Summary Transcriptome-wide association studies (TWAS) have been increasingly applied to identify (putative) causal genes for complex traits and diseases. TWAS can be regarded as a two-sample two-stage least squares method for instrumental variable (IV) regression for causal inference. The standard TWAS (called TWAS-L) only considers a linear relationship between a gene’s expression and a trait in stage 2, which may lose statistical power when not true. Recently, an extension of TWAS (called TWAS-LQ) considers both the linear and quadratic effects of a gene on a trait, which however is not flexible enough due to its parametric nature and may be low powered for nonquadratic nonlinear effects. On the other hand, a deep learning (DL) approach, called DeepIV, has been proposed to nonparametrically model a nonlinear effect in IV regression. However, it is both slow and unstable due to the ill-posed inverse problem of solving an integral equation with Monte Carlo approximations. Furthermore, in the original DeepIV approach, statistical inference, that is, hypothesis testing, was not studied. Here, we propose a novel DL approach, called DeLIVR, to overcome the major drawbacks of DeepIV, by estimating a related but different target function and including a hypothesis testing framework. We show through simulations that DeLIVR was both faster and more stable than DeepIV. We applied both parametric and DL approaches to the GTEx and UK Biobank data, showcasing that DeLIVR detected additional 8 and 7 genes nonlinearly associated with high-density lipoprotein (HDL) cholesterol and low-density lipoprotein (LDL) cholesterol, respectively, all of which would be missed by TWAS-L, TWAS-LQ, and DeepIV; these genes include BUD13 associated with HDL, SLC44A2 and GMIP with LDL, all supported by previous studies.

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Transcriptome-Wide Association Studies (TWASs) have become increasingly popular in identifying genes (or other endophenotypes or exposures) associated with complex traits. In TWAS, one first builds a predictive model for gene expressions using an expression quantitative trait loci (eQTL) data set in stage 1, then tests the association between the predicted gene expression and a trait based on a large, independent genome-wide association study (GWAS) data set in stage 2. However, since the sample size of the eQTL data set is usually small and the coefficient of multiple determination (i.e., R 2 ) of the model for many genes is also small, a question of interest is to what extent these factors affect the statistical power of TWAS. In addition, in contrast to a standard (univariate) TWAS (UV-TWAS) considering only a single gene at a time, multivariate TWAS (MV-TWAS) methods have recently emerged to account for the effects of multiple genes, or a gene's nonlinear effects, simultaneously. With the absence of the power analysis for these MV-TWAS methods, it would be of interest to investigate whether one can gain or lose power by using the newly proposed MV-TWAS instead of UV-TWAS. In this paper, we first outline a general method for sample size/power calculations for two-sample TWAS, then use real data-the Alzheimer's Disease Neuroimaging Initiative (ADNI) expression quantitative trait loci (eQTL) data and the Genotype-Tissue Expression (GTEx) eQTL data for stage 1, the International Genomics of Alzheimer's Project Alzheimer's disease (AD) GWAS summary data and UK Biobank (UKB) individual-level data for stage 2-to empirically address these questions. Our most important conclusions are the following. First, a sample size of a few thousands (~8000) would suffice in stage 1, where the power of TWAS would be more determined by cis-heritability of gene expression. Second, as in the general