## Introduction

Prostate cancer (PrCa) affects ~1 in 7 men during their lifetime and is one of the most common cancers worldwide, with up to 58% of risk due to genetic factors1,2. Genome-wide association studies (GWAS) have identified over 100 genomic regions harboring risk variants for PrCa which explain roughly one-third of familial risk3,4,5,6,7. With few exceptions8, the causal variants and target susceptibility genes at most GWAS risk loci have yet to be identified. Multiple studies have shown that PrCa- and other disease-associated variants are enriched near variants that correlate with gene expression levels9,10,11,12,13. In fact, recent approaches have integrated expression quantitative trait loci (eQTLs) with GWAS to implicate several plausible genes for PrCa risk (e.g., IRX4, MSMB, NCOA4, NUDT11, and SLC22A3)5,14,15,16,17,18,19,20,21. While overlapping eQTLs and GWAS is powerful, the high prevalence of eQTLs22 coupled with linkage disequilibrium (LD) renders it difficult to distinguish the true susceptibility gene from spurious co-localization at the same locus23. Therefore, disentangling LD is critical for prioritization and causal gene identification at risk loci.

Gene expression imputation followed by a transcriptome-wide association study24,25,26 (TWAS) has been recently proposed as a powerful approach to prioritize candidate risk genes underlying complex traits. By taking LD into account across SNPs, the resulting association statistics reflect the underlying effect of steady-state gene or alternative splicing expression levels on disease risk25,27, which can be used to identify new regions or to rank genes for functional validation at known risk regions24,25,26,27,28. Here we perform a multi-tissue transcriptome-wide association study24,25,26 to identify new risk regions and to prioritize genes at known risk regions for PrCa. Specifically, we integrate gene expression data from 48 panels measured in 45 tissues across 4448 individuals with GWAS of prostate cancer from the OncoArray in 142,392 men29. Notably, we include alternatively spliced and total gene expression data measured in tumor prostate to identify genes contributing to prostate cancer risk or to continued oncogenesis. We identify 217 gene-trait associations for PrCa with 23 (11) genes identified uniquely using models of alternative spliced (total) expression in tumor. Significant genes were found in 84 independent 1 Mb regions, of which 9 regions are located more than 2 Mb away from any OncoArray GWAS significant variants, thus identifying new candidate risk regions. Second, we use TWAS to investigate genes previously reported as susceptibility genes for prostate cancer identified by eQTL-based analyses. We find a significant overlap with 56 out of 102 previously reported genes assayed in our study also significant in TWAS. Third, we use a novel Bayesian prioritization approach to compute credible sets of genes and prioritize 109 genes that explain at least 90% of the posterior density for association signal at TWAS risk regions. One notable example, IRX4, had 97% posterior probability to explain the association signal at its region with the remaining 3% explained by 9 neighboring genes. Overall, our findings highlight the power of integrating gene expression data with GWAS and provide testable hypotheses for future functional validation of prostate cancer risk.

## Results

### Overview of methods

To identify genes associated with PrCa risk, we performed a TWAS using 48 gene expression panels measured in 45 tissues22,30,31,32,33,34,35,36 integrated with summary data from the OncoArray PrCa GWAS of 142,392 individuals of European ancestry (81,318/61,074 cases/controls; Methods)29. We performed the summary-based TWAS approach as described in ref. 25 using the FUSION software (Methods). Briefly, this approach uses reference linkage disequilibrium (LD) and reference gene expression panels with GWAS summary statistics to estimate the association between the cis-genetic component of gene expression, or alternative splicing events, and PrCa risk25. First, for each panel, FUSION estimated the heritability of steady-state gene and alternative splicing expression levels explained by SNPs local to each gene (i.e., 1 Mb flanking window) using the mixed-linear model (see Methods). Genes with nominally significant (P < 0.05) estimates of SNP-heritability (cis-$$h_g^2$$), are then put forward for training predictive models. Genes with non-significant estimates of heritability are pruned, as they are unlikely to be accurately predicted. Next, FUSION fits predictive linear models (e.g., Elastic Net, LASSO, GBLUP37, and BSLMM38) for every gene using local SNPs. The model with the best cross-validation prediction accuracy (significant out-of-sample R2; nominal P < 0.05) was used for prediction into the GWAS cohort. This was repeated for all expression datasets, resulting in 109,170 tissue-specific models spanning 15,383 unique genes using total expression and 4990 using alternatively spliced introns for a combined 16,389 unique genes. The average number of models per expression panel was 2228 (Supplementary Data 1). Gene expression measured in normal prostate tissue from GTEx22 resulted in only 710 gene models, which can be explained due to smaller sample size (N = 87) compared with the average (N = 234; Supplementary Data 1). Indeed, the number of gene models per panel was highly correlated with sample size, which implies that statistical power to detect genes with cis-regulatory control is limited by sample size (Supplementary Figure 1). Focusing only on models capturing total gene expression, genes on average had heritable levels of expression in 6.1 different panels (median 3) with 10,628/15,383 genes having heritable expression in at least two panels (Fig. 1). We found R2 for predictive models was largely consistent across genomic locations, and predominantly affected by the number of non-zero weights used for prediction (Supplementary Figure 2). Predictive power of linear gene expression models is upper-bounded by heritability; thus, we use a normalized R2 to measure in-sample prediction accuracy (R2/ cis-$$h_g^2$$). We found the average R2/ cis-$$h_g^2$$ across all tissue-specific models was 65%, which indicates that most of the signal in cis-regulated total expression and alternative splicing levels is captured by the fitted models (Fig. 1). To assess the predictive stability for models of normal prostate gene expression, we compared measured and predicted gene expression for TCGA36,39 normal prostate samples using models fitted in GTEx22 normal prostate. We found a highly significant replication (mean R2 = 0.07; P = 1.5 × 10−29), explaining 41% of in-sample cross-validation R2 (Supplementary Figure 3), which is consistent with previous out-of-sample estimates24,25. We performed a cross-tissue analysis within TCGA and found tumor prostate gene expression models replicated in normal prostate (total expression R2 = 0.06; splicing R2 = 0.05; Supplementary Table 1). Given the large number of genes having evidence of genetic control across multiple tissues, we next aimed to measure the similarity of different tissue models (Methods). Across all reference panels for each gene we observed an average R2 = 0.64 (Supplementary Figure 4). Similarly, when averaging across genes, reference panels displayed an average cross-tissue R2 = 0.52 (Supplementary Figure 5). Together, these results suggest that trained models predict similar levels of cis-regulated expression on average, despite reference panels measuring expression in different tissues, with varying QC, and differing capture technologies. Next, we performed simulations to measure the statistical power of TWAS under a variety of trait architectures (Supplementary Note 1). Consistent with previous work, we found TWAS to be well-powered at various effect-sizes and heritability levels for gene expression. Importantly, we found no inflation under the null when cis-regulated gene expression has no effect on downstream trait (Supplementary Figure 6).

### TWAS identifies 217 genes associated with PrCa status

In total, we tested 109,170 tissue-specific gene models of expression for association with PrCa status and observed 892 reaching transcriptome-wide significance (PTWAS < 4.58 × 10−7; two-tailed Z-test), resulting in 217 unique genes, of which 114 were significant in more than one panel (Supplementary Data 2; Fig. 2). On average, we found 18.2 tissue-specific models associated with PrCa per reference expression panel (Supplementary Data 1). In 1 Mb regions with at least 1 transcriptome-wide significant gene, we observed 10.6 tissue-specific associated models on average, and 2.6 associated genes on average, indicating that further refinement of association signal at TWAS risk loci is necessary. To quantify the overlap between non-HLA, autosomal risk loci in the OncoArray PrCa GWAS and our TWAS results, we partitioned GWAS summary data into 1 Mb regions and observed 131 harboring at least one genome-wide significant SNP. Of these, 127/131 overlapped at least one gene model in our data and 68/131 overlapped at least one transcriptome-wide significant gene (Supplementary Figure 7). Associated genes were the closest gene to the top GWAS SNP 20% of the time when using 26,292 RefSeq genes. This result is consistent with previous reports9,25,26 and suggests that prioritizing genes based on distance to index SNPs is suboptimal. We found gene model associations were largely consistent, further supporting the predictive stability of models using cis-SNPs (Supplementary Figure 8; Supplementary Note 1). We observed little evidence of prediction accuracy introducing biased results (Supplementary Figure 9; Supplementary Note 1). As a partial control, we compared TWAS results with S-PrediXcan, a related method for predicting gene expression into GWAS summary statistics, using independently trained models and observed a strong correlation (R = 0.90; see Supplementary Figure 10; Supplementary Note 1), further supporting the validity of the TWAS approach.

Most of the gene models captured total expression levels in normal tissues, however as a positive control we included models for total expression in tumor prostate tissue (Methods). Predicted expression using tumor prostate models accounted only for 43/217 significant genes compared with 6/217 in normal prostate which is likely due to the large difference in sample size between the original reference panels (Supplementary Data 1). Given this, we found no significant increase in proportion of tumor prostate associated models compared with normal prostate (Fisher’s exact P = 0.22). Of the 309 genes with models trained in both reference panels a single shared gene, MLPH (OMIM: 606526, a gene whose function is related to melanosome transport40), was associated with PrCa risk. In all, 11/43 genes were significant only in tumor prostate models of total expression. We found, 7/11 genes were modeled in other panels but did not reach transcriptome-wide significance while the other 4/11 were not significantly heritable, and thus not testable, in other panels. We also tested models of alternatively spliced introns for association to PrCa risk. We identified predicted expression of alternatively spliced introns in tumor prostate accounted for 68/217 genes, with an average of 2.5 (median 1) alternatively spliced intron associations per significant gene. We next quantified the amount of overlap between results driven from models of alternative splicing events versus models of total gene expression. In all, 23/68 genes were found only in alternatively spliced introns, and 14/23 genes had models of total gene expression but did not reach transcriptome-wide significance. The remaining 9/23 were tested solely in alternatively spliced introns, due to heritability of total gene expression not reaching significance. Together these results emphasize earlier work demonstrating that sQTLs for a gene commonly capture signal independent of eQTLs41.

### TWAS analysis increases power to find PrCa associations

Most of the power in the TWAS approach can be attributed to large GWAS sample size. However, two other factors can increase power over GWAS. First, TWAS carries a reduced testing burden compared with that of GWAS, due to TWAS having many fewer genes compared with SNPs. In all, 9/217 genes were located at nine novel independent 1 Mb regions (i.e., no overlapping GWAS SNP), all of which remained significant under a summary-based permutation test (P < 0.05/9; Table 1; Supplementary Data 2; Methods). We found this result was stable to increasing region sizes (Supplementary Data 3) and unlikely to be the result of long-range tagging with known GWAS risk (Supplementary Data 4; Supplementary Note 1). We observed increased association signal for SNPs at these regions compared to the genome-wide background after accounting for similar MAF and LD patterns (Supplementary Figure 11), which, together with observed TWAS associations, suggests that GWAS sample size is still a limiting factor in identifying PrCa risk SNPs. As a partially independent check, we performed a multi-tissue TWAS using summary data from an earlier PrCa GWAS (N = 49,346)7 and found 2 novel regions. We found both regions to overlap a genome-wide significant SNP within 1 Mb in this data further supporting the robustness of TWAS (Supplementary Table 2). Second, we expect to observe increased association signal when expression of a risk gene is regulated by multiple local SNPs25. We observed 88/892 instances across 28 genes where TWAS association statistics were stronger than the respective top overlapping GWAS SNP statistics (one-sided Fisher’s exact P < 2.2 × 10−16; 6.5% higher χ2 statistics on average). For example, GRHL3 (OMIM:608317; a gene associated with suppression of squamous cell carcinoma tumors42) exhibited stronger signal in TWAS using expression in prostate tumor (PTWAS = 9.38 × 10−10) compared with the lead SNP signal (PGWAS = 1.49 × 10−5). Similarly, POLI (OMIM:605252, a DNA repair gene associated with mutagenesis of cancer cells43,44) resulted in larger TWAS associations (PTWAS = 2.29 × 10−8) compared with the best proximal SNP (PGWAS = 5.44 × 10−7).

### TWAS replicates previously reported genes

We next sought to quantify the extent of overlapping results between TWAS and previous studies that integrated eQTL data measured in normal and tumor prostate tissues at PrCa risk regions (Methods; Supplementary Table 3)5,14,15,16,17,18,19,20. We considered only autosomal, non-HLA genes which resulted in 130 previously reported genes. We found a significant overlap between reported genes, with 102/130 assayed in our study and 56/102 reaching transcriptome-wide significance in at least one of our panels (Fisher’s exact P < 2.2 × 10−16; Supplementary Table 3, Supplementary Data 5). For example, MLPH was reported in 4/8 studies. We found significant associations suggesting that decreased expression of MLPH in normal and tumor prostate tissue increases risk for PrCa (e.g., GTEx prostate MLPH ZTWAS = −5.80; PTWAS = 6.69 × 10−9; TCGA prostate ZTWAS = −6.77; PTWAS = 1.25 × 10−11). Predicted MLPH in tumor prostate remained significant under permutation, which suggests that chance co-localization with GWAS risk is unlikely (Supplementary Data 2). To assess the amount of residual association signal due to genetic variation in the GWAS risk region after accounting for predicted expression of MLPH, we performed a summary-based conditional analysis (Methods). We found MLPH to explain most of the signal at its region (lead SNP PGWAS = 4.03 × 10−11; conditioned on MLPH lead SNP PGWAS = 1.13 × 10−3; Fig. 3). Our findings are consistent with recent work that found decreased expression levels of MLPH to be associated with increased PrCa risk45. Despite previous eQTL data focusing on normal and tumor prostate tissue, we observed associations in 45 expression panels overlapping the 56 observed genes in total, underscoring earlier works demonstrating the consistency of cross-tissue cis-regulatory effects46.

### Prioritization pinpoints a single gene for most risk regions

TWAS genes are indicative of association and do not necessarily reflect causality (e.g., due to co-regulation at the same region). To prioritize genes at regions with multiple TWAS signals (Fig. 2), we used a Bayesian formulation to estimate 90%-credible gene sets (Methods). We found 109 unique genes across 84 non-overlapping 1 Mb regions comprising our 90% credible sets (Supplementary Data 6, 7). In all, 68/84 credible sets contained either a single gene or the same gene in multiple tissues. The average number of unique genes per credible set was 1.29 (median 1). We observed that 28/109 prioritized genes were previously reported in eQTL analyses5,14,15,16,17,18,19,20, which supports the hypothesis that TWAS followed by Bayesian prioritization refines associations to relevant disease genes. For example, MLPH was the sole gene defining its region’s 90% credible set with a posterior probability of 94%. Similarly, SLC22A3 (OMIM: 604842; a gene involved in poly-specific organic cation transporters47 and previously implicated in PrCa risk18) exhibited >94% posterior probability to be causal.

### Prostate tissue genes have largest average effect

Given the large number of significant associations observed for non-prostate tissues in our data, we wanted to quantify which tissue is most relevant for PrCa risk. We first grouped TWAS PrCa associations into prostate/non-prostate and tested for enrichment in normal and tumor prostate expression models. Predicted expression and splicing events in normal and tumor prostate made up 221/892 associations with PrCa (Supplementary Data 2) which was highly significant compared to the grouping of all other tissues (Fisher’s exact P = 7.3 × 10−9). This measure only quantifies the total amount of observed associations and neglects average association strength. Next, we computed the mean TWAS association statistic using all genes predicted from each expression reference panel (Fig. 4). We observed the largest average TWAS associations in genes predicted from normal and tumor prostate tissue, which reaffirms our intuition of expression and splice events in prostate being the most relevant for PrCa risk. We re-ranked mean associations using only genes found to be transcriptome-wide significant and observed a similar ordering with total expression in normal prostate ranked highest (average χ2 = 176.2; Supplementary Figure 12).

## Discussion

Prostate cancer is a common male cancer that is expected to affect more than 180,000 men in the United States in 2017 alone48. While GWAS has been successful in localizing risk for PrCa due to genetic variation, the underlying susceptibility genes remain elusive. Here we have presented results of a transcriptome-wide association study using the OncoArray PrCa GWAS summary statistics for over 142,000 case/control samples. This approach utilizes imputed expression levels and splicing events in the GWAS samples to identify and prioritize putative susceptibility genes. We identified 217 genes whose expression is associated with PrCa risk. These genes localized at 84 genomic regions, of which nine regions do not overlap with a genome-wide significant SNP in the OncoArray GWAS. We found 23 genes using predictive models for alternatively spliced introns in tumor prostate, which supports the its role in continued risk for tumor oncogenesis. A large fraction of identified genes was confirmed in earlier work, with 56 genes previously reported in eQTL/PrCa GWAS overlap studies. We used a novel Bayesian prioritization approach to refine our associations to credible sets of 109 genes with statistical evidence of causality under standard assumptions. Our results provide a functional map for PrCa risk which can be explored for follow-up and validation.

In this study, we compared our reported TWAS results with genes identified in previous works focusing on expression measured in normal and tumor prostate tissue. Several of these studies considered an eQTL and GWAS risk SNP to overlap if they are in linkage at a specified threshold. While these approaches are sound, they may be limited in statistical power for several reasons. First, if multiple local SNPs independently contribute to risk, overlap studies relying only on the top risk SNP will lose power. Second, earlier overlap studies used thresholds for association signal (i.e., GWAS P < 5 × 10−8) and linkage strength (i.e., LD > 0.5) to consider pairs of SNPs for evidence of expression influencing risk of PrCa. TWAS is largely agnostic to both issues as it jointly considers all SNPs in the region, regardless of reported GWAS association strength. However, when expression of a risk gene is regulated by a single causal SNP, we expect TWAS and earlier overlap approaches to have similar levels in power25.

Previous works have strongly implicated expression of certain genes in PrCa risk that were not assayed in our study (e.g., MSMB18,49) due to non-significant heritability estimates. TWAS operates by fitting predictive linear models of gene expression based on local genotype data, followed by prediction into large cohorts and subsequent association testing. Expression of genes that are not significantly heritable at current sample sizes are not included in the pipeline. This is the consequence of heritability providing an upper bound on the predictive accuracy under a linear model for genotype; therefore, if a gene has undetectable heritability at a given sample size, it will be difficult to predict using linear combinations of SNPs. To compute TWAS weights for normal prostate tissue, we used samples collected in the GTEx v6 panel (n = 87). Thus, our inability to detect heritable levels of gene expression can be explained due to the relatively small number of samples compared with other tissues. Indeed, previous work has shown a strong correlation between sample size in expression panels and the number of identified eGenes27; therefore, as sample size increases for relevant tissues, we expect the number of genes included in the TWAS framework to increase. TWAS will lose power in situations where gene expression is a nonlinear function of local SNPs, or when trans (or distal) regulation is a major component in modulating expression levels.

We conclude with several caveats and possible future directions. First, while TWAS associations are consistent with models of steady-state gene expression levels altering risk for PrCa, they may be the result of confounding25,26. Imputed gene expression levels are the result of weighted linear combinations of SNPs, many of which may tag non-regulatory mechanisms driving risk and result in inflated association statistics. Second, our results relied on validating prediction models using multiple approaches: within-reference methods (i.e., cross-validation), cross-reference methods (e.g., GTEx into TCGA), and external-reference methods (i.e., 1000 Genomes predictive stability). While results from these approaches support our models generalizing out-of-sample, we still lack within-GWAS replication of predictive models. Third, since genes with eQTLs are common, associations may be the result of chance co-localization between eQTLs and PrCa risk. Finally, we note recent work has extended TWAS-like methods to expose regulatory mechanisms for susceptibility genes by incorporating chromatin information50. An extension to our work would be to pinpoint chromatin variation regulating expression levels at identified risk genes, thus describing a richer landscape of the molecular cascade where SNP → chromatin → expression → PrCa risk.

## Methods

### OncoArray GWAS summary statistics

Genome-wide association summary statistics for the OncoArray PrCa study were obtained from ref. 29. Summary statistics were computed using a fixed-effect meta-analysis for 142,392 total samples of European ancestry from the OncoArray (81,318/61,074 cases/controls), UK stage 1 (1854/1894) and UK stage 2 (3706/3884), CaPS 1 (474/482) and CaPS 2 (1458/512), BPC3 (2068/3011), NCI PEGASUS (4600/2941) and iCOGS (20,219/20,440). The initial summary data contained association statistics for 19,726,430 variants. We filtered out summary statistics for SNPs with MAF <0.01 and any SNPs with ambiguous alternative alleles (e.g., A → T; C → G; or vice-versa). Finally, we kept only SNPs with rsIDs defined by dbSNP144. Our QC pipeline resulted in association statistics at 10,516,237 SNPs for downstream TWAS analyses.

### Previous prostate-tissue eQTL studies

We collected previous studies that investigated the overlap of eQTLs in normal and tumor prostate tissue at known PrCa risk loci5,14,15,16,17,18,19,20. We compared TWAS statistics versus reported eQTL overlap results as aggregated in refs. 14,15. Across these studies, overlap of eQTLs and PrCa risk loci are computed by one of two possible methods. The first method tests known PrCa risk SNPs for association with expression levels of nearby genes/transcripts. The second method takes a two-step approach. First, genes nearby PrCa risk loci are tested for harboring eQTLs at some significance level. Next, genes with identified eQTL SNPs are tested to be in LD with known PrCa risk variants at some level (e.g., r2 > 0.5).

### Reference gene expression data and predictive models of expression

We downloaded the FUSION software (see URLs) along with its prepackaged weights for gene expression data. FUSION is an R package that implements the TWAS scheme described in ref. 25. Weights for gene expression measured using RNA sequencing data were obtained from the CommonMind Consortium30 (dorsolateral prefrontal cortex, n = 452), the Genotype-Tissue Expression Project22 (GTEx; 44 tissues; n = 449), the Metabolic Syndrome in Men study32,33 (adipose, n = 563), and The Cancer Genome Atlas (TCGA; prostate adenocarcinoma, n = 483)39. Expression microarray data were obtained from the Netherlands Twins Registry35 (NTR; blood, n = 1247), and the Young Finns Study31,34 (YFS; blood, n = 1264). All non-TCGA expression panel individuals were PrCa controls. Detailed description of quality control procedures on measured gene expression and genotype information for all non-TCGA reference panels are described in refs. 25,27. TCGA genotype, gene expression, and exon-junction data for 525 samples were downloaded using the Broad GDAC FireHose version 2016_1_28 (see URLs). Genotypes were imputed to the Haplotype Reference Consortium51 and restricted to well-imputed (INFO > 0.9) HapMap352 sites. Genes (exon junctions) missing in more than half of samples were removed. RPKM and log-adjusted gene expression levels were estimated in a generalized linear model controlling for three gene expression PCs. The estimated log-abundances were quantile-normalized and inverse-normal rank-normalized. We estimated alternatively spliced introns using the software MapSplice version 2 (see URLs). A total of 482 samples passed quality control procedures in both genotype and gene expression data. We note that batch effects from measurement biases (e.g., RNA-degradation) should be uncorrelated with SNPs local to a gene body definition, and therefore not impact prediction accuracy. By maximizing the sample size, predictive power when using cis-SNPs should increase and be largely unbiased. This is evidenced by the fact that models are largely stable across and within TCGA PRAD datasets (Supplementary Table 1).

We filtered genes that did not exhibit cis-genetic regulation at current samples sizes by keeping only genes with nominally significant (P < 0.05) estimates of cis-SNP heritability (cis-$$h_g^2$$), which resulted in 117,459 total tissue-gene pairs from 17,023 unique genes. We refrain from reporting genes from the HLA region due to complicated LD patterns.

To train predictive models, FUSION defines gene expression for n samples (yGE) as a linear function of p SNPs (X) in a 1 Mb region flaking the gene as

$${\mathbf{y}}_{{\mathrm{GE}}} = {\mathbf{C\beta}} + {\mathbf{Xw}}_{{\mathrm{GE}}} + {\bf{\epsilon }},$$

where wGE are the p SNP weights, are covariates (e.g., sex, age, genotype principal components, genotyping platform, and PEER factors) and their effects, and $$\epsilon$$ is random environmental noise. FUSION estimated weights for expression of a gene in a tissue using multiple penalized linear models. Generally, FUSION optimizes for

$$\left[ {\begin{array}{*{20}{c}} {{\hat{\mathbf w}}_{{\mathrm{GE}}}} \\ {{\hat{\mathbf \beta }}} \end{array}} \right] = \arg \mathop {{\min }}\limits_{{\mathbf{w}}_{{\mathrm{GE}}},{\mathbf{\eta }}} \|{\mathbf{y}}_{{\mathrm{GE}}} - {\mathbf{Xw}}_{{\mathrm{GE}}} - {\mathbf{C\beta }}\|_2^2 + f\left( {{\mathbf{w}}_{{\mathrm{GE}}}} \right),$$

where f(wGE) is a parameterized penalty function specific to each model (e.g., GBLUP37, LASSO, the Elastic Net). The exception to this optimization criterion is the Bayesian sparse linear mixed model (i.e., BSLMM)38 which fits the posterior mean for wGE using MCMC in the GEMMA v 0.94 software (see URLs) to obtain weights. To determine which model has the best prediction accuracy for a given gene-tissue pair, FUSION computes out-of-sample R2 by performing fivefold cross-validation for each model. We compute the normalized prediction accuracy for a gene as $${\mathrm{min}}\left( {\frac{{R^2}}{{h_g^2}},1} \right)$$. Weights from the model with the largest R2 that was also nominally non-zero (P < 0.05) were used to compute TWAS association statistics. This resulted in a final tally of 109,170 tissue-specific models at 16,389 unique genes.

### Cis-heritability of gene expression

FUSION reports the estimated SNP-heritability (i.e., $$h_g^2$$) for measured gene expression levels explained by SNPs in the cis-region (1 Mb region surrounding the TSS). This is modeled under a mixed-linear model as

$${\mathrm{var}}\left( {{\mathbf{y}}\prime _{{\mathrm{GE}}}} \right) = {\mathbf{A}}{{\sigma }}_{{g}}^2 + {\mathbf{I}}{{\sigma }}_{{e}}^2,$$

where $${\mathbf{y}}\prime _{{\rm GE}}$$ is the residual gene expression after regressing out fixed-effect covariates C, A is the estimated kinship matrix from SNPs in the cis-region and $${\mathrm{\sigma }}_g^2$$ ($${\mathrm{\sigma }}_e^2$$) is the variance explained by the cis-SNPs (environment). SNP-heritability is then defined to be ratio of genotypic variance and total trait variance as, $$h_g^2 = \frac{{{\mathrm{\sigma }}_g^2}}{{{\mathrm{\sigma }}_g^2 + {\mathrm{\sigma }}_e^2}}$$. Variance parameters are estimated using the AI-REML algorithm implemented in GCTA v1.26 (see URLs) with the top 3 genotypic principal components, sex, age, genotyping platform, and PEER factors as covariates.

### Measuring cross-tissue similarity in predicted expression

We took an unbiased approach to identify susceptibility genes for PrCa by using gene expression panels measured in various tissues. To quantify how similar predicted expression levels are for the same gene across different tissues, we measured the squared Pearson correlation (R2). This value represents how well predicted expression from one tissue predicts expression in another tissue. To dissect similarities and differences of tissue-specific models, the ideal scenario would be to inspect effects at individual SNPs defining the models. In practice this is not possible due to predictive models not including the same set of SNPs due to QC and technological differences in the original studies. Therefore, as a proxy we predict gene expression into the 489 samples of European ancestry from 1000 Genomes53 and compute R2 across shared genes for pairs of tissues (Supplementary Note 1).

### Transcriptome-wide association study using GWAS summary statistics

FUSION estimates the strength of association between predicted expression of a gene and PrCa (zTWAS) as function of the vector of GWAS summary Z-scores at a given cis locus zGWAS (i.e., vector of SNP association Wald statistics) and the LD-adjusted weights vector learned from the gene expression data wGE as

$${\rm z}_{{\rm TWAS}} = \frac{{{\mathbf{{w}}}_{{\rm GE}}^\prime {\mathbf{{z}}}_{{\rm GWAS}}}}{{\sqrt {{\mathop{\rm{var}}} ({\mathbf{{w}}}_{{\rm GE}}^\prime {\mathbf{{z}}}_{{\rm GWAS}})} }} = \frac{{{\mathbf{{w}}}_{{\rm GE}}^\prime {\mathbf{{z}}}_{{\rm GWAS}}}}{{\sqrt {{\mathbf{{w}}}_{{\rm GE}}^\prime {\mathbf{{Vw}}}_{{\rm GE}}} }},$$

where V is a correlation matrix across SNPs at the locus (i.e., LD) and “‘” indicates transpose. A P-value for zTWAS is obtained using a two-tailed test under N(0,1). In this work, we estimated V using 489 samples of European ancestry in 1000 Genomes53. To account for the large number of hypotheses tested, we perform Bonferroni correction at α = 0.05/M, where M = 109,170 is the number of predictive models, which is conservative as many gene models are correlated. As reported by ref. 25, there may be inflation at GWAS risk loci, due to chance co-varying of SNP effects between expression and PrCa. The same work described a permutation procedure that assesses likelihood of observing association by chance conditioned on GWAS signal. The algorithm works by permuting the eQTL weights wGE while keeping zGWAS fixed and computing zTWAS,perm. FUSION implements an adaptive procedure that stops once enough scores (i.e. |zTWAS,perm|≥|zTWAS|) have been observed such that the empirical null cannot be rejected at a specified level. We define novel risk regions as a flanking region around a transcriptome-wide significant gene (splicing event; PTWAS < 4.58 × 10−7; two-tailed Z-test) that does not harbor a genome-wide significant SNP (PGWAS < 5 × 10−8; two-tailed Z-test). We consider 2 Mb windows by default (i.e. TSS ± 1 Mb) and show that the results are robust to the choice of window size (Supplementary Data 3).

### GWAS analyses conditional on predicted expression

To assess the extent of residual association of SNP with PrCa risk after accounting for predicted gene expression levels, FUSION estimates conditional SNP association scores using GWAS summary statistics. Namely, define V as LD for SNPs in the region, VGE as the correlation between predicted expression levels, and C as the correlation between SNPs and predicted expression. The least-squares estimates of zGWAS|zTWAS are determined by,

$${\mathbf{z}}_{{\mathrm{GWAS}}}|{\mathbf{z}}_{{\mathrm{TWAS}}} = {\mathbf{z}}_{{\mathrm{GWAS}}} - {\mathbf{CV}}_{{\mathbf{GE}}}^{ - 1}{\mathbf{z}}_{{\mathrm{TWAS}}}.$$

The variance of the residual association strength is given by,

$${\mathrm{var}}\left[ {{\mathbf{z}}_{{\mathrm{GWAS}}}|{\mathbf{z}}_{{\mathrm{TWAS}}}} \right] = {\mathrm{var}}\left[ {{\mathbf{z}}_{{\mathrm{GWAS}}}} \right] - {\mathrm{var}}\left[ {{\mathbf{CV}}_{{\mathbf{GE}}}^{ - 1}{\mathbf{z}}_{{\mathrm{TWAS}}}} \right] = {\mathbf{V}} - {\mathbf{CV}}_{{\mathbf{GE}}}^{ - 1}{\mathbf{C}}^{\prime}.$$

This results in the final conditional association score for the ith SNP as,

$${{z}}_{{i}} = \left[ {{\mathbf{z}}_{{\mathrm{GWAS}}} - {\mathbf{CV}}_{{\mathbf{GE}}}^{ - 1}{\mathbf{z}}_{{\mathrm{TWAS}}}} \right]_{{i}}/\surd {\mathrm{diag}}\left[ {{\mathbf{V}} - {\mathbf{CV}}_{{\mathbf{GE}}}^{ - 1}{\mathbf{C}}^\prime } \right]_{{{ii}}}.$$

### Bayes factors and posterior inference of causal genes

Complex correlations between predicted expression levels at a given region can yield multiple associated genes in TWAS (Fig. 2). Thus, for the vast majority of risk regions it remains unclear which gene is causally influencing PrCa risk. Here, modeling under the assumption of a single causal gene per risk region and relying on the central limit theorem for normality, we can compute the Bayes Factor that the ith gene in a region is causal as,

$${\rm BF}_i = \frac{{N\left( {{\rm {z}}_{{\mathrm{TWAS}},{{i}}}|0,1 + n\sigma _\alpha ^2} \right)}}{{N\left( {{\rm {z}}_{{\mathrm{TWAS}},{{i}}}|0,1} \right)}} = \left( {1 + n\sigma _\alpha ^2} \right)^{ - 1/2}\exp \left( {\frac{{{\rm {z}}_{{\mathrm{TWAS}},{{i}}}^2}}{2}\frac{{n\sigma _\alpha ^2}}{{1 + n\sigma _\alpha ^2}}} \right),$$

where $${\rm z}_{{\mathrm{TWAS}},{{i}}}^2$$ is the squared TWAS association statistic for the ith gene, n is the GWAS sample size, and $$\sigma _\alpha ^2$$ is prior effect-size variance for gene expression on PrCa risk (Supplementary Note 1). This model is structurally similar in form to earlier works54,55,56 describing Bayes Factors for fine mapping SNPs at GWAS risk regions. The important distinction is that here, we formulate a Bayes Factor for genes at TWAS risk regions. The Bayes Factor for each gene quantifies the amount of evidence in favor of the causal model (ith gene drives risk) versus the null (ith gene has no causal effect). We extend individual Bayes Factors for k genes at a PrCa risk region to compute the posterior probability that a gene is causal as,

$$\Pr \left( {{\mathrm{gene}}\,i\,{\mathrm{is}}\,{\mathrm{causal}}|{\mathbf{{z}}}_{{\mathrm{TWAS}}},n\sigma _\alpha ^2} \right) = \frac{{{\rm BF}_{{i}}}}{{\mathop {\sum }\nolimits_{\mathrm{k}} {\rm {BF}}_{\mathrm{k}}}}.$$

Equipped with our definition of posterior probability for each gene being causal, we define ρ-credible gene sets for a PrCa risk region. Formally, a set of indices $$i \in I$$ defines a ρ-credible gene set if

$$\rho = \mathop {\sum}\limits_{i \in I} {\Pr \left( {{\mathrm{gene}}\,i\,{\mathrm{is}}\,{\mathrm{causal|}}{\mathbf{{z}}}_{{\mathrm{TWAS}}},\,n\sigma _\alpha ^2} \right).}$$

For a fixed ρ we optimize over k genes at a region by greedily adding genes until the total density is at least ρ.

To ensure that our ρ-credible sets are well-calibrated we performed simulations by predicting expression levels into 489 samples of European ancestry from 1000 Genomes53 and estimating the local correlation structure to sample TWAS Z-scores directly (Supplementary Note 1). Under the assumption of a single causal gene at a risk region, we sampled TWAS Z-scores for 1000 independent regions. We then performed Bayesian prioritization at each region and computed ρ-credible sets for various levels of ρ while counting the proportion of causal genes identified across all simulations.

### Pathway analyses

To determine which pathways may be enriched with genes identified from our Bayesian prioritization approach, we used the R package GOseq57 which internally links gene identifiers to GO terms (GO db: 2017-09-02). We categorized all 16,389 genes into prioritized/not-prioritized and ran the analysis using custom R scripts linking GOseq. GOseq obtains P-values for overrepresented genes using the Wallenius approximation to the non-central hypergeometric distribution. We limited analysis to Gene Ontology Biological Pathways (GO:BP). GOSeq drops genes without GO annotations from analysis. We observed 4711 genes dropped from analyses resulting in 11,678 genes put forward for enrichment tests (Supplementary Data 8; Supplementary Note 1).

### URLs

1000 Genomes Phase3: http://www.internationalgenome.org/