null score vectors for and the *P*-value of the test is Then, the *P*-value for the aSPU test is calculated as with and

### Association testing with summary statistics

One practical way to increase the sample size is to form large consortia, aiming for meta-analysis of multiple GWAS, for which often only summary statistics for single SNP-single trait associations, rather than individual-level genotypic and phenotypic data, are available (and practically feasible for many cohorts with possibly different study designs). Hence, it is extremely useful to develop methods like TWAS that are applicable to GWAS summary statistics as well as to GWAS individual-level data. The aSPU test is easily extended to GWAS summary statistics without individual-level data. Suppose that is the Z-statistic for association between the GWAS trait and SNP *j*, where is the estimated (marginal and signed) association effect and is its SE. We just need to simply redefine with then proceed as before. We use a reference sample (*e.g.*, the 1000 Genomes Project data) to estimate linkage disequilibrium (LD) among the SNPs, and thus the correlation matrix for *Z* and *U* (Gusev *et al.* 2016; Kwak and Pan 2016).

### Association testing with multiple sets of weights

Now, we extend the aSPU test to the case with multiple sets of eQTL datasets, or more generally, multiple sets of weights. This is important because of the existence of multiple eQTL datasets measured from different populations or different tissues; it is in general unclear which one is most suitable. After applications with each eQTL dataset separately, it may gain statistical power and biological insights to combine the results across multiple eQTL datasets. Suppose we have *K* sets of weights, for each estimated from a separate eQTL dataset. To avoid the results depending on the varying scales of the sets of weights, we first standardize the weights to have for each *k*. Based on the score vector (with individual-level data) or Z-statistics *Z* (with GWAS summary data) and the weights we define or accordingly. As before, for a fixed *γ*, we first apply SPU(*γ*) to yielding its test statistic and *P*-value We then Z-transform each *P*-value to a Z-statistic where is the CDF of a standard normal distribution. To recover the sign of each statistic, for an odd *γ*, we have for an even *γ* or we use We combine the *K* sets of weights through combining the *K* statistics to form an omnibus SPU(*γ*) test:where and are the sample mean vector and covariance matrix of under which can be calculated along with other *P*-values inside the single layer of simulations. Then, as usual, we combine the omnibus SPU(*γ*)-O tests into an omnibus aSPU test:where is the *P*-value of SPU-O. As before, the *P*-values of all the SPU(γ)-O and aSPU-O can be calculated in a single layer of Monte Carlo simulations.

The omnibus test is not sensitive to weight standardization. For example, scaling by yielded similar results in our experiments.

It is easy to verify that SPU(1)-O is equivalent to the omnibus TWAS, denoted TWAS-O. Again, by combining SPU(1)-O and other SPU(*γ*)-O tests, we obtain the adaptive and omnibus aSPU-O test that may be more powerful across a wide range of scenarios.

### Data availability

The WTCCC data can be found at https://www.wtccc.org.uk. The original 2010 lipid GWAS summary data can be downloaded at http://csg.sph.umich.edu/abecasis/public/lipids2010/, while the original 2013 lipid GWAS summary data at http://csg.sph.umich.edu/abecasis/public/lipids2013/; the two preprocessed datasets that we used are downloadable from https://figshare.com/articles/Lipid_2010_summary_data/5373370 and https://figshare.com/articles/Lipid_2013_summary_data/5373382. The LD reference data and eQTL-based weights can be obtained from http://gusevlab.org/projects/fusion/ and https://github.com/hakyimlab/PrediXcan. The related computer scripts and examples can be found at https://github.com/ChongWu-Biostat/TWAS, and the online manual about how to use our proposed methods can be also found at www.wuchong.org/TWAS.html or https://github.com/ChongWu-Biostat/TWAS.

## Results

### Application to the WTCCC data

We first applied the aSPU test and PrediXcan to the WTCCC individual-level data with the weights downloaded from the PrediXcan database, demonstrating the equivalence of the SPU(1) test and PrediXcan, and more importantly, that the aSPU test could identify more associated genes than PrediXcan in many cases. Specifically, first, following the same procedure of quality control (Burton *et al.* 2007), we lifted the annotation of the WTCCC genotype data from hg18 to hg19 via the UCSC browser (http://genome.ucsc.edu/cgi-bin/hgLiftOver); second, we imputed the genotype data via the Michigan Imputation Server with the following specifications: 1000G Phase 1 v3 as the reference panel, SHAPEIT as the phasing algorithm and EUR (European) as the target population. After imputation, the variants with a minor allele frequency (MAF) 0.05, the Hardy-Weinberg equilibrium exact test *P*-value and *R* were kept. As Gamazon *et al.* (2015), we kept only the HapMap Phase 2 subset of SNPs. We considered seven traits/diseases: bipolar disorder (BD), coronary artery disease (CAD), inflammatory bowel disease (CD), rheumatoid arthritis (RA), hypertension (HT), type 1 diabetes (T1D), and T2D. The weights based on the Depression Genes and Networks (DGN; 922 whole-blood samples) whole blood expression were downloaded from the PrediXcan database (https://app.box.com/s/gujt4m6njqjfqqc9tu0oqgtjvtz9860w). There were 8917 genes whose expression levels could be predicted by elastic net with a cross-validated we thus tested on these 8917 genes with a conservative Bonferroni adjustment with a genome-wide significance level at

As most of the genes were not expected to be significantly associated with a trait, we used a step-up procedure to increase the number of simulations when calculating the *P*-values of aSPU and aSPU-O in the subsequent data analysis. We started with a relatively small and reran the tests with for the genes with *P*-values (but stopped otherwise); we repeated this process by increasing *B* to 10 times of its previous value for the genes with *P*-values up to finally, to be more accurate for a *P*-value around the significance cut-off, we reran the tests on the genes with *P*-values between and with

Here are the main results. First, as shown in Supplemental Material, Figure S1 in File S1, as expected, PrediXcan gave essentially the same results (*i.e.*, *P*-values) as those of the SPU(1) test for each of the seven traits. Hence, we confirmed and would treat the SPU(1) test to be equivalent to PrediXcan. Second, as shown in Figure S2 in File S1, the aSPU test identified more significant genes than the SPU(1) test (or equivalently, PrediXcan) for traits CD, BD, and T1D [*i.e.*, (10, 3, 38) *vs.* (8, 2, 29)], while it was the opposite for HT (*i.e.*, 0 *vs.* 1), and they were tied [with (1, 4, 0)] for CAD, RA, and T2D; note the large difference for T1D, which was statistically significant with an exact *P*-value = 0.0039 by McNemar’s test (Fagerland *et al.* 2013) (while other differences were not). Table 1 lists the significant genes identified by the aSPU test but not by the SPU(1) test (and PrediXcan) at the genome-wide significance level; some of the significant genes were confirmed in later studies. In total, we identified 15 new genes, five of which have been reported by other studies. For CD, we identified four new genes, all of which contained some genome-wide significant SNPs identified by other studies, constituting a highly significant validation of our discoveries. Specifically, genes IRGM, P4HA2, PTGER4, and RBM22 contain significant SNPs rs11741861 ( de Lange *et al.* 2017), rs2188962 ( de Lange *et al.* 2017), rs11742580 ( Franke *et al.* 2010), and rs11741861 ( de Lange *et al.* 2017), respectively. Importantly, gene IRGM is related to CD pathogenesis with the involvement in the process of autophagy (Liu *et al.* 2015). However, our newly identified genes JAKMIP1 and PDK1, associated with BD and CAD respectively, have not yet been reported elsewhere. For T1D, we identified nine new genes, of which one contains some genome-wide significant SNPs as reported by another study (Plagnol *et al.* 2011). In summary, these 15 significant genes identified by the aSPU test would have been missed by PrediXcan, some of which have been confirmed by other studies, while the remaining ones are to be validated in the future.

### Application to the lipid GWAS summary data

We next applied our new methods and TWAS to a 2010 lipid GWAS summary dataset [100,000 samples, Teslovich *et al.* (2010)], while using its follow-up with a larger sample size (189,000) for partial validation. To facilitate comparison, we used the three sets of weights and the 1000 Genomes Project data (1000 Genomes Project Consortium 2010) as the reference sample, all downloaded from the TWAS database (http://gusevlab.org/projects/fusion/#reference-functional-data) (on January 11, 2017). The three sets of weights were based on three eQTL datasets: microarray gene expression data of peripheral blood from 1245 unrelated subjects from the Netherlands Twin Registry (NTR) (Wright *et al.* 2014), microarray expression data of blood from 1264 subjects from the Young Finns Study (YFS), and RNA-seq measured in adipose tissue from 563 individuals from the Metabolic Syndrome in Men study (METSIM); for each pair of gene-eQTL dataset, we used the set of the optimal weights estimated by TWAS. For each trait, there were 1264, 3555, and 2295 significant *cis*-heritable genes, with weights drawn from the NTR, YFS, and METSIM eQTL datasets, respectively, resulting in a total of 7114 genes being tested; when combining across three sets of weights, there were 1223 genes being tested by the omnibus TWAS and thus considered as the candidate genes by any omnibus test. Thus, we used a conservative Bonferroni adjustment with as the genome-wide significance level. The GWAS Z-scores were imputed for any missing SNPs using the IMPG algorithm (Pasaniuc *et al.* 2014).

### The new test identified more associations

We numerically confirmed the equivalence between the SPU(1) test and TWAS (Figure S3 in File S1). Hence, we used the results of the SPU(1) test to represent those of TWAS in the following. More importantly, the aSPU test could identify a larger number of significant genes than TWAS in every case across the four traits (HDL, LDL, TC, and TG) and three sets of weights (NTR, YFS, and METSIM); the same conclusion holds for the omnibus aSPU and omnibus TWAS tests (Table 2). As a partial validation, a high proportion of the identified genes covered at least one genome-wide significant SNP in the 2010 (100,000 samples) or the larger 2013 data (189,000 samples; Global Lipids Genetics Consortium 2013). In Table S1, Table S2, Table S3, Table S4, Table S5, Table S6, and Table S7 in File S2, we list the significant genes identified by aSPU and TWAS based on analyzing the 2010 lipid data, including novel ones not overlapping with any known risk loci covering any significant SNPs () in the 2010 data (Table S1 and Table S2 in File S2) or in the larger 2013 lipid GWAS dataset (Table S3 in File S2).

Compared to TWAS, the aSPU test can still maintain high power if many of the SNPs in a gene are not associated with a trait. For example, based on the 2010 data, for trait HDL and gene DR1 with the YFS-based weights, among the 17 SNPs with nonzero weights, there were only one SNP with a *P*-value < (but larger than the genome-wide significance level ), resulting in a nonsignificant *P*-value (= ) by TWAS, or equivalently by SPU(1). Since an SPU(*γ*) test with a larger relied more on the SNPs with the smaller *P*-values (*i.e.*, more strongly associated with the trait), it yielded a more significant *P*-value with a larger *γ*: the SPU(2) and SPU(5) tests gave *P*-value = and respectively, leading to the significant *P*-value = of aSPU in the end. Furthermore, based on the 2013 data, several SNPs in the locus were genome-wide significant, and both aSPU and TWAS confirmed the significance of gene DR1. Two locusZoom plots for the locus based on the 2010 and 2013 data are shown in Figure 1. Another similar example is with the 2010 data for trait TG and gene NEIL2 with the NTR-based weights: although all the five weighted score elements for the SNPs were negative, their absolute values varied from 0.07 to 2.89, leading to the *P*-values for any SPU(*γ*) with more significant than the *P*-value = of SPU(1).

An SPU(*γ*) test with an even *γ* could be more powerful than SPU(1) with varying SNP association directions. In the 2010 data, for trait LDL and gene NTN5 with the METSIM-based weights, among the 450 weighted score elements of the individual SNPs, 43% were positive while the remaining 57% were negative, leading to a much more significant *P*-value of SPU(2) over that of SPU(1): 8.2 *vs.* 4.9

Generally, as expected from statistical theory and as shown in Figure S4 and Figure S5 in File S1, since the SPU(1) test (*i.e.*, TWAS) might not be powerful for a given gene, the aSPU test could gain statistical power through other more powerful SPU tests like SPU(2).

### The new test identified novel associations

Finally, we applied the aSPU and TWAS (and their omnibus versions) to the larger 2013 lipid dataset (Global Lipids Genetics Consortium 2013), listing the numbers of the significant genes identified by each method in Table 3. Again the aSPU test identified a much larger number of significant associations. The Manhattan plots for the pooled results of aSPU for each set of the weights and of aSPU-O combining the three sets of the weights for each trait are shown in Figure 2 and Figure 3; a comparison between aSPU/aSPU-O *vs.* TWAS/TWAS-O for trait LDL is shown in Figure S6 and Figure S7 in File S1. In total, aSPU and TWAS identified 17 and 14 new associations not overlapping with known risk loci, respectively; among the six new associations uniquely identified by aSPU test, gene PFAS was reported to be associated with LDL in a later meta-analysis (Below *et al.* 2016). The new associations identified by aSPU or/and TWAS are listed in Table 4, while all other ones are in Table S8, Table S9, Table S10, and Table S11 in File S2. It is noteworthy that in Table 4, with the *P*-values close to the significance cut-off, the aSPU test barely missed the three significant genes uniquely identified by the SPU(1) test (*i.e.*, TWAS); in contrast, the SPU(1) test gave the much larger *P*-values for several significant genes uniquely identified by aSPU.

### Simulations

It was shown previously that the aSPU test could control its type I error rate effectively in the context of unweighted association testing (Pan *et al.* 2014), which is expected to hold in the current context. Nevertheless, we conducted a simulation study to confirm it. We used the individual-level (imputed) genotypic data of the WTCCC control and T2D samples with a combined sample size of We randomly generated a binary trait with an equal probability 0.5 (of being in either category) for each subject, and calculated a summary Z statistic for each SNP. We then applied the aSPU test along with the asymptotic SPU(1) and SPU(2) tests to the individual-level data with the same PrediXcan-constructed weights based on the DGN whole blood gene expression data; in addition, we also applied the tests to the summary Z-statistics with the TWAS-constructed weights based on the NTR, YFS, and METSIM gene expression data, respectively; finally, we applied the omnibus aSPU-O test to the summary statistics to combine results across the three sets of NTR, YFS, and METSIM weights. Similar to that in the real data analysis, we used a step-up procedure to adaptively determine the number of simulations when calculating the *P*-values of aSPU and aSPU-O. We started with a relatively small we reran the tests with for the genes with *P*-values but stopped otherwise; we repeated this process by increasing *B* to 10 times of its previous value for the genes with *P*-values ) up to As shown in the Q-Q plots in Figure S8 in File S1, in each case each test controlled the type I error rate satisfactorily.

## Discussion

We note that the algorithm in PrediXcan and TWAS (for quantitative GWAS traits) is similar to two-stage least squares (2SLS) with the SNPs as instrumental variables, which is related to the Mendelian Randomization (MR) approach of Zhu *et al.* (2016). However, since only *cis*-eQTLs are considered in the above approaches (while *trans*-eQTLs are ignored), the used SNPs may be weak instrumental variables, and the assumptions underlying MR are likely to be violated. Hence, we have so far avoided a causal interpretation of detected associations.

In summary, we have developed a powerful adaptive test (aSPU) to integrate GWAS and eQTL data. We have demonstrated its improved power over the existing methods; in fact, the same association test underlying the two existing methods, PrediXcan and TWAS, can be regarded as a special case of our proposed test, explaining why our proposed test may have improved power. Importantly, our new formulation of PrediXcan and TWAS in the general framework of weighted multi-SNP association testing suggests other possible extensions, *e.g.*, applications not only to other informative endophenotypes (Xu *et al.* 2017), but also to incorporate other sources of information like SNP functional annotation, previous linkage scans (Roeder *et al.* 2006) and multiple phenotypes (Kim *et al.* 2015; Zhu *et al.* 2015), which may be worth investigating in the future.

The proposed statistical tests are implemented in R package aSPU2, which along with some example R code is publicly available at www.wuchong.org/TWAS.html.

## Acknowledgments

The authors are grateful to the reviewers and the editor for helpful and constructive comments. The authors thank Sasha Gusev for providing and helping with the use of the TWAS database. This research was supported by National Institutes of Health grants R21AG057038, R01HL116720, R01GM113250, and R01HL105397, and by the Minnesota Supercomputing Institute; Z.X. was supported by a University of Minnesota MnDRIVE Fellowship and C.W. by a University of Minnesota Dissertation Fellowship.

## Footnotes

Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.117.300270/-/DC1.

*Communicating editor: G. Churchill*

- Received August 2, 2017.
- Accepted September 5, 2017.

- Copyright © 2017 by the Genetics Society of America