Linking epigenetic marks to clinical outcomes improves insight into molecular processes, disease prediction, and therapeutic target identification. Here, a statistical approach is presented to infer the epigenetic architecture of complex disease, determine the variation captured by epigenetic effects, and estimate phenotype-epigenetic probe associations jointly. Implicitly adjusting for probe correlations, data structure (cell-count or relatedness), and single-nucleotide polymorphism (SNP) marker effects, improves association estimates and in 9,448 individuals, 75.7% (95% CI 71.70–79.3) of body mass index (BMI) variation and 45.6% (95% CI 37.3–51.9) of cigarette consumption variation was captured by whole blood methylation array data. Pathway-linked probes of blood cholesterol, lipid transport and sterol metabolism for BMI, and xenobiotic stimuli response for smoking, showed >1.5 times larger associations with >95% posterior inclusion probability. Prediction accuracy improved by 28.7% for BMI and 10.2% for smoking over a LASSO model, with age-, and tissue-specificity, implying associations are a phenotypic consequence rather than causal. Linking epigenetic marks to clinical outcomes promises insight into the underlying processes. Here, the authors introduce a statistical approach to estimate associations between a phenotype and all epigenetic probes jointly, and to estimate the proportion of variation captured by epigenetic effects. The identification of clinically relevant epigenetic loci can provide insight into the molecular underpinning of disease3, leading to identification of biologically relevant therapeutic targets4 and potentially epigenetic-guided clinical decision making5.Data characterizing gene expression, protein structure, or epigenetic modifications such as DNA methylation, histone marks and nucleosome positioning are becoming increasingly available. Epigenetic marks reflect a wide range of environmental exposures and genetic influences, are critical for regulating gene and non-coding RNA expression6. This methodology does not account for correlations among probes and leads to model over-fitting, poor effect size estimation, and poor calibration of prediction owing to omitted variable bias7. In addition, data structure such as intra-sample cellular heterogeneity, sample relatedness, population stratification, or experimental design effects are a major challenge8 and result in more cross-chromosome correlation than genetic data. This structure, in conjunction with the fact that cases and controls typically differ in their cell-type composition, can result in spurious associations and many statistical algorithms have been proposed to tackle these potential biases10. However, all current statistical approaches rely upon corrections for structure that require a choice of either a suitable reference profile of representative cell types, or a limited number of pre-selected variables computed from the methylation data , with the underlying assumption that all confounders are reflected by a sparse set of latent covariates and methylation sites.Most studies testing for association between genomic data and complex traits utilize methodology from genome-wide associations studies, meaning that probe effects are tested one at a time13, given that we can assign different prior distributions to different sources of variance . We demonstrate properties (i) through (vii) with theory, simulation and then empirical analysis of body mass index (BMI) and smoking behavior for 9448 individuals with methylation probe measures from whole blood14.Here, we present an alternative approach, based on Bayesian inference, that: (i) estimates probe effects on an outcome jointly whilst adjusting for other covariates such as sex and age, avoiding model over-fitting and controlling for both data structure (including cell-count effects) and correlations among probes; (ii) does not require any knowledge of cell-type composition or any selection of proxy confounder variables ; (iii) estimates the total proportion of disease risk accounted for by the probe effects (cumulative proportion of variance explained); (iv) estimates probe effects conditional on other sources of data such as single-nucleotide polymorphism data, enabling a determination of the unique contribution of different data; (v) gives an in-depth understanding of the genome-wide range of probe effects in terms of the likely number of independent effects and their variance explained; (vi) can incorporate genomic annotation information into the analysis when estimating probe effects, facilitating unique genome-wide enrichment analyses, describing the variance explained and number of trait-associated probes of each annotation; and (vii) provides improved estimation of biomarker effects, which could be used for disease risk assessment. The approach is similar to, but more flexible than linear mixed model analyses recently proposed in genome-wide association studiesy is reflected by a linear combination of genetic effects (βG) estimated from single-nucleotide polymorphism (SNP) data, epigenetic effects (βcpg) estimated from probes on an array, along with age and sex specific effects , such that:βG, βcpg, α, and γ being estimated in a Bayesian statistical model. Unlike previous approaches that assign a mixture of Gaussian distributions and a discrete spike at zero as a prior for all effects, we assign a new set of mixtures to each group, effectively augmenting the number of hyperparameters proportionally to the number of groups of variables (two in the case of SNPs and methylation probes). This allows for non-identifiable effects to be excluded from the model, whereas the rest are estimated jointly. Adjusting for all different covariates and their effects while estimating each individual probe effect, better alleviates problems related to correlations and structure in the marker data as we show in our simulation study , negating the need to add additional controls for cell counts or principal components within the model.Our approach assumes that the observed phenotype Our software implementation of this modeling framework,  is entirely flexible. Unlike existing methods, any number of data sources can be modeled together, each with separate mixtures, making it applicable to any kind of genetic or epigenetic data. Owing to efficient computational implementation (see Methods), it is also entirely scalable to future data sizes. Furthermore, if only epigenetic probe data are available, estimates of the probe effects would still be obtained jointly, avoiding model over-fitting and controlling for both data structure (including cell-count effects) and correlations among probes , which cumulatively explained 60% of the phenotypic variance in the trait. We focus in the main text on two scenarios where probes are associated with cell-type proportion variation and the norm of the correlation vector between the phenotype and the cell-type proportions is either 0.08 or 0.25. These scenarios reflect different degrees of confounding between phenotype and cell-type proportions. We then conduct additional simulations with a wide a range of settings, varying the cell-type proportions, the proportions of differentially methylated probes, the variance of differentially methylated probes, and the variance of the measurement noise and we present these within the Supplementary Information .We simulated methylation data for 2000 individuals at 103,638 probes with five different cell types. We reproduced cell-type proportion variation present in real methylation probe data, using a recently proposed simulation model12), single-probe mixed linear model association test that estimates probe associations one-by-one conditional on a relationship matrix estimated from the probe data (OSCA-moa13), a multi-probe ridge regression that estimates all probe associations jointly and conditionally on latent factors (LFMM2-ridge11), and a multimarker LASSO model, which estimates all probe associations jointly and conditionally on latent factors (LFMM2-lasso11). Our BayesRR approach outperforms these approaches as it estimates phenotype-probe associations more accurately with higher correlation of the estimated effects with the true simulated values and with lower mean square error , all other approaches only enable estimation of the proportion of variance attributable probes identified as genome-wide significant and thus do not provide an estimate of the total association between the phenotype and the probe data  of the phenotypic variance was captured by methylation probes, with 39.5% 95% CI 28.3–49.7) of this attributable to 509.3 (95% CI 348–663) probes that each explain ~0.1% of the phenotypic variance . We repeated the analysis but excluded close relatives and modeled only the methylation probe effects in 2,614 unrelated individuals, finding 69.55% (95% CI 57.35–78.32) of the phenotypic variance for BMI and 73.83% (95% CI 54.33–88.26) for smoking, respectively. We also used a linear mixed effects model where probe values are used to calculate a co-variance matrix, which is then used in a restricted maximum likelihood estimation (REML) approachWe then proceeded to derive annotation information from the posterior distribution over effects to provide some biological inference. First, we looked for Gene Ontology (GO) enrichment for the probes whose IP was  >95% (see Methods). From the 20 top-enriched terms, we find those corresponding to thrombin-related pathways and cerebral cortex development for smoking . Under our enrichment statistic, we can measure those GO terms, which explain a greater proportion of phenotypic variance than expected, given the proportion of probes that map to the GO term  specifically to a given tissue if the absolute value of its t score for that tissue was in the 0.001 or 0.999 quantiles of the t distribution. We then mapped the probes to genes and genes to tissues, probes that did not mapped to a gene in the GTEx data were put in a different group. In order to achieve non overlapping associations of probes to tissues (as required for our model), we assigned each gene to the tissue they were the most specific to . The probes were therefore put in 46 different groups (one for each tissue available in GTEx) when feasible. Some tissues had too few probes associated to them, we therefore lumped them together in group 47 (others) so that every group contained at least 200 probes. Probes that were mapped to genes that do not appear in the array were put in group 48, age and sex were assigned to group 49, and SNPs to group 50. Each one of these groups were assigned the prior mixture variances of . We estimated the variance attributable to each group of probes and further decomposed these estimates by the mixture they belonged  for adult BMI in LBC and 30.83% with a slope of 0.86 (0.041 SE) in TwinsUK (Table R2 of 3.34% with a slope of 0.38 (0.061 SE) for birth weight, 2.05% with slope of 0.3 (0.069 SE) for BMI at age 7, 9.65% with a slope of 0.6 (0.071 SE) for BMI age 15 and up, 24.43% with a slope of 0.84 (0.069 SE) for BMI in adult males, and 18.36% with a slope of 0.7 (0.069 SE) for BMI in adult females (Table R2 of 47.9% with slope of 1.02 (0.058 SE) for LBC and 38.49% with slope of 1.01 (0.057 SE) for males in ARIES, 13.7% more than the LASSO predictor , the Accessible Resource of Integrated Epigenomics Studies (ARIES) data set and the UK Adult Twin (TwinsUK) Registry (see Methods). We compared the prediction accuracy gained from our approach to recently obtained estimates from another model based on the LASSO estimator, using the R2 with a slope of 0.4  an R2 of ~25% is expected for BMI within our prediction samples, which is in-line with the values we obtain here. If the initial study sample size increased to 100,000 individuals than an R2 of over 60% is expected, which in combination with SNP array data, would lead to a predictor of BMI with an R2 of ~80% from a single blood test. However, the theory described above does not account for the fact that probe variation is likely a consequence of the phenotype and thus in a regression equation, phenotypic variance will appear on both sides. If the consequential effects are large and there are considerable changes with age that occur29, or differences in effects across tissues, then the prediction accuracy obtained from methylation probe data will likely differ across cohorts as demonstrated here.It is important to note that the sample size and variance captured by all of the probe effects in the training data govern the prediction accuracy obtaining in the testing data. FollowingWe present BayesRR, a statistical model for joint inference of genetic and epigenetic effects over complex phenotypic traits. Using simulation, we show that BayesRR outperforms other approaches as it has the advantage of controlling for all factors at once and performing statistical inference jointly on all of the model parameters and adjusting estimates conditionally on each other. By working in a Bayesian framework, we derive a rich representation of the estimated effects through probability distributions, where all markers are taken into account and for which we can assess genome-wide enrichment of relevant biological features. From these distributions, we conclude that from the same set of probes in the same individuals, two example phenotypes show different architecture in the distribution of their effects, with the distribution of effects for cigarette consumption being more concentrated in a few epigenetic markers (15 markers with 95% IP explaining ~26.78%), whereas for BMI, we have more probes associated with the phenotype (17 markers with 95% IP explain only 9.70%). Our genome-wide enrichment analyses, identified blood cholesterol, lipid transport, and sterol metabolism pathways for BMI, and response to xenobiotic stimulus for smoking, all with  >95% posterior IP of having methylation probes with effects sizes  >1.5 times larger than the average. For both BMI and cigarette consumption, a large amount of phenotypic variance is captured by epigenetic markers in the training data set, which may be expected as trait-associated DNA methylation probe variation is likely to a large degree to be a consequence of the phenotype, as evidenced in our enrichment analyses and the prediction results from the ARIES study. These consequential effects lead to the expectation that if applied to common complex disease, the model we present may enable accurate characterization of disease progression and better identification of individuals who are on a path to disease where future diagnosis is likely . It remains to be seen whether such large amounts of phenotypic variance can be captured by a methylation array for common complex disease, but our prediction results shows that our approach can better describe the overall distribution of associations leading to improved phenotypic prediction.29, it is highly unlikely that the phenotypic variance attributable to probes is stable across cohorts and with age. Determining biomarkers for future disease outcomes requires a different experimental design, for example, longitudinal studies with a baseline, along with methodological extensions for causal inference within this framework, which our future work will focus on. In addition, while we extend the model to ask how much additional phenotypic variance of each trait can be captured by methylation probes from whole blood above that captured by a SNP array, partitioning the phenotypic variance explained exactly may be difficult in data sets where factors are highly correlated. Furthermore, although we present a whole-genome enrichment approach, identifying novel pathways is currently limited and technological improvements are required to improve our ability to capture, define, and understand epigenetic marker variation. Finally, Bayesian inference comes at increased computational cost and requires the specification of prior distributions, for example here, that effects can be well described by a series of Gaussian distributions. A Student-t likelihood could be a path worth exploring as its inferences could be more robust to outliers30 and additionally, although a Gaussian model may still be applied to categorical disease-or-not measurements, developing an extension to model binary response variables and explore performance in unbalanced case–control settings will likely be worthwhile.There are a number of important considerations and caveats. It is important to punctuate that the inferred associations only relate to the present state of the biomarkers and are not intended to capture any causality between methylation status and outcome. Given the highly variable nature of “omics” measures, the variation across data sets in the degree of confounding by experimental biases or unwanted biological variation that will contribute to the variation captured by probes, and the considerable changes with age that occurIn conclusion, our model can be applied to any kind of genomics data providing unbiased estimates of marker effects, conditional on other markers, covariates and on the data structure, without the need for specific cell-type proportion control. By operating in a Bayesian framework, the uncertainties over the estimates given the data are represented explicitly, helping the researcher to interpret and draw conclusions over the architecture of the variance in the trait. We provide freely available software with source code available to facilitate further replication and potential applications of the methodology (see “Code Availability”).α, γ associated over a vector of measurements over a trait Xcpg and the genotype matrix XG have been centered and scaled to unit variance. We assume that only a subset of y, as such, and proceeding in a Bayesian framework, we assign a sparsity inducing prior over Θ. The chosen prior follows the formulation of ref. 31, which is a mixture of L Gaussian probability densities and a discrete “spike” at zero. As such, each Θi ∈ Θ is distributed according to:δ0 is a discrete probability mass at zero.We assume additive probe effects σ2, and the component-specific variances are proportional to σ2, that isWe further constrain the prior by assuming a single parameter representing the total variance explained by the effects σ2 and π parameters, our contention is that in cases where one data source’s effect are in a different scale than on the other, the extra degrees of freedom will allow to better resolve the smaller effects, whereas the group specific variance parameters will pool information of the effects within a group and keep the model identifiable. In our case, we assign to the genetic effects βG a set of mixture variances πG and variance parameter Our main contribution consists of allowing different subsets of Θ to have specific 31 but with additional parameters for groups G and ϕ.The rest of the model follows the prior hierarchy of ref. y and observed covariates Inference of the probabilistic model follows a Gibbs sampling algorithm. Here, the joint posterior probability density of parameters mG and mϕ are the number of markers in each respective category in a sample and #KG, #Kϕ are vectors that contain the number of markers in each mixture for the respective categories.Given the prior distributions presented in Eq. 2, the conditional distributions are:m and co-variance Σ such thati\ represents all the effects except effect Θi. Then each individual update consists of:The most computationally expensive step of sampling from the distribution in Eq. 2 involves drawing from the conditional distributionϵ = y − μ − ZΘ, then we can compute efficiently y − μ − ZiΘi by the update i to update, as seen in successful implementations of the algorithm31.This obviates the necessity of inverting matrix Σ, if in addition we keep in memory the vector of residuals l from which to draw the effect Θi, we must evaluate the likelihood ratio between all the mixtures. Using the log-likelihood l is given by:To select the mixture 33 and Intel’s Threading Building Blocks library34. Source code available in https://github.com/ctggroup/BayesRRcmd.The algorithm was implemented in C++−11.0, with the help of the templated matrix algebra library Eigen12, where the matrix M is decomposed ini of R is distributed according toWe use a generative model of methylation levels as in ref. 12. We performed a slight modification to be able to change the variance of the cell proportions matrix P. For each simulation we generated a matrix of N = 2000 and M = 103,638. The simulation parameters were in accordance to12:p).Proportion of differentially methylated probes (τ).Variance of the DMPs(sp).Variance of the cell proportions (ϑ)Variance of the measurement noise , for 20,000 samples with 10,000 samples of burn-in after which a thinning of 10 samples was used to select samples of the posterior for the simulation. We selected trait-associated probes as those that were in the model in  >95% of the posterior samples, which we define as 95% posterior IP.For each of the 270 simulations, we ran our method for the observed phenotype MSEsig), the false discovery rate (FDR), the norm of the correlation vector between a individual-level predictor made from the probe effects and the cell-type proportions ∣∣) and the phenotypic variance attributable to the probes : We conducted a set of 103,638 linear regressions using the function lm from R version 3.4.2, and accepted the effect sizes whose p value (t test) was <0.05/103638 (Bonferroni correction).Single-probe least squares regression with sparse latent factors (ReFACTor): Using the approach outlined in ref. 12, we selected five sparse components (one for each cell type in the simulation) and estimated probe associations conditional upon these one-by-one. We selected associated probes whose p value (t test) was less than 0.05/103638.Single-probe mixed linear model association analysis (OSCA): Using the approach outlined in13, we used a linear mixed effects model where probe values are used to calculate a co-variance matrix, which is then used in a REML approach to estimate the proportion of phenotypic variance attributable to the probes, and conditional upon this probe effects are estimated one-by-one. Although this estimates probe associations conditional on the co-variance matrix, it does not account for genome-wide co-variance across probes. We selected associated probes whose p value (t test) was <0.05/103,638.Multi-probe penalized regression with latent factors (LFMM): Using the approach outlined in ref. 11, we ran LASSO and ridge regression with 10-fold cross-validation using the default settings , whereas also fitting latent factors within the model that are intended to control for cell-type proportion confounding. We used the lfmm_function and selected probes based on the calibrated p value (t test) whose p value (t test) was <0.05/103,638.Multi-probe penalized regression without latent factors (glmnet): We ran LASSO and ridge regression with 10-fold cross-validation using the default settings of package glmnet16 version 2.0-16. Previous experiments suggested that best performance was achieved by leaving the phenotype vector y un-scaled.M with parameters p = 0.15, τ = 0.07, sp = 1000, ϑ = 0.01 as above, with the same number of SNPS as the methylation matrix . Then, we generated a genotype matrix M we select column j of X to have a correlated genotype by sampling its elements from the distribution:We simulated a methylation matrix O (the noisy observations over matrix M) and X centered and scaled, we generated 100 methylation effects y   Oβcg + Xβg + ϵ, with Having both matrices 16, with a baseline of single marker regression (GWAS) where we first adjusted the phenotype by the first 10 principal components of the genotype matrix and then regressed the residuals against the scaled methylation matrix. The methods were compared over the estimation of the true genetic and epigenetic VE and ability to estimate the true effects. Results shown in Supplementary Fig. We repeated the process 15 times to have 15 simulated data sets. For each of the simulated data sets we ran our method for 200,00 samples, with a burn-in of 100,00 and a thinning of 10 samples. The mixture variances were set to  for methylation effects and to  for genotype effects. We compared out approach with LASSO and Ridge regression implemented in glmnet29. In briefly, outliers based on the visual inspection of methylated to unmethylated log intensities were excluded, along with poorly performing probes and samples, and sex mismatches (predicted based on genetics versus questionnaire data) yielding an analysis data set of 9448. As reported in McCartney et al.29, further filtering was performed to exclude non-autosomal CpG sites and sites that were exclusive to the EPIC array. This allowed for the predictors to be applied to data sets that collected DNA methylation using an earlier version of the Illumina arrays (450 k array) giving a total of 370,262 probes.Generation Scotland: the Scottish Family Health Study is a large population-based, family-structured cohort of over 24,000 individuals aged 18–99 years. The study baseline took place between 2006 and 2011 and included detailed cognitive, physical, and health questionnaires, along with sample donation for genetic and biomarker data. DNA methylation data from whole blood was obtained on a subset of ~10,000 participants. The Illumina HumanMethylationEPIC Bead Chips array was used to measure methylation and quality control details have been reported previously35 and the 36, with the help of the R package ggmcmc37, see Supplementary Figs. R2 of 0.97 slope of 0.99 for BMI and R2 of 0.99 slope of 1 for smoking.After the quality control steps described above, we integrated the SNP marker and methylation matrices, along with the log-transformed age and the sex of the individuals . All matrices and phenotypes were centered and scaled to variance 1. The data were used as input for our Bayesian model, with parameters  for genetic effects mixtures variances and  for the epigenetic effects, age, and sex. Four chains for each trait with different starting values were executed. We assessed the convergence of the hyperparameters entclass1pt{minimaWe then contrasted the variance explained by the first five PCs of the probes with 95% PIP in BMI and smoking, to predict their respective phenotypes. We found that these first five PCs in BMI probes do not have predictive power for BMI, thus, this suggests that the main axis of variation of these probes with 95% PIP do not explain as much of the phenotype as the whole set of probes with 95% PIP for BMI. For smoking the five PCs derived from the probes with 95% PIP for the same phenotype explain ~44% of the variance. We further verified that the predictive power of the 46 covariates  explain 7% variance in BMI and 10% variance in smoking. We observed that the first PC for the probes with 95% PIP for BMI is highly correlated with cell counts  in the current sample, we havep.Term being the proportion of probes mapping to the term among all the probes mapping to a term in the current sample. Having #probesTerm being the number of probes mapping to a term and #probesModel the number of probes in the model in the current sample, we haveThen, we associated the mapped genes with their respective terms in the Gene Onthology (GO) using the R package23, for which, we accept the hypothesis that a term is significantly over/under-enriched if 95% of the posterior mass for the enrichment value is outside the interval  and the term has an IP  >95%. Significantly enriched GO terms are presented in the Supplementary Information.We also computed the IP for a term by counting the times a term appears in the model and dividing by the number of samples. Finally, given that we have a posterior distribution over enrichment values, we adopt the ROPE decision ruleR2 statistic.For both BMI and smoking, the posterior samples over effects where averaged and associated to their respective probe and SNP identities. For each replication cohort, a predictor was built by multiplying the posterior mean effects by the corresponding centered and scaled genetic and epigenetic marker readings, and predictive ability measured over the scaled and centered cohort trait was measured using the 41. It follows 1091 members of the 1947 Scottish Mental Survey, who were recontacted in later life, when they were living in the Edinburgh area of Scotland. The cohort members were all born in 1936 and have been assessed for a wide variety of health and lifestyle outcomes at ages 70, 73, 76, 79, and 82 years. DNA has been collected at each clinical visit. In the present study, we considered DNA methylation data (Illumina 450 k array) from whole blood, taken at mean age 70, for analysis. Details of the collection and processing of the data have been reported previously29. In brief, after quality control to remove poorly performing methylation sites, samples, and individuals with mismatching genotypes or predicted sex, a sample of 906 individuals was available for prediction analysis. The genotype and methylation matrices were processed as with GS, given that the posterior effect sizes for age and sex were equal to zero for both traits, they were not included.The Lothian Birth Cohort 1936 is a longitudinal study of aging42. Whole-blood DNA methylation profiles have previously been described43. Additional data processing steps for this project included use of ENmix44 for quality control and minfi45 to exclude samples with median methylated and unmethylated signals below 10.5. After data-quality control, covariate assessments, and sample processing, downstream analyses were carried out in 540 adipose  and 977 whole blood  TwinsUK samples. Ethical approval was granted by the National Research Ethics Service London-Westminster, the St Thomas’ Hospital Research Ethics Committee (EC04/015 and 07/H0802/84). All research participants provided written informed consent prior to taking part in the study.The TwinsUK registry consists of over 14,000 research volunteer twin participants from the United Kingdom who have joined since 1992, with equal numbers of same-sex monozygotic and dizygotic twin pairs who are predominately female (84%). Data are collected through longitudinal questionnaires and clinical visits. The registry collects biological samples and further data collected through analysis of biological samples. DNA methylation profiles were generated using the Infinium HumanMethylation450 BeadChip in adipose tissue biopsies and whole-blood samples from TwinsUK participants. Adipose tissue DNA methylation data were generated from subjects who were free from severe diseases, as previously described46,47. Blood from 1018 mother–child pairs (children at three time points and their mothers at two time points) were selected for analysis as part of the Accessible Resource for Integrative Epigenomic Studies 48. Following DNA extraction, samples were bisulphite converted using the Zymo EZ DNA Methylation kit . Following conversion, genome-wide methylation was measured using the Illumina Infinium HumanMethylation450 (HM450) BeadChip. The arrays were scanned using an Illumina iScan, with initial quality review using GenomeStudio. ARIES was preprocessed and normalized using the meffil R package49. ARIES consists of 5469 DNA methylation profiles obtained from 1022 mother–child pairs measured at five time points . Low-quality profiles were removed from further processing, and the remaining 4593 profiles were normalized using the Functional Normalization algorithm50 with the top 10 control probe principal components. Full details of the preprocessing and normalization of ARIES have been described previously49.Samples were drawn from the Avon Longitudinal Study of Parents and ChildrenFurther information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


N = 90K) and Europeans (average N = 267K) with an average trans-ethnic genetic correlation of 0.85. We determine that squared trans-ethnic genetic correlation is 0.82× (s.e. 0.01) depleted in the top quintile of background selection statistic, implying more population-specific causal effect sizes. Accordingly, causal effect sizes are more population-specific in functionally important regions, including conserved and regulatory regions. In regions surrounding specifically expressed genes, causal effect sizes are most population-specific for skin and immune genes, and least population-specific for brain genes. Our results could potentially be explained by stronger gene-environment interaction at loci impacted by selection, particularly positive selection.Many diseases exhibit population-specific causal effect sizes with trans-ethnic genetic correlations significantly less than 1, limiting trans-ethnic polygenic risk prediction. We develop a new method, S-LDXR, for stratifying squared trans-ethnic genetic correlation across genomic annotations, and apply S-LDXR to genome-wide summary statistics for 31 diseases and complex traits in East Asians (average  Trans-ethnic genetic correlation is significantly less than 1 for many diseases. Here, the authors stratify this correlation by genomic annotations, finding that loci whose causal disease effect sizes differ between ethnicities are likely impacted by selection, particularly positive selection. However, current methods for estimating genome-wide trans-ethnic genetic correlations assume the same trans-ethnic genetic correlation for all categories of SNPs13, providing little insight into why causal disease effect sizes are population-specific. Understanding the biological processes contributing to population-specific causal disease effect sizes can help inform polygenic risk prediction in non-European populations and alleviate health disparities15.Trans-ethnic genetic correlations are significantly less than 1 for many diseases and complex traits16); we stratify the squared trans-ethnic genetic correlation across functional categories to robustly handle noisy heritability estimates. S-LDXR analyzes GWAS summary statistics of HapMap317 SNPs with minor allele frequency (MAF) greater than 5% in both East Asian (EAS) and European (EUR) populations (regression SNPs) to draw inferences about causal effects of all SNPs with MAF greater than 5% in both populations (heritability SNPs). We confirm that S-LDXR yields robust estimates in extensive simulations. We apply S-LDXR to 31 diseases and complex traits with GWAS summary statistics available in both East Asian (EAS) and European (EUR) populations, leveraging recent large studies in East Asian populations from the CONVERGE consortium and Biobank Japan20; we analyze a broad set of genomic annotations from the baseline-LD model23, as well as tissue-specific annotations based on specifically expressed gene (SEG) sets24. Most results are meta-analyzed across the 31 traits to maximize power , as we expect to see similar patterns of enrichment/depletion across traits . We also investigate trait-specific enrichments/depletions for the tissue-specific annotations .Here, we introduce a new method, S-LDXR, for estimating enrichment of stratified squared trans-ethnic genetic correlation across functional categories of SNPs using GWAS summary statistics and population-matched linkage disequilibrium (LD) reference panels (e.g. the 1000 Genomes Project (1000G)22 (S-LDSC), a method for partitioning heritability from GWAS summary statistics while accounting for LD. S-LDXR determines that a category of SNPs is enriched for trans-ethnic genetic covariance if SNPs with high LD to that category have a higher product of Z-scores than SNPs with low LD to that category. Unlike S-LDSC, S-LDXR models per-allele effect sizes (accounting for differences in MAF between populations).Our method (S-LDXR) for estimating stratified trans-ethnic genetic correlation is conceptually related to stratified LD score regressionj in two populations, Zj1Zj2, has the expectationNp is the sample size for population p; ℓ× = ∑krjk1rjk2σj1σj2aC(k) is the trans-ethnic LD score of SNP j with respect to annotation C, whose value for SNP k, aC(k), can be either binary or continuous; rpjk is the LD (Pearson correlation) between SNP j and k in population p; σpj is the standard deviation of SNP j genotypes in population p; and θC represents the per-SNP contribution to trans-ethnic genetic covariance of the per-allele causal disease effect size of annotation C. Here, rpjk and σpj can be estimated from population-matched reference panels (e.g. 1000 Genomes Project16). We estimate θC for each annotation C using weighted least square regression. Subsequently, we estimate the trans-ethnic genetic covariance of each binary annotation C (ρg(C)) by summing trans-ethnic genetic covariance of each SNP in annotation C as squared trans-ethnic genetic correlation, defined asλ2(C) can be meta-analyzed across traits with different 17 SNPs (regression SNPs) to estimate λ2(C) for  all common SNPs (heritability SNPs). Further details  of the S-LDXR method are provided in the Methods section and the Supplementary Notes; we have publicly released open-source software implementing the method (see “Code availability”).In detail, the product of Z-scores of SNP 23 (v1.1), and 53 SEG annotations (Supplementary Data We apply S-LDXR to 62 annotations (defined in both EAS and EUR populations) from our baseline-LD-X model  (“Methods”); we did not have access to individual-level EAS data at sufficient sample size to perform simulations with real genotypes. For each population, we randomly selected a subset of 500 simulated samples to serve as the reference panel for estimating LD scores. We performed both null simulations  = 1) and causal simulations (λ2(C) ≠ 1). In our main simulations, we randomly selected 10% of the SNPs as causal SNPs in both populations, set genome-wide heritability to 0.5 in each population, and adjusted genome-wide genetic covariance to attain a genome-wide rg of 0.60 (unless otherwise indicated). In the null simulations, we used heritability enrichments from analyses of real traits in EAS samples to specify per-SNP causal effect size variances and covariances. In the causal simulations, we directly specified per-SNP causal effect size variances and covariances to attain λ2(C) ≠ 1 values from analyses of real traits, as these were difficult to attain using the heritability and trans-ethnic genetic covariance enrichments from analyses of real traits.We evaluated the accuracy of S-LDXR in simulations using genotypes that we simulated using HAPGEN2rg); we note that S-LDXR does not use the shrinkage estimator for genome-wide estimates. Across a wide range of simulated rg values (0.0 to 1.0), S-LDXR yielded approximately unbiased estimates and well-calibrated jackknife standard errors  in quintiles of the 8 continuous-valued annotations of the baseline-LD-X model. We performed both null simulations (λ2(C) = 1) and causal simulations (λ2(C) ≠ 1). Results are reported in Fig. λ2(C) for most annotations, validating our analytical bias correction. As a secondary analysis, we tried varying the S-LDXR shrinkage parameter, α, which has a default value of 0.5. In null simulations, results remained approximately unbiased; in causal simulations, reducing α led to less precise (but less biased) estimates of λ2(C), whereas increasing α biased results towards the null (λ2(C) = 1), demonstrating a bias-variance tradeoff in the choice of α  for the 28 main binary annotations of the baseline-LD-X model (inherited from the baseline model of ref. 21). We discarded λ2(C) estimates with the highest standard errors (top 5%), as estimates with large standard errors  are uninformative for evaluating unbiasedness of the estimator . Results are reported in Fig. λ2(C), further validating our analytical bias correction. In causal simulations, estimates were biased towards the null (λ2(C) = 1)—particularly for annotations of small size (proportion of SNPs < 1%)—due to our shrinkage estimator; increasing the shrinkage parameter above its default value of 0.5 further biased the estimates towards the null (λ2(C) = 1) in causal simulations , well-calibrated standard errors and conservative p-values in these simulations . S-LDXR yielded robust estimates of 28, defined as architectures in which the level of MAF-dependence is annotation-dependent, to ensure that estimate of λ2(C) remains unbiased. We disproportionately sampled low-frequency causal variants from the top quintile of background selection statistic, and set the variance of per-allele effect sizes of a causal SNP to be inversely proportional to its maximum MAF across both populations (Methods). Results are reported in Supplementary Figs. λ2(C) for the 28 binary functional annotations  for most quintiles of continuously valued annotations . We again determined that the small systematic biases were generally on the same order of magnitude as for NEAS = 18K, NEUR = 37K  . We simulated GWAS summary statistics based on the baseline-LD-X model as well as the model with annotation-dependent MAF-dependent genetic architectures. We determined that the small systematic biases in null simulations of continuous-valued annotations were on the same order of magnitude as for 500 reference samples  (as well as population-specific heritabilies) for each trait. Results are reported in Supplementary Fig. rg across 31 traits was 0.85 (s.e. 0.01) ; the lowest rg was 0.34 (s.e. 0.07) for Major Depressive Disorder (MDD), although this may be confounded by different diagnostic criteria in the two populations29. Several other complex traits, including Age at Menopause (rg = 0.57 (s.e. 0.09)) and LDL (rg = 0.66 (s.e. 0.11)) also had low trans-ethnic rg, likely due to pervasive gene-environment interaction across the genome. These estimates were consistent with estimates obtained using Popcorn2  heritability greater than 0.2 in both populations), which are expected to have different genetic architectures.We applied S-LDXR to 31 diseases and complex traits with summary statistics in East Asians (average N = 90K) and Europeans (average N = 267K) available from Biobank Japan, UK Biobank, and other sources (Supplementary Table λ2(C)) in quintiles of the 8 continuous-valued annotations of the baseline-LD-X model, meta-analyzing results across traits; these annotations are moderately correlated  in all analyses. Results are reported in Fig. λ2(C) < 1, implying more population-specific causal effect sizes) in functionally important regions. For example, we estimated λ2(C) = 0.82 (s.e. 0.01) for SNPs in the top quintile of background selection statistic ; λ2(C) estimates were less than 1 for 29/31 traits, including 2 traits (Height and EGFR) with two-tailed p < 0.05/31. The background selection statistic quantifies the genetic distance of a site to its nearest exon; regions with high background selection statistic have higher per-SNP heritability, consistent with the action of selection, and are enriched for functionally important regions22. We observed the same pattern for CpG content and SNP-specific Fst  = 0.87 (s.e. 0.03) for SNPs in the top quintile of average LLD (which is positively correlated with background selection statistic), although these SNPs have lower per-SNP heritability due to a competing positive correlation with predicted allele age22. We caution that average LLD was the annotation most susceptible to bias in our simulations; see “Simulations”. Likewise, we estimated λ2(C) = 0.84 (s.e. 0.02) for SNPs in the bottom quintile of recombination rate (which is negatively correlated with background selection statistic), although these SNPs have average per-SNP heritability due to a competing negative correlation with average LLD22. However, λ2(C) < 1 estimates for the bottom quintile of GERP (NS) (which is positively correlated with both background selection statistic and recombination rate) and the middle quintile of predicted allele age are more difficult to interpret. For all annotations analyzed, heritability enrichments did not differ significantly between EAS and EUR, consistent with previous studies31. Results were similar at a more stringent shrinkage parameter value  for the 28 main binary annotations of the baseline-LD-X model (Supplementary Data λ2(C)  within these annotations: 17 annotations had λ2(C) < 1, and 5 annotations had λ2(C) significantly less than 1 after correcting for 20 annotations tested (P < 0.05/20). These annotations included Conserved (λ2(C) = 0.93 (s.e. 0.02)), Promoter (λ2(C) = 0.85 (s.e. 0.04)) and Super Enhancer (λ2(C) = 0.93 (s.e. 0.02)), each of which was significantly enriched for per-SNP heritability, consistent with ref. 21. For all annotations analyzed, heritability enrichments did not differ significantly between EAS and EUR  across 10 MAF bin annotations; we did not observe variation in λ2(C) estimates across MAF bins  < 1) within the 20 binary annotations could be explained by the 8 continuous-valued annotations. For each binary annotation, we estimated its expected λ2(C) based on values of the 8 continuous-valued annotations for SNPs in the binary annotation (Methods), meta-analyzed this quantity across traits, and compared observed vs. expected λ2(C)  < 1) within binary annotations are largely explained by corresponding values of continuous-valued annotations.Since the functional annotations are moderately correlated with the 8 continuous-valued annotations (Supplementary Data (C) Fig. . We obseIn summary, our results show that causal disease effect sizes are more population-specific in functionally important regions impacted by selection. Further interpretation of these findings, including the role of positive and/or negative selection, is provided in the “Discussion” section.24 as ±100kb regions surrounding the top 10% of genes specifically expressed in each of 53 GTEx32 tissues ) is standardized with respect to heritability (i.e. increase in heritability in the denominator would lead to increase in trans-ethnic genetic covariance in the numerator (Eq. (λ2(C) estimates across the 31 diseases and complex traits (trait-specific estimates are assessed below).We analyzed 53 SEG annotations, defined in ref. λ2(C) estimates were less than 1 for all 53 tissues and significantly less than 1 (p < 0.05/53) for 37 tissues, with statistically significant heterogeneity across tissues . The strongest depletions of squared trans-ethnic genetic correlation were observed in skin tissues (e.g. λ2(C) = 0.83 (s.e. 0.02) for Skin Sun Exposed (Lower Leg)), Prostate and Ovary (λ2(C) = 0.84 (s.e. 0.02) for Prostate, λ2(C) = 0.86 (s.e. 0.02) for Ovary) and immune-related tissues (e.g. λ2(C) = 0.85 (s.e. 0.02) for Spleen), and the weakest depletions were observed in Testis (λ2(C) = 0.98 (s.e. 0.02); no significant depletion) and brain tissues (e.g. λ2(C) = 0.98 (s.e. 0.02) for Brain Nucleus Accumbens ; no significant depletion). Results were similar at less stringent and more stringent shrinkage parameter values  estimates  results were consistent with the higher background selection statistic30 in Skin Sun Exposed (Lower Leg) (R = 0.17), Prostate (R = 0.16), and Spleen (R = 0.14) as compared to Testis (R = 0.02) and Brain Nucleus Accumbens  (R = 0.08)  or CpG content (R = −0.20) (in ±100kb regions) implies limited confounding. We note the well-documented action of recent positive selection on genes impacting skin pigmentation38, the immune system39, and Ovary40; we are not currently aware of any evidence of positive selection impacting Prostate. We further note the well-documented action of negative selection on fecundity- and brain-related traits41, but it is possible that recent positive selection may more closely track differences in causal disease effect sizes across human populations, which have split relatively recently42 (see “Discussion”).These λ2(C) results could be explained by the 8 continuous-valued annotations  within SEG annotations are explained by corresponding values of continuous-valued annotations.More generally, since SEG annotations are moderately correlated with the 8 continuous-valued annotations . We observed a significant depletion (λ2(C) = 0.88 (s.e. 0.03)), further implicating positive selection  conditional on other annotations; see “Discussion”). In addition, we observed a high genome-wide trans-ethnic genetic correlation for schizophrenia (rg = 0.95 (s.e. 0.04) vs. average of 0.85 (s.e. 0.01) across traits), a psychiatric disorder hypothesized to be strongly impacted by negative selection46, suggesting that negative selection may play a limited role in population-specific causal effect sizes. As noted above, these estimates pertain to parameters that were defined based on common variants (see “Overview of methods”); we note that although negative selection has the strongest impact on low-frequency variants26, common variants are also impacted by negative selection and can inform inferences about negative selection22. The role of positive selection (as opposed to negative selection) in population-specific causal effect sizes is discussed further in the Discussion section.The strong depletion of squared trans-ethnic genetic correlation in tissues impacted by positive selection (as opposed to negative selection) suggests a possible connection between positive selection and population-specific causal effect sizes. To further assess this, we estimated the enrichment/depletion of squared trans-ethnic genetic correlation in SNPs with high integrated haplotype score (iHS)λ2(C) in the 53 SEG annotations for each individual trait  estimates were generally consistent across traits. Results for BMI and height, two widely studied anthropometric traits, are reported in Fig. λ2(C) = 0.84 (s.e. 0.05)) in Pituitary. Previous studies have highlighted the role of Pituitary in obesity49; our results suggest that this tissue-specific mechanism is population-specific. For height, we observed significant depletion of squared trans-ethnic genetic correlation for Transformed fibroblasts (λ2(C) = 0.87 (s.e. 0.03)), a connective tissue linked to human developmental disorders50; again, our results suggest that this tissue-specific mechanism is population-specific. Although Pituitary was significantly depleted for BMI but not height, and Transformed fibroblasts were significantly depleted for height but not BMI, we caution that for both tissues our λ2(C) estimates did not differ significantly between BMI and height.We investigated the enrichment/depletion of In summary, our results show that causal disease effect sizes are more population-specific in regions surrounding SEGs.30 have substantially depleted squared trans-ethnic genetic correlation (vs. the genome-wide average), implying that causal effect sizes are more population-specific. Accordingly, squared trans-ethnic genetic correlations were substantially depleted for SNPs in many functional categories and enriched in less functionally important regions . In analyses of SEG annotations, we observed substantial depletion of squared trans-ethnic genetic correlation for SNPs near skin and immune-related genes, which are strongly impacted by recent positive selection, but not for SNPs near brain genes. We also observed trait-specific depletions of squared-trans-ethnic genetic correlation for SEG annotations, which indicate population-specific disease mechanisms.We developed a new method (S-LDXR) for stratifying squared trans-ethnic genetic correlation across functional categories of SNPs that yields approximately unbiased estimates in extensive simulations. By applying S-LDXR to East Asian and European summary statistics across 31 diseases and complex traits, we determined that SNPs with high background selection statistic51, and evidence that G × G interaction and dominance variation explain limited heritability54, we hypothesize that depletion of squared trans-ethnic genetic correlation in the top quintile of background selection statistic and in functionally important regions may be primarily attributable to stronger G × E interaction in these regions. Interestingly, a recent study on plasticity in Arabidopsis observed a similar phenomenon: lines with more extreme phenotypes exhibited stronger G × E interaction55. Although depletion of squared trans-ethnic genetic correlation is often observed in regions with higher per-SNP heritability, which may often be subject to stronger G × E, depletion may also occur in regions with lower per-SNP heritability that are subject to stronger G × E; we hypothesize that this is the case for SNPs in the top quintile of average LLD and the bottom quintile of GERP (NS)  interaction, gene–gene (G × G) interaction, and dominance variation . Given the increasing evidence of the role of G × E interaction in complex trait architecturesNS) Fig. .56 . Applying S-LDXR to the 31 diseases and complex traits using a joint model incorporating baseline-LD-X model annotations and the nearest exon annotation, the background selection statistic remained highly conditionally informative for trans-ethnic genetic correlation, whereas the nearest exon annotation was not conditionally informative . In particular, when multiple SNPs have a similar level of significance, the SNPs enriched for trans-ethnic genetic correlation should be prioritized. Analogously, when applying more recent methods that estimate posterior mean causal effect sizes66 , these estimates should subsequently be weighted according to the expected enrichment/depletion for squared trans-ethnic genetic correlation based on their functional annotations. Second, modeling population-specific genetic architectures may improve trans-ethnic fine-mapping. Our results suggest that causal effect sizes and/or causal variants are likely to differ across different populations, contrary to standard assumptions67. Thus, incorporating information about trans-ethnic genetic correlations in trans-ethnic fine-mapping may lead to more accurate identification of both population-specific and shared causal variants68. Third, modeling population-specific genetic architectures may also increase power in trans-ethnic meta-analysis69, e.g. by adapting MTAG70 to two populations (instead of two traits), leveraging trans-ethnic (instead of cross-trait) genetic correlation between pairs of populations to improve the estimation of SNP effect sizes in both populations. Fourth, it may be of interest to stratify G × E interaction effects51 across genomic annotations. Fifth, modeling and incorporating environmental variables, where available, may provide additional insights into population-specific causal effect sizes. In our simulations, we did not explicitly simulate G × E. However, G × E would induce population-specific causal effect sizes, which we did explicitly simulate. Sixth, the S-LDXR method could potentially be extended to stratify squared cross-trait genetic correlations71 across genomic annotations72.Our study has several implications. First, PRSs in non-European populations that make use of European training data73 (e.g. African Americans or admixed Africans from UK Biobank74), analogous to LDSC and its published extensions75. However, a recently proposed extension of LDSC to admixed populations76 could be incorporated into S-LDXR, enabling its application to the growing set of large studies in admixed populations10. Second, S-LDXR estimates of enrichment of stratified squared trans-ethnic genetic correlation (λ2(C)) are slightly downward biased in null simulations of the top quintile of the background selection statistic and average LLD annotations, especially in simulations involving annotation-dependent MAF-dependent genetic architectures. However, these biases are small compared to the depletions of λ2(C) observed in the analysis of real traits. We further note that our estimates are unbiased in null simulations of binary annotations, implying that our results on real traits for binary annotations are robust. Third, since S-LDXR applies shrinkage to reduce standard error in estimating stratified squared trans-ethnic genetic correlation and its enrichment, estimates are conservative—true depletions of squared trans-ethnic genetic correlation in functionally important regions may be stronger than the estimated depletions. However, we emphasize that S-LDXR is approximately unbiased in null data. Fourth, the optimal value of the shrinkage parameter α may be specific to the pair of populations analyzed. In our simulations, we determined that α = 0.5 provides a satisfactory bias-variance tradeoff across a wide range of values of polygenicity and power. Thus, α = 0.5 may also be satisfactory for other pairs of populations. However, we recommend that one should ideally perform simulations on the pair of populations being analyzed to selection the optimal value of α. Fifth, it is difficult to assess whether a focal annotation contains unique information about λ2(C) conditional on other annotations, as squared trans-ethnic genetic correlation is a non-linear quantity defined by the quotient of squared trans-ethnic genetic covariance and the product of heritabilities in each population.We note several limitations of this study that pertain to the S-LDXR method. First, S-LDXR is designed for populations of homogeneous continental ancestry (e.g. East Asians and Europeans) and is not currently suitable for analysis of admixed populations24. We hypothesize that results based on SEG annotations defined in East Asian populations would likely be similar, as heritability enrichment of functional annotations (predominantly defined in Europeans) are consistent across continental populations31, despite the fact that gene expression patterns and genetic architectures of gene expression differ across diverse populations78. Thus, SEG annotations derived from gene expression data from diverse populations may provide additional insights into population-specific causal effect sizes. Third, we restricted our analyses to SNPs that were relatively common (MAF > 5%) in both populations (estimating parameters that were defined based on common SNPs), due to the lack of a large LD reference panel for East Asians. Extending our analyses to lower-frequency SNPs may provide further insights into the role of negative selection in shaping population-specific genetic architectures, as negative selection has the strongest impact on variants with low frequency27. Fourth, we did not consider population-specific variants in our analyses, due to the difficulty in defining trans-ethnic genetic correlation for population-specific variants5, a more fundamental challenge than analyzing low-frequency SNPs; a recent study79 has reported that population-specific variants substantially limit trans-ethnic genetic risk prediction accuracy. Fifth, estimates of genome-wide trans-ethnic genetic correlation may be confounded by different trait definitions or diagnostic criteria in the two populations, particularly for major depressive disorder. However, this would not impact estimates of enrichment/depletion of squared trans-ethnic genetic correlation (λ2(C)), which is defined relative to genome-wide values. Sixth, we have not pinpointed the exact underlying phenomena  that lead to population-specific causal disease effect sizes at functionally important regions. Despite these limitations, our study provides an improved understanding of the underlying biology that contribute to population-specific causal effect sizes, and highlights the need for increasing diversity in genetic studies.We also note several limitations of this study that pertain to our analysis of real traits. First, we focused on comparisons of East Asians and Europeans, due to the limited availability of very large GWAS in other populations. For other pairs of continental populations, if differences in the environment are similar, then we would expect similar genome-wide trans-ethnic genetic correlation and similar enrichment/depletion of squared trans-ethnic genetic correlation, based on our hypothesis that imperfect trans-ethnic genetic correlation is primarily attributable to G × E. We also note that different sets of SNPs, with different MAF and LD patterns, would be analyzed for different pairs of populations. However, we expect that these differences would not contribute to differences in trans-ethnic genetic correlation, if G × E is the fundamental factor impacting trans-ethnic genetic correlation. Second, the SEG annotations analyzed in this study are defined predominantly (but not exclusively) based on gene expression measurements of EuropeansY1 = X1β1 + ϵ1 and Y2 = X2β2 + ϵ2, where Y1 and Y2 are vectors of phenotype measurements of population 1 and population 2 with sample size N1 and N2, respectively; X1 and X2 are mean-centered but not normalized genotype matrices at M SNPs in the two populations; β1 and β2 are per-allele causal effect sizes of the M SNPs; and ϵ1 and ϵ2 are environmental effects in the two populations. We assume that in each population, genotypes, causal effect sizes, and environmental effects are independent of each other. We assume that the per-allele effect size of SNP j in the two populations has variance and covariance,aC(j) is the value of SNP j for annotation C, which can be binary or continuous-valued; τC1 and τC2 are the net contribution of annotation C to the variance of βj1 and βj2, respectively; and θC is the net contribution of annotation C to the covariance of βj1 and βj2.We model a complex phenotype in two populations using linear models, C  asC; and C in population p. Here, squared stratified trans-ethnic genetic correlation,C compared with the genome-wide average, λ2(C) instead of λ2(C) is challenging, as squared correlation is a non-linear term involving a quotient of squared covariance and a product of variances; we elected to instead estimate λ2(C) for quintiles of continuous-valued annotations . We note that the average value of λ2(C) across quintiles of continuous-valued annotations is not necessarily equal to 1, as squared trans-ethnic genetic correlation is a non-linear quantity.We define stratified trans-ethnic genetic correlation of a binary annotation We provide a more detailed definition of the estimands in the Supplementary Notes.22 (S-LDSC), a method for stratifying heritability from GWAS summary statistics, to two populations. The S-LDSC method determines that a category of SNPs is enriched for heritability if SNPs with high LD to that category have higher expected χ2 statistic than SNPs with low LD to that category. Analogously, the S-LDXR method determines that a category of SNPs is enriched for trans-ethnic genetic covariance if SNPs with high LD to that category have higher expected product of Z-scores than SNPs with low LD to that category.S-LDXR is conceptually related to stratified LD score regressionθC, where Zpj is the Z-score of SNP j in population p; ℓ× = ∑krjk1rjk2σj1σj2aC(k) is the trans-ethnic LD score of SNP j with respect to annotation C, whose value for SNP k, aC(k), can be either binary or continuous; rpjk is the LD between SNP j and k in population p; and σpj is the standard deviation of SNP j in population p. We obtain unbiased estimates of ℓ× using genotype data of 481 East Asian and 489 European samples in the 1000 Genomes Project16. To account for heteroscedasticity and increase statistical efficiency, we use weighted least square regression to estimate θC. We use regression equations analogous to those described in ref. 21 to estimate τC1 and τC2. We include only well-imputed (imputation INFO > 0.9) and common (MAF > 5% in both populations) SNPs that are present in HapMap 317 (irrespective of GWAS significance level) in the regressions (regression SNPs), analogous to our previous work75. We use all SNPs present in either population in 1000 Genomes16 to estimate trans-ethnic LD scores ℓ× , so that the resulting coefficients θC also pertain to these SNPs. However, we estimate λ2(C)  for all SNPs with MAF > 5% in both populations (heritability SNPs), accounting for tagging effects .S-LDXR relies on the regression equationτC1, τC2, and θC, respectively. For each binary annotation C, we estimate the stratified heritability of annotation C in each population, ρg(C), asheritability SNPs), using coefficients (λ2(C) primarily depends on the total allelic-scale heritability of SNPs in the annotation , which appears as the denominator (λ2(C) indirectly depends on the size of the annotation, because larger annotations tend to have larger total heritability. However, estimates of λ2(C) for a large annotation may have a large standard error if the annotation is depleted for heritability.Let C analogous to ref. 22,C; τC1, τC2, and θC are the corresponding unstandardized effect sizes, defined in Eq.  in simulations. We note that S-LDXR does not use the shrinkage estimator when estimating genome-wide rg and We shrink heritability and trans-ethnic genetic covariance of annotation ) in Eq.  to obtaiC is enriched or depleted of squared trans-ethnic genetic correlation for a trait, we test the null hypothesis 80, we instead test the equivalent null hypothesis B − 1 degrees of freedom, where B is the number of jackknife blocks. Since the To assess whether an annotation 21, which consists of 28 main annotations including conserved annotations  and epigenomic annotations  derived from ENCODE81 and Roadmap82, 24 500-base-pair-extended main annotations, and 1 annotation containing all SNPs. We note that although chromatin accessibility can be population-specific, the fraction of such regions is small83. Following ref. 22, we created an additional annotation for all genomic positions with number of rejected substitutions84 greater than 4. Further information for all functional annotations included in the baseline-LD-X model is provided in Supplementary Data We include a total of 54 binary functional annotations in the baseline-LD-X model. These include 53 annotations introduced in ref. 22 (see “Data availability”), without modification: background selection statistic30, CpG content (within a ± 50 kb window), GERP (number of substitutation) score84, nucleotide diversity (within a ± 10 kb window), and Oxford map recombination rate (within a ± 10 kb window)85. Second, we include 2 minor allele frequency (MAF) adjusted annotations introduced in ref. 22, with modification: level of LD (LLD) and predicted allele age. We created analogous annotations applicable to both East Asian and European populations. To create an analogous LLD annotation, we estimated LD scores for each population using LDSC75, took the average across populations, and then quantile-normalized the average LD scores using 10 average MAF bins. We call this annotation “average level of LD”. To create analogous predicted allele age annotation, we quantile-normalized allele age estimated by ARGweaver86 across 54 multi-ethnic genomes using 10 average MAF bins. Finally, we include 1 continuous-valued annotation based on FST estimated by PLINK287, which implements the Weir & Cockerham estimator of FST88. Further information for all continuous-valued annotations included in the baseline-LD-X model is provided in Supplementary Data We also include a total of 8 continuous-valued annotations in the baseline-LD-X model. First, we include 5 continuous-valued annotations introduced in ref. 25 (see “Code availability”), which preserves population-specific MAF and LD patterns, starting from phased haplotypes of 481 East Asians and 489 Europeans individuals available in the 1000 Genomes Project16 (see “Data availability”), restricting to ~2.5 million SNPs on chromosome 1 – 3 with minor allele count greater than 5 in either population. Since the direct output of HAPGEN2 includes substantial relatedness2, we used PLINK287 (see “Code availability”) to remove simulated individuals with genetic relatedness greater than 0.05, resulting in 35,378 EAS-like and 36,836 EUR-like individuals. From the filtered set of individuals, we randomly selected 500 individuals in each simulated population to serve as reference panels. We used 18,418 EAS-like and 36,836 EUR-like individuals to simulate GWAS summary statistics, capturing the imbalance in sample size between EAS and EUR GWAS in the analysis of real traits. In our secondary simulations, we also decreased or increased the reference panel size or decreased the GWAS sample size, to evaluate the robustness of our method with respect to reference panel size and GWAS sample size.We used simulated East Asian (EAS) and European (EUR) genotype data to assess the performance of our method, as we did not have access to real EAS genotype data at a sufficient sample size to perform simulations with real genotypes. We simulated genotype data for 100,000 East-Asian-like and 100,000 European-like individuals using HAPGEN2λ2(C), is 1 across all functional annotations, and causal simulations, where λ2(C) varies across annotations, under various degrees of polygenicity . In the null simulations, we set τC1, τC2, θC to be the meta-analyzed τC in real-data analyses of EAS GWASs, and followed Eq.  ≠ 1, as these were difficult to attain using the coefficients from analyses of real traits.We performed both null simulations, where enrichment of squared trans-ethnic genetic correlation, owed Eq.  to obtaiβj1], Var[βj2], and Cov to be 0 for all remaining non-causal SNPs. We scaled the trans-ethnic genetic covariance to attain a desired genome-wide rg. Next, we drew causal effect sizes of each causal SNP j in the two populations from the bi-variate Gaussian distribution,p as Xpβp, where Xp is column-centered genotype matrix, and drew environmental effects, ϵp, from the Gaussian distribution, p, Zp, as σpj is the standard deviation of SNP j in population p. We have publicly released Python code for simulating GWAS summary statistics for 2 populations (see “Code availability”). Fifth, we performed additional null simulations with annotation-dependent MAF-dependent genetic architectures28, defined as architectures in which the level of MAF-dependence is annotation-dependent.We randomly selected a subset of SNPs to be causal for both populations, and set Var[28, defined as architectures in which the level of MAF-dependence is annotation-dependent, to assess the impact on estimates of enrichment of stratified squared trans-ethnic genetic correlation, (λ2(C)). In these simulations, we set the variance of causal effect size of each SNP j in both populations to be proportional to pj,max is the maximum MAF of SNP j in the two populations. (We elected to use maximum MAF because a SNP that is rare in one population but common in the other is less likely to be impacted by negative selection.) We set α to − 0.38, as previously estimated for 25 UK Biobank diseases and complex traits in ref. 28. We sampled causal effect sizes using Eq.  than common variants , capturing the action of negative selection across low-frequency and common variants27.We also performed null simulations with annotation-dependent MAF-dependent genetic architecturesmentclasspt{minima74, Biobank Japan20, and CONVERGE18. All summary statistics were based on genotyping arrays with imputation to an appropriate LD reference panel , except those of the MDD GWAS in the East Asian population, which was based on low-coverage whole genome sequencing data18. These include: atrial fibrillation (AF)92, age at menarche(AMN)94, age at menopause (AMP)94, basophil count(BASO)95, body mass index (BMI)96, blood sugar(BS)96, diastolic blood pressure (DBP)96, eosinophil count(EO)96, estimated glomerular filtration rate (EGFR)97, hemoglobin A1c(HBA1C)96, height (HEIGHT)98, high density lipoprotein (HDL)96, hemoglobin (HGB)95, hematocrit (HTC)95, low density lipoprotein (LDL)96, lymphocyte count(LYMPH)96, mean corpuscular hemoglobin (MCH)96, mean corpuscular hemoglobin concentration (MCHC)95, mean corpuscular volume (MCV)95, major depressive disorder (MDD)99, monocyte count (MONO)96, neutrophil count(NEUT)95, platelet count (PLT)96, rheumatoid arthritis(RA)100, red blood cell count (RBC)96, systolic blood pressure (SBP)96, schizophrenia (SCZ)101, type 2 diabetes (T2D)103, total cholesterol (TC)96, triglyceride (TG)96, and white blood cell count (WBC)96. Further information for the GWAS summary statistics analyzed is provided in Supplementary Table 71) less than 0.25 in both populations: AF, AMN, AMP, BASO, BMI, EGFR, EO, HBA1C, HEIGHT, HTC, LYMPH, MCHC, MCV, MDD, NEUT, PLT, RA, SBP, TC, TG.We analyzed GWAS summary statistics of 31 diseases and complex traits, primarily from UK BiobankC, λ2(C), from 8 continuous-valued annotations, we first fit the S-LDXR model using these 8 annotations together with the base annotation for all SNPs, yielding coefficients, βj1 and βj1, substituting τC1, τC2, θC with α = 0.5), and use Eqs. (λ2(C).To obtain expected enrichment of squared trans-ethnic genetic correlation of a binary annotation  use Eq.  to obtaiuse Eqs.  to obtai24 as ±100k-base-pair regions surrounding genes specifically expressed in each of 53 GTEx32 tissues. A list of the SEG annotations is provided in Supplementary Data We obtained 53 SEG annotations, defined in ref. λ2(C), across the 53 SEG annotations, we first computed the average λ2(C) across the 53 annotations, Ci is the ith SEG annotation, and λ2(C). We computed a p-value for this test statistic based on a χ2 distribution with 53 degrees of freedom.To test whether there is heterogeneity in enrichment of squared trans-ethnic genetic correlation, 104 (see “Data availability”). This annotation is moderately correlated with the background selection statistic annotation22 (R = −0.21), defined as (1 - McVicker B statistic/1000), where the McVicker B statistic quantifies a site’s genetic distance to its nearest exon30. We have publicly released this annotation (see “Data availability”).We created a continuous-valued annotation, named “distance to nearest exon annotation”, based on a SNP’s physical distance (number of base pairs) to its nearest exon, using 233,254 exons defined on the UCSC genome browserλ2(C)) and standardized annotation effect size  have highly constrained functionality, and therefore mutations in these genes are subject to negative selection. We included SNPs within a 100kb-base-pair window around each gene, following ref. 24. A correlation heat map between pLI decile gene annotations and the 8 continuous-valued annotations is provided in Supplementary Fig. We created 10 annotations based on genes in deciles of the probability of being loss-of-function intolerant (pLI) (see “Data availability”), defined as the probability of assigning a gene into haplosufficient regions, where protein-truncating variants are depleted44 (see “Data availability”) across all 1000 Genomes Project EAS and EUR sub-populations are greater than 2.0, the recommended threshold to detect positive selection in ref. 43. This annotation is positively correlated with the top quintile of the background selection statistic annotation (R = 0.077). We note that although the iHS is a recombination-rate-adjusted quantity to detect the action of recent positive selection, it may also capture actions of negative selection44.We created a binary annotation (proportion of SNPs: 6.3%) that includes all SNPs whose maximum absolute value of the integrated haplotype score (iHS)Further information on research design is available in the Supplementary InformationPeer Review FileReporting SummaryDescription of Additional Supplementary FilesSupplementary Data 1-2Supplementary Data 3-5Supplementary Data 6-8Supplementary Data 9-10Supplementary Data 11Supplementary Data 12Supplementary Data 13Supplementary Data 14Supplementary Data 15-16Supplementary Data 17Supplementary Data 18-20Supplementary Data 21


Transcriptome-wide association studies (TWAS and PrediXcan) have been increasingly applied to detect associations between genetically predicted gene expressions and GWAS traits, which may suggest, however do not completely determine, causal genes for GWAS traits, due to the likely violation of their imposed strong assumptions for causal inference. Testing colocalization moves it closer to establishing causal relationships: if a GWAS trait and a gene’s expression share the same associated SNP, it may suggest a regulatory  role of the SNP mediated through the gene on the GWAS trait. Accordingly, it is of interest to develop and apply various colocalization testing approaches. The existing approaches may each have some severe limitations. For instance, some methods test the null hypothesis that there is colocalization, which is not ideal because often the null hypothesis cannot be rejected simply due to limited statistical power . Some other methods arbitrarily restrict the maximum number of causal SNPs in a locus, which may lead to loss of power in the presence of wide-spread allelic heterogeneity. Importantly, most methods cannot be applied to either GWAS/eQTL summary statistics or cases with more than two possibly correlated traits. Here we present a simple and general approach based on conditional analysis of a locus on multiple traits, overcoming the above and other shortcomings of the existing methods. We demonstrate that, compared with other methods, our new method can be applied to a wider range of scenarios and often perform better. We showcase its applications to both simulated and real data, including a large-scale Alzheimer’s disease GWAS summary dataset and a gene expression dataset, and a large-scale blood lipid GWAS summary association dataset. An R package “jointsum” implementing the proposed method is publicly available at github. In the post-GWAS era, colocalization testing has been playing an increasingly important role in inferring causal genetic variants and causal genes from GWAS trait-associated loci. However, colocalization testing is challenging. We first discuss some severe limitations of the existing methods, thus motivating our development of a general and powerful approach. We use extensive simulations to demonstrate the advantages of our approach over other existing methods. To further demonstrate the performance differences, we apply our and other methods (when possible) for colocalization analyses of multiple correlated GWAS traits and that of a GWAS trait and gene expression. Genome-wide association studies (GWAS) have identified many loci associated with complex traits or diseases, but causal relationships cannot be established, leading to lack of mechanistic understandings ,2. TestiTo test colocalization in a formal way, there have been extensive efforts in developing and applying various approaches. However, they all have some severe limitations as briefly discussed below. First, the popular coloc  and HEIDWe develop a simple and general approach overcoming the above shortcomings of the existing methods. We call this approach the conditional method, due to the central role of conditional modeling in fine mapping; in addition, more general than most existing methods, our approach applies to multiple traits with only summary statistics based on one or more possibly dependent/overlapping samples. We demonstrate that, in certain scenarios, the new method has power advantages over the existing methods using simulated data. Also, to show our method's flexibility as well as its ability to detect significant loci, we compare different methods using the large-scale GWAS lipid data  with fouWe will compare our method with some major representatives for the existing methods, namely JLIM, coloc, coloc.abf, HEIDI and eCAVIAR, the details of which can be found in the p traits and q SNPs, and we want to test whether at least one of the SNPs is causal for all the traits. We first assume the availability of individual-level data, and will discuss its extension to GWAS summary data at the end. The kth SNP and the jth trait in the individual level data are Xk = (Xk1…Xnk)′ and Yj = (Yj1…Ynj)′, where n is the sample size. With the individual level data, we can easily build a conditional model for each trait asX = (X1…Xq), bj = (bj1…bjq)′, and ej = (ej1…enj)′; ej is assumed to be independently normal with mean 0. The null and alternative hypotheses for our colocalization test areSuppose we are interested in k = 1,…,q and j = 1,…,p. We can also get the p-value for Pjk. To test H0, we can apply the principles of the Intersection-Union Test (IUT) and the Union-Intersection Test (UIT)  is a valid test statistic for H0 = ⋂kHk0. If mink[maxj(Pjk)] is small enough, it suggests there is colocalization since at least one SNP's maxj(Pjk) is small enough. Thus we can simply define our test statistic for H0 asα. The colocalization test can be carried out by rejecting H0 if P*<α, which means for at least one SNP, its effects on all of the traits are nonzero . However, this approach involves multiple testing, which will lead to inflated type I errors. Hence, we propose using the Bonferroni correction: we reject H0 if and only if P*<α/q. In this way, the type I error rate is guaranteed to be controlled.Suppose the given nominal statistical significance level is kth SNP on the jth trait and its variance by If we have only GWAS summary statistics, instead of individual-level data, we can also perform the colocalization test. Denote the estimated marginal effect size of the ribed in , to estiCMC), to test colocalization, which is expected to be less conservative. For convenience, we denote the previously described conditional method with Bonferroni correction by CB. Suppose Z = (Zjk) is the Z-statistic matrix with j and SNP k. Assume that the covariance/correlation matrices are constant across the rows and columns, denoted as Cov(Zj.) = corr(Zj.) = R1, and Cov(Zk.) = corr(Zk.) = R2. R1 and R2 are usually estimated by and regarded as the correlations among the SNPs and among the traits respectively. Our test statistic isSince the Bonferroni correction may be quite conservative, we propose another way to calculate the p-value, called the conditional method with Monte Carlo approximation ≤t) by using multivariate normal tail probabilities or Monte Carlo simulations. The challenge here is that the means of Zjk's are unknown under H0, which is a composite null hypothesis. For instance, it is possible that the first SNP has some effect on the first trait but not the second trait, which means Z11 has a nonzero mean while Z21 has a zero mean. It is also possible that the first SNP does not have any effect at all, meaning that Z11 and Z21 both have a zero mean. In different situations, the distribution of maxk(minj|Zjk|) can vary a lot, which we will show numerically in the simulation section. What we know is for each SNP k, at most (p−1) of the Zjk's have a nonzero mean under the null and at least one has a zero mean. A conservative way to get around is to assume all Z11,…,Zq1 have a zero mean and useP is now a calculable tail probability. The problem of this approach is that it is quite conservative, due to its use of the above inequality.If we know the mean of Zjk's have nonzero means. For each SNP k, we compare its |Zjk|'s with |Φ−1(θ/2)|, where Φ is the probability function of the standard normal distribution and θ is a tuning parameter, usually chosen as 0.05 or 0.1. We will discuss more about it later. If |Zj*k| is greater than |Φ−1(θ/2)|, we assume under the null, SNP k is associated with trait j*; otherwise SNP k has no effect on trait j*. Since each SNP must have no effect on at least one trait under the null , we always assume the effect corresponding to the smallest |Zjk| to be 0 regardless of how large this |Zjk| is. Let △ be the set of all the trait-SNP pairs  whose corresponding effects are assumed to be nonzero. Then we assume for each k,Zjk|'s with nonzero means are always larger than the other |Zjk|'s under the null. If none of the |Zjk|'s is greater than |Φ−1(θ/2)|, we assume SNP k has no effect on any trait under the null, so Zjk's all have a zero mean. In this way, we can estimateWe propose to use the current data to roughly estimate which of the Zjk's on the right-hand side have a zero mean, and their covariance matrix can be easily constructed with R1 and R2, becauseNow all of the q) due to the limitation of the function, which makes it computationally infeasible for large q's. A better way is to use a Monte Carlo approximation. For each iteration b, we can simulate Zjkb)( 's that satisfy Zjkb) and calculate tb)( = maxk minj|Zjkb)′. Now μjk = 0 if Zjk∉△ and μjk = Inf otherwise, for defining μjk = Inf results in μjk . For convenience, we denote this method by CMC. The choice of B depends on the significance level α. In the simulations where we aim at α = 0.05,B = 104 is usually more than enough. For the real data application where α = 5×10−8 is often considered, we can try B = 104 first. If the p-value of one locus turns out to be less than 0.005, we can try B = 106. If the new p-value turns out to be less than 5×10−5, we can use B = 109 to get a final p-value for that locus. We will follow this proposal in the next sections unless otherwise specified.Note that it is equivalent to simulate all of the Zjk|'s to determine the null distribution and then uses Monte Carlo simulations to estimate a p-value. Nevertheless, its performance may be affected if the null distribution is not optimally determined (i.e. the assumed △ is far off from the truth). Following the model averaging idea from [The previously described process uses |dea from , we can Zjk|'s. Suppose  has the largest |Zjk|,  has the second largest |Zjk|, etc.(1) Order |0 is ϕ, the empty set, meaning that all of the means are zero.(2) △j1,k1), , etc. Suppose currently the biggest △ we have is △t and the trait-SNP pair we are looking at is .(3) Examine each trait-SNP pair  satisfies bothCheck whether  into △t, it still satisfies the null hypothesis 's with the same kl).After adding  satisfies these two conditions, then △t+1 = △t∪{}. If not, move to the next pair  and check the conditions again. Continue this process until one of the following conditions is met:If . Note that when we use the above process to get different △'s, if the number of SNPs is large (e.g. 100), instead of adding one SNP at a time like described in step (3), it may be beneficial to add multiple SNPs at a time. For example, if we add one SNP at a time, the largest size of △ (or number of nonzero effects) is u. If we add s SNPs at a time, the largest number of nonzero effects that can be considered is su.For each △Cc value  as in [3Cc value , detailsθ. When θ is small, CMC will be less conservative, but it can have inflated type I errors as well. When θ is large, CMC will be more conservative with loss of power. As a result, we recommend choosing a θ that is not too large or too small (e.g. 0.1). One way to interpret θ is that if the mean of Zjk is zero, the probability of getting |Zjk|>|Φ−1(θ/2)| will be θ. More extensive simulation studies and discussions on choosing θ are provided in the θ = 0.1 by default. MA involves two more tuning parameters s and u. We recommend choosing u = 10 and modifying s according to the number of SNPs in each locus . This choice gives us a fairly reasonable su, which means the largest number of nonzero effects that can be considered under the null. When su is fixed, choosing a larger u and a smaller s means adding more models. This usually leads to higher power but also larger type I errors, partly because the more models we add, the more likely one or more of them have significant results with none-negligible weights. Based on our experience and to keep it simple, we suggest using u = 10 by default, though cautions have to be taken .As mentioned before, CMC depends on a tuning parameter http://cnsgenomics.com/software/smr/#SMR&HEIDIanalysis, while the eCAVIAR method is available at http://genetics.cs.ucla.edu/caviar/download.html. The JLIM method was originally implemented as the "jlim.test" function in the "jlimR" package, which requires input files in specific formats (e.g. need to include unnecessary SNP information); we re-implemented JLIM in our own R function, which only needs the p-values, a reference panel and permuted genotypes as input. This function for JLIM and the functions for CB, CMC and MA are included in the R package "jointsum", publicly available at https://github.com/yangq001/conditional.The coloc method uses the "coloc.test" function (for hypothesis testing) and “coloc.abf” (for drawing ROC curves) from the R "coloc" package. The HEIDI method was implemented by the original authors in the "smr" software available at Tconditional under different null scenarios. In the first scenario, no SNP has any effect on any trait. In the second scenario, the first two SNPs have nonzero effects on the first two traits. In the third scenario, the first four SNPs have nonzero effects on the first two traits. Since no SNP has nonzero effects on all of the three traits, the null hypothesis holds . As shown in First, we use a simple example to demonstrate that the null distribution of the test statistic of the conditional method largely depends on the unknown truth of which effects are nonzero under the composite null hypothesis. Suppose we have 5 independent SNPs and 3 independent traits. Next, we did some simulation studies to compare the performances of different methods .q SNPs in a small region in chromosome 19, making sure none of the correlations' absolute values were greater than 0.9. For the first trait, we selected the first 2000 subjects from the 4136 subjects in the Lung Health Study (LHS) data, and then used a linear model Y = Xβ+ε to obtain the trait, where β = (β1…βq), and ε was independently normally distributed with mean 0 and variance 1. βk was nonzero when the kth SNP was causal for the first trait. For the second trait, we used the rest of the subjects and a similar model Y* = X*β*+ε*. Note that X and X* represented the same set of SNPs but for different samples, so the two traits, based on two non-overlapping samples, were independent. We regressed Y and Y* separately on each SNP using the corresponding subjects to get the summary statistics of marginal associations for each trait-SNP pair. The individual level genotype data were also used as our reference panels.We simulated two traits for k is causal for both traits. Simply reject the null hypothesis if both p-values of this SNP are smaller than α. The p-values can be from the marginal models or the conditional models. For our simulations, we only included the case looking at the first SNP using the conditional p-values. To distinguish testing one SNP and testing a region in conditional analysis, we indicate the former by "1st" in the tables.We compared the conditional method with various approaches  along with JLIM, coloc and HEIDI, using the suggested tuning parameters for the methods  curves of JLIM, CB, CMC and coloc, as well as eCAVIAR . This time we used the more recent and popular Bayesian version (coloc.abf) of coloc, which gives the posterior probability of having one common causal variant ,37. We cNote that JLIM's ROC curve seemed to have an uncommon shape in Y = Xβ+ε and Y* = Xβ*+ε*. For subject i, the correlation between εi and ε*i was 0.5.In the above scenarios, we assumed the data for the two traits came from different subjects. We also looked at some other situations where the two traits were correlated from the same subjects. This time we only used the 2000 subjects previously used for trait 1. The models for two traits were As shown in Y = Xβ+ε and X is the genotypes from the LHS data. β and β* were obtained by shrinking smaller effects (<τ) in τ) by 2. Then we applied the methods in the same way as previous. The "true" effect sizes in each region can be found in We did some other simulations that were closer to real scenarios, involving larger numbers of SNPs and various effect sizes. The genotypes were obtained from all the subjects (around 4000) in the LHS data. We used the Lipid data from the Global Lipids Genetics Consortium GWAS study  to get sThe first part of We looked at the Lipid data from the Global Lipids Genetics Consortium GWAS study , containAccording to Using the conditional method with Bonferroni correction or CMC, we are also able to test colocalization of more than 2 traits, while the other methods we discussed can only be applied to 2 traits. We tested colocalization for LDL, HDL and TG with the same SNPs and loci as in the previous section. As shown in α). Note that the reference data (1000G) only have 503 subjects while the window size is as large as 100. The joint models we build may be quite inaccurate. Hence, we applied the Sum-MI method in [In addition, we tested colocalization of all four traits, LDL, HDL, TG and TC with the new methods. This time we used the same gap length and window size to obtain large non-overlapping windows, and then applied the new methods to each window. We also applied the marginal analysis looking at one SNP at a time to see whether a SNP was associated with all the traits  , along wWe have presented a new method to test colocalization with conditional analysis, which plays a central role in fine mapping to distinguish causal SNPs from marginally associated ones due to the latter’s LD with the former. The proposed conditional method offers some distinct advantages over other existing methods in many scenarios. First of all, unlike coloc and HEIDI, the conditional method's null hypothesis is that there is no colocalization, which seems more natural and desirable for the purpose of detecting colocalization. Besides, compared to methods like JLIM, the conditional method does not put any restriction on the number of causal SNPs in a locus. It also considers conditional effects of SNPs in a locus, rather than marginal effects, which could be quite different. As shown in the simulation study and discussed in the literature ,18, whenH0) is unknown (with some unknown nuisance parameters); specifically, under H0, although we have the asymptotic distribution of (Zjk) as a matrix normal variate, but for any given j and k, the mean of Zjk is unknown that may not equal to 0. In CMC, we have proposed a simple method to estimate these non-zero means/components; it can be made more conservative with the choice of a larger tuning parameter value (closer to 1) to estimate an upper bound of the true p-value, yielding a type I error rate under the nominal significance level with possible loss of power; we proposed a default value that seems to be working well across a wide range of simulations. However, CMC may also suffer from power loss when the estimation of non-zero components is too difficult to be good. To deal with this problem, we proposed to use MA, which looks at multiple estimates/choices of non-zero components and takes the weighted average of different results. As long as one of those choices is close to the truth, it can benefit MA's power, making it even less conservative than CMC. One limitation of MA is that it not only requires a cut off value as CMC does, but also needs a pre-specified number of effects to be added each time as well as the total number of models. Furthermore, MA requires more computing time. In addition, we tried the harmonic mean p-value (HMP) by [We have developed two new methods called CMC and MA-CMC (or simply MA), which are less conservative than the Bonferroni correction, in conditional analysis of multiple SNPs in a candidate locus. The performances of CMC and MA have been shown to improve over many existing methods both in simulations and real data applications. A technical challenge in colocalization testing with conditional analysis is the nature of the composite null hypothesis: the null distribution of the test statistic (under (HMP) by  for the (HMP) by . FinallyS1 Text(DOCX)Click here for additional data file.