Increasingly large Genome-Wide Association Studies (GWAS) have yielded numerous variants associated with many complex traits, motivating the development of “fine mapping” methods to identify which of the associated variants are causal. Additionally, GWAS of the same trait for different populations are increasingly available, raising the possibility of refining fine mapping results further by leveraging different linkage disequilibrium (LD) structures across studies. Here, we introduce multiple study causal variants identification in associated regions (MsCAVIAR), a method that extends the popular CAVIAR fine mapping framework to a multiple study setting using a random effects model. MsCAVIAR only requires summary statistics and LD as input, accounts for uncertainty in association statistics using a multivariate normal model, allows for multiple causal variants at a locus, and explicitly models the possibility of different SNP effect sizes in different populations. We demonstrate the efficacy of MsCAVIAR in both a simulation study and a trans-ethnic, trans-biobank fine mapping analysis of High Density Lipoprotein (HDL). Genome-Wide Association Studies (GWAS) have successfully identified numerous genetic variants associated with a variety of complex traits in humans. However, most variants that are associated with traits do not actually cause those traits, but rather are correlated with the truly causal variants through Linkage Disequilibrium (LD). This problem is addressed by so-called “fine mapping” methods, which attempt to prioritize putative causal variants for functional follow-up studies. In this work, we propose a new method, MsCAVIAR, which improves fine mapping performance by leveraging data from multiple studies, such as GWAS of the same trait using individuals with different ethnic backgrounds (“trans-ethnic fine mapping”), while taking into account the possibility that causal variants may affect the trait more or less strongly in different studies. We show in simulations that our method reduces the number of variants needed for functional follow-up testing versus other methods, and we also demonstrate the efficacy of MsCAVIAR in a trans-ethnic, trans-biobank fine mapping analysis of High Density Lipoprotein (HDL). Genome-Wide Association Studies (GWAS) have successfully identified numerous genetic variants associated with a variety of complex traits in humans –3. HowevThe classic approach to fine mapping involves simply selecting a given number of SNPs with the strongest association statistics for follow-up, but this performs sub-optimally because it does not account for LD structure . BayesiaThere is growing interest in improving fine-mapping by leveraging information from multiple studies. One of the most important examples of this is trans-ethnic fine mapping, which can significantly improve fine mapping power and resolution by leveraging the distinct LD structures in each population –14, as sIn this paper, we present MsCAVIAR, a novel method that addresses these challenges. We retain the Bayesian MVN framework of CAVIAR while introducing a novel approach to explicitly account for the heterogeneity of effect sizes between studies using a Random Effects (RE) model. Our method requires only summary statistics and LD matrices as input, allows for multiple causal variants at a locus, and models uncertainty in association statistics and between-study heterogeneity. The output is a set of SNPs that, with a user-set confidence threshold (e.g. 95%), contains all causal SNPs at the locus.https://github.com/nlapier2/MsCAVIAR.We show in simulation studies that MsCAVIAR outperforms existing trans-ethnic fine mapping methods  and exteρ, contains all causal SNPs, and ideally contains far fewer SNPs than the set of significant SNPs obtained via meta-analysis  and linkage disequilibrium (LD) matrix for SNPs at the same locus in each study . The LD analysis .C. For each configuration C compatible with the causal set, we compute its (posterior) probability in a Bayesian manner: the probability of a configuration of SNPs being causal given the association statistics can be computed by modeling a prior probability for that configuration and a likelihood function for the association statistics given the assumed causal SNPs given by C , β is the vector of causal variant effect sizes, and e is a vector of environmental noise terms where each βi for different causal variants allows for the modeling of heterogeneity.We then simulated causal SNPs and their effect sizes sed GCTA  to simulFinally, we run a linear regression using fastGWA  to generρ* = 0.95, so methods with 95% or higher sensitivity were considered “well-calibrated”  using summary statistics from the UK Biobank (UKB) , 24 and −8), discarding all SNPs that did not reach even marginal significance (p > 0.05), as they were highly unlikely to be informative and would slow down analyses. We also excluded all loci with fewer than ten SNPs in each study after filtering SNPs with p > 0.05, as fine mapping may not be seen as necessary or may even be trivial for existing methods when there are only a few strongly associated SNPs. Two very large loci were excluded for computational reasons. We excluded loci from chromosome six, where there were numerous statistically significant SNP effect sizes due to the presence of human leukocyte antigen (HLA) regions.To generate loci for fine mapping, we centered 1 megabase windows around genome wide-significant peak SNPs  matrices were generated from the 1000 Genomes project , with “Eτ2 , CAVIAR’s sets using the UKB data totaled 11,538 SNPs , and CAVIAR’s sets using the BBJ data totaled 18,520 SNPs . Thus, similarly to our simulation study’s findings, MsCAVIAR and PAINTOR generally returned smaller causal set sizes than CAVIAR, and MsCAVIAR’s median causal set size was slightly smaller than PAINTOR’s. In contrast with the simulation study, MsCAVIAR’s average causal set size was slightly larger than that of PAINTOR’s. A full list of the loci we identified and the causal set sizes returned can be found in Table A in As an additional way of viewing the results, we generated scatter plots of the causal set sizes at each locus for MsCAVIAR compared to those of PAINTOR and CAVIAR . This vihttps://github.com/nlapier2/MsCAVIAR.In this work, we introduced MsCAVIAR, a method for identifying causal variants in associated regions while leveraging information from multiple studies. Our approach requires only summary statistics as opposed to genotype data and handles heterogeneity of effect sizes, differing sample sizes, and different LD structures between studies, making trans-ethnic fine mapping an ideal application. We demonstrated that our method is well-calibrated and improves fine-mapping resolution in simulation studies. MsCAVIAR is available as free and open source software at We make several important assumptions in this model, which may not always be true. It has been shown that many causal SNPs are shared across populations , 17, 18.τ2. This is unlikely to be true if multiple studies are from a single population while another study is from a different population. In such a scenario, we recommend grouping the studies by population, running fixed effects meta-analysis on each group, and then running MsCAVIAR on the results for the different groups. Concretely, the input summary statistics for MsCAVIAR should be the results from the meta-analysis of each population, and the input LD matrices should be derived from either the genotype data (if available) or the appropriate reference panels for each population. However, it is still possible that even ostensibly different populations may be more similar to each other at certain loci than other populations. Therefore, we plan to extend our method to handle this case in future work.We also assume that all studies are drawn with equal heterogeneity τ2 parameter to a fixed value, which was chosen to give power to detect both small and large amounts of heterogeneity  so that not every variant has the same prior probability of being causal, as described in the CAVIAR paper  according tom is non-causal and We now model the observed summary statistics C can be defined as:C = {0, 1}M is an M × 1 binary vector indicating whether each variant is causal, andϵ is a small constant to ensure that the matrix ΣC is full rank. (We later relax the need for ΣC to be full rank in “Handling Low Rank LD Matrices”). Here, and below, we use the shorthand σ2 to represent the variance of the C are zero because the effect sizes of causal variants are independent of one another.The distribution of ΛC to refer to the non-centrality parameters (NCPs) of the statistics of all SNPs, which are induced by Linkage Disequilibrium (LD) with the causal SNPs. Thus, C and LD structure is symmetric (Σ = ΣT), we have the following distribution for Λ|C:We use the shorthand Λ = ΣΛγ as the probability of a variant being causal, which makes the causal status for the mth variant a Bernoulli random variable with the following probability mass function: γ of having a causal effect, the prior can then be written as follows:f is the probability density function shown in We will now define We determine which variants are causal by calculating the posterior probability of each configuration C* given S, we need to integrate over all possible values for the non-centrality parameters of the causal variants in Λ in order to get the values of Λ that makes observing S most probable.For us to calculate the posterior probability of S is thenThe integral above is intractable in the absence of parametric assumptions about the data. Fortunately, a closed-form solution is available due to the fact that, when a conjugate prior is multivariate normally distributed, its predictive distribution is also multivariate normal. As shown above, S with this distribution is still computationally expensive. Consider the multivariate normal probability density function, assuming the variable Z below is MVN distributed with mean μ and covariance matrix Σ:However, computing the likelihood of S, the covariance matrix is Σ + ΣΣC Σ, which has dimension (M × M), where M is the number of SNPs in each study. Taking the determinant or inverse of this covariance matrix, as required by the above likelihood function, would take O(M3) time. Here, we demonstrate how to compute this likelihood efficiently, leveraging insights from several studies that have explored this topic We are sorry that we cannot be more positive about your manuscript at this stage. Please do not hesitate to contact us if you have any concerns or questions.Yours sincerely,Eleftheria ZegginiAssociate EditorPLOS GeneticsDavid BaldingSection Editor: MethodsPLOS GeneticsReviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: Summary: Minority population GWAS and trans-ethnic finemapping have increased in popularity due to the diversity in LD patterns across populations and how this diversity can be exploited to improve fine-mapping resolution. To this end, the authors present the statistical finemapping method “MsCAVIAR”, which is an extension of the previously developed Bayesian fine-mapping software CAVIAR. The primary advantage and novelty of the proposed approach is that it simultaneously addresses both effect heterogeneity and multiple causal variants at a given locus of interest for trans-ethnic finemapping. The authors’ perform a number of simulation studies as well as real data application to demonstrate the performance of their approach against leading methods that accommodate multiple causal variants. The method performs as well or better than its primary competitor (PAINTOR) in many regards. Overall, the paper is clear and well-written. The authors’ carefully outline their statistical methodology, addressing a number of computational hurdles. They address a number of limitations in their assumptions and highlight the strengths/weaknesses of their method. The authors also make code publicly available on Github, which appears to be complete and well-documented. I do have some questions regarding computational efficiency, and there are some issues with respect to consistency of notation – although most of my comments are minor.As the authors’ note, there are a few limitations on their assumptions regarding tau and how it is fixed a priori in application. This corresponds both across loci and across studies, where imbalance in contributing studies vis a vis ancestral population may be a concern (as published GWAS are in predominantly European populations). Regarding the latter, one advantage of MR-MEGA in trans-ethnic finemapping is how it leverages meta-regression to account for distribution of genetic ancestry across the contributing individual studies, thus decomposing the effect heterogeneity into ancestral and random components. To this end, the authors suggest selecting one study from each population for inclusion. However, this naturally reduces finemapping efficiency depending on the distribution of study sizes. Would an alternative strategy using MsCAVIAR be to initially conduct fixed-effects meta-analyses within homogenous populations, where warranted, and then combine diverse population results for trans-ethnic finemapping?Speaking of which – the authors initially mention MR-MEGA in the introduction but do not discuss it any further or include it in their comparisons. Given that the MR-MEGA paper demonstrates improved finemapping over PAINTOR in their paper, I assume the rationale for its exclusion for comparative performance analyses is due to its limiting assumption regarding number of underlying causal variants?How would the authors additionally integrate functional annotation into their fine-mapping method, similar to PAINTOR?It may be useful to quickly mention in “Parameter setting in practice” the relevance of 5.2 as a Z-score in relation to the traditional genome-wide significance criterion . Similarly - I’m not finding any clear justification for MsCAVIAR’s default value of tau2 = 0.52 – I assume the connection lies with just taking forcing a mean/variance relationship in effect heterogeneity for Z-scores at the genome-wide significance threshold?Equation 10. It’s kind of confusing to use T to denote both the number of studies as well as the transpose operator. Similarly, later in the methods n is used to denote the number of studies, but also the number of subjects within a study. And then again it seems that ‘m’ is used to index study in “Extending MsCAVIAR to different sample sizes”. Some care should be used in maintaining consistent notation.It’s not immediately clear how the computational cost scales with respect to number of included studies. Given the single study setting is O(k^3 ) plus some O(mk^2) – do we replace k with (k*n) to get the computational costs, where n = # of studies? Thus, is # of studies a similarly strong limiting factor in computational burden as k, which would potentially motivate the population-wise study aggregation study mentioned above?Reviewer #2: LaPierre and colleagues present a novel approach for fine-mapping loci using summary statistics from multiple studies whilst accounting for heterogeneity in the effects of causal variants between them. The methodology can allow for multiple causal variants at a locus, and can be applied in the context of trans-ethnic fine-mapping by allowing for study-specific patterns of LD between variants. The methodology tackles an important challenge in human genetics, is extremely timely, and likely to be of great interest to the readership of PLoS Genetics. I am looking forward to trying the software! The manuscript is generally well written, although some additional details of the simulation study and the applications to trans-ethnic GWAS of type 2 diabetes and cholesterol would be useful.Comments on methodology1. Presumably it would be straightforward to incorporate a non-uniform prior of causality? Given the availability of enrichment in associations in specific annotations, it would be really useful to allow this flexibility in the method/software.2. In the context of trans-ethnic meta-analysis, could the authors provide some guidance as to whether each study should be included separately in the meta-analysis, or whether a fixed-effects meta-analysis of each ethnic group should undertaken first, and then used as input to msCAVIAR? Presumably this would be computationally more demanding, but are there advantages in allowing for heterogeneity between studies from the same ancestry?3. Heterogeneity is modelled under a random-effects model, but would it be feasible  to consider alternative models ? Could the tau2 parameter be considered as a hyperparameter to be estimated, and could this give some intuition as to the extent of heterogeneity?4. It wasn’t totally clear to me whether the (maximum) number of causal variants needs to be specified in advance.5. Some details of computational efficiency would be beneficial. In simulations and data applications, SNPs are thinned by various criteria. Is this because the methodology/software does not work well if there are SNPs in strong LD, or is computationally demanding if the number of SNPs is large?6. One of the nice features about Susie is that it effectively gives a credible set for each causal variant . If we were then keen to colocalise association signals with eQTLs, this could be done for each credible set separately. However, for msCAVIAR, it does not seem that we could extract equivalent information. For example, if there were two signals at a locus , the first signal might be easy to fine-map, so that we are clear of the causal variant for that signal, but for the other signal, there might be several variants with equivalent fine-mapping support. Is there anyway to distinguish the fact that the variants in the credible set are somehow grouped by distinct associations, and if not, do the authors view this as a disadvantage?Comments on the simulation study1. Details of the simulation study are rather scant. How large are the two regions ? I can understand removing SNPs in perfect LD to select causal variants , but then this only leaves 48/38 SNPs, which does not seem like a “realistic” fine-mapping scenario. I also was not clear if the SNPs included in the analysis were also LD pruned, or if all SNPs in the region were considered as potentially causal. Could there every be a situation where a causal variant was specific to just one population (i.e. monomorphic in the other) – or did causal variants have to be present in both populations (at some frequency)?2. What is Sigma_i on line 118? It might be described later, but it is not clear what it is at this point in the manuscript.3. The authors suggest that the improved performance of msCAVIAR over PAINTOR could be because of the modelling of heterogeneity. Could the authors investigate this further by simulating effects of causal SNPs that are homogenous across studies?4. Line 151. The authors state that the fact that msCAVIAR performs better than CAVIAR applied to each population is an indication of the improved fine-mapping resolution offered by trans-ethnic meta-analysis. However, could this actually be a reflection of the larger total ample size used by msCAVIAR? Could the authors also run simulations where the sample size of the population-specific studies are the same as a trans-ethnic study ?Comments on data applications1. Effective sample size is a more useful way of representing the sample size for a disease phenotype – would actually better to give the number of diabetes cases and controls for each study.2. It would be good to give information on the loci used in comparisons in supplementary information (for both applications), together with the numbers of credible causal variants for each locus with the different methods.3. Centering the loci 50kb up- and down-stream of the lead SNP seems rather restrictive – we know that LD often extends over greater distances, and I think using 500kb up- and down-stream would be much more realistic. Was this done for computational reasons?4. I didn’t follow the motivation for removing SNPs with p>0.0001. There could be examples where a causal variant does not have strong association in a single SNP analysis, and is only revealed when considering multiple causal variants . Was this done for computational reasons?5. Figures 4 and 5. I did not find the violin plots useful in Figure 4. I understand that it is hard to summarise results when there are just five loci in Figure 4. It would be useful to have three scatter plots where the x-axis was the credible set size in msCAVIAR and the y-axis was the credible set size in one of each of the three other methods (i.e. each point is a locus) – this will provide useful information on within locus comparisons that could not be assessed with the current presentation. The box and whisker plots in Figure 5 are more useful, but I think could also be presented alongside scatter plots as described above.6. I wasn’t totally clear about the final sentence of the last paragraph (line 259). In particular, I wasn’t clear about why it mattered that msCAVIAR models heterogeneity as being the same at each locus. As far as I am aware, PAINTOR also models heterogeneity as being the same at each locus . So do these results imply that msCAVIAR does not perform well if effects are homogenous across studies? I think this emphasizes the importance of running some simulations under a model in which effects are the same in the two studies.Comments on the discussion1. I think it would be beneficial to expand somewhat on the comment about “equal heterogeneity” – presumably this is just the underlying assumption of a random effects model? I agree that this model is not appropriate when several studies are from one population, and one study is from another (because less heterogeneity would be expected between studies from the same population). However, I disagree with the recommendation to use a single study from each population. It would be much better to meta-analyse studies from the same population/ethnicity together, and use those as input to msCAVIAR .Reviewer #3: Review of LaPierre et alThe paper presents an extension of the fine-mappingmethod (CAVIAR) to deal with multiple studies, allowingfor heterogeneity in effects among studies.The paper uses simulation to show that this extension(MsCAVIAR) produces better localization, in thatit reduces the size of the "causal set" produced compared with CAVIAR(and other methods) run on the individual studies.Results on real data show similar trends This is a potentially useful -- if conceptually fairly straightforward --extension of the CAVIAR method. However, there are several importantissues that would need addressing to make it suitable for publication.Main Issues1. While the main text is very clearly written and nicely presented,the Methods section is confusing and difficult to follow.I suggest the following:First, there needs to be a very simple and clear statement of the model andprior distribution used. This should be separated from any "derivation" of thismodel (which can likely be mostly justified by appropriate citations of previous work)and also separated from the computational tricks .At the moment the model is very hard to extract from the text.Lambda is used in different places in different ways: at thetop of p11 lambda_i = beta_i sqrt(n_i)/sigma_e , but thenlater in the same paragraph it is used for the mean of S, which isnot the same thing. Maybe because of this there appear to becircular definitions (Lambda_C is defined in terms of Lambda at(2), and then Lambda is defined as \\Sigma Lambda_C at the topof p15). The extension to different sample sizes is describedimprecisely in words (top of p23), and needs equations to make it precise.I would suggest just giving the model for different sample sizesdirectly, since the case where they are the same are then a special case.The model seems to be a "matrix normal" model, and making that explicit could help.Second, the definitions of the summary data S need to be madeclear. At the moment they are defined as beta-hat_i \\sqrt(n_i)/sigma_ebut sigma_e is unknown. And at line 444 you say "we now operate... thatsigma_e has been standardized (sigma_e=1)". But there is noway to standardize to ensure sigma_e=1 because we do not know the true residual variance.It is common to standardize $y$ to have unit variance, but this does not imply the residualvariance is 1.  ? Or using an estimateof the residual variance? In any cae these kinds of details, assumptionsand approximations need to be more precise.)2. The simulation study is rather too favorable to the method, and shouldbe made more realistic. In particular i) the simulation is performed underthe assumed summary data model, rather than under a more realisticfull data model (ie the regression model (1)); ii) the simulationis performed with "effect sizes" with a narrow range centered on 5.2, which not onlyseems also to be used in the prior, but also seems unrealistic - it will seldomproduce either small difficult-to-detect effects or very large effects,both of which are likely to occur frequently in practice;iii) the simulation is done assuming the same LD structure inboth the study and the inference, whereas the real data analysisuses a panel to approximate the study LD.It would seem easy to generate more realistic simulated databy simulating outcomes Y from the full data model (1), using real genotypedata (X) an a range of beta values for some sigma^2_g) so that both small and big effects occur.3. As argued in Wang et al, the idea of outputting a "causalset" that, with high probability, contains *all* causal SNPs isflawed. There are two reasons for this. First, it can ignore a lot of useful information.For example, suppose there are two causal SNPs, and that one is in LD withjust itself, but the other is in LD with 50 others. Then the causal set willcontain (at least) 52 SNPs, and does not include the information that one SNP is veryprecisely mapped (which a user could clearly find helpful!)Second, if we allow that SNPs may have small effects, which is realistic,then it becomes impossible for any method to be confident to include all the causal SNPs in a set. The paper sidesteps thisissue by avoiding simulations with small effects, which shouldbe rectified (see 2 above).In light of this it seems unsatisfactory to rely on the size of the causal setas the only indicator of improved performance, and I think the paper shouldalso provide other evidence for the superiority of the multi-study approach.One possibility would be to demonstrate the benefits in terms of PIPs (posteriorinclusion probabilities). For example, does MsCAVIAR show a betterprecision-recall curve as PIP threshold is modified?4. The filtering in the real data analysis seems very ad hoc. and it is not clearwhy it is done or whether it is necessary. Is it necessary to make the method'sperformance look good compared with other methods? If so, this seems worrying.If not, why not present results on much less filtered data? .To comment in more detail on the filters: i) Discarding SNPs with marginalp values >0.0001 could miss signals as SNPs can become more significantonce one controls for other SNPs in LD. ii) The logic that if the peak SNP is genome-widesignificant in one population and >0.0001 in another then the second populationwon't help with localization isn't clear: first,the potentially different patterns of LD in the twopopulations mean that a second population could still help with localization evenwithout a genome-wide significant association; second maybe there are secondary SNPsthat will only show up as significant when one analyzes both populations.iii) You say "fine mapping is not as useful when there are few strongly associated SNPs".Why? I would think these loci may give the potential to fine map quite precisely!Surely the problem cases are whether there are many SNPs in strong LD, all strongly associated,which makes fine mapping difficult?iv) "As a final step, we pruned groups of SNPs that were perfectly correlated witheach other in both studies... would cause the LD matrix to be low rank". Doesthis mean you pruned if they were perfectly correlated in the *study* samplesor in the LD panel? It could make sense to pool together SNPs that are perfectlycorrelated in the study, but if they are perfectly correlated in the panelbut not in the study then it seems you would want to keep them both.Other issues- the caption to Figure 2 should include some explanation of the fact that thecalibration of SuSie can't be compared to the other methods because its setshave a different goal.- at line 110-111, I understand you pruned SNPs in perfect LD to reduce computationand possibly to reduce low-rank issues for MsCAVIAR. However, while pruning may initiallyseem innocuous, it raises several concerns. For example, a group of 10 SNPs in complete LD shouldhave approximately 10 times the probability that at least one of themis causal compared with a single SNP that is in LD only with itself.Most analysis methods would take that into account if the SNPs were simplyin "very high LD", but this is hard to do ifyou have pooled/pruned the SNPs. And the pruning will understatethe size of typical causal sets (and so overstate performance) for all methods.Also posterior quantities of interest (eg posterior inclusion probabilities)may be difficult to correct for this pooling. The bottom line: ifpooling is just a way to reduce computation, can you showthat results of analyzing data with pooling are similar to resultswithout pooling?- line 131-3 suggest that Fig 2 boxplots are for only a subset ofthe simulations .That seems dangerous, and I do not see whynot to include all simulations in the boxplot.- line 148-9; they are only equivalent under the *assumption* that there is 1 causal SNP,and not when methods are applied to data where the truth is only 1 causal SNP but theydo not assume only 1 causal SNP.- line 218: Do the reported set sizes include all SNPs that were prunedfor being in complete LD withselected SNPS? It seems that they should in order to give an accurate impressionof the effectiveness of fine mapping in practice.- In the real data, how many causal SNPs do you estimate/identify at each locus?- l273: this advice seems useless because how would one know? How much worse is MsCAVIARthan CAVIAR if effects are unique to one population? One might hope that itwould be robust to this because of the heterogeneity in the model, andthis robustness seems worth assessing in simulations.- Please provide a stable link to the code used to perform the simulations and data analysisDetails:- l99: "dividing by the sum of posterior probabilities of all configurations": isn't this necessarily 1?- l101: "continue increasing the size of the causal set" how is this done?- l107-8: do you mean 20%/80% of SNP *pairs*?- l114: don't use effect sizes and non-centrality parameter interchangeably as they are different.- l115: casual -> causal- l153 effect size -> non-centrality parameter- l190: Should "White European" be "White British"?- In the methods section using (lower-case) sigma for a variance (eg in equation (4)and subsequently) is confusing. Use sigma^2 for a variance. Related to this, line 294 I think should besigma_e^2, and line 448 \\sigma_g should be squared.- l338 "the integral above is intractable..." but then you say closed form is available!- l339,341 posterior predictive -> predictive- l352: " rows ... are zero" isn't this only true if you set epsilon=0?It would be helpful to be more consistent throughout about treatmentof epsilon and what value it takes.- in equation before l438 sqrt(n_m) should be n_m?- line 459, say where 5.2 comes from**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Geneticsdata 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: YesNo: I don't think the specific details of the data in Figures 4 and 5 have been presented - I've made some suggestions in my comments about providing additional supplemental information to deal with this.Reviewer #2: No: Reviewer #3: **********what does this mean?). 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You will be contacted if needed following the screening process.PLOS has incorporated To resubmit, you will need to go to the link below and 'Revise Submission' in the 'Submissions Needing Revision' folder.[LINK]Please let us know if you have any questions while making these revisions.Yours sincerely,Eleftheria ZegginiAssociate EditorPLOS GeneticsDavid BaldingSection Editor: MethodsPLOS GeneticsReviewer's Responses to QuestionsComments to the Authors:Reviewer #1: The authors have largely addressed all of my concerns. I have no further comment.Reviewer #2: The authors have made extensive revisions to the manuscript and have dealt with all comments. I look forward to trying out the software!Reviewer #3: Reveiew of LaPierre et al (revised)The authors have been very responsive to the previous reviews, and the manuscript is much improved.I have just one major remaining issue, and a few remaining minor comments I would ask the authors to address.Main issue:1. Although the new simulation study is much improved, it still shows results only for "in sample" LD matrices.As the authors note, when using summary data it is very common to use "out of sample" LD matrices (eg LD matrices from a similar population reference panel like 1000 Genomes), and indeed the real dataanalysis used that out-of-sample strategy. It is important for readers and users to understand whether this impacts accuracy, and so the simulation study should show results for both in-sample and out-of-sample LD matrices.Other issues:2. The MSCaviar model is essentially specified bymthe equation after l296, S|C \\sim N. I suggest to number that equation, since it is key. Further, please add at this point the precise descriptions of \\Sigma and \\Sigma_C. \\Sigma is a block-diagonal matrix  with each block containing the LD matrix from a study. And \\Sigma_C is given in the equation after l429, which you couldgive here. It might seem a minor point, but it took me quite a while to collect those pieces all together to convince myself the model had been fully specified. Putting them all in the same place will help the reader see the model immediately, and numbering the key equations would help future writers refer to them if necessary.3. I believe it would be further useful to state here that this model comes from assuming S_q | \\Lambda_q \\sim N independently for each q where S_q are the summary data in study q,Sigma_q is the LD matrix in study q and Lambda_q is the NCP in study q. . This is combined with a prior on Lambdavec(\\Lambda) \\sim N where vec(\\Lambda) is the KM vector of concatenated NCPs from the K different studies to give the final model.4. In equation (10) I would assume S_q denotes the data for study q. But then this equation seems irrelevant as you want p(C* | S) [ie posterior given all the data] and not p(C* | S_q).********** Have all data underlying the figures and results presented in the manuscript been provided?PLOS Geneticsdata 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: YesReviewer #3: None********** what does this mean?). 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PLOS Genetics may also choose to issue a press release for your article. If there's anything the journal should know or you'd like more information, please get in touch via  3 Sep 2021PGENETICS-D-20-01186R2 Identifying Causal Variants by Fine Mapping Across Multiple Studies Dear Dr Eskin, We are pleased to inform you that your manuscript entitled "Identifying Causal Variants by Fine Mapping Across Multiple Studies" has been formally accepted for publication in PLOS Genetics! Your manuscript is now with our production department and you will be notified of the publication date in due course.The corresponding author will soon be receiving a typeset proof for review, to ensure errors have not been introduced during production. Please review the PDF proof of your manuscript carefully, as this is the last chance to correct any errors. 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Pisum sativum L.) accessions—genotyped with 30,600 single nucleotide polymorphic (SNP) markers and phenotyped for seed yield and yield-related components—for enhancing selection of accessions from the USDA Pea Germplasm Collection. Genomic prediction models and several factors affecting predictive ability were evaluated in a series of cross-validation schemes across complex traits. Different genomic prediction models gave similar results, with predictive ability across traits ranging from 0.23 to 0.60, with no model working best across all traits. Increasing the training population size improved the predictive ability of most traits, including seed yield. Predictive abilities increased and reached a plateau with increasing number of markers presumably due to extensive linkage disequilibrium in the pea genome. Accounting for population structure effects did not significantly boost predictive ability, but we observed a slight improvement in seed yield. By applying the best genomic prediction model , we then examined the distribution of genotyped but nonphenotyped accessions and the reliability of genomic estimated breeding values (GEBV). The distribution of GEBV suggested that none of the nonphenotyped accessions were expected to perform outside the range of the phenotyped accessions. Desirable breeding values with higher reliability can be used to identify and screen favorable germplasm accessions. Expanding the training set and incorporating additional orthogonal information  into the genomic prediction framework can enhance prediction accuracy.Phenotypic evaluation and efficient utilization of germplasm collections can be time-intensive, laborious, and expensive. However, with the plummeting costs of next-generation sequencing and the addition of genomic selection to the plant breeder’s toolbox, we now can more efficiently tap the genetic diversity within large germplasm collections. In this study, we applied and evaluated genomic prediction’s potential to a set of 482 pea ( Pisum sativum L.) is a vitally important pulse crop that provides protein (15.8–32.1%), vitamins, minerals, and fibers. Pea consumption has cardiovascular benefits as it is rich in potassium, folate, and digestible fibers, which promote gut health and prevent certain cancers (http://www.choosemyplate.gov/). In 2019, more than 446,000 hectares of edible dry pea were planted with production totaling 1,013,600 tonnes in the USA, making it the fourth largest legume crop (http://www.fao.org) (https://www.nasdaq.com/articles/heres-why-nows-thetime-to-buy-beyond-meat-stock-2019-12-05). Another product made from pea, Ripptein, is a non-dairy milk product of pea protein that is gaining tremendous interest as an alternative dairy product (https://www.ripplefoods.com/ripptein/). Additionally, peas are gaining attention in the pet food market as it is grain-free and an excellent source of essential amino acids required by cats and dogs  . Growingfao.org) . RecentlPisum collection, held at the Western Regional Plant Introduction Station at Washington State University, is a good starting point to investigate functional genetic variation useful for applied breeding efforts. To date, this collection consists of 6,192 accessions plus a Pea Genetic Stocks collection of 712 accessions. From this collection, the USDA core collection, comprised of 504 accessions, was assembled to represent ∼18% of all USDA pea accessions at the time of construction  takes advantage of high-density genomic data that holds a promise to increase the rate of genetic gain . As genoTo date, methods to sample and utilize an extensive genetic resource like germplasm collections remain a challenge. In this study, a genomic prediction approach targeting complex traits, including seed yield and phenology, was evaluated to exploit diversity contained in the USDA Pea Germplasm Collection. No research has been conducted before on genomic prediction for the genetic exploration of the USDA Pea Germplasm Collection. Different cross-validation schemes were used to answer essential questions surrounding the efficient implementation of genomic prediction and selection, including determining best prediction models, optimum population size and number of markers, and impact of accounting population structure into genomic prediction framework. We then examined the distribution of all nonphenotyped accessions using SNP information in the collection by applying genomic prediction models and estimated reliability criteria of genomic estimated breeding values for the assessed traits.A total of 482 USDA germplasm accessions were used in this study, including the Pea Single Plant Plus Collection (Pea PSP) comprised of 292 accessions . The USDApeKI (http://www.bioinformatics.babraham.ac.uk/projects/fastqc/). Sequencing adapter remnants were clipped from all raw reads. Reads with final length <50 bases were discarded. The high-quality reads were aligned to the reference genome of Pisum sativum . Missing data were imputed using a k-nearest neighbor genotype imputation method . Genomic libraries for the Single Plant Plus Collection were prepped at the University of Minnesota Genomics Center (UMGC) using genotyping-by-sequencing (GBS). Four hundred eighty-two (482) dual-indexed GBS libraries were created using restriction enzyme ApeKI . A NovaSn method  implemenn method . SNP dat−1. All seeds were treated with fungicides; mefenoxam (13.3 ml a.i. 45 kg-1), fludioxonil (2.4 ml a.i. 45 kg -1), and thiabendazole (82.9 ml a.i.45 kg -1), insecticide; thiamethoxam (14.3 ml a.i. 45 kg -1), and sodium molybdate (16 g 45 kg -1) prior to planting. Thirty seeds were planted per plot; each plot was 152 cm long, having double rows with 30 cm center spacing. The dimensions of each plot were 152 × 60 cm. Standard fertilization and cultural practices were used.Pea germplasm collections (Pea PSP) were planted following augmented design with standard checks  at the USDA Central Ferry Farm in 2016, 2017, and 2018 . The central Ferry farm is located at Central Ferry, WA at 46°39′5.1″N; 117°45′45.4″ W, and elevation of 198 m. The Central Ferry farm has a Chard silt loam soil  and was irrigated with subsurface drip irrigation at 10 min d−1) is the plot weight converted to seed yield in kg per hectare.The following traits were recorded and are presented in this manuscript. Days to first flowering are the number of days from planting to when 10% of the plot’s plants start flowering. The number of seeds per pod is the number of seeds in each pod. Plant height (cm) is defined as when all plants in a plot obtained full maturity and were measured in centimeters from the collar region at soil level to the plants’ top. Pods per plant is the number of recorded pods per plant. Days to maturity referred to physiological maturity when plots were hand-harvested, mechanically threshed, cleaned with a blower, and weighed. Plot weight (gm) is the weight of each plot in grams after each harvest. Seed yield , j is number of years), A mixed linear model was used to extract best linear unbiased predictors (BLUPs) for all traits evaluated using the following model:H2) on an entry-mean basis for each assessed trait was calculated to evaluate the quality of trait measurements following the equation . We also calculated heritability proposed by g refers to the genotypic variance, md is the mean values from the diagonal of the relationship matrix, which is an identity matrix.For the purpose of estimating heritability, we fit the same model above. The heritability in broad sense , partial least squares regression model (PLSR), random forest (RF), BayesCpi, and Reproducing Kernel Hilbert Space (RKHS).u ∼ Z is the genotype matrix e.g.,  =  for biallelic single nucleotide polymorphisms (SNPs) that relates to phenotype y  used to predict the worth of entries in the context of mixed models . The RR-notype y . The RR-notype y .Partial least square regression is a reduction dimension technique that aims to find independent latent components that maximize the covariance between the observed phenotypes and the markers (predictor variables) . The numRandom forest is a machine learning model for genomic prediction that uses an average of multiple decision trees to determine the predicted values. This regression model was implemented using the “randomForest” package . The numBayesCpi was used to verify the influence of distinct genetic architectures of different traits on prediction accuracy. The BayesCpi assumes that each marker has a probability y of 0.5 .The last model used was the RKHS. The method is a regression where the estimated parameters are a linear function of the basis provided by the reproducing kernel (RK). RKHS considers both additive and non-additive genetic effects . In thisth kernel the proportion of sharing alleles between pairs of individuals (ii′) was given by k is a bandwidth parameter associated with rth reproducing kernel and ested by . Those vGenomic prediction models RR-BLUP, PLSR, RF were carried out using “GSwGBS” package  while thThe influence of training population size on predictive ability was evaluated using a validation set comprising of 50 randomly selected lines and training sets of variable sizes. The validation set was formed by randomly sampling 50 lines without replacement. The training population of size n was formed sequentially by adding 25 accessions from the remaining accessions such that its size ranged between 50 and 175. We subset the collection into subgroups of 50, 75, 100, 125, 150, and 175 individuals each. The RR-BLUP model was used to predict each trait. This procedure was repeated 20 times, and accuracies of each training population size were averaged across 20 replicates. To predict a particular subpopulation with increasing population size, a similar procedure was followed to using variable training population size ranged from 50 to 175 with an increment of 25.To evaluate the effects of GBS marker selection on predictive ability, we randomly sampled markers five times with the following subset: one thousand (1 K), five thousand (5 K), ten thousand (10 K), fifteen thousand (15 K), twenty thousand (20 K), twenty-five thousand (25 K), and thirty thousand (30 K). A random sampling of SNP was implemented to minimize or avoid any possible biases on sampling towards a particular distribution. Using the RR-BLUP model, a five-fold cross validation approach was used to obtain predictive ability in each marker subset. This procedure was repeated 20 times and predictive ability for each subset of SNPs were averaged across 20 replicates.We explored the confounding effect due to population structure on predictive ability. We investigated subpopulation structure on 482 accessions genotyped with 30,600 SNP markers using the ADMIXTURE clustering-based algorithm . ADMIXTUTo account for the effect of population structure, we included the top 10 PC, or the Q-matrix from ADMIXTURE into the RR-BLUP model and performed five-fold cross-validation repeated 20 times. Alternatively, we also used the subpopulation (SP) designation identified by ADMIXTURE as a factor in the RR-BLUP model. Albeit a smaller population size, we also performed a within-subpopulation prediction. As stated above, a subpopulation was defined based on >60% ancestry cut-off. Only three subpopulations with this cut-off were identified and used with reasonable number of entries : SP5 (N = 51), SP7 (N = 58), and SP8 (N = 41). A leave-one-SP-out was used to predict individuals within the subpopulation with the RR-BLUP model. We also used increasing population sizes to predict specific subpopulation (e.g. SP8) using the RR-BLUP model.Nonphenotyped entries were predicted based on the RR-BLUP model using SNP markers only. The reliability criteria for each of the nonphenotyped lines were then calculated using the formula  as folloH2Cullis = 0.66). The heritability for plant height was 0.81 (H2Cullis = 0.68), with an average height of 74 cm. The number of pods per plant had a heritability estimate of 0.50 (H2Cullis = 0.27) with a mean of 18 number of pods per plant and ranged from 15 to 23 pods. Days to physiological maturity had a mean of 104 days with an estimated heritability of 0.51 (H2Cullis = 0.38). Seed yield per hectare ranged widely from 1734 to 4,463 kg ha−1 with a mean yield of 2,918 kg ha−1 and a heritability value of 0.67 (H2Cullis = 0.46). The number of pods per plant was highly and positively correlated with seed yield. Correlation estimation also suggested seed yield was positively correlated with plant height, days to physiological maturity, and days to first flowering  and seed yield (17%). Using either ADMIXTURE or PCA to account for the effect due to population structure, we improved the predictive ability. We observed a 6% improvement for days to first flowering and 32% for seed yield compared over models that did not account for population structure.We also performed within subpopulation predictions. Presented here are the predictive abilities for subpopulations 5, 7, and 8, as they had at least 40 entries. Subpopulation 8 had the highest predictive ability for days to first flowering (0.68), plant height (0.33), days to maturity (0.43), and seed yield (0.37). The highest predictive abilities for the number of seeds per pod (0.40) and pods per plant (0.12) were obtained from subpopulation 7 . Notably−1) was not significantly different from the mean seed yield of observed genotypes . The mean of observed and predicted entries were non-significant for the other five traits , the highest reliability was obtained from the bottom 50  into genomic prediction framework  coupled


Technological advances and decreasing costs have led to the rise of increasingly dense genotyping data, making feasible the identification of potential causal markers. Custom genotyping chips, which combine medium-density genotypes with a custom genotype panel, can capitalize on these candidates to potentially yield improved accuracy and interpretability in genomic prediction. A particularly promising model to this end is BayesR, which divides markers into four effect size classes. BayesR has been shown to yield accurate predictions and promise for quantitative trait loci (QTL) mapping in real data applications, but an extensive benchmarking in simulated data is currently lacking. Based on a set of real genotypes, we generated simulated data under a variety of genetic architectures and phenotype heritabilities, and we evaluated the impact of excluding or including causal markers among the genotypes. We define several statistical criteria for QTL mapping, including several based on sliding windows to account for linkage disequilibrium (LD). We compare and contrast these statistics and their ability to accurately prioritize known causal markers. Overall, we confirm the strong predictive performance for BayesR in moderately to highly heritable traits, particularly for 50k custom data. In cases of low heritability or weak LD with the causal marker in 50k genotypes, QTL mapping is a challenge, regardless of the criterion used. BayesR is a promising approach to simultaneously obtain accurate predictions and interpretable classifications of SNPs into effect size classes. We illustrated the performance of BayesR in a variety of simulation scenarios, and compared the advantages and limitations of each. In particular, genomic prediction models are widely used as an evaluation tool for genomic selection in plant , to predict phenotypes, in plant , there hπ , where SNP effects are modeled using a four-component normal mixture model. The related BayesRC model further allows for SNPs to be grouped into disjoint categories , for which the BayesR model is subsequently fit independently.Most models used in routine genomic selection are based on linear models, notably best linear unbiased prediction (BLUP) and genomic BLUP (GBLUP). These models assume that all SNPs contribute equally to the genomic variance, with each SNP effect following a normal distribution with common variance. Although the assumption about common SNP effects allows for great computational efficiency, it is quite strong and can limit the biological interpretability of results. To address this limitation, although deep learning models have recently started to appear  methods, SNP effects are estimated simultaneously and make use of variable selection within the model itself, rather than relying on univariate hypothesis tests and corrections for multiple testing. As the quantity and quality of prior biological knowledge continues to improve and the identification of causal mutations from WGS data  is not available. Using simulated data, we evaluate the robustness of BayesR under a wide variety of genetic architectures, phenotype heritabilities, and polygenic variances, and we illustrate the conditions under which BayesR successfully identifies known QTLs while maintaining high accuracy for phenotypic prediction. Finally, we describe and compare several statistical criteria that can be used to perform QTL mapping using BayesR output. Based on the results of our simulation study, we discuss the optimal framework for an accurate and interpretable analysis using BayesR, as well as its limitations.To address this gap, our goal in this study is to identify the coherence between the BayesR model specification and known QTL effects in simulated data under a variety of conditions. The BayesR approach is of particular interest here, as it has been shown in the literature to improve prediction accuracy , with the 80% oldest bulls  structure among SNPs, we generated simulated data based on a set of genotypes assayed using Illumina Bovine SNP50 BeadChip arrays from y for the n = 2605 bulls, we made use of a standard linear model:μ denotes the intercept, p SNPs, and X is the marker matrix, centered and scaled as: ijw the number of copies of the reference allele {0,1,2} and jf the frequency of the reference allele. Parameters for this linear model were set as follows. For each simulated dataset we sampled from the available SNPs a set of k of total genetic additive variance μ = 0, and SNP heritability To simulate phenotypes k of genetic variance attributed to the QTLs, with 10 independent datasets created for each , iβ was set to 0. We varied the proportion of genetic variance attributed to each QTL between k, the same We constructed 13 scenarios with different proportions for each . For theried see . As the To investigate the sensitivity of BayesR to a different underlying genetic architecture, we also simulated a secondary set of simulations in an analogous manner with 5 large QTLs as well as 5 additional intermediate QTLs, whose percentage of genetic variance was set to 10% of that of the large QTLs. The settings for these additional simulations are described in Supplementary Table S1.iβ follow a four-component normal mixture, making it well-aligned to our simulations . The effect of SNP i is assumed to be distributed asi.e., the cumulative variance of all SNP effects) and π are assumed to follow a Dirichlet prior, α representing a vector of pseudocounts and γ the cardinality of each component. In this study, we used a flat Dirichlet distribution, with The models of the Bayesian alphabet are all based on the linear model in As exact computation of the posterior distribution is intractable for this model, Bayesian inference is performed by obtaining draws of the posterior using a Gibbs sampler; full details of the algorithm can be found in https://github.com/syntheke/bayesR. We made a few modifications to this code, notably adding the posterior variance of estimated SNP effects at each iteration to the output; our modified BayesR code may be found at https://github.com/fmollandin/BayesR_Simulations.In this study, we used the open source Fortran 90 code described in y) and those estimated using BayesR  each SNP individually; (2) neighborhoods, or sliding windows, around each marker; and (3) those used for ranking potential QTLs.Mapping criteria for individual SNPs: BayesR is unique in the Bayesian alphabet, in that it assigns SNPs to one of four effect size classes at each iteration by weighting according to their likelihood of belonging to each. We thus have access to the posterior frequency with which SNPs were assigned to each class, which can be interpreted as an inclusion probability. We denote the posterior inclusion probability (PIP) of SNP i belonging to class j as j = 1, 2, 3, and 4, respectively. The PIP provides a straightforward method for classifying SNPs as having a null, small, medium, or large effect. We define the maximum a posteriori (MAP) rule for SNP i asi is thus included in the model if Based on this criterion, SNP i corresponds toX, the per-SNP posterior variance can be estimated usingn is the number of iterations and i as The BayesR model definition explicitly allows for some SNPs to have larger estimated variances than methods such as GBLUP, which tends to shrink the variance of causal markers due to the assumption of a common variance , heritability, and use of 50k or 50k custom genotype data.The mean validation correlation (over the ten independent datasets simulated for each) for each simulation scenario illustrates the expected drop in prediction quality for decreasing heritabilities, whether 50k or 50k custom data are used . For theh2=0.01, the average difference in validation correlation was 0.003 (±0.009), and in some cases, the use of the 50k custom data actually corresponded to a slightly worse prediction. Similar results are observed at this level of heritability regardless of the simulated effect size of the QTLs. However, for Although the trends of the mean validation correlation are nonlinear as the QTL effects take on an increasing percentage of genetic variance for both types of data, we do remark an increasing disparity in performance between the 50k and 50k custom data, particularly as the heritability itself increases . In particular, as expected the potential gain in including the true causal mutations among genotypes (as is the case of the 50k custom data) appears to be particularly strong for moderate to large heritabilities and QTL effects. For A natural first tool to investigate for QTL mapping is the neighborhood PIP defined in k, and they are exhaustively assigned to the large effect class for The neighborhood PIP results provide a preview of how QTLs are grouped into nonnull effect classes according to the neighborhood MAP rule . In all i.e., when the neighborhood inclusion probability >0.5). The Given these results, it is not surprising that the neighborhood i.e., a QTL window correctly identified for k = 0.02 but not 0.025). With the exception of Because the same five QTLs are simulated in each independent dataset across effect size scenarios, gσ for each were 10, 30, 50 and 80. In the case of the 50k custom data, this same parameter was estimated to be 9.11 (±3.27), 31.01 (±3.97), 50.27 (±4.32), and 77.54 (±4.49), respectively.Beyond the assignment of SNPs to effect classes using the neighborhood PIPs (and corresponding MAP rules), BayesR also provides posterior estimates of variability at several levels, including the additive genetic variance Given that the total additive genetic variance appears to be well-estimated for both types of genotype type, we turn our attention to the posterior variance 2.5%,5%} ; similarAs a final criterion, we investigate the weighted cumulative inclusion probability statistic iV (top 10) and neighborhood iV ranking were all also flagged by the To evaluate this point, we compared the QTL mapping performance of BayesR using the following three criteria: the neighborhood criteria . SimilarFor all three criteria, the number of detected QTLs increases with the simulated effect size and heritability, as well as with their inclusion among the genotypes (50 k custom data), with the exception of the lowest considered heritability, i.e., intermediate QTLs with 10% of the proportion of genetic variance assigned to large QTLs), particularly for moderate to large heritabilities. Although the estimated posterior variances of intermediate QTLs do tend to increase with larger simulated effect sizes and heritabilities, the pattern is muted compared to that of large QTLs. In addition, with the exception of the most favorable scenario , intermediate QTLs were not assigned to the small or medium BayesR class. This suggests that the mapping criteria discussed above would be unlikely to prioritize such intermediate QTLs in all but the most highly favorable scenarios.An additional point of interest is to investigate the extent to which more intermediate QTL effects can be identified using these QTL mapping criteria. Using our secondary set of simulations (Supplementary Table S1), in which a set of five large and five intermediate QTLs were included, we calculated the estimated proportion of genetic variance for each QTL . Unsurprisingly, the posterior variances for intermediate QTLs are smaller than those for large QTLs; however the discrepancy is considerably larger than the differential that was in fact simulated  of BayesR for identifying QTLs. A critical related issue to contextualize these results is the corresponding error rate. We will focus on an evaluation of the neighborhood-based criteria, which generally led to more detections across scenarios compared to the i.e., QTLs and their immediate neighborhoods are known); however, the quantification of negatives and false positives for the neighborhood-based criteria can lead to some ambiguity. In particular, the cumulative nature of the CIP, which sums weighted inclusion probabilities across a window, leads to highly dependent values for contiguous SNPs whose neighborhoods overlap a large-effect SNP. In cases where the signal of a QTL is carried by a neighboring SNP due to LD structure, any neighborhood overlapping the latter (even those not containing the true QTL) will thus tend to have inflated CIP values. Similarly, for the posterior variance iV, LD structure can also lead to cases where neighboring SNPs have large estimated values; determining whether and how these should each be individually counted or aggregated is not clear-cut. For simplicity here, each marker located in the 15-SNP window centered on one of the 5 QTLs was individually considered to represent a positive, while all others were considered to represent negatives.In the previous section, our primary interest was in the detection power . As our focus was on the ranking of the top SNPs, we instead calculated AUROC values based on the 10 and 150 most highly ranked SNPs using iV and the CIP, respectively. AUROC values were averaged across the 10 datasets for each simulation setting, and undefined values  were set to be zero. As expected, AUROC values were very small in cases of low heritability or small QTL effect sizes (h2 > 0.3) and QTL effect sizes increase (k > 2%), a marked increase in AUROC can be observed. For example, for sufficiently large values of heritability and QTL effect sizes, . When considering genome-wide results, there is a considerable imbalance between positive and negative cases . This is due to the fact that BayesCπ has a single nonnull class that tends to include a larger number of SNPs, each assigned a small proportion of the genetic variance. This implies that the use of the nonnull MAP criterion generally lacks interpretability in the case of BayesCπ. In the case of the posterior variance iV , we remark that although the ranking of SNPs by BayesCπ generally mirrors that of BayesR, the estimated proportion of genetic variance is largely under-estimated. This phenomenon is another reflection of the consequences of a single, nonnull class made up of many markers with small variances. Finally, we note that the CIP criterion previously defined in π as the variance of the nonnull class is itself a random variable and not fixed as in BayesR; a similar statistic could be defined for BayesCπ, but such an adaptation is out of the scope of this study.With respect to the predictive performance of BayesCvs 50k custom genotype data. Simulated phenotypes were generated using SNPs from a real set of genotype data in cattle that were divided into three categories , with variable corresponding shares of the additive genetic variance. In our study, polygenic SNPs were simulated to have the same share of genetic additive variance as the default BayesR small effect class, i.e., i.e., In this study, we evaluated the performance of the BayesR Bayesian genomic prediction model for prediction quality and QTL mapping performance on simulated data under a variety of scenarios, including varying QTL effect sizes, heritabilities, and the use of 50k e.g., number and magnitude of nonnull effect classes) can be adapted for different applications. In this study, we investigated the sensitivity of BayesR results based on the magnitude of the large effect class, and we found that the performance of BayesR  was relatively robust. This suggests a limited benefit to modifying the priors based on prior biological knowledge. A more promising approach to integrate such prior knowledge is the related BayesRC model . As this flexibility can help prioritize informative SNP subsets that contain a larger proportion of QTLs, it would be of great interest to evaluate the impact of the choice of SNP subsets on QTL mapping with BayesRC, using the criteria we investigated here.The specific parameterization of BayesR ; this increase is particularly marked for highly heritable phenotypes as well as for QTLs with large effects. We note that the predictive power of the BayesR model varied both across simulated scenarios, as well as within a given scenario, suggesting that the specific position of simulated QTLs and polygenic SNPs appears to have an influence on the behavior of BayesR.With the exception of very low heritability . While the e.g., large heritabilities and QTL effect sizes) that would be rare in real applications. However, studying these extreme scenarios enables the behavior of the BayesR model to be established in ideal cases. All of our simulations made use of a constant number of individuals in both the training and validation sets, but a future study evaluating the impact of the training population sample size on QTL mapping ability, particularly for cases with low heritability (e.g., There are several limits to our current study that should be taken into consideration. First, we note that some of our simulation scenarios could be considered to represent optimal conditions (BayesR is a powerful tool for simultaneously providing accurate phenotypic predictions and mapping causal regions. Our simulation results illustrate the flexibility of BayesR for different genomic architectures for all but very low heritabilities (


In high-throughput genetics studies, an important aim is to identify gene–environment interactions associated with the clinical outcomes. Recently, multiple marginal penalization methods have been developed and shown to be effective in G×E studies. However, within the Bayesian framework, marginal variable selection has not received much attention. In this study, we propose a novel marginal Bayesian variable selection method for G×E studies. In particular, our marginal Bayesian method is robust to data contamination and outliers in the outcome variables. With the incorporation of spike-and-slab priors, we have implemented the Gibbs sampler based on Markov Chain Monte Carlo (MCMC). The proposed method outperforms a number of alternatives in extensive simulation studies. The utility of the marginal robust Bayesian variable selection method has been further demonstrated in the case studies using data from the Nurse Health Study (NHS). Some of the identified main and interaction effects from the real data analysis have important biological implications. The ideRecently, substantial efforts have been devoted to novel penalized variable selection methods for G×E studies  regression, which has been a popular choice in frequentist G×E studies but seldom investigated in a similar context from the Bayesian perspective. We consider the Bayesian LAD LASSO for regularized identification of interaction effects. As Bayesian LAD LASSO does not lead to zero coefficients, the spike-and-slab priors  be the p genetic variants, E =  be the q environmental factors and C =  be the m clinical factors. We denote the ith subject with i. Let   be independent and identically distributed random vectors. For Xij , the measurement of the jth genetic factor on the ith subject considers the following marginal model:We use k's and γt's are the regression coefficients corresponding to effects of environmental and clinical factors, respectively. For the jth gene Xj , the G×E interactions effects are defined with Wj = , j's and ηjk's are the regression coefficients of the genetic variants and G×E interactions effects, correspondingly. Let us denote where αThe necessity of accounting for robustness in interaction studies has been increasingly recognized  are i.i.d. random variables following the Laplace distribution with density:Here, we propose the robust marginal Bayesian variable selection based on LAD. As the Laplace distribution is equivalent to the mixture of an exponential distribution and a scaled normal distribution  = ϵ{θ − I(ϵ < 0)} is the check loss function. The random errors can be written aswhere ρwherezi∽N.with quantile level θ ∈ , 1 = 0 and Yi can be written as:The Bayesian LAD regression is a special case of Bayesian quantile regression , the coefficients βjth gene , the marginal LAD LASSO model is given byFor the 1 = τλ1 and φ2 = τλ2. Then the conditional Laplace prior on the coefficient of main effect βj can be expressed as scale mixtures of normals:Let φj can be written as:The conditional Laplace prior on the coefficients of interaction effect ηTherefore, we consider the following hierarchical formulation for the marginal G×E model:0(βj) and δ0(ηjk) denote the spike at 0, respectively, and the slab distributions are represented by two normal distributions, N and N. Here, π1 ∈  and π2 ∈ . The mixture of the spike and slab components facilitate the selection of main and interaction effects. Instead of setting π1 and π2 to a fixed value such as 0.5, we assign conjugate beta priors on them as π1∽Beta and π2∽Beta, which account for the uncertainty in π1 and π2. In this paper, we choose r1 = u1 = r2 = u2 = 1 as it gives a prior mean with 0.5 and it also allows a prior to spread out.where δk and clinical factor γt as:In addition, the normal prior has been placed on the coefficients of environmental factor αWe also assume conjugate Gamma priors on τ, In typical G×E studies, the environmental and clinical factors are of low dimensionality and the selection of them is not of interest. Therefore, the sparsity-inducing priors have not been adopted for these factors. We consider the Bayesian LAD LASSO type of regularization in the proposed study as published studies have demonstrated that baseline penalty such as MCP and LASSO work well for marginal variable selection (− = E(yi) − Eikαk, , , representing the mean effect without the contribution of Eikαk. The posterior distribution of the coefficient of environmental factor αk conditional on all other parameters can be expressed as:Let μk is normal distribution Hence, the full conditional distribution of αand variancet conditional on all other parameters can be obtained in similar way. Let μγt)(− = E(yi) − Citγt, i = 1, …, n, thenThe posterior distribution of the coefficient of clinical factor γwhereβj)(− = E(yi) − Xijβj and l1 = π(βj = 0|rest), the conditional posterior distribution of the coefficient of genetic factor βj is a spike-and-slab distribution:Let μwhereWe can show thatj is a mixture of a normal distribution and a point mass at 0. That is, at each iteration of MCMC, βj is drawn from l1) and is set to 0 with probability l1.The posterior distribution of βjth gene and environmental factors ηjk is also a spike-and-slab distribution. Denote μηjk)(− = E(yi) − Wikηjk and lk2 = π(ηjk = 0|rest), ηjk follows this distribution:Similarly, the posterior distribution of the interaction of the whereAnds1 is:The full conditional posterior distribution of j = 0, equation (8) is proportional to s1 is When βj ≠ 0, equation (8) is proportional toWhen βj ≠ 0, the posterior distribution for Therefore, when βsk2, when ηjk = 0, the posterior distribution of sk2 is jk ≠ 0, the posterior distribution for Similarly, for The full conditional posterior distribution of c1+1, s1/2+d1). Similarly, the posterior distribution for Therefore, the posterior distribution for 1 is given as:The full conditional posterior distribution of π1 is Beta (1 + r1 − I(βj ≠ 0), u1 + I(βj ≠ 0)).Then, the posterior distribution for π2 is given as:The full conditional posterior distribution of π2 is Beta So, the posterior distribution for πThe full conditional posterior distribution of τ is given as:Therefore, the posterior distribution for τ is Gamma with three alternatives, LAD Bayesian Lasso (denoted as LADBL), Bayesian Lasso with spike-and-slab priors (denoted as BLSS) and Bayesian Lasso (denoted as BL). LADBL is similar to the proposed method, except that it does not adopt the spike-and-slab prior. The details of posterior inference are given in the n = 200, and the number of G factors is p = 500 with q = 4, m = 3. For environmental factors, we simulate four continuous variables from multivariate normal distributions with marginal mean 0, marginal variance 1 and AR1 correlation structure with ρ = 0.5. In addition, three clinical factors are generated from a multivariate normal distribution with marginal mean 0 and marginal variance 1 and AR1 structure with ρ = 0.5. Among the p main G effects and pq G×E interactions, 8 and 12 effects are set as being associated with the response, respectively. All the environmental and clinical factors are important with nonzero coefficients, which are randomly generated from a uniform distribution Unif. The random error are generated from: (1) N(Error 1), (2) t-distribution with 2 degrees of freedom (t(2)) (Error2), (3) LogNormal(Error3), (4) 90%N+10%Cauchy(Error4), (5) 80%N+20%Cauchy(Error5). All of them are heavy-tailed distribution except the first one.Under all settings, the sample size is set as In addition, the genetic factors are simulated in the following four settings.Setting 1: In simulating continuous genetic variants, we generate multivariate normal distributions with marginal mean 0 and variance 1. The AR structure is considered in computing the correlation of G factors, under which gene j and k have correlation ρj−k|| with ρ = 0.5.Setting 2: We assess the performance under single-nucleotide polymorphism (SNP) data. The SNPs are obtained by dichotomizing the gene expression values at the 1st and 3rd quartiles, with the 3 levels  for genotypes . Here, the gene expressions are generated from the first setting.Setting 3: Consider simulating the SNP data under a pairwise linkage disequilibrium (LD) structure. For the two minor alleles A and B of two adjacent SNPs, let q1 and q2 be the minor allele frequencies (MAFs). The frequencies of four haplotypes are as pAB = q1q2 + δ, pab = (1 − q1)(1 − q2) + δ, pAb = q1(1 − q2) − δ, and paB = (1 − q1)q2 − δ, where δ denotes the LD. Assuming Hardy–Weinberg equilibrium and given the allele frequency for A at locus 1, we can generate the SNP genotype  from a multinomial distribution with frequencies r = 0.6, we have jth genetic factor, Xj, as an example. Suppose we have collected H posterior samples from MCMC after burn-ins. The jth G factor is included in the marginal G×E model at the hth MCMC iteration if the corresponding indicator is 1, i.e., jth genetic main effect in the final marginal model is defined as the average of all the indicators for the jth G factor among the H posterior samples. That is,We collect the posterior samples from the Gibbs Sampler with 10,000 iterations and discard the first 5,000 samples as burn-ins. The posterior medians are used to estimate the coefficients. For approaches incorporating spike-and-slab priors, we consider computing the inclusion probability to indicate the importance of predictors. Here, we use a binary indicator ϕ to denote that the membership of the non-spike distribution. Take the main effect of the pj indicates a stronger empirical evidence that the jth genetic main effect has a non-zero coefficient, i.e., a stronger association with the phenotypic trait.A larger posterior inclusion probability To comprehensively assess the performance of the proposed and alternative methods, we consider a sequence of probabilities as cutting-offs in inclusion probability for methods with spike-and-slab priors. Given a cutoff probability, the main or interaction is included in the final marginal model if its posterior inclusion probability is larger than the cutoff, and is excluded otherwise. Provided with a sequence of cutting-off probabilities from small to large, we can investigate the set of identified effects and calculate the true/false positive rates (T/FPR) as the ground truth is known in simulation. For the sequence of cut-offs, we are able to compute the area under curve (AUC) as a comprehensive measure. Besides, for methods without spike-and-slab priors, the confidence level of the credible intervals can be adopted as the cut-off to compute TPR and FPRs. Therefore, all the methods under comparison can be evaluated on the same ground.In addition, we also consider Top100, which is defined as the number of true signals when 100 important main effects (or interactions) are identified. For methods with spike-and-slab priors, 100 main effects or interactions are chosen with the highest inclusion probabilities. For methods without spike-and-slab priors, the indicators of all effects are computed for a sequence of credible levels. The top 100 main effects or interactions are chosen in terms of the highest average identification values.Simulation results for the gene expression data in the first setting are tabulated in n = 2,000 under setting 1. While the difference among the 4 methods significantly diminishes with such a large sample size, we can still observe the superior performance of LADBLSS by using a shorter list of top ranked effects. The results are provided in the table from Similar patterns can be observed in Tables 4, 5 in n = 200, p = 500 and standard normal error, the CPU time (in minutes) for fitting all the 500 marginal models through 10,000 MCMC iterations on a laptop with standard configurations are 1.27(BL), 1.75(BLSS), 6.16(LADBL), and 5.95 (LADBLSS) minutes, respectively. The source codes of implementing all the methods under comparison are included in the In simulation, the hyperparameters for the Gamma priors and Beta priors specified in section Bayesian LAD LASSO With Spike-and-slab Priors are set to 1. In addition, the initial values of the regression parameters are also set to 1. Based on our experiments, the results and convergence of the MCMC algorithm are not sensitive to the choice of these parameters. We have observed satisfactory convergence for all of our simulations. For one simulated dataset under the first setting with p < 0.05 are kept.The scale of working data is generally not a major concern for marginal analysis, as the computation can be done in a highly parallel manner. Here, we focus on chromosome 10 which has been reported to harbor interesting genes in existing studies.In this study, we analyze the type 2 diabetes (T2D) data from Nurses' Health Study (NHS), which is a well-characterized cohort study of women with high-dimensional SNP data, as well as measurements on lifestyle and dietary factors. We consider SNPs on chromosome 10 to identify main and gene–environment interactions associated with weight, which is an important phenotypic trait related to type 2 diabetes. Here, weight is used as response and five environment factors, age (age), total physical activity (act), trans fat intake (trans), cereal fiber intake (ceraf), and reported high blood cholesterol (chol), are considered. Data are available on 3,391 subjects and 17,016 gene expressions after cleaning the raw data through matching phenotypes and genotypes and removing SNPs with MAF <0.05. A prescreening is done before downstream analysis. We use a marginal linear model with weight as response and age, act, trans, ceraf, and chol as environment factors. Note that 10,000 SNPs that have at least two main or interaction effects with We use Top 100 method to identify 100 most important main and interaction effects. The proposed method LADBLSS identifies 20 main SNP effects and 80 gene–environment interactions, which are listed in Table 8 in Analysis with alternatives BL, BLSS, and LADBL has also been conducted. To compare the alternative methods with the proposed method, we provide the numbers of main effects and interactions identified by these methods with pairwise overlaps in In the past, G×E interaction studies have been mainly conducted through marginal hypothesis testing, based on a diversity of study designs utilizing parametric, nonparametric, and semiparametric models .Publicly available datasets were analyzed in this study. This data can be found here: Authorized access should be granted before accessing the data. Applications to access the data should be sent to dbGap (accession number phs000091.v2.p1). For more information, please refer to NIH dbGap (https://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/study.cgi?study_id=phs000091.v2.p1). In the dataset, the patient information has been de-identified. As indicated from the dbGap website under section Authorized Access/Use Restrictions, IRB is not required for accessing and using the data. According to the original publication, The study was approved by the institutional review board of Brigham and Women's Hospital in Boston; completion of the self-administered questionnaire was considered to imply informed consent. For more information regarding study population, please refer to the original publication: Hu et al. (The studies involving human participants were reviewed and approved by This study is a secondary data analysis. The dataset has been applied through NIH dbGap (u et al. . The patXL and CW: conceptualization and writing—original draft preparation. XL, KF, JR, and CW: methodology and writing—review and editing. XL: data analysis. All authors contributed to the article and approved the submitted version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.