Bayesian large-scale multiple regression with summary statistics from genome-wide association studies

Abstract

Bayesian methods for large-scale multiple regression provide attractive approaches to the analysis of genome-wide association studies (GWAS). For example, they can estimate heritability of complex traits, allowing for both polygenic and sparse models; and by incorporating external genomic data into the priors, they can increase power and yield new biological insights. However, these methods require access to individual genotypes and phenotypes, which are often not easily available. Here we provide a framework for performing these analyses without individual-level data. Specifically, we introduce a “Regression with Summary Statistics” (RSS) likelihood, which relates the multiple regression coefficients to univariate regression results that are often easily available. The RSS likelihood requires estimates of correlations among covariates (SNPs), which also can be obtained from public databases. We perform Bayesian multiple regression analysis by combining the RSS likelihood with previously proposed prior distributions, sampling posteriors by Markov chain Monte Carlo. In a wide range of simulations RSS performs similarly to analyses using the individual data, both for estimating heritability and detecting associations. We apply RSS to a GWAS of human height that contains 253,288 individuals typed at 1.06 million SNPs, for which analyses of individual-level data are practically impossible. Estimates of heritability (52%) are consistent with, but more precise, than previous results using subsets of these data. We also identify many previously unreported loci that show evidence for association with height in our analyses. Software is available at https://github.com/stephenslab/rss.

ACAT: A Fast and Powerful p Value Combination Method for Rare-Variant Analysis in Sequencing Studies

“…In addition, compared to other methods for analyzing summary statistics, ACAT does not need the LD information that is often estimated from a population reference panel, which greatly speeds up the computation and avoids the potential issues caused by the estimation accuracy of the LD structure. 30 For example, it is convenient and simple to use ACAT to perform gene-based analysis to complement the standard single-variant analysis in GWAS. The p values from single-variant analysis can be directly used and are the only input required by ACAT for gene-, pathway-, or network-based analysis, and therefore the computation can be done efficiently.…”

Improved genetic prediction of complex traits from individual-level data or summary statistics

“…However, while Zhao et al restrict to independent SNPs, and subsequently derive V = I + X T YY T X/n 2 , where I is an identity matrix, we instead use V = X T X, as proposed by Zhu and Stephens. 44 If the matrix X' denotes the genotypes of the reference panel (size n' x m), we can achieve this sampling by setting X A T Y A /n A = X T Y/n + (n E /n A ) 1/2 X' T /n' 1/2 G, where G is a vector of length n' with elements drawn from a standard Gaussian distribution. Additionally, we calculate (X E T Y E )/n E = (X T Y -X A T Y A )/n E , the complementary estimate of γ, which we use when measuring the accuracy of the partial models (as explained above).…”

Multi-scale Inference of Genetic Trait Architecture using Biologically Annotated Neural Networks

“…GBJ attempts to improve upon the previously mentioned methods by generalizing the Berk-Jones statistic to account for complex correlation structures and adaptively adjust the size of annotated SNP-sets to only SNPs that maximize power [49]. Lastly, RSS is a Bayesian linear mixed model enrichment method which places a likelihood on the observed SNP-level GWA effect sizes (using their standard errors and LD estimates), and assumes a spike-and-slab shrinkage prior on the true SNP effects to derive a probability of enrichment for genes or other annotated units [50]. It is worth noting that, while RSS and the BANNs framework are conceptually different, the two methods utilize very similar variational approximation algorithms for posterior inference [46] (Methods and Supplementary Notes).…”