Integrating association evidence across multiple traits can improve the power of gene discovery and reveal pleiotropy. Most multi-trait analysis methods focus on individual common variants in genome-wide association studies. Here, we introduce multi-trait analysis of rare-variant associations (MTAR), a framework for joint analysis of association summary statistics between multiple rare variants and different traits. MTAR achieves substantial power gain by leveraging the genome-wide genetic correlation measure to inform the degree of gene-level effect heterogeneity across traits. We apply MTAR to rare-variant summary statistics for three lipid traits in the Global Lipids Genetics Consortium. 99 genome-wide significant genes were identified in the single-trait-based tests, and MTAR increases this to 139. Among the 11 novel lipid-associated genes discovered by MTAR, 7 are replicated in an independent UK Biobank GWAS analysis. Our study demonstrates that MTAR is substantially more powerful than single-trait-based tests and highlights the value of MTAR for novel gene discovery. Methods to integrate association evidence across multiple traits often focus on individual common variants GWAS. Here the authors present multi-trait analysis of rare-variant associations (MTAR), a framework for joint analysis of association summary statistics between multiple rare variants and different traits. Multi-trait analyses that combine association evidence across traits can boost statistical power over single-trait analyses in detecting risk variants, especially for those traits that have weak associations with the variants. Many multi-trait methods are designed for testing the single-variant association8. However, the statistical power of single-variant tests is low for rare-variant association studies (RVAS)9. In light of this limitation, gene-based tests have been developed for RVAS to aggregate mutation information across several variant sites within a gene to enrich association signals and reduce the penalty resulting from multiple testing9. Although several methods are available for multi-trait multi-variant tests, most of them require individual-level genotype and phenotype data15 or are designed for common variants19 (Supplementary Table Rich genome-wide association study (GWAS) findings have suggested the sharing of genetic risk variants among multiple complex traits20 and burden tests24 are the most widely used gene-based tests for RVAS and represent two main patterns of genetic effects across rare variants. Burden tests assume effects across variants are largely homogeneous and SKAT assumes they are heterogeneous. SKAT-O25 is a test that achieves robustness by combining tests with various degrees of effect heterogeneity, including the SKAT and burden tests as special cases. Specifically, SKAT-O assumes rare-variant effects are random variables with a uniform (exchangeable) correlation and different levels of heterogeneity can be considered by changing the correlation coefficient.The genetic architecture of complex traits is unknown in advance and is likely to vary from one gene to another across the genome and from one trait to another. Therefore, the main challenge of multi-trait multi-variant analyses is to flexibly accommodate a variety of genetic effect patterns among traits and variants such that the test is robust and has high power. The effect structures among rare variants within a gene have been well-studied when numerous gene-based tests were developed. The sequence kernel association test (SKAT)26. Hence, it is not adequate to use a uniform correlation coefficient to model the degree of similarity for all trait pairs. Many recent studies have investigated the genetic overlap for many pairs of complex traits and diseases and estimated genetic correlation as a global measure of genetic similarity for trait pairs28. Although a genetic correlation is calculated using common variants across the genome and RV association tests are performed on the gene level, the idea of utilizing genetic correlation to guide the specification of gene-level effect heterogeneity across traits is intriguing and has not been considered in existing multi-trait methods.The effects on multiple traits may also exhibit homogeneous and heterogeneous patterns. However, the degree of genetic effect similarity/heterogeneity are likely to vary from one trait pair to another. As an example, for the pair of traits that are biologically related  and high-density lipoprotein cholesterol (HDL)), we expect they share more causal variants and have a higher level of genetic similarity than the pair of traits less relevant 30 to combine single-trait RVAS P-values. To achieve robustness and improve overall power, we combine the P-values of iMTAR, cMTAR, and cctP, and refer to this omnibus test as MTAR-O. To demonstrate the usefulness of MTAR empirically, we analyze summary statistics from the Global Lipids Genetics Consortium (GLGC) on low-density lipoprotein cholesterol (LDL), HDL, and TG. MTAR discovers more lipid-associated genes than single-trait-based analyses and many novel association signals are replicated in an independent UK Biobank data. Moreover, our simulation results show that MTAR methods have well-preserved type I error rate and greater power over single-trait-based methods across a wide range of effect patterns across traits and variants. Finally, we compare MTAR with two existing multi-trait methods that outperform other competing methods. We find that MTAR is more powerful in almost all simulation settings and discovers more genes in the application to the GLGC data.Here we develop multi-trait analysis of rare-variant association (MTAR), a framework for the multi-trait analysis of RVAS. MTAR is built upon a random-effects meta-analysis model that uses different correlation structures of the genetic effects to represent a wide spectrum of association patterns across traits and variants. To model genetic effects across variants, MTAR employs the same strategy as SKAT-O. To model the rare-variant effect heterogeneity on multiple traits, MTAR leverages the genetic correlation. Specifically, we propose two correlation structures on the among-trait genetic effects. The first structure allows the between-trait effect similarity to change from the value of the genetic correlation to completely heterogeneous as an extreme and we term the resulting multi-trait association test iMTAR. The second structure allows the between-trait effect similarity to change from the value of the genetic correlation to homogeneous as an extreme and we term the resulting test cMTAR. Besides the aforementioned association patterns across traits, we also consider the scenario in which only a small number of traits are associated with the set of rare variants. This association pattern naturally occurs for the genes that have very specific biological functions and do not affect many traits. To accommodate this pattern, we construct another test, cctP, which uses the Cauchy methodm variants in a gene on K traits. For k = 1, …, K, we let βk = T denote the effects of the m genetic variables on trait k. To perform MTAR tests, we first obtain the vector of variant-level score statistics for testing βk = 0 denoted by Uk = T and the covariance estimate for Uk denoted by Vk. The Uk and Vk can be easily constructed using the information routinely shared in public domains (Methods). We let β and covariance ∑32, where ∑ are not zeros. For any given traits k and k′ with sample overlap, the formula for estimating the covariance between Suppose that we are interested in the effects of m variants are not associated with any of the K traits: H0 : β1 = β2 = ⋯ = βK = 0. Multivariate test for this hypothesis has a large degrees of freedom and low statistical power. In MTAR, we further assume that the genetic effects β are zero-mean random effects with covariance matrix σB, where σ is an unknown scalar and B is a pre-specified matrix dictating the covariances of genetic effects among traits and variants. Under this random-effects model, the equivalent null hypothesis is H0 : σ = 0 and we test this hypothesis using a variance-component score test . By specifying different values of ρ1, this structure allows various degrees of among-variant effect heterogeneity. As the two extremes, the effects across variants are homogeneous when ρ1 = 1; the effects are completely heterogeneous and vary independently when ρ1 = 0.We are interested in testing the null hypothesis that the est (Eq. ). The teW2 is a diagonal matrix with each diagonal element being a trait-specific weight and Ω2 is a between-trait effect correlation matrix. By setting the diagonal elements in W2 to 0 or 1, we can choose to focus on any subset of the traits and consider any degree of association sparsity across traits . It is not sensible to assume the exchangeable correlation structure for B2, because some pairs of traits are more similar in the rare-variant effects than other pairs . Here we propose to leverage the genetic correlation27 to inform the similarity of rare-variant effects among traits. Genetic correlation is a single number measure that quantifies the overall genetic similarity between a pair of traits. Recent advancement of methods enables us to conveniently estimate genetic correlation based on GWAS summary statistics28 and there are web portals to query genetic correlations among many complex traits26. We hypothesize that the genetic correlation is also informative to measure the similarity/heterogeneity of the gene-level rare-variant effects among traits for most genes in the genome. Specifically, let Ckk′ denote the genetic correlation between traits k and k′. We propose two types of correlation structures for Ω2. In both structures, we specify a parameter ρ2 (0 ≤ ρ2 ≤ 1) to control the contribution of genetic correlation Ckk′ to the degree of effect heterogeneity between traits k and k′. The iMTAR structure assumes the correlation coefficient is ρ2Ckk′. Under this structure, the rare-variant effects across traits are heterogeneous and the degree of heterogeneity can change from Ckk′ (when ρ2 = 1) to completely heterogeneous (strongest level of heterogeneity as effects across traits can vary independently when ρ2 = 0). The cMTAR structure assumes the correlation coefficient is ρ2Ckk′ + (1 − ρ2). Under this structure, the degree of heterogeneity can change from Ckk′ (when ρ2 = 1) to homogeneous (no heterogeneity when ρ2 = 0).For the between-trait effect covariance, we set ρ1 and ρ2 are unknown, we propose to search a grid of different values of ρ1 and ρ2 and use the Cauchy method30 to combine multiple P-values (Methods). The resulting tests are named after the two aforementioned iMTAR and cMTAR structures. The Cauchy method is a fast and powerful approach to combine multiple correlated P-values without the need for estimating and accounting for their correlation. To accommodate the situation where the gene is associated with a small number of traits, we develop a test called cctP that uses the Cauchy method to combine single-trait P-values from SKAT and burden tests. As we demonstrate in the GLGC data analysis and simulation studies, the cMTAR, iMTAR, and cctP cover different effect patterns among traits. To achieve further robustness, we use the Cauchy method to combine P-values of the three complementary tests and term this omnibus test as MTAR-O. The summary of the proposed iMTAR, cMTAR, cctP, and MTAR-O methods are presented in Fig. P-values are described in Methods.As the optimal values of ∑ and B are covariance matrices among traits and variants, it is important to note the difference. Matrix ∑ reflects the correlation due to the residual relatedness among traits in the presence of sample overlap and linkage disequilibrium (LD) among variants. An inaccurate estimate of ∑ yields inflated type I error in the association testing. On the other hand, the matrix B = B2 ⊗ B1 reflects the similarity of the true gene-level rare-variant effects among traits and variants. This information is unknown a priori; hence, B needs to be pre-specified. The power of the tests can be greatly improved if the specification reflects the truth. MTAR utilizes the genetic correlation, a global measure of cross-trait genetic similarity, to guide the specification of B2. The effectiveness of this strategy in gaining power has been demonstrated in the following sections.Although both 33. The participants were from 73 different studies and single-variant association summary statistics were combined across studies via fixed-effects meta-analysis32. The acquisition of the GLGC summary statistics is described in Methods.We performed multi-trait RVAS for three plasma lipid traits: LDL, HDL, and TG. The GLGC data set includes ~300,000 individuals of primarily European ancestry genotyped with the HumanExome BeadChip (exome array)33, we considered 179,884 rare variants with minor allele frequency (MAF) < 5% and the highest priority according to their functionality and deleteriousness. We focused on 15,378 genes that contain at least two rare variants. In our analysis, we used the previously reported genetic correlation estimates among the three lipid traits27 in MTAR. Specifically, the genetic correlation is −0.61 for the pair , 0.35 for , and 0.09 for . For comparison, we performed the single-trait-based analysis by combining SKAT and burden test P-values across traits using either the cctP or the Bonferroni-corrected minimal P-value .Following Liu et al.33, the slightly elevated genomic control lambdas in the quantile–quantile plots suggest the polygenic inheritance of the lipid traits , a total of 140 genes were identified by at least one test  is close to the single-trait-based P-values (1.8 × 10−6).Similar to the previous gene-based RVAS of GLGC dataes Table . There iP-value when ρ2 is large (ρ2 ≥ 0.5), highlighting the informativeness of using genetic correlations to guide the among-trait effect correlation  have the smallest P-values than other tests. When the effects between-trait effects are largely homogeneous  and genes with known association with lipid-related Mendelian disorders . Gene set enrichment analysis of the 139 genes highlighted the gene sets related to lipid metabolism and transport  or Genotype-Tissue Expression (GTEx) as reference sets demonstrated enrichment of liver-specific genes ; ATGL is involved in the breakdown of TG. Although variants associated with PNPLA2 have not previously been directly linked with any of the three lipid traits in humans, ATGL-knockout mice display altered very-low-density lipoprotein, HDL, and TG levels36.Among the 41 genes exclusively identified by MTAR tests, 27 (66%) genes have previously reported association evidence with at least one of the three lipid traits and 20 (74%) of them are associated with at least two lipid traits  that have different patterns of sample overlap for the three traits  or bidirectional . We first evaluated the empirical type I error rates based on 10α = 2.5 × 10−6 based on 104 replicates  overlapping samples. Failure to account for the correlation between summary statistics induced by the overlapping samples can greatly inflate type I error39. Sample overlap is prevalent in the multi-trait analysis. Sometimes the overlap pattern is clear , but other times is often elusive—public domains only have combined summary statistics across many studies for each trait and study-specific summary statistics are not available7. MTAR can handle these scenarios and use a simple approach to accurately estimate the correlation between summary statistics for the traits with sample overlap. Third, MTAR is computationally fast. The MTAR P-value calculation is analytical and does not require time-consuming procedures such as permutation and Monte Carlo simulation.In comparison with existing multi-trait multi-variant methods (Supplementary Table 10 (MultiSKAT R package) and MTaSPUsSet17 (aSPU R package) in numerical studies. These two existing methods have demonstrated superior performance to other competing multi-trait multi-variant methods such as metaCCA18, MGAS19, DKAT11, MAAUSS13, MSKAT15, and GAMuT14. Similar to MTAR, Multi-SKAT and MTaSPUsSet proposed several tests to accommodate different patterns of associations across traits and variants, and omnibus tests to gain robustness. We compared their omnibus tests with MTAR-O. In the simulation study, we let all cohorts have complete trait values as it is required by Multi-SKAT. Empirical power was estimated at the α = 10−4 level due to the speed of MTaSPUsSet. MTAR-O has greater power than Multi-SKAT and MTaSPUsSet in almost all scenarios, especially when the genetic correlation reflects the heterogeneity of effects among traits  as long as the score statistics from the models are unbiased and their covariance estimates are accurate. For binary traits, the normal approximation to the score statistics could be inaccurate in the unbalanced case-control settingWith the increasing number of complex traits available in large-scale whole exome/genome sequencing studies and electronic health record linked biobank data, multi-trait analysis based on summary statistics of multiple rare variants will become an important tool to boost the power of discovering genetic components of complex traits and unravel their shared genetic architectures. We envision that MTAR will facilitate the accumulation of adequately large sample sizes to accelerate discoveries in complex trait genetics and provide new biological insights by revealing pleiotropic genes.B1 is a m × m covariance matrix for the effects among variants. We set W1 is a diagonal matrix with each element being a variant-specific weight and Ω1 is a between-variant effect correlation matrix of exchangeable structure with correlation coefficient ρ1 (0 ≤ ρ1 ≤ 1). Specifically, the single-trait analysis becomes SKAT20 if ρ1 = 0 and burden tests46 if ρ1 = 1. Burden tests are more powerful when the association effects are similar across the aggregated variants, whereas SKAT is more powerful when the effects are in opposite directions or the number of causal variants is small relative to neutral variants. As for the variant-specific weights (in W1), by default, we set them based on the MAF through a beta distribution density function Beta  as in SKAT. Other weighting schemes can be employed as well.k  and subject i , when the individual-level phenotype (Yik), genotypes (Gik), and covariates (Xik) are available, the score statistics Uk and their covariance estimate Vk can be obtained from the generalized linear model with the likelihood function βk and γk are regression parameters, ϕk is a dispersion parameter, and a, b, and c are specific functions. Specifically, we have γk and ϕk under H0 : βk = 0, and b′ and b″ are the first and second derivatives of function b. For the linear regression model, we have b′(z) = z, and b″(z) = 1. For the logistic regression model, we have a(b′(z) = ez/(1 + ez), and b″(z) = ez/(1 + ez)2.For each trait Uk and Vk can also be derived from different forms of summary statistics shared in public domains38. When the score statistics Uk and their variances ) are available, the covariance matrix of Uk can be approximated as Vk ≈ {diag(Vk)}1/2R{diag(Vk)}1/2, where m variants based on the working genotypes or external reference. In another case, when the effect estimates The Uk among traits k = 1, …, K are correlated. Assume trait k is from cohort A with sample size nA and trait k′ is from cohort B with sample size nB, and there are nC overlapping subjects in these two cohorts. For any SNP j not associated with the traits, the correlation matrix of Z-score j47. In particular, if both traits k and k′ are quantitative, we havek and k′ are binary, let nC0 (nC1) represent the number of overlapping samples with trait value of 0 (or 1), nA0 (nA1) denotes the number of subjects with trait k and takes the value of 0 (or 1) and nB0 (nB1) denotes the number of subjects with trait k′ and takes the value of 0 (or 1), then we have39ζkk′ using the independent null variants across the whole genome. Specifically, we first perform LD pruning using LD threshold r2 < 0.01 in 500 kb region to obtain a set of independent common variants. We then remove variants with association test P-values < 0.05 and only keep variants that are not associated with any traits. For any traits k and k′, we calculate the between-trait sample correlation of the Z-scores on the remaining variants and denote it as Z-scores and confirmed the accuracy of the estimate R is the SNP LD matrix defined in the previous subsection.If all the traits are from the same study or multiple studies with overlapping samples, the summary statistics m genetic effects across K traits and Uk and Vk. The MTAR framework assumes the hierarchical model∑ reflects the correlation due to the residual relatedness among traits in the presence of sample overlap and LD among variants, and B reflects the correlation among the rare-variant effects across traits and variants. The B matrix contains two coefficients ρ1 and ρ2, where ρ1 controls the effect correlation among variants and ρ2 controls the contribution of the genetic correlation to the among-trait rare-variant effect correlation.We let ρ = , we test H0 : σ = 0 against H1 : σ ≠ 0 by a variance-component score test48:χ2 distribution under the null hypothesis. Davies method can be used to accurately estimate the P-value49. In addition, rare variants often show polymorphisms in some but not all traits, the adjustment of the formula for this case is described in the For a fixed set of B), the Cauchy P-value combination method is utilized to combine results from various ρ1 and ρ2. Similar to the minimum P-value method, the Cauchy method mainly focuses on a few smallest P-values30. The advantage of the Cauchy method over the minimum P-value method is that the Cauchy method is computationally fast because it does not rely on the Monte Carlo simulation to account for the correlation of the individual tests29. Specifically, the iMTAR or cMTAR test statistic is defined asp(Qρ) is the P-value of Qρ, ρ = , and ρ1 and ρ2 such that there are nine combinations. We have shown in the Supplementary Fig. P-value of QiMTAR/cMTAR can be accurately approximated by 0.5 − arctan (QiMTAR/cMTAR)/π29.In the cMTAR and iMTAR tests /π.In addition, the MTAR framework reduces to single-trait analysis when we set a single diagonal element of matrix P-values from cMTAR, iMTAR, and cctP aspcMTAR, piMTAR, and pcctP are the P-values of the cMTAR, iMTAR, and cctP tests. The P-value of the MTAR-O test can be approximated by 0.5 − arctan(QMTAR−O)/π.Finally, the Cauchy method is used to construct MTAR-O test by combining http://csg.sph.umich.edu/abecasis/public/lipids2017/. For each trait k, the web portal contains variant-level genetic effect estimates sek. We obtained Uk and Vk by using sek as described in Summary statistics subsection of Methods. As the original genotypes from the study are not publicly available, we estimated the LD matrix R based on the genotypes of the European population from the NHLBI Exome Sequencing Project (ESP)37. To account for possible sample overlap among traits, we used Eq. .Gene set enrichment analysis was conducted using the one-sided hypergeometric test against Reactome Pathways and Gene Ontology Biological Processes, as implemented in the GENE2FUNC from FUMA53  exclusively identified by MTAR tests but without any annotation evidence, we applied the MTAR methods to an independent study with association summary statistics from the UK Biobank GWAS data set. The GWAS summary statistics were released by the Neale Lab with the re-release of UK Biobank genotype imputation (termed imputed-v3). The three related traits LDL direct (mmol/L), HDL direct (mmol/L), and TG (mmol/L) were jointly analyzed in a similar manner as the analysis of GLGC data.55. These haplotypes were used to form the genotypes of 8500 subjects across three cohorts. To simulate the genotypes for a data set, we randomly selected one thousand 3 KB regions in each haplotype and focused on rare variants with MAF < 0.05.For all simulations, we generated 100 haplotypes of length 1 MB under a calibrated coalescent model to mimic the LD structure and local combination rate of the European populationi, three traits were generated based on a multi-response regression modelβkj is the genetic effect for trait k at variant j, Gij is the genotype at variant j, Xi1 is a binary covariate simulated from Bernoulli(0.5), Xi2 is a continuous covariate simulated from a standard normal distribution. The covariance matrix of the error term used here is based on the estimated residual correlations among the lipid traits LDL, HDL, and TG in the ESP data37. The reduced model was used when we needed to generate only one or two traits for subject i.For each subject m = 5, 10, 20, 50, 100 and traits K = 3, 6, or 9 (Supplementary Fig. We estimated the computation time of MTAR tests by considering different numbers of variants https://cran.r-project.org/web/packages/SKAT MultiSKAT R package v1.0: https://github.com/diptavo/MultiSKAT aSPU R package v1.48: https://cran.r-project.org/web/packages/aSPU FUMA v1.3.5: http://fuma.ctglab.nl TissueEnrich v1.8.0: https://tissueenrich.gdcb.iastate.edu Open Targets: https://genetics.opentargets.org STOPGAP: https://github.com/StatGenPRD/STOPGAP.SKAT R package v1.3.2.1: Further information on research design is available in the Supplementary InformationPeer Review FileDescription of Additional Supplementary FilesSupplementary Data 1Supplementary Data 2Supplementary Data 3Reporting Summary


Polygenic scores (PGS) have been widely used to predict disease risk using variants identified from genome-wide association studies (GWAS). To date, most GWAS have been conducted in populations of European ancestry, which limits the use of GWAS-derived PGS in non-European ancestry populations. Here, we derive a theoretical model of the relative accuracy (RA) of PGS across ancestries. We show through extensive simulations that the RA of PGS based on genome-wide significant SNPs can be predicted accurately from modelling linkage disequilibrium (LD), minor allele frequencies (MAF), cross-population correlations of causal SNP effects and heritability. We find that LD and MAF differences between ancestries can explain between 70 and 80% of the loss of RA of European-based PGS in African ancestry for traits like body mass index and type 2 diabetes. Our results suggest that causal variants underlying common genetic variation identified in European ancestry GWAS are mostly shared across continents. Polygenic scores (PGS) are often based on GWAS data from individuals of European ancestry, thus limiting their use in populations of non-European ancestry. Here, the authors predict the relative accuracy of PGS across ancestries and suggest that causal variants are mostly shared across continents. However, the over-representation of European ancestry in the majority of GWASs has been shown to yield an unbalanced improvement of PGS prediction accuracy in non-European ancestry populations3. For example, Duncan et al.2 report the average accuracy of PGS across multiple traits to be ~64% lower in individuals of African ancestry as compared with that in individuals of European ancestry. Similarly, Martin et al.3 report, across multiple traits, reductions of PGS accuracy of ~37%, ~50% and ~78% in individuals of South-Asian, East-Asian and African ancestries, respectively, relative to individuals of European ancestry.Polygenic scores  are now routinely utilised to predict complex traits and risk of diseases from findings of genome-wide association studies (GWASs). Over recent years, the predictive performances of PGS have steadily increased with GWASs sample sizes, as predicted by theory5, consistent with the observation of limited transferability of GWAS findings across diverse human populations7. These studies also highlight major factors influencing that loss such as differences between populations in causal variants effect sizes, in alleles frequencies and in linkage disequilibrium (LD) between causal variants and SNPs assayed in GWAS9. To illustrate the latter point, let us consider a SNP which has an LD r2 with a causal variant of 0.8 in the discovery population and 0.6 in the target population. Such a SNP would therefore explain 25% = (1 − 0.6/0.8) less trait variation and thus be less predictive in the target population as compared with the discovery population, even when causal variants and their effect sizes are shared between ancestries. More generally, previous empirical and simulation studies have shown that accuracy of genetic predictors decays monotonically with increased genetic differentiation (FST) and LD differences between ancestries10. Other factors such as population specific causal variants12, gene × environment interaction14 have been implicated as potential explanations of the loss of PGS prediction accuracy.Although increasingly emphasised in the recent GWAS literature, it is worth noting that the loss of accuracy problem is not utterly new. Indeed, a number of studies in the animal breeding literature have previously reported lower accuracy of genomic selection across genetically distant breeds16. However, these deterministic formulas mostly apply to best linear unbiased predictors17, which are not classically used in human studies. Consequently, a theoretical understanding of the trans-ancestry predictive capacities of standard PGS is still missing. Although the key factors causing the loss of accuracy have been enumerated in previous studies, quantification of their relative contributions has not been done systematically. Note that quantifying the relative contributions of all these factors is critical for understanding aetiological differences between ancestries, which may have important clinical implications. Here, we develop an approximation of the theoretical relative accuracy of PGS in an ancestry divergent sample as a function of population genetics parameters. Our method only requires GWAS summary statistics and ancestry-specific reference panels. We evaluate the performances of our theory through extensive simulations and apply it to GWAS of 5 quantitative traits and 3 common diseases with different genetic architectures in ~350,000 unrelated UK Biobank participants.In addition, deterministic formulas have been derived to predict the accuracy of genomic prediction across breeds as a function of population parameters  and also using selection index theoryy, for which the genetic component is underlain by random additive effects of MC causal variants. Without loss of generality, we assume causal variants to be shared between ancestries but allow their effect sizes to vary from one ancestry to another. Therefore, ancestry-specific causal variants are a special case with non-zero effect sizes in only one ancestry. We then assume that a GWAS of y has been performed in a discovery sample of a given ancestry, hereafter denoted by Population 1 and that a PGS, defined as the sum of minor allele counts weighted by their estimated effects from the discovery GWAS, is used to predict y in a target sample of another ancestry, hereafter denoted by Population 2. The study design is shown in Supplementary Fig. We consider a quantitative trait pk,1 and pk,2 at the kth PGS-SNP (i.e. SNPs included in the PGS) in Populations 1 and 2, respectively, the LD between the jth causal SNP and the kth PGS-SNP in Population 1 and 2, respectively , the heritabilities y in Populations 1 and 2, respectively, and the correlation ρb of causal SNP effects between Population 1 and Population 2. It is worth underlining here that direct attempts to predict MC). Unfortunately, no method to date can provide estimates of MC with high enough precision. However, under the assumption that causal variants are shared between ancestries, focusing on the ratio MT denotes the number of GWS SNPs used to calculate PGSs. Note that a special case of Eq.  between cohorts of the same ancestry.We derived the expected accuracy of such PGS in Population 2  with a GWS SNP and located within 100 kb of the latter  and propose to quantify other input parameters such as allele frequencies and LD correlations using data from an ancestry diverse reference panel like that of Phase 3 of the 1000 Genomes Project (1KGP)We ran computer simulations to evaluate the performances of Eqs.  and 2)  2) underNEUR = 333,263), East-Asian ancestry , South-Asian ancestry  and African ancestry . The European ancestry group was further divided into a discovery set of N = 313,284 participants in which GWS SNPs were identified (Methods), a validation set in which the accuracy of PGS within-European-ancestry was quantified and a reference group in which we predicted the accuracy of PGS. A thorough description of how these groups were defined is given in the Methods section. As our main focus is to predict the fraction of the RA that can be attributed to alleles frequencies and LD differences between populations, we therefore assumed that effect sizes of causal variants are perfectly correlated across populations, i.e. ρb = 1 and that heritability is constant across populations, i.e. MC = 1000, 5000 and 10,000; and two values of trait heritability h2 = 0.25 and 0.5 (Methods).Our simulations utilise existing genotypes at ~1.1 million common HapMap3 SNPs imputed in 351,983 unrelated UK Biobank (UKB) participants. These participants were categorised into four ancestry homogeneous groups corresponding to European ancestry , which has an average FST of ~0.14 with EUR , we found their sign to be inconsistent between ancestries. In fact, predicted RA from Eq. . Similarly, we found our heuristic approach based on candidate causal variants to yield unbiased predictions of the RA in 7/18 simulated scenarios. More specifically, differences across scenarios between the mean observed RA and that predicted from Eq.  the RA attributable to allele frequencies and LD differences between ancestries under various scenarios.β) and their minor alleles frequencies (hereafter denoted by p). This relationship has been modelled in many studies28 using a parameter S such that β2 is assumed to be proportional to [2p(1−p)]s. Values of S determine the relative contributions of common versus rare variants to the genetic variance in the population and thus have been used as an indirect measure of the strength of natural selection29. Our model assumes that the variance explained by each causal SNP is constant regardless of allele frequencies. This assumption is consistent with a strong negative selection on causal variants shared between populations and corresponds to a value of S = −1. Although previous studies30 have reported pervasive negative selection on complex traits and diseases, these studies often report estimates of S with less extreme magnitudes than that assumed in our model. Moreover, given that little is known on the strength of negative selection in non-European populations, we next investigated through additional simulations the impact of violations of this assumption.In addition to heritability and polygenicity, another important aspect of the genetic architecture of complex traits and diseases is the relationship between effect sizes at causal variants  S1 = S2 = −0.5, i.e. equal strength of negative selection in both populations, (ii) S1 = −0.75 and S2 = −0.5, i.e. stronger selection in Population 1 and (iii) S1 = −0.5 and S2 = −0.75, i.e. stronger selection in Population 2.We adopted a similar framework (Methods) as in our first simulation. However, for the sake of simplicity, we fixed the number of causal variants to p-value >0.05). This somewhat surprising observation is suggestive that when heritability is constant and effect sizes of causal variants are perfectly correlated, differences in strengths of selection between ancestries might have a negligible impact on the RA of PGS. As a consequence, we can expect differences in strengths of selection between ancestries to mainly impact the term In non-European ancestries, we found over 100 replicates that our theory based on Eq. . This th31 (Methods). Note that using ancestry-specific prevalence from previous population studies35 did not change our results  also differed between ancestries  and AFR  ancestries as compared with participants of SAS ancestry . We further investigated this observation and found this pattern to be explained by a single large effect variant  can be quantified separately. We have shown through simulations that the contribution of differences in LD and MAF to the RA of PGS can be predicted with little bias (<5%) using a simple heuristic approach that models local correlation of LD and MAF across ancestries in the close vicinity of GWS SNPs. Our approach only requires GWAS summary statistics as well as data from a globally-diverse reference panel such as the 1KGP.In this study, we developed a new theory to predict the relative prediction accuracy of PGS across populations of different ancestries. Our theory overcomes the challenge of predicting accuracy directly by modelling the relative prediction accuracy instead. Under the assumption that causal variants are shared but allowing their effects to differ between populations, we have shown that the contribution of MAF and LD differences at causal SNPs and that of other parameters of the genetic architecture  of these traits. It is noteworthy that AFR participants of the UK Biobank reside in the UK and therefore are likely to share similar environments as EUR participants. As consequence, the relative contribution of ρb and h2, which partially reflects the effects of gene by environment interactions (if any), might be underestimated. Similarly, the contribution of 12 suggested that ~50% of the heritability is captured by European specific variants when a trait is under moderate negative selection, thus limiting the upper bound of prediction accuracy in AFR for example when using European-derived GWAS summary statistics. Our results based on common variants also show a RA <50% in AFR but we demonstrate that this reduction is mostly explained by MAF and LD differences at common causal variants and not necessarily by population specific causal variants. The latter conclusion is further supported by large genetic correlations observed between EUR and non-EUR ancestries36, which overall suggests that causal variants underlying common genetic variation identified in European ancestry GWASs are mostly shared across continents.We further assessed the ability of our theory to predict the accuracy of PGS of multiple traits and diseases in non-European UKB participants. Altogether, we found that between ~70% (S.E. 7.0%) and 100% (S.E. 20.6%) of the reduction of RA of PGS in AFR ancestry could be explained by differences in LD and MAF. Importantly, we found that the fraction of the RA attributable to MAF and LD varied between traits, which mostly reflects differences in the genetic architecture  in all populations, which limits the generalization of our conclusions to rarer variants. Indeed, rare variants are more likely to be population specific and are usually poorly imputed using a small imputation reference panel. Thirdly, our study has focused on PGS calculated from GWS SNPs alone while other methods such as LDpred37 or SBayesR38, which utilise information from the entire genome, have been shown to be more accurate within ancestry. Although we acknowledge that characterizing the theoretical RA of PGS based on genome-wide methods like LDpred or SBayesR deserves further investigation, we also emphasise that the gap in prediction accuracy between GWS SNPs and genome-wide methods is destined to shrink as the sizes of GWAS continue to grow. Nevertheless, we re-analysed our simulated data using these two methods  approved the study and all participants in the UKB study analysed here provided written informed consent. Additional study and quality control details are shown in Bycroft et al.44. The approach to infer the ancestry of each individual is described in Yengo et al.45. We firstly projected each individual of UKB onto the genotypic principal components (PCs) calculated in 2000 participants of 1KGP22. We only extracted individuals from four ancestries in the 1KGP, namely, European ancestry , South-Asian ancestry , East-Asian ancestry  and African ancestry . We excluded African Caribbeans in Barbados (ACB) and Americans of African Ancestry in SW USA (ASW) populations from AFR and all individuals of American ancestry (AMR) considering their complex admixture patterns. We then assigned each of those genotyped participants of UKB to the closest ancestry based on the first three PCs, resulting in 463,795 EUR, 11,906 SAS, 2486 EAS and 9184 AFR. To remove cryptic relatedness in the UKB, we used the GCTA software to calculate the genomic relationship matrix (GRM)46 based on genotyped SNPs in each of the aforementioned populations. With one of each pair of individuals with estimated relatedness larger than 0.05 being removed, a subset consisting of unrelated individuals was generated in each ancestry. For the European ancestry, we only extracted those self-reported British and Irish participants. After randomly sampling 10,000 individuals from British subset, we created the discovery dataset using the remaining 313,284 individuals. As for the target populations, we used an independent dataset of ~39,000 UKB individuals. Those individuals included the 10,000 randomly sampled participants who identified themselves as British, 9979 participants of EUR who identified themselves as Irish, the 9448 participants of SAS, the 2257 participants of EAS and the 7015 participants of AFR. Data from the 1KGP were used as the reference panel in this study. We generated subsets of unrelated individuals in 1KGP with the same strategy as described above, resulting in 495 EUR, 457 SAS, 498 EAS and 484 AFR.The UK Biobank (UKB) comprises of ~500,000 individuals recruited from the UK, aged from 40 to 69 years old. Participants were genotyped using two genotyping arrays, the Affymetrix UK BiLEVE Axiom47 and UK10K48, used in the UKB are predominant by European descents thus a large number of missing SNPs are observed when using hard call (<0.1) thresholds on dosage data. In each ancestry we firstly extracted genotyped SNPs such that Hardy–Weinberg equilibrium (HWE) p-value >0.001 and missing rates <0.05 and also excluded individuals with genotype call rates <0.9. Filtered SNPs in each ancestry were then phased using SHAPEIT249 and imputed to 1KGP by IMPUTE250. In each ancestry, stringent quality control procedures were performed separately. We removed SNPs with imputation quality scores <0.30, MAF < 0.01, HWE p-value <10−6, or missing genotype call rates >0.05. HapMap3 SNP set, which has been well designed for human genome-wide common genetic variants51, was then extracted from imputed data to run follow-up analyses. A total of 990,395 filtered HapMap3 SNPs in common between populations were selected.For participants of non-European ancestries in the UKB, we further imputed the SNP array data to the 1KGP given that the imputation reference panels, Haplotype Reference Consortium (HRC)SNPs in the 1KGP reference panel were restricted to a total of 6,877,707 SNPs common to all four ancestries after excluding those SNPs with minor allele count (MAC) <5 in each ancestry. The above described HapMap3 SNP set was further intersected with 1KGP, thus limiting the number to 978,783.l = 1) and the target/test sample as Population 2 (l = 2).We derived a deterministic equation to quantify the relative prediction accuracy of PGS across ancestries. To evaluate this equation, we performed various simulations (each scenario with 100 replicates) using the real genotypes in the UKB cohort. To simplify our notations, we denoted the training/discovery sample as Population 1 . In each simulation replicate, we simulated a trait with a heritability of 0.25 or 0.5 in all ancestries. Traits were simulated from MC  causal variants sampled at random from the HapMap3 SNPs. We assumed the effect sizes of causal variants (β) were perfectly correlated across populations, i.e. ρb = 1. For each causal variant, β was sampled from a normal distribution with mean 0 and variance pjl is the MAF in jth causal variant in population l. For each individual, the genetic value g was defined such as xjl denotes the minor allele count  at the jth causal variant in population l. The environmental effect (e) was simulated using a normal distribution with mean 0 and variance equal to (1 − h2): The phenotypes were simulated based on the additive model 52 was used to run GWAS for the simulated phenotypes in Population 1 using the simple linear association testing on HapMap3 SNPs. To mimic the imperfect LD between the causal variants and the SNP markers used in GWAS, the causal variants were always left out of the analysis.PLINK1.90 (note version 20 Mar was used in this study)β from a multi-normal distribution with mean 0 and variance S1 = S2 = −0.5), (ii) stronger selection in Population 1 (S1 = −0.75 and S2 = −0.5) and (iii) stronger selection in Population 2 (S1 = −0.5 and S2 = −0.75). The phenotypes were generated in the same way as described above. For simplicity, we focused on a trait with a heritability h2 = 0.5 and controlled by MC = 5000 causal variants.To further explore the impact of negative selection, we sampled 52. We used the following command: --clump-p1 5e-8 --clump-p2 5e-8 --clump-kb 2000 --clump-r2 0.01. The genotypes of the training population were used as LD reference for clumping. We used here a more stringent LD threshold than classically used (e.g. 0.1 or 0.2) because SNPs with LD r2 as large as 0.1 can still reflect the same signal when GWAS sample size is large .After running GWAS, we selected approximately independent SNPs associated with the trait (referred to as GWS SNPs or PGS-SNPs), using the LD clumping algorithm implemented in PLINK1.90We calculated the deterministic accuracy in the target populations based on selected GWS SNPs. Since we focused on in this study the contribution of LD and allele frequency differences between populations to the RA, the term mentclass2pt{minimmentclass2pt{minimpred1 in the simulations). It was done by constraining the window centred at each GWS SNP as 100 kb and then selecting those pairs including known causal variants. This window was based on the report that ~95% top lead SNPs (with MAF >0.01) identified from GWASs are within 100 kb distance from the causal variants in European ancestry21. Although the causal variants are often unknown or unobserved in a classical GWAS, they are usually tagged by numerous SNPs. Therefore, we took advantage of the information regarding fine-mapping precision of GWAS studies and selected candidate causal variants as those SNPs in LD r2 >0.45 with GWS SNPs and located within 100 kb window21. Those GWS SNPs and candidate causal variants pairs were then used in Eq. . The LD correlations were estimated using PLINK1.9052 (−r).When using Eq.  with knoThe final predicted parameters to evaluate predictive ability were calculated as the mean of the estimates across 100 replicates, respectively. To explore the impact of imputation on our model, we used both 1KGP WGS data and UKB imputed data as the reference in the simulated data. We used the same approaches, except for the one assuming causal variants were known, to analyse GWAS summary statistics of complex traits and diseases in UKB  as the validation, in which the accuracy of PGS within-European-ancestry was quantified and a reference group (self-reported Irish with N = 9979), in which we predicted the accuracy of PGS, separately.The relative accuracy (RA) was calculated as the ratio of R2) between the true phenotypes and the PGS. For diseases, we calculated the liability-scale R2 as described in Lee et al.31. We estimated the disease prevalence as the proportion of cases in each ancestry given that UK Biobank is a population-based study. We combined the two target populations of EUR ancestry  to calculate the corresponding R2 for traits/diseases in UKB. The empirical relative accuracies (RA) were calculated as the ratio of the R2 in UKB participants of non-European ancestry over the same R2 estimated in ~20,000 independent UKB participants of European ancestry.After selecting approximately independent GWS SNPs using LD clumping, we then generated PGS in each target population by adding up the product of minor allele counts times effect sizes of GWS SNPs estimated from GWAS summary statistics. The prediction accuracy was then estimated using the squared correlation (Further information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


Genome Wide Association Studies (GWAS) have successfully identified thousands of loci associated with human diseases. Bayesian genetic fine-mapping studies aim to identify the specific causal variants within GWAS loci responsible for each association, reporting credible sets of plausible causal variants, which are interpreted as containing the causal variant with some “coverage probability”. Here, we use simulations to demonstrate that the coverage probabilities are over-conservative in most fine-mapping situations. We show that this is because fine-mapping data sets are not randomly selected from amongst all causal variants, but from amongst causal variants with larger effect sizes. We present a method to re-estimate the coverage of credible sets using rapid simulations based on the observed, or estimated, SNP correlation structure, we call this the “adjusted coverage estimate”. This is extended to find “adjusted credible sets”, which are the smallest set of variants such that their adjusted coverage estimate meets the target coverage. We use our method to improve the resolution of a fine-mapping study of type 1 diabetes. We found that in 27 out of 39 associated genomic regions our method could reduce the number of potentially causal variants to consider for follow-up, and found that none of the 95% or 99% credible sets required the inclusion of more variants—a pattern matched in simulations of well powered GWAS. Crucially, our method requires only GWAS summary statistics and remains accurate when SNP correlations are estimated from a large reference panel. Using our method to improve the resolution of fine-mapping studies will enable more efficient expenditure of resources in the follow-up process of annotating the variants in the credible set to determine the implicated genes and pathways in human diseases. Pinpointing specific genetic variants within the genome that are causal for human diseases is difficult due to complex correlation patterns existing between variants. Consequently, researchers typically prioritise a set of plausible causal variants for functional validation—these sets of putative causal variants are called “credible sets”. We find that the probabilistic interpretation that these credible sets do indeed contain the true causal variant is variable, in that the reported probabilities often underestimate the true coverage of the causal variant in the credible set. We have developed a method to provide researchers with an “adjusted coverage estimate” that the true causal variant appears in the credible set, and this has been extended to find “adjusted credible sets”, allowing for more efficient allocation of resources in the expensive follow-up laboratory experiments. We used our method to reduce the number of genetic variants to consider as causal candidates for follow-up in 27 genomic regions that are associated with type 1 diabetes. Genome-Wide Association Studies (GWAS) have identified thousands of disease-associated regions in the human genome, but the resolution of these regions is limited due to linkage disequilibrium (LD) between variants . ConsequP value, called the lead-SNP. However, it is generally acknowledged that this SNP may not be the causal variant in a given region due to correlations with the true causal variants  A letter containing a detailed list of your responses to all review comments, and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).Important additional instructions are given below your reviewer comments.Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.Sincerely,Ilya IoshikhesAssociate EditorPLOS Computational BiologyJian MaDeputy EditorPLOS Computational Biology***********************ploscompbiol@plos.org immediately:A link appears below if there are any accompanying review attachments. If you believe any reviews to be missing, please contact [LINK]Reviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: The authors argue that Bayesian credible sets of causal SNPs, computed during fine mapping of GWAS hits, tend to overestimate their coverage. Bias arises both by the fixed nature of a genetic effect at a given locus, and from selection of regions for fine mapping according to statistically significant association. An algorithmic method is proposed to estimate the magnitude of bias and adjust the coverage probability accordingly. An algorithm is presented to re-estimate credible sets with pre-specified coverage, and is shown to reduce the size of credible sets in several regions associated with type-1 diabetes. The paper raises some interesting points and should be of broad interest. It is refreshingly well written and low on typos.Major point1. The paper hinges on the understanding of the term “coverage”. While a frequentist understanding is alluded to in the introduction, the authors take a particular interpretation that may not be universally held. Specifically, the authors ask how often the credible set for the specific region under study covers the causal variant, given that it has been followed up. Here the fixed effects view is justifiable, and the bias of the standard calculation is shown (fig 1A right). However, given that credible sets are generated for many regions, both within and between studies (as the authors have done so themselves in the applied example), we can equally ask how often the credible set covers the causal variant, on average across all regions. This is an equally valid frequentist version of the question “what is the probability that the causal variant is contained in the credible set?” Here there is much less bias (fig 1A left). The bias by selecting significant regions is also open to interpretation. For regions not followed up, we effectively have estimated PP of zero for all SNPs and an empty credible set, which does not include the causal variant. Averaging over these regions as well as those explicitly followed up, the coverage probabilities would be unbiased. Although not a declared Bayesian, I am inclined to favour these broader understandings of coverage since the random effects scenarios are more common than the repeated sampling ones in practice. The authors are welcome to their view but should discuss this key point of interpretation in the introduction and discussion. I also suggest a change in terminology from “corrected coverage estimate” to “conditional coverage estimate” to reflect that it is only a correction under an interpretation that conditions on the specific region, phenotype and population being studied.Minor points2. P5 L139 “effect size” – it’s an expected Z-score that is derived, which is not the same as an effect size.3. P5 L151 “Our method relies on MAF and SNP correlation data” – I don’t see how MAF information is needed to simulate correlated Z scores. It was used to simulate test data for the method, but doesn’t seem necessary for the method itself.4. P7 L222 “the claimed coverage estimates are too low” – this seems to be based simply on comparing the sizes of the standard and corrected credible sets. Without knowledge of the actual causal variants, we can’t be sure that the corrected sets have the desired coverage.5. In fig S1 the N actually looks like quite a poor fit to the GWAS data. Would smaller bin sizes improve the visual fit, or is this just a poor model? The choice of prior for the ABF is another factor that could bias the coverage estimate, and this ought to be discussed.Typos6. P2 L10 “loci” -> “locus”7. P12 L406 “was” -> “were”8. P12 L410 “it’s” -> “its”9. P13 L429 and L431 “these” -> “them”Reviewer #2: This was an interesting and topical manuscript. My overall enthusiasm for the manuscript was slightly diminished by the fact it was unclear how serious a problem the issue raised (namely that of the conservative coverage of fine-mapped variants) would be in practice. In particular, Figure 1C (rightmost plot) suggested that for signals reaching the de-facto genome-wise significance threshold of around 10-8, there is in fact little discrepancy between the claimed and true coverage.Given this fact, I am slightly unclear which analyses presented in the manuscript support the statements made in the Abstract: “We show that this is because fine-mapping data sets are not randomly selected from amongst all causal variants, but from amongst causal variants with larger effect sizes” and in the Discussion: “We show that this is because fine mapped regions are not a random sample as assumed by fine mapping methods, but the subset of regions with the most extreme P values.”? Surely Figure 1C (rightmost plot) suggests that the regions with the most extreme P values (which will likely correspond to those with the larger effect sizes) are exactly those regions that will NOT be affected by the problem?! As these regions in fact show little discrepancy between the claimed and true coverage?Additional comments:https://onlinelibrary.wiley.com/doi/full/10.1002/gepi.20543 here, who I think also investigated the use of penalised regression for fine mapping ?Page 2, Line 23: reference [7] – it might also be worth referencing Ayers and Cordell (2010) Page 3, lines 70-81 : this issue of simulating from the prior as opposed from a fixed beta requires more detailed discussion/explanation. By “simulating from the prior” do you mean you resample the value of beta within each simulation replicate? I cannot see this as a correct approach  Within any given data set, the effect size at any particular causal variant surely takes a specific value (that may be considered to have been drawn from some distribution before the study started). If one is trying to investigate the properties of some analysis approach under repeated sampling of the dataset, surely one should indeed leave the effect size at the variant fixed at that sampled value?This is a bit different from the idea of simulating the effect sizes of multiple different causal variants across the genome from a distribution , Depending on context one might want to resample the random effects, although I think would argue that for analysis of a specific data set, the effects (one sampled) should be considered fixed. It is also a bit different from a random effects meta-analysis whereby one could consider different studies to have genuinely different effect sizes . Anyway, I think this all needs a bit more clarification and discussion.Page 6, lines 190-191: This finding  is presumably specific to this particular example? i.e. this calculation would need to be repeated for any given GWAS that one was analysing, if one wished to calculate the appropriate threshold? Could be clarified.Page 7 lines 215-225: It would be informative to state what significance levels were achieved in the 40 genomic regions interrogated. Given that your approach is successful in reducing the size of the 95% and 99% credible sets, it suggests there must indeed be some degree of conservative coverage for this data set when using standard approaches. This, combined with the observation made above  suggests that many of the the 40 regions must not in fact have reached genome-wide significance levels?Page 8 lines 261-270: You are correct that methods such as GUESSFM and FINEMAP and JAM produce more complicated outputs in terms of posterior probabilities for various configurations of causal variants. However I am not sure that *that much* post-processing is necessary to investigate these approaches. In particular, FINEMAP versions 1.3 and 1.4 already produces lists of credible SNPs for the best single-SNP configuration, the best two-SNP configuration, the best 3-SNP configuration etc. etc. It would certainly be interesting to investigate whether these lists of credible SNPs are subject to the same issue that you have highlighted for single-variant fine mapping approaches.Page 10 line 326-327: I am not sure that I follow why defining coverage for regions containing > 2 CVs is particularly problematic? Would you not just define it as the frequency with which a credible set contained at least one of the CVs (as you did for regions with 2 CVs)? I could imagine that the more CVs there are in a region, the easier it would be for *any one* of them to “get chosen” into the credible set – I’m not sure what effect this would have on the coverage? However this issue would presumably pertain to the 2 CV situation as well?Page 11 line 368: Rephrase “at the causal SNP’s position which takes the value $\\mu$” as “at the causal SNP’s position where the value of $z_i$ takes the value $\\mu$” ? .**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Computational Biologydata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: YesReviewer #2: Yes**********what does this mean?). If published, this will include your full peer review and any attached files.PLOS authors have the option to publish the peer review history of their article  digital diagnostic tool, Data Requirements:http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001908#s5.Please note that, as a condition of publication, PLOS' data policy requires that you make available all data used to draw the conclusions outlined in your manuscript. Data must be deposited in an appropriate repository, included within the body of the manuscript, or uploaded as supporting information. This includes all numerical values that were used to generate graphs, histograms etc.. For an example in PLOS Biology see here: Reproducibility:http://journals.plos.org/ploscompbiol/s/submission-guidelines#loc-materials-and-methodsTo enhance the reproducibility of your results, PLOS recommends that you deposit laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see  18 Mar 2020Attachmentresponses.docxSubmitted filename: Click here for additional data file. 27 Mar 2020Dear Miss Hutchinson,We are pleased to inform you that your manuscript 'Improving the coverage of credible sets in Bayesian genetic fine-mapping' has been provisionally accepted for publication in PLOS Computational Biology.Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.Should you, your institution's press office or the journal office choose to press release your paper, you will automatically be opted out of early publication. We ask that you notify us now if you or your institution is planning to press release the article. All press must be co-ordinated with PLOS.Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology. Best regards,Ilya IoshikhesAssociate EditorPLOS Computational BiologyJian MaDeputy EditorPLOS Computational Biology***********************************************************Reviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: I'm satisfied with these revisions. Thanks.Reviewer #2: The authors have satisfactorily addressed all my previous comments.**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Computational Biologydata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: NoneReviewer #2: Yes**********what does this mean?). If published, this will include your full peer review and any attached files.PLOS authors have the option to publish the peer review history of their article  1223-442824 | ploscompbiol.org | @PLOSCompBiolPLOS Computational Biology | Carlyle House, Carlyle Road, Cambridge CB4 3DN | United Kingdom


The data also included contemporary group, technician, imaging lab, and a three-generation pedigree for each animal. Variance components for ultrasound carcass measurements were first estimated with univariate animal models for each breed and imaging laboratory using derivative-free restricted maximum likelihood. Then, treating data from each imaging laboratory as separate traits, genetic correlations between laboratories for LMA, percentage of IMF, and subcutaneous fat were estimated with trivariate animal models. The technician explained 12–27%, 5–23%, and 4–26% of the variance for IMF, SFD, and LMA, respectively, across all three breeds. Variance due to technician was often greater than variance due to additive genetic effects but almost always less than that explained by the contemporary group. Within breeds, estimates of additive genetic variance for LMA, SFD, and IMF differed (range divided by mean) among laboratories by 4.5%, 21.5%, and 39.4 % (Angus); 31.6%, 15.0%, and 49.1% (Hereford); and 19.9%, 46.6%, and 55.3% , respectively. Likewise, estimates of residual variance for LMA, SFD, and IMF differed among laboratories by 43.4%, 22.9%, and 43.3% (Angus); 24.9%, 15.2%, and 79.2% (Hereford); and 26.4%, 32.5%, and 46.2% , respectively. Genetic correlations between labs across breeds ranged from 0.79 to 0.95 for IMF, 0.26 to 0.94 for SFD, and 0.78 to 0.98 for LMA. The impact of the observed heterogeneity of variance between labs on genetic evaluation requires further study.Ultrasound technology provides cattle breeders with a quick, noninvasive, and inexpensive way to measure carcass data on live animals. Ultrasound data are used as indicator traits in cattle genetic evaluations for economically relevant carcass traits. Ultrasound cattle genetic evaluations assume homogeneous additive genetic and residual variance. Thus, the objective was to partition phenotypic variance in ultrasound carcass measurements into components for additive genetic effects, technicians, contemporary groups within technicians, and residual and to examine the homogeneity of these variances among image interpretation laboratories. Records of longissimus muscle area (LMA), percentage of intramuscular fat (IMF), and subcutaneous fat depth (SFD), measured using ultrasound, were provided by the American Angus Association ( Because field technicians are certified by the UGC, it is assumed that technician does not affect carcass traits measured by ultrasound and, therefore, the variance due to technician is negligible. However, this assumption has not been rigorously evaluated and very little is known about the contribution of the technician to the variation of ultrasound carcass phenotypes. Similarly, the contribution of imaging lab to the variation of ultrasound carcass phenotypes is also unknown and technicians often report data to the same laboratory. An assumption of the NCE is that additive genetic and residual variances are homogeneous , yet thiThe hypothesis was that the technician and laboratory contribute to the variation in carcass traits measured by ultrasound. To test this hypothesis, the variance was partitioned into technicians, additive genetics, contemporary groups within technicians, and residual effects for LMA, subcutaneous fat depth (SFD), and percentage of IMF as measured by ultrasound. If the technician did not contribute to variation in carcass traits measured with ultrasound, then technician variance should be negligible. Second, the heterogeneity of residual variance among labs interpreting each of the above three traits was tested. Third, ultrasound carcass traits interpreted by each lab were treated as different traits. This allowed genetic correlations between labs for each trait to be estimated. If the lab was not contributing to variation in ultrasound carcass measurements, then these genetic correlations should not differ from 1.n = 281,982), American Hereford Association , and American Simmental Association  for a total of 391,160 records. The carcass ultrasound data received by the breed associations were the interpretations made by imaging laboratories that would be used to calculate EBVs for carcass traits. The data used in this evaluation were exactly as provided by the breed association and without any preadjustment. Contemporary group, technician, imaging laboratory, LMA, SFD, and IMF were provided when available on each animal in the data set  and lab combination from each breed for variance component estimation . For the AHA and ASA data, pedigrees included all sires, dams, grandsires, and granddams. A total of 87,339 animals and 5,008 sires were included in the pedigree for the AHA data. For the ASA data, 79,513 animals and 3,902 sires were included in the pedigree file. Only 157 (mean F = 0.17) and 228 (mean F = 0.028) animals had nonzero inbreeding coefficients in the AHA and ASA pedigrees, respectively. The analyses did not include inbreeding. Unlike the pedigrees for AHA and ASA, the AAA pedigree was formulated using sires and maternal grandsires. This allowed more Angus animals with records to be included in the analysis because the number of maternal grandsires was less than the number of dams. A total of 78,149 animals and 5,007 sires were included in the AAA pedigree file. None of these animals had nonzero inbreeding coefficients. Variance components were estimated fitting the model:Variance components were estimated within lab because technicians often reported ultrasound images to the same laboratory, resulting in a lack of independence between technician and lab. Homogeneity of residuals within technicians, after correction for contemporary group effects, was tested using Bartlett’s test. Contemporary groups were nested within technicians because only one technician scanned each contemporary group. Multiple-trait derivative-free restricted maximum likelihood was utilized for the estimation of (co)variance components and genetic correlations . Convergwhere t is a random effect of the technician; c is a random effect of the contemporary group; a is a random effect of the animal; I is the identity matrix; A is the animal additive numerator relationship matrixwhere , i = 1–3), contemporary group within technicians , and animal  to the data and the random residual effects  estimates of ultrasound carcass traits among labs were calculated as follows.where The technician was not included in the denominator of the above equation because only one technician would collect scans for each contemporary group. The ultrasound technician and contemporary group were not independent of the imaging laboratory; therefore, variance components due to imaging laboratory were unable to be partitioned.The total numbers of records for each trait used were 65,950, 43,380, and 48,300 from AAA, AHA, and ASA, respectively . The numFor all traits analyzed in all breeds and labs, the contemporary group consistently explained the greatest fraction of phenotypic variance –4. The tDifferences in the percentage of variation explained by the technician were observed among imaging laboratories. The technician explained the least amount of phenotypic variation for SFD for lab 2 across all breeds  and IMF n = 9). Tests of homogeneity of variance consistently demonstrated that residuals were not homogeneous across technicians for all traits and breeds (Bartlett’s Test P < 0.0001). This test of homogeneity confirmed heterogeneous residual variance among imaging labs, which was consistent with the estimates of residual variation among labs across all breeds and traits  among laboratories for all breeds by 4.5%, 21.5%, and 39.4% (Angus); 31.6 %, 15.0%, and 49.1% (Hereford); and 19.9%, 46.6%, and 55.3% , respectively. Likewise, the estimates of residual variance for LMA, SFD, and IMF differed among laboratories by 43.4%, 22.9%, and 43.3 % (Angus); 24.9%, 15.2%, and 79.2% (Hereford); and 26.4%, 32.5%, and 46.2% , respectively. The homogeneity of residuals across technicians was tested for each trait and breed combination  genetic correlations between imaging laboratories were ≥0.80, suggesting a lesser contribution of imaging laboratory to variation in the EBVs. Still, 6 of the 27 genetic correlations between laboratories were ≤0.80, with Hereford SFD between labs 2 and 3 having the lowest genetic correlation (0.26). This low correlation may be partly attributed to the different technology utilized by lab 2. Generally, the technician explained a lower percentage of phenotypic variation for SFD and IMF interpreted by lab 2. Furthermore, genetic correlations between labs were generally highest between labs 1 and 3. Although the genetic correlations were high, in many instances, these correlations were different from 1. Taken together, some evidence exists that the imaging laboratory contributes to ultrasound carcass variation, although the contribution of the lab was less than the contribution of the variance of the technician. The cause of Hereford genetic correlations among imaging laboratories being substantially less than anticipated and than those observed for the other breeds cannot be definitively determined from these data. The technician contributed more variance for SFD from lab 3 than the other two labs within data submitted to the AHA, which is in contrast to results for IMF and LMA, where technicians reporting to lab 3 did not contribute more variation than the other two labs. Furthermore, heritability estimates for SFD were consistently lower in Herefords than the other breeds. Based on the estimates of heritability, the imaging laboratories would appear to be internally consistent in their measurement technique for SFD. However, Hereford hides have long been known to be heavier than the hides from other breeds , and theBos taurus cattle breeds found heritability estimates ranging from 0.29 to 0.42 for LMA, 0.18 to 0.51 for SFD, and 0.20 to 0.33 for IMF (Heritability estimates were moderate to high, which is expected for carcass traits. Previous literature describing carcass ultrasound traits of  for IMF . These v for IMF –4, the aWhile these results found variation among ultrasound technicians and laboratories, the underlying causes of this variability cannot be identified. Technician and laboratory variance can stem from a number of factors. Ultrasound machines, transducers, and scanning techniques have been shown to contribute to technician variance . UltrasoUltrasound technician and laboratory are potential sources of variation when estimating carcass merit by ultrasound. This study demonstrates that the technician is an important source of variation in carcass phenotypes measured by ultrasound. Technician variance often equaled or exceeded additive genetic variation, showing that improvements in the technician certification process may be needed. Variance among ultrasound technicians should be addressed by future work into the causes of variation among technicians. Reducing this variance will help improve the accuracy of ultrasound carcass measurements and genetic predictions. Genetic correlations between laboratories were generally high, which suggests that they play a lesser role in the contribution of variance to ultrasound measurements than the technician. However, these correlations, especially for SFD in Herefords, were often statistically different from 1, which is the assumption in the beef cattle genetic evaluations. Residual variation was also not homogeneous among laboratories. Further investigation into the consistency of lab performance when interpreting ultrasound carcass images is warranted. The results of this research indicate potential areas for improvement of U.S. beef cattle evaluations using carcass ultrasound measurements, specifically in the certification process for ultrasound technicians and image processing laboratories. Breed associations can use these results to guide the collection of carcass data using ultrasound, resulting in improved genetic evaluations for carcass traits. Until the technician and imaging lab variances can be made more homogeneous, breed associations may consider fitting a heterogeneous variance model to account for differences among technicians and laboratories. Imaging lab contributed to a lesser degree to breeding value estimation and, with the exception of SFD in Hereford cattle, little change in sire rankings based on EBVs was observed when different labs interpreted the ultrasound images for the progeny of the same sires.