An efficient Bayesian meta-analysis approach for studying cross-phenotype genetic associations
[['Given the summary statistics for a SNP across multiple traits , CPBayes estimates two different measures evaluating overall pleiotropic association and an optimal subset of associated traits underlying a pleiotropic signal .', 'The evidence for aggregate-level pleiotropic association is given by the local false discovery rate ( locFDR ) and the Bayes factor ( BF ) .', 'Let H0 denote the global null hypothesis of no association with any trait and H1 denote the global alternative hypothesis of overall association with at least one of the traits .', 'Then locFDR is defined as the posterior probability of H0 being true conditioned on the summary statistics and BF is the ratio of the likelihood of summary statistics under H1 versus H0 .', 'A small value of locFDR ( e . g . , 0 . 05 ) or a large value of Bayes factor ( e . g . , BF > 1 ) indicates an evidence for the overall pleiotropic association .', 'We estimate the locFDR and BF based on the MCMC posterior sample of model parameters obtained by Gibbs sampling .', 'The subset of traits which is selected as the set of non-null traits most often in the MCMC posterior sample is defined as the maximum a posteriori ( MAP ) estimate of the optimal subset of associated traits .', 'CPBayes explicitly accounts for possible correlation between the effect estimates across traits .', 'To estimate the correlation structure of the effect estimates , we use two different approaches—one based on the number of overlapping cases and controls between studies and another based on genome-wide summary statistics across traits .', 'The correlation formulae based on the number of shared cases and controls ( Eq 6 in Material and methods ) is accurate when the genetic variant and environmental covariates are not associated with the traits of interest .', 'However in real data , environmental covariates are expected to be associated with the primary traits .', 'In such scenario , the GW summary statistics based approach is more robust with respect to estimating a reasonably accurate correlation structure .', 'CPBayes also provides some more insights into a pleiotropic signal , e . g . , marginal trait-specific posterior probability of association ( PPAj ) , direction of associations , credible interval of true genetic effects , etc .', 'Detailed description of CPBayes and a brief outline of ASSET , BH0 . 01 and GPA are provided in the “Material and methods” section and supporting information .', 'We compare CPBayes and ASSET with respect to correctly detecting a signal of pleiotropy and the accuracy of selection of non-null traits underlying a pleiotropic signal in various simulation scenarios .', 'We consider multiple case-control studies with or without shared controls [7] and a cohort study where the data on multiple disease states are available for a group of individuals [31 , 32] .', 'First , we specify the simulation model to generate the phenotype and genotype data .', 'After computing the summary statistics based on the simulated data , we assume that only the summary-level data are available .', 'For case-control studies with overlapping subjects or a cohort study , we estimate the correlation structure of summary statistics based on Eq 6 ( Material and methods ) .', 'For non-overlapping case-control studies , we consider a separate group of 7000 cases and 10000 controls in each study .', 'For overlapping case-control studies , we consider a distinct set of 7000 cases in each study , and a common set of 10000 controls shared across all the studies .', 'For each disease , we assume an overall disease prevalence of 10% in the whole population .', 'While simulating the genotype data for multiple case-control studies , we assume the standard logistic model of disease probability conditioning on the genotype: P ( case | G ) = exp ( α + β G ) 1 + exp ( α + β G ) , where G is the genotype at the SNP of interest coded as the minor allele count ( 0 , 1 , 2 ) .', 'We assume that the SNP is in Hardy-Weinberg equilibrium ( HWE ) .', 'Let A ( minor ) and a be the two alleles at the SNP and p = P ( A ) .', 'Under HWE , the genotype probabilities are: P ( AA ) = p2 , P ( Aa ) = 2p ( 1 − p ) , P ( aa ) = ( 1 − p ) 2 .', 'Given the log odds ratio ( β ) and disease prevalence , we compute the probability of observing each genotype conditioned on the case/control status using Bayes theorem .', 'Based on these conditional probabilities , we simulate the genotypes in cases and controls .', 'For the cohort study , we consider 15000 individuals .', 'First , we generate the genotype data at a quantitative trait locus ( QTL ) and continuous multivariate phenotype data using the simulation model in Majumdar et al . [31] that was adopted from Galesloot et al . [32] .', 'Then we dichotomize each continuous phenotype ( liability ) to case-control status subject to an overall disease prevalence of 10% .', 'We describe the simulation model in more details in S1 Text in supporting information .', 'We emphasize that the simulation models used here are general in nature and independent of the modeling assumptions underlying CPBayes which are also very general and only require that the sample sizes of the participating GWAS should be sufficiently large to satisfy the standard asymptotic properties ( Material and methods ) .', 'These simulation models were also used in Bhattacharjee et al . [7] , Galesloot et al . [32] , Majumdar et al . [31] .', 'For overlapping case-control studies and a cohort study when the summary statistics are expected to be correlated , a combined strategy of CPBayes is implemented ( Material and methods ) .', 'If the effect estimates are strongly correlated and a majority of the traits are associated with the risk locus ( non-sparse scenario ) , the Gibbs sampler underlying CPBayes may sometimes be trapped in a local mode of the posterior distribution of model parameters .', 'To increase robustness for correlated summary statistics , CPBayes considers a joint strategy combining its uncorrelated and correlated versions .', 'First , we implement CPBayes considering the correlation structure of the effect estimates .', 'If the selected subset of non-null traits have the smallest univariate association p-values among all the traits , we accept the results; otherwise , we employ CPBayes assuming that the effect estimates are uncorrelated and accept the results obtained .', 'To investigate the performance of CPBayes using real data , we analyzed multiple traits in the large Northern California Kaiser Permanente “Resource for Genetic Epidemiology Research on Adult Health and Aging” ( GERA ) cohort obtained from dbGaP [dbGaP Study Accession: phs000674 . v1 . p1] .', 'We also analyzed this dataset using ASSET for an empirical comparison of the methods .', 'We restricted our analysis to 62 , 318 European-American individuals , who constitute more than 75% of the dbGaP data .', 'We tested 657 , 184 SNPs genotyped across 22 autosomal chromosomes for their potential pleiotropic effects on 22 case-control phenotypes in the GERA cohort ( S8 Table ) .', 'Note that in the dbGaP data , the cancers are collapsed into a single variable ( any cancer ) .', 'Therefore , we could only use an overall cancer categorization even though the genetic architecture is likely heterogeneous across different cancers .', 'The phenotypes are correlated modestly with a maximum absolute value of correlation as 0 . 36 observed between Hypertension and Dyslipidemia .', 'Before our analysis , we undertook the following QC steps .', 'First , we removed individuals with: over 3% of genotypes missing; any missing information on covariates ( described below ) ; genotype heterozygosity outside six standard deviations; first degree relatives; or discordant sex information .', 'This left us with 53 , 809 individuals .', 'Next , we removed SNPs with: MAF < 0 . 01; 10% or more missingness; or deviation from HWE at a level of significance 10−5 .', 'This leaves 601 , 175 SNPs that were tested for pleiotropic association by CPBayes and ASSET .', 'We adjusted the analysis for the following covariates: age , gender , smoking status , BMI category and 10 principal components of ancestry ( PCs ) .', 'We tested the single-trait association for each of 22 phenotypes by a logistic regression of the case-control status on the genotype incorporating the same set of adjusting covariates .', 'We used SNP-trait effect estimates ( log odds ratios ) and their standard errors in CPBayes and ASSET .', 'As the summary statistics are correlated here , we used the combined strategy of CPBayes and the correlated version of ASSET .', 'Since we have environmental covariates in the GERA study , we estimated the correlation matrix of the effect estimates under the null using the GW summary statistics data [34] .', 'First , we extracted all of the SNPs for which the trait-specific univariate p-value across 22 traits are > 0 . 1 .', 'This ensures that each SNP is either weakly or not associated with any of the 22 phenotypes .', 'Then we selected a set of 24 , 510 independent SNPs from the initial set of null SNPs by using a linkage disequilibrium ( LD ) threshold of r2 < 0 . 01 ( r: the correlation between the genotypes at a pair of SNPs ) .', 'Finally , we computed the correlation matrix of the effect estimates as the sample correlation matrix of β ^ 1 , … , β ^ 22 across the selected 24 , 510 independent null SNPs .', 'We also considered different SNP filtering thresholds and compared the resulting correlation matrices .', 'We provide numerical results demonstrating that the estimated matrix was not sensitive to our primary choice of the thresholds .', 'We also give numerical results indicating that the estimated correlation matrix based on the sample overlap counts ( Eq', '6 ) may be biased .', 'These numerical results and their interpretation are provided in more detail in S9 Table and S1 Text .', 'We apply the conventional GW level of statistical significance 5 × 10−8 for ASSET .', 'For CPBayes , a pre-fixed significance threshold of locFDR needs to be considered here .', 'While concluding that a locFDR threshold of 5% indicates good evidence of association , selecting the most promising pleiotropic variants may require a more stringent threshold ( as with the frequentist p-value threshold 5 × 10−8 ) .', 'Liley and Wallace [12] also highlighted this point and suggested using a more stringent threshold of conditional false discovery rate ( cFDR ) than the nominal levels ( e . g . , 0 . 05 or 0 . 01 ) .', 'They used a FDR and cFDR cut-off in the order of 10−5 or 10−6 to make the analysis analogous to using a stringent threshold of p-value ( 5 × 10−8 ) .', 'We applied various thresholds of locFDR and detected 610 ( locFDR < 10−2 ) , 537 ( locFDR < 10−3 ) , 523 ( locFDR < 10−4 ) , 481 ( locFDR < 10−5 ) , 442 ( locFDR < 10−6 ) , 417 ( locFDR < 10−7 ) and 380 ( locFDR < 10−8 ) SNPs , respectively .', 'We note that locFDR is not used as extensively as p-value in practice .', 'Unlike the commonly used p-value threshold 5 × 10−8 , different thresholds of locFDR ( or other FDR related measures ) have been used in different applications [8 , 9 , 12] .', 'Since Liley and Wallace [12] used a FDR and cFDR cut-off in the order of 10−5 or 10−6 , here we report the results based on a similar threshold of locFDR as 10−6 which detected 442 SNPs .', 'Of note , locFDR , FDR and cFDR are distinct by definition .', 'Theoretically , locFDR is an upper bound of FDR [27] .', 'ASSET detected 394 SNPs based on the p-value threshold 5 × 10−8 .', 'We note that CPBayes and ASSET identified a common set of 322 SNPs based on these chosen significance thresholds .', 'Many of the associated SNPs are expected to be in LD .', 'It is challenging to report independent pleiotropic variants since the LD pattern across a chromosome is irregular and converting the conditional analysis approach used in single phenotype GWAS to select independently associated variants in multi-phenotype context is difficult .', 'Hence for the sake of convenience , we undertook the following simplified approach to identify correlated LD blocks .', 'For CPBayes ( ASSET ) on each chromosome , we first chose the associated SNP that has the minimum locFDR ( ASSET p-value ) and created a LD block around it using a threshold of r2 = 0 . 25 .', 'Then we implement the same strategy on the remaining set of associated SNPs to identify the next LD block , and so on .', 'A major limitation of this approach is that defining such discrete LD blocks may not be on par with the irregular LD pattern across a chromosome and choosing an appropriate threshold of r2 is also difficult .', 'For CPBayes , 442 GW associated SNPs comprised 59 LD blocks , and for ASSET , 394 GW associated SNPs comprised 30 LD blocks .', 'For each of 394 SNPs detected by ASSET , the optimal subset of non-null traits always included more than one trait .', 'Within each LD block detected by ASSET , we chose the SNP associated with the maximum number of traits as the lead SNP of the block .', 'If multiple SNPs in a LD block satisfy this criterion , the one with minimum p-value of pleiotropic association was selected .', 'We present the results for the lead SNPs detected by ASSET only on chromosome 1 and 2 ( to save space ) in S11 Table .', 'CPBayes selected more than one trait for 93 among 442 SNPs .', 'Within each LD block identified by CPBayes , we chose the SNP associated with the maximum number of phenotypes as the lead SNP of the block .', 'If multiple SNPs satisfy this criterion , we chose the one having the minimum locFDR .', 'In addition , if every SNP in a LD block is associated with one trait , we chose the SNP which provided the maximum number of traits having the marginal trait-specific posterior probability of association ( PPAj ) > 20% ( these traits were termed important phenotypes ) .', 'Again , if multiple SNPs satisfy this criterion , we chose the one having the minimum locFDR .', 'We note that the strategy of choosing the lead SNP in a LD block identified by CPBayes and ASSET are similar but technically different .', 'As CPBayes locFDR and ASSET p-value are not directly comparable , it is very difficult to contrast the two methods with respect to the power of detecting aggregate-level pleiotropic association in a real data application .', 'However , it makes sense to contrast the selection of non-null traits at a pleiotropic variant detected by both the methods .', 'In Table 3 , we present the results for the independent pleiotropic SNPs at which CPBayes selected at least two phenotypes .', 'At some of the lead SNPs detected by CPBayes , some phenotypes produced a non-negligible value of PPAj but were left out of the optimal subset of non-null traits .', 'In Table 4 , we list these SNPs and the corresponding important phenotypes having a PPAj > 20% .', 'In S10 Table , we report the independent SNPs at which CPBayes selected one trait .', 'In the tables for CPBayes , we present PPAj and the direction of association ( genotype was coded as the number of the wild allele ) for the selected phenotypes ( Material and methods ) .', 'In all the tables for CPBayes and ASSET , we also provide the trait-specific univariate association p-values .', 'Many of the pleiotropic variants detected by CPBayes and ASSET are already reported in the NHGRI-EBI GWAS catalog .', 'For example , rs6025 at 1q24 . 2 ( Table', '3 ) has been associated with inflammatory bowel disease and venous thromboembolism; rs10455872 at 6q25 . 3 ( Table', '3 ) has been associated with myocardial infarction , response to statins ( LDL cholesterol change ) , coronary artery disease , and aortic valve calcification; rs1410996 at 1q31 . 3 ( Table', '4 ) is reported to be associated with Post bronchodilator FEV1/FVC ratio in COPD , End-stage coagulation and Age-related Macular Degeneration; rs4506565 at 10q25 . 2 ( Table', '4 ) has been associated with Fasting glucose-related traits and Type 2 Diabetes .', 'For a majority of the SNPs detected by ASSET , the subset of non-null traits included many phenotypes that have large univariate association p-values .', 'For example , rs77394225 at 1q31 . 3 was detected by both the methods ( S10 and S11 Tables ) ; ASSET selected 10 traits , 9 of which had univariate p-value ≥ 0 . 3 and one ( Macular Degeneration ) had p-value = 6 . 04 × 10−16 ( S11 Table ) .', 'In contrast , CPBayes only selected Macular Degeneration ( S10 Table ) .', 'This suggests that CPBayes selects only those phenotypes with strong genetic associations , while ASSET may select many more traits with lower specificity as seen in our simulation study .', 'For the independent pleiotropic SNPs identified by both the methods , for contrast’s sake , we applied BH0 . 01 .', 'At rs77394225 , BH0 . 01 only selected Macular Degeneration which is consistent with CPBayes .', 'Among the nine independent pleiotropic SNPs on chromosome 1 and 2 detected by ASSET , BH0 . 01 selected more than one trait only for two SNPs , whereas ASSET selected multiple traits for each of them ( S11 Table ) .', 'This again indicates lower specificity of ASSET .', 'CPBayes detected six independent pleiotropic SNPs that were associated with at least two phenotypes ( Table 3 ) .', 'For example , at rs6025 ( 1q24 . 2 ) , it selected a maximum of 5 phenotypes: Dermatophytosis , Hemorrhoids , Iron Deficiency , Osteoporosis and Peripheral Vascular Disease , which have univariate p-values equal to 0 . 0018 , 0 . 0014 , 0 . 0004 , 0 . 0002 and 6 . 81 × 10−14 , respectively ( Table 3 ) .', 'Interestingly , this SNP was positively associated with Dermatophytosis , Hemorrhoids and Iron Deficiency , but negatively associated with Osteoporosis and Peripheral Vascular Disease ( Table 3 ) .', 'rs10455872 at 6q25 . 3 appeared to be the lead SNP for both the methods in their corresponding LD block on chromosome 6 .', 'In Fig 4 , we contrast the selection of traits at rs10455872 between the methods .', 'CPBayes selected Cardiovascular Disease , Dyslipidemia and Peripheral Vascular Disease ( Fig 4 ) .', 'ASSET selected these three traits and five more phenotypes with large univariate p-values indicating weak genetic effects ( Fig 4 ) .', 'CPBayes detected two independent pleiotropic SNPs in the chromosomal region 6p21 . 32: rs13211628 which was associated with Asthma , Cancers and Dyslipidemia , and rs3957148 which was associated with Asthma , Type 2 Diabetes and Macular Degeneration ( Table 3 ) .', 'The r2 value between rs13211628 and rs3957148 was 0 . 03 .', 'We provide a circos plot in Fig 5 presenting 23 pair-wise trait-trait pleiotropic association signals detected by CPBayes .', 'It shows that CPBayes detected rs7601401 at 2p16 . 1 associated with Osteoarthritis and Abdominal Hernia; rs687289 at 9q34 . 2 associated with Dyslipidemia , Type 2 Diabetes and Peripheral Vascular Disease .', 'We also present forest plot for some of the independent pleiotropic signals ( at rs6025 , rs13211628 , rs10455872 , rs3957148 , rs687289 ) detected by CPBayes in S11–S15 Figs .', 'For these six independent pleiotropic SNPs , BH0 . 01 selected the same subset of traits as CPBayes .', 'For CPBayes , the marginal trait-specific posterior probability of association ( PPAj ) provides a better insight into the relative strength of association between a pleiotropic variant and the selected non-null traits .', 'For example at rs6025 , PPAj for Dermatophytosis , Hemorrhoids , Iron Deficiency , Osteoporosis and Peripheral Vascular Disease are 67% , 71% , 97% , 94% and 100% , respectively ( Table 3 ) .', 'This implies that the association with Peripheral Vascular Disease is the strongest among the five selected phenotypes .', 'At some of the GW significant SNPs detected by CPBayes , it produced a non-negligible value of PPAj for some traits; but these traits were left out of the optimal subset of non-null traits ( Table 4 ) .', 'For example at rs4506565 ( 10q25 . 2 ) , CPBayes only selected Type 2 Diabetes , but Dyslipidemia also produced a PPAj of 52% .', 'Thus , even though the effect of rs4506565 on Dyslipidemia was not strong enough to make it into the optimal subset , a further consideration of the pleiotropic effect of rs4506565 on Type 2 Diabetes and Dyslipidemia looks promising .', 'We observed similar pattern across the other pleiotropic variants listed in Table 4 .', 'We note that the combined strategy of CPBayes used the uncorrelated version only for one SNP among all the 601 , 175 SNPs analyzed .'], ['We have proposed a Bayesian meta-analysis approach CPBayes for pleiotropic association analysis based on summary-level data .', 'It simultaneously evaluates the evidence of aggregate-level pleiotropic association and estimates an optimal subset of traits associated with the risk locus under a unified Bayesian statistical framework .', 'The method is implemented by Gibbs sampling designed for both uncorrelated and correlated summary statistics .', 'We have conducted an extensive simulation study and analyzed the large GERA cohort for evaluating the performance of CPBayes .', 'An appealing feature of CPBayes is that , in addition to locFDR , Bayes factor , and an optimal subset of non-null traits , it simultaneously provides other interesting insights into an observed pleiotropic signal .', 'For example , it estimates a trait-specific posterior probability of association ( PPAj ) , the direction of association , posterior mean/median and the credible interval of the unknown true genetic effect across traits .', 'PPAj quantifies the marginal probability of each trait being associated with a pleiotropic variant .', 'As demonstrated in the real data application , even if CPBayes does not select a phenotype in the optimal subset of non-null traits , PPAj for the phenotype may not be negligible .', 'This may help an investigator to better explain a pleiotropic signal .', 'One can also define the optimal subset of associated traits as {Yj: PPAj > p} , where p can be chosen as 0 . 5 ( known as the median model ) , or other values .', 'Moreover , the joint posterior probability of association for a particular subset of traits can be calculated .', 'Such flexibility in making inference on pleiotropy is mainly due to the MCMC construction underlying CPBayes .', 'CPBayes selects the non-null traits underlying a pleiotropic signal with higher accuracy than ASSET .', 'CPBayes performs the selection probabilistically through updating the latent association status by MCMC .', 'ASSET selects that subset of traits as non-null which maximizes the observed value of a weighted linear combination of the normalized univariate association statistics corresponding to the phenotypes belonging to a subset .', 'So given the summary statistics , ASSET does not select the non-null traits probabilistically based on the distribution of the summary statistics .', 'For ASSET , even if a trait having a small genetic effect contributes a little to a pleiotropic signal , it is included in the optimal subset of associated traits .', 'But CPBayes considers only those traits as non-null which substantially contribute to a pleiotropic signal .', 'For example in the real data application , at rs10455872 , allergic rhinitis had an estimated odds ratio 1 . 05 ( univariate association p-value 0 . 07 ) .', 'ASSET included this phenotype in the optimal subset of non-null traits as it might contribute to the overall signal of pleiotropic assocition .', 'However , CPBayes estimated PPAj for this trait as 2 . 4% ( S13 Fig ) .', 'So its contribution to the pleiotropic signal was not necessarily null , but CPBayes did not include it in the optimal subset of non-null traits as the effect was weak .', 'We also note that ASSET is based on the framework of a fixed effects meta-analysis and assumes that the effects in a given direction ( positive/negative ) have the same size .', 'But we observed in our real data application that , in a given direction , the effects of a variant across phenotypes may often be heterogeneous .', 'CPBayes allows for heterogeneity simultaneously in the direction and size of the effects .', 'S2 Table summarizes the key features of CPBayes and ASSET .', 'While assessing the selection accuracy , we have placed more emphasis on specificity than sensitivity .', 'This was because a higher sensitivity at the expense of a lower specificity can lead to a false selection of too many traits as non-null .', 'CPBayes consistently maintained a very good level of specificity while offering a good level of sensitivity across a wide range of simulation scenarios .', 'While CPBayes produced a limited number of independent pleiotropic SNPs associated with more than one phenotype in the analysis of GERA cohort , these pleiotropic signals seem very promising .', 'The subsets of non-null traits selected by BH0 . 01 in the GERA cohort were consistent with CPBayes but not with ASSET which indicates that the non-null traits for a pleiotropic variant selected by CPBayes may be more reliable than ASSET .', 'Here we note that , BH0 . 01 only facilitates the selection of associated traits underlying a pleiotropic signal but can not test for the evidence of overall pleiotropic association as CPBayes and ASSET .', 'While CPBayes and ASSET estimate the measure of aggregate-level pleiotropic association and subset of non-null traits simultaneously under the same framework , BH0 . 01 has to be implemented in a separate step for a GW significant pleiotropic variant .', 'Hence , CPBayes is a substantially more complete statistical tool for pleiotropy analysis than BH0 . 01 .', 'While evaluating the selection accuracy of different approaches by simulations , we simultaneously obtained the measures of overall pleiotropic association provided by CPBayes and ASSET across 500 replications .', 'We present various summary measures of these in some selected simulation scenarios ( to save space ) and corresponding brief interpretation in S3 , S4 and S5 Tables and S1 Text .', 'We also carried out simulations for 50 traits .', 'Since ASSET is computationally very slow for 50 traits due to an extremely large number of possible subsets of traits , we only implemented CPBayes and BH0 . 01 .', 'CPBayes performed consistently well similarly as for smaller number of traits .', 'See S1 Text and S6 and S7 Tables for more details .', 'Note that the continuous spike inherits the assumption that a SNP contributes to the variation of all traits under consideration , and the distinction is made between a negligible and a significant contribution .', 'In contrast , the Dirac spike assigns the null effects explicitly to zero .', 'We conducted a simulation study ( see S1 Text and S10 Fig ) to compare the continuous spike and Dirac spike .', 'We found that the continuous spike offers better accuracy in the selection of non-null traits than the Dirac spike .', 'The continuous spike is also computationally much faster ( 2-3 times ) than the Dirac spike .', 'Hence , we adopted the continuous spike for constructing CPBayes .', 'In a related work , Han and Eskin [35] proposed a modified random effects meta-analysis for combining heterogeneous studies coupled with a Bayesian approach to provide a better interpretation of an observed signal of aggregate-level association .', 'They investigated how to combine heterogeneous genetic studies across different populations/ethnicities .', 'However , they did not address how to account for a possible correlation between the summary statistics while selecting the most important studies underlying an observed signal of aggregate-level association .', 'Moreover , they assumed that the non-null effects are similar across studies which is less likely to hold in the context of pleiotropy .', 'Hence we compared CPBayes with ASSET and GPA .', 'We note that the CPBayes locFDR ( Bayes factor ) and ASSET p-value are not directly comparable .', 'In our simulation study , we adopted the strategy outlined by Coram et al . [33] to compare CPBayes and ASSET with respect to the number of true positives detected while maintaining the FDR at a pre-specified threshold .', 'Another possible approach to quantify the false positive rate could be the permutation-based strategy suggested by Servin and Stephens [36] .', 'However in our context , such an approach is computationally too expensive as it requires the MCMC underlying CPBayes to be implemented for each permuted dataset .', 'Of note , we did not conduct a replication study and all the pleiotropic association signals obtained from the Kaiser cohort are reported based on the analysis only in the discovery sample .', 'Even though we demonstrated CPBayes only for binary traits , we note that CPBayes can also be applied to non-binary traits .', 'We carried out simulations for continuous traits ( distributed as multivariate normal ) , and observed a similar pattern between the performance of CPBayes and ASSET as for binary traits .', 'Another useful approach to pleiotropy analysis is MultiPhen [37] which can accommodate general types of traits .', 'However , we chose ASSET as the main competing method because it simultaneously provides an optimal subset of associated traits along with a measure of aggregate-level pleiotropic association , which provides a direct comparison to CPBayes .', 'In contrast , MultiPhen does not facilitate the simultaneous selection of the optimal associated traits underlying a pleiotropic signal .', 'Of note , MultiPhen requires individual level phenotype and genotype data .', 'In the analysis of GERA cohort , we assumed that the correlation matrix of estimated effect sizes is the same across SNPs , for each of which , we converted this correlation matrix to its covariance matrix by incorporating its standard error across traits .', 'Hence , if the standard error vector varies across SNPs , the covariance matrix of the effect estimates also varies .', 'We did some simulations to assess this assumption of constant correlation matrix .', 'Consider the simulation framework of a cohort study with five case-control phenotypes designed to evaluate the partial ROC curves for CPBayes .', 'Here the continuous traits ( liability ) underlying the binary traits were generated at random for every SNP following the simulation model in S1 Text; hence the binary traits dataset varied across SNPs .', 'So the correlation matrix of effect estimates obtained by Eq 6 also varied across SNPs .', 'We implemented CPBayes for each SNP using the corresponding effect estimates’ correlation matrix obtained by Eq 6 which is accurate for the null SNPs as there are no environmental covariates here .', 'In the simulated data for a single risk SNP , we anticipate that the real correlation matrix of the effect estimates will be very close to the sample overlap correlation matrix ( Eq 6 ) .', 'Because , a single risk SNP usually explains a very small proportion of total heritability for a complex trait and the genetic correlation between two traits due to a single risk SNP is expected to be very small .', 'Next we estimated a constant correlation matrix as the sample correlation matrix of the observed effect estimates for the null SNPs ( following the GW strategy in GERA cohort analysis ) , and implemented CPBayes for all the SNPs using this constant correlation matrix .', 'We observed that the partial ROC curve for CPBayes obtained by using varying correlation matrices and constant correlation matrix across SNPs had almost the same AUC .', 'We also found by simulations that using the constant correlation matrix provides a very similar selection accuracy compared to using varying correlation matrix across SNPs .', 'We repeated these simulations for a cohort study with five continuous phenotypes distributed as multivariate normal .', 'Since the phenotypes are normally distributed , the real correlation matrix of effect estimates can be analytically computed .', 'Again , CPBayes performed very similarly using constant and varying correlation matrices .', 'If a set of binary traits are measured on separate independent group of individuals , one would expect the summary statistics across traits be independent .', 'However , if these traits are mutually exclusive because of competing risks , risk SNPs for one trait may be underrepresented among the cases of the other trait , leading to a correlation in the summary statistics , especially for high penetrance variants .', 'For common and complex traits , competing risks may not result in mutual exclusivity , and may lead to very limited correlation among subjects from independent samples .', 'Moreover , since such traits arise from many different low risk SNPs , any co
['Introduction', 'Discussion', 'Materials and methods']
['Simultaneous analysis of genetic associations with multiple phenotypes may reveal shared genetic susceptibility across traits ( pleiotropy ) .', 'For a locus exhibiting overall pleiotropy , it is important to identify which specific traits underlie this association .', 'We propose a Bayesian meta-analysis approach ( termed CPBayes ) that uses summary-level data across multiple phenotypes to simultaneously measure the evidence of aggregate-level pleiotropic association and estimate an optimal subset of traits associated with the risk locus .', 'This method uses a unified Bayesian statistical framework based on a spike and slab prior .', 'CPBayes performs a fully Bayesian analysis by employing the Markov Chain Monte Carlo ( MCMC ) technique Gibbs sampling .', 'It takes into account heterogeneity in the size and direction of the genetic effects across traits .', 'It can be applied to both cohort data and separate studies of multiple traits having overlapping or non-overlapping subjects .', 'Simulations show that CPBayes can produce higher accuracy in the selection of associated traits underlying a pleiotropic signal than the subset-based meta-analysis ASSET .', 'We used CPBayes to undertake a genome-wide pleiotropic association study of 22 traits in the large Kaiser GERA cohort and detected six independent pleiotropic loci associated with at least two phenotypes .', 'This includes a locus at chromosomal region 1q24 . 2 which exhibits an association simultaneously with the risk of five different diseases: Dermatophytosis , Hemorrhoids , Iron Deficiency , Osteoporosis and Peripheral Vascular Disease .', 'We provide an R-package ‘CPBayes’ implementing the proposed method .']
['Genome-wide association studies ( GWAS ) have detected shared genetic susceptibility to various human diseases ( pleiotropy ) .', 'We propose a Bayesian meta-analysis method CPBayes that simultaneously evaluates the evidence of overall pleiotropy while determining which traits are pleiotropic .', 'This approach investigates pleiotropy using GWAS summary statistics and allows for overlapping subjects across traits .', 'It performs a fully Bayesian analysis and offers a flexible inference .', 'CPBayes also provides additional information about a pleiotropic signal , such as the trait-specific posterior probability of association and the credible interval of unknown true genetic effects .', 'Using computer simulations and an application to a large GWAS cohort , we demonstrate that CPBayes can offer improved accuracy compared to the existing subset-based meta-analysis approach ASSET .', 'We provide a user-friendly R-package ‘CPBayes’ for general use of this approach .']
['genome-wide association studies', 'medicine and health sciences', 'simulation and modeling', 'diabetes mellitus', 'research design', 'dyslipidemia', 'endocrine disorders', 'genome analysis', 'type 2 diabetes', 'research and analysis methods', 'case-control studies', 'endocrinology', 'peripheral vascular disease', 'genetic loci', 'metabolic disorders', 'phenotypes', 'genetics', 'biology and life sciences', 'genomics', 'vascular medicine', 'computational biology', 'vascular diseases', 'human genetics']