Abstract
Genomic prediction has the potential to contribute to precision medicine. However, to date, the utility of such predictors is limited due to low accuracy for most traits. Here theory and simulation study are used to demonstrate that widespread pleiotropy among phenotypes can be utilised to improve genomic risk prediction. We show how a genetic predictor can be created as a weighted index that combines published genomewide association study (GWAS) summary statistics across many different traits. We apply this framework to predict risk of schizophrenia and bipolar disorder in the Psychiatric Genomics consortium data, finding substantial heterogeneity in prediction accuracy increases across cohorts. For six additional phenotypes in the UK Biobank data, we find increases in prediction accuracy ranging from 0.7% for height to 47% for type 2 diabetes, when using a multitrait predictor that combines published summary statistics from multiple traits, as compared to a predictor based only on one trait.
Introduction
Personalised medicine, in which genetic testing is the basis for informing future health status and determining intervention, is effectively applied for a number of monogenic disorders^{1}. For common complex disorders, which are those that are underlain by multiple genetic and environmental factors^{2}, predictive genetic testing that can discriminate individuals who are most at risk is currently limited, mainly because much of the genetic variation remains poorly understood^{3,4}. The potential of genetic risk prediction to (i) inform early interventions and (ii) aid diagnosis by identifying individuals with an increased genetic risk of disease could be improved substantially by increasing the accuracy of genetic risk predictors^{5}. While genomewide association studies (GWASs) of increased sample size will continue to unravel the role of genetic factors for complex diseases^{6}, improved prediction models are also required to maximise the accuracy of a risk predictor.
GWASs use linear regression to independently estimate the effects of singlenucleotide polymorphisms (SNPs) across the genome, and commonly, these estimated SNP effects are then used to create a genetic risk predictor in independent samples^{7,8,9}. However, this approach is not optimal because it either ignores linkage disequilibrium (LD) between markers or accounts for LD by discarding potentially informative SNPs^{10}. Prediction accuracy of complex phenotypes can be improved by methods that jointly estimate the SNP associations to obtain SNP effect estimates with best linear unbiased predictor (BLUP) properties within a linear mixed model (LMM) approach, a model termed genomic BLUP (GBLUP)^{7,11,12}. A multitrait extension of the LMM approach, yielding multivariate BLUP (MTBLUP) predictors of the SNP effects, can further improve prediction accuracy when phenotypes are genetically correlated, because measurements on each trait provide information on the genetic values of the other correlated traits^{13,14,15,16}. MTBLUP has been shown to improve prediction accuracy for genetically correlated common psychiatric disorders when combining individuallevel data across independent data sets^{16,17}. However, the application of MTBLUP to complex common disorders is limited as combining individuallevel genotypephenotype data across case–control studies of all complex diseases is generally not feasible due to data protection concerns and restrictions on data sharing.
Here we overcome this limitation by developing a framework that combines publically available GWAS summary statistics across multiple studies of different traits together in a weighted index to generate approximate multitrait summary statistic BLUP (wMTSBLUP) predictors (Supplementary Table 1). We show through theory and simulation study that MTBLUP predictors, which traditionally require individuallevel phenotype–genotype data for all traits, can be approximated accurately by wMTSBLUP predictors in a computationally efficient manner using only summary statistic data and an independent genomic reference sample. We also show how multitrait summary statistic predictors can be created directly from GWAS summary statistics (wMTGWAS) or from predictors obtained using the software LDPred^{18} that extends a singletrait summary statistic BLUP model (SBLUP) by assuming that marker effects come from a mixture of distributions. We apply our approach to multiple phenotypes in the Psychiatric Genomics Consortium (PGC) to compare summary statistic approaches to direct estimation on individuallevel data. We further apply our approach to summary statistics of several other phenotypes to create predictors that we evaluate using the UK Biobank data. We show that, for most traits, our multitrait predictors improve prediction accuracy as compared to a singletrait predictors.
Results
Overview of the approach
Standard GWAS summary statistics are ordinary least squares (OLS) estimates of the SNP effects and do not have optimal properties for prediction^{11}. Even when LMM association analysis is used, the estimated SNP effects still represent marginal effects and not effects conditional on other SNPs, which is what is desirable for prediction^{19}. Previous studies have shown how OLS summary statistics can be reanalysed in a mixed model framework to produce approximate BLUP predictors (summary statistic BLUP: SBLUP, implemented in the most recent release of GCTA)^{18,20,21} or approximate mixture model predictors (LDPred). We first extend the SBLUP approach to a multitrait framework (MTSBLUP) and find a computational limitation associated with the inversion of a SNPbySNPbytrait matrix. To overcome this, we then derive theory to show how singletrait predictors with BLUP properties can be combined together in a weighted index to generate predictors with equivalent properties to those gained from a MTBLUP analysis (Fig. 1).
Consider two genetically correlated traits for which we have individuallevel genetic predictors with BLUP properties. For each individual, i, and focal trait of interest, f, we have a genetic prediction \(\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_{i,k}}} \right)\) for each trait, k, that we can combine together using the index weights, w_{i,k}, for each \(\widehat {\bf{g}}_{{\mathrm{BLUP}}_{i,k}}\) effect to produce a weighted multitrait BLUP genetic predictor:
In the Methods section, we show that the optimal index weights can be calculated as:
where \(h_k^2\) is the SNP heritability of trait k (proportion of phenotypic variance explained by genomewide SNPs), r_{G} is the genetic correlation between trait k and the focal trait and \(R_k^2\) is the expected squared correlation between a phenotype and a BLUP predictor, calculated as:
where M_{eff} is the effective number of chromosome segments and N_{ k } is the sample size of trait k. These weights will ensure that the contribution of each added trait is approximately proportional to the square root of its sample size, its SNP heritability and its genetic correlation with the focal trait (trait 1), while accounting for different variances of singletrait BLUP predictors.
Both \(h_k^2\) and r_{G} can be estimated from GWAS summary statistics using LD score regression^{22,23}. Following^{20}, individuallevel genetic predictors with BLUP properties can also be obtained from GWAS summary statistics (\(\widehat {\bf{g}}_{{\mathrm{SBLUP}}_k}\), where SBLUP represents summary statistic approximate BLUP). Therefore, for any given trait, genetic predictors with BLUP properties \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}_k}} \right)\) can be created from GWAS summary statistics and these can then be placed in a weighted index to produce approximate multitrait summary statistic BLUP (wMTSBLUP) predictors, using only LD score regression and an independent reference sample. This approach, provided in the freely available software SMTPred (see Code availability section), approximates MTBLUP predictors without the need for individuallevel phenotype–genotype data for all traits, enabling prediction accuracy to be improved by fully utilising all of the publically available GWAS summary statistic data. We also show how weighted indices can be calculated for GWAS summary statistics (wMTGWAS) or from predictors obtained using the software LDPred^{18} (wMTLDPred), therefore depending upon the genetic architecture of the trait approximate multitrait summary statistics can be created to maximise genomic prediction accuracy.
Simulation study
We first conducted a simulation study using observed SNP genotype data to confirm the expectations from our theory. We show through theory (see Methods section) that a wMTSBLUP genetic predictor has the same expected prediction accuracy as one created from a multivariate mixedeffects model (multitrait BLUP: MTBLUP) if the linkage disequilibrium among SNP markers in the individuallevel analysis is well approximated by a reference genotype panel (see Methods section). We demonstrate that a wMTSBLUP predictor increases prediction accuracy over a singletrait predictor, with the magnitude of increase being proportional to the ratio of the SNP heritability of the added traits relative to that of the predicted trait, the sample size of the added traits relative to that of the predicted trait and the genetic correlation between the added traits and the predicted trait (Fig. 2, Supplementary Figs. 1 and 2). We also demonstrate how genetic predictors generated by LDPred^{18} can be combined in an approximate multitrait weighting (Supplementary Fig. 3).
We also provide a theoretical expectation for the loss in prediction accuracy that occurs when using an independent reference sample to compute SBLUP effects compared to a predictor based on BLUP effects (see Methods section), and we detail the loss of prediction accuracy in our simulation study (Fig. 2b, Supplementary Figs. 1 and 4).
Application to psychiatric disorders
We then applied our approach to the PGC schizophrenia^{24,25} and bipolar data, two psychiatric disorders known to have a high genetic correlation^{26}. The availability of combined individuallevel data for both disorders enabled a direct comparison of the MTBLUP^{16} and wMTSBLUP approaches. We calculated all predictors for the previously used^{16} PGC wave 1 (PGC1) data sets^{24} and compared the prediction accuracy (correlation between predicted values and phenotypes adjusted for sex, cohort and the first 20 principal components) across diseases and approaches. We find comparable but slightly lower accuracies in the wMTSBLUP predictors as compared to the MTBLUP predictors (0.151 vs 0.156 in bipolar disorder and 0.217 vs 0.219 in schizophrenia) and an increase in prediction accuracy as compared to the singletrait (BLUP) predictors (0.128 in bipolar disorder, 0.198 in schizophrenia) (Fig. 3). Our results demonstrate that creating SBLUP genetic predictors using an independent LD reference sample and combining these in a weighted sum results in prediction accuracy comparable to a full MTBLUP prediction for common complex disease traits, at a much lower computational burden.
We then applied our approach to the larger PGC wave 2 (PGC2) data sets for schizophrenia^{25} and bipolar disorder (see Methods section), which included the PGC1 data. To test whether the addition of more cohorts improved prediction accuracy, we estimate wMTSBLUP predictors in the PGC2 data. Having shown the resemblance of wMTSBLUP and MTBLUP by theory, simulation and in the PGC1 data, we refrained from running a MTBLUP model in the PGC2 data to avoid the computational burden of analysing the combined schizophrenia bipolar data set. For schizophrenia, there were 36 cohorts (26,412 cases and 32,440 controls in total) and for bipolar disorder there were 23 cohorts (18,865 cases and 30,460 controls in total). We conducted a cohortwise leaveoneout crossvalidation approach to examine variation in prediction accuracy across cohorts.
For schizophrenia, we find that prediction accuracy increases in 20 of the 36 cohorts of the PGC2 data when using a wMTSBLUP predictor as compared to a SBLUP predictor (Supplementary Fig. 5). However, the median correlation (0.300 with an SBLUP predictor, and 0.304 with a wMTSBLUP predictor) and mean correlation (0.295 with a SBLUP predictor and 0.294 with a wMTSBLUP predictor) across the 36 PGC2 cohorts did not improve with a wMTBLUP predictor. For bipolar disorder, we find an improvement of the wMTSBLUP predictor over the SBLUP predictor in 17 out of the 23 cohorts (Supplementary Fig. 6), with a mean correlation increase from 0.212 to 0.229 and a median correlation increase from 0.210 to 0.225. To evaluate whether this is because the weights we used for schizophrenia and bipolar disorder do not represent the mixing proportions that lead to the highest accuracy in this data set or whether other factors explain the variable results across cohorts, we created multitrait predictors using not only weights calculated from Eq. (17) but also weights corresponding to any other mixing proportion of the two disorders (Supplementary Figs. 5, 6 and 7). This demonstrates (i) that our calculated weights are very close to the empirically optimal weights when averaged across cohorts (Supplementary Fig. 7), (ii) that there is substantial heterogeneity across cohorts as shown by the variable prediction accuracies of singletrait and crosstrait predictors across cohorts, which is supported by previous studies^{25}, and (iii) that, for some test set cohorts, there is no mixing proportion that will lead to a multitrait predictor which outperforms a singletrait predictor. The larger gain in accuracy that results from supplementing a bipolar disorder predictor with schizophrenia data compared to supplementing a schizophrenia predictor with bipolar disorder data is consistent with greater power of the schizophrenia discovery sample. We find that for both singletrait and multitrait predictors the SBLUP predictors outperform the OLS predictors in almost all cohorts (Supplementary Figs. 5 and 6).
Application to traits recorded in a large population study
In principle, any number of traits can be combined into a multitrait predictor at almost no computational cost. We therefore extended our approach to create wMTSBLUP predictors from 34 phenotypes for which we could access summary statistics. In order to calculate wMTSBLUP weights, we used LD score regression to estimate SNP heritability and genetic correlations of the 34 summary statistics traits. The results are mostly in line with previous reports^{23} (Supplementary Fig. 8, Supplementary Data 1). As test set, we used 112,338 individuals in the UK Biobank data. We matched 6 of the 34 discovery traits to traits in the UK Biobank (Supplementary Table 1) and created wMTSBLUP predictors. For the wMTSBLUP predictor of each focal trait, we included predictor traits with genetic correlation pvalue < 0.05. For all traits, wMTSBLUP genetic predictors were more accurate than any singletrait (SBLUP) predictor (Fig. 4). wMTSBLUP predictors generally improved prediction accuracy over singletrait GWAS OLS predictors (Supplementary Fig. 9) and were similar to wMTLDPred predictors (Supplementary Figs. 10 and 11.) We observe the largest increases in accuracy for Type 2 diabetes (47.8%) and depression (34.8%). Accuracy for height (0.7%) and body mass index (BMI) (1.4%) increase only marginally. As shown in our theory and simulation study, the magnitude of increase in prediction accuracy of a wMTSBLUP predictor over a singletrait SBLUP predictor depends upon the prediction accuracies of all the traits included in the index and the genetic correlation among phenotypes. As GWAS sample sizes increase and genomic predictors increase in accuracy, a wMTSBLUP approach will likely become increasingly beneficial.
Discussion
In summary, we demonstrate that multivariate predictors derived from GWAS summary statistics can increase prediction accuracy in a wide range of traits. This approach has particular utility in risk prediction of traits for which it is hard to generate large sample sizes for GWAS, as SNP heritability and sample size are the two factors that determine prediction accuracy of a polygenic trait, when using a singletrait predictor. The increase in prediction accuracy of a multitrait over a standard singletrait genetic predictor is therefore greatest when the additional traits included in the predictor have higher SNP heritability and sample size than the trait to be predicted, as well as a high genetic correlation with the trait to be predicted. We show how genetic predictors from GWAS OLS effects, LDPred effects or SBLUP effects can be combined, yielding an approach that is general across different phenotypes.
Special consideration should be given to the risk of sample overlap between the summary statistics data used to create the predictor and the prediction target. Sample overlap will lead to inflated estimates of accuracy, and while here we were able to take steps to avoid individuals being recorded across multiple data sets, further work is required to negate these effects within this framework. In principle, assuming perfect homogeneity between training and test set and perfect estimates of SNP heritability and genetic correlation, there is no limit to the number of traits that can be combined using our approach. In practice, however, there will be little benefit of combining traits with low genetic correlation, as they will not influence the predictor much. Some added traits might even reduce accuracy, if the genetic correlation is not estimated accurately. The focus of our analysis was the prediction of genetic risk and we aimed to provide a fast, computationally efficient, general framework for genomic prediction. This sets it apart from other multitrait approaches like phenomewide association studies, which focus on the effects of individual SNPs on multiple phenotypes. We note, however, that a multitrait testing approach can in principle also be used to increase the power to identify loci associated with specific traits as demonstrated in the recently developed MTAG method^{27}. Another potential caveat of our analysis is that prediction accuracy increases for a focal trait may come from the addition of traits that are standardly measured on patients, and improved frameworks are required to identify marker effects conditionally on known health risk factors. Despite these limitations, current evidence suggests that genetic correlations among phenotypes are pervasive^{23}, sample sizes of GWAS are increasing^{6} and public availability of genomewide summary statistics is becoming the norm^{28}, meaning that genomic prediction of complex common disease will continually improve especially when predictors of multiple phenotypes are integrated across studies within this framework.
Methods
General model
We consider a general linear mixed model:
where y is the phenotype, W a matrix of SNP genotypes, where values are standardised to give the ij^{th} element as: \(w_{ij} = \left( {x_{ij}  2p_j} \right){\mathrm{/}}\sqrt {2p_j\left( {1  p_j} \right)}\), with x_{ ij } the number of minor alleles (0, 1 or 2) for the i^{th} individual at the j^{th} SNP and p_{ j } the minor allele frequency. b are the genetic effects for each SNP, and \({\bf{\epsilon }}\) the residual error. The dimensions of y, W, b and \({\bf{\epsilon }}\) are dependent upon the number of phenotypes, k, the number of SNP markers, M, and the number of individuals, N, and are described in the sections below. We denote the distributional properties var(b) = B, var(\({\bf{\epsilon }}\)) = R and var(y) = WBW′ + R.
For human complex diseases and quantitative phenotypes, GWASs have typically estimated the solutions for b of Eq. (1) one SNP at a time using OLS regression^{29} as:
where diag[W′W] has diagonal elements \(w_j\prime w_j\) and offdiagonal elements of zero. However, by analysing one SNP at a time, GWAS effect size estimates do not account for the covariance structure among SNPs and they are not unbiased in the sense that \({\mathit{E}}\left[ {{\bf{b}}{\hat{\bf b}}} \right] = {\hat{\bf b}}\)^{12}. BLUP of the SNP effects have the property \({\it{E}}\left[ {{\bf{b}}{\hat{\bf b}}} \right] = {\hat{\bf b}}\), are used in genomic prediction in animal and plant breeding^{30} and more recently in human medical genetics, yielding improved prediction accuracy for a number of traits over genetic predictors created from OLS SNP estimates^{16,17}. In a general form, BLUP solutions for b of Eq. (1) can be written using Henderson’s mixed model equations^{31} as:
and if R is diagonal, then Eq. (6) can be reduced to:
Below, we describe how Eqs. (6) and (7) can be used to estimate BLUP SNP effects for a single trait and for multiple traits jointly from individuallevel phenotype–genotype data. We then show how Eqs. (6) and (7) can be approximated to obtain BLUP SNP effects for single and multiple traits in the absence of individuallevel data from publically available GWAS summary statistics and an independent reference sample.
Estimation of BLUP SNP effects for a single trait
For a univariate analysis of trait k, y of Eq. (4) is a column vector of length N × 1 and W has dimension N × M. Assuming b is an M × 1 vector of random SNP effects for trait k, with distribution \({\bf{b}}\sim N\left( {0,{\bf{I}}_M\sigma _{b_k}^2} \right)\), then \({\bf{B}} = {\bf{I}}_M\sigma _{b_k}^2\) with I_{ M } is an identity matrix of dimension M. \({\bf{\epsilon }}\) of Eq. (1) is a column vector of independent residual effects, with distribution \({\bf{\epsilon }}\sim N\left( {0,{\bf{I}}_N\sigma _{{\it{\epsilon }}_k}^2} \right)\), giving \({\bf{R}} = {\bf{I}}_N\sigma _{\epsilon _k}^2\), with I_{ N } an identity matrix of dimension N. Substituting these expressions into Eq. (6) means that Eq. (7) can then be written as:
with \(\lambda _k = \sigma _{\epsilon _k}^2{\mathrm{/}}\sigma _{{\it{b}}_k}^2\).
Joint estimation of BLUP SNP effects for multiple traits
Phenotypic measurements of a trait can be informative for the genetic values of other traits, if the traits are genetically correlated with one another^{14,15,32}. Recent studies have shown that prediction accuracy of common complex disease can be improved by estimating SNP effects for multiple traits jointly within a multivariate mixedeffects model^{16,17}.
If k traits are measured on different individuals, with N_{ k } observations for trait k, the elements of Eq. (4) become: \({{\bf y}^ \prime} = [{\bf y}\prime_{\bf 1}...{{\bf y}^ \prime_{k}}], {\bf{W}} = \left[ {\begin{array}{*{20}{c}} {{{\bf W}}_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\bf{W}}_k} \end{array}} \right]\), and R = diag[R_{ k }] = diag\(\left[ {{\bf{I}}_{N_k}\sigma _{\epsilon _k}^2} \right]\), a diagonal matrix of length \(N = \mathop {\sum }\limits_k N_k\). \({\bf{B}} = {\bf{\Sigma }}_b \otimes {\bf{I}}_M\), where Σ_{ b } is a k × k matrix, with diagonal elements \(\sigma _{b_k}^2\) and offdiagonal elements the covariances of SNP effects between traits k and l, \(\sigma _{b_{k,l}}\). For Kronecker products, \({\bf{B}}^{  1} = {\bf{\Sigma }}_b^{  1} \otimes {\bf{I}}_M\) and substituting these expressions directly into Eq. (6) means that multitrait BLUP solutions for b can be obtained in Eq. (7) as:
with \({\bf{\Sigma }}_\epsilon = {\mathrm{diag}}\left[ {\sigma _{\epsilon _k}^2} \right]\), a diagonal k × k matrix. For a twotrait example, Eq. (9) expands to:
Multitrait BLUP SNP effects from summary statistics
Estimating SNP effects for multiple traits jointly in Eq. (9) requires individuallevel genotype and phenotype data across a range of common complex diseases and quantitative phenotypes, which are not readily available in human medical genetics due to privacy concerns and data sharing restrictions. Additionally, Eq. (9) requires a series of computationally intensive M × k equations to be solved. However, these issues can be overcome by approximating Eq. (9) using publically available GWAS summary statistic data and an independent genomic reference sample.
Singletrait approximate BLUP SNP effects can be obtained from GWAS summary statistics (SBLUP: summary statistic approximate BLUP) by replacing \({\bf{W}}_k\prime {\bf{W}}_k\) and \({\bf{W}}_k\prime {\bf{y}}_k\) of Eq. (8) by their expectation, which are \({\Bbb E}\left[ {{\bf{W}}_k\prime {\bf{W}}_k} \right] = N_k{\bf{L}}\) and \({\Bbb E}\left[ {{\bf{W}}_k\prime {\bf{y}}_k} \right] = N_k\widehat {\bf{b}}_{{\mathrm{OLS}}_k}\), respectively, where L is an M × M scaled SNP LD correlation matrix estimated from a reference SNP data set and \(\widehat {\bf{b}}_{{\mathrm{OLS}}_k}\) are obtained from publically available GWAS summary statistics^{20}. GWAS summary statistics report effect estimates of SNPs on an unstandardised scale and not \(\widehat {\bf{b}}_{{\mathrm{OLS}}}\) as it is defined here. To obtain \(\widehat {\bf{b}}_{{\mathrm{OLS}}}\) from GWAS summary statistics, the effect of each SNP must be multiplied by the standard deviation of each SNP: \(\widehat {\bf{b}}_{{\mathrm{OLS}}_{\mathrm{j}}}\) = \(\widehat {\bf{b}}_{{\mathrm{OLS}}  {\mathrm{UNSCALED}}_{\mathrm{j}}} \times \sqrt {2p_j\left( {1  p_j} \right)}\). Equation (8) can then be written as:
The shrinkage parameter is \(\lambda _k = \sigma _{{\it{\epsilon }}_k}^2{\mathrm{/}}\sigma _{b_k}^2\) = \(M\sigma _{{\it{\epsilon }}_k}^2{\mathrm{/}}h_{{\rm SNP}_k}^2\) = \(M\left( {1  h_{{\rm SNP}_k}^2} \right){\mathrm{/}}h_{{\rm SNP}_k}^2\), under the assumption of phenotypic variance of 1 that makes the proportion of phenotypic variance of trait k attributable to the SNPs \(h_{{\rm SNP}_k}^2 = M\sigma _{b_k}^2\).
This approach was implemented in ref. ^{21} and is similar to the LDpred model presented by Vilhjálmsson et al.^{18} but with a few differences. The first is that it only considers the infinitesimal case, where all SNPs are considered to be causal and their effect sizes follow a normal distribution. This corresponds to the LDpredInf model. The second difference is that the shrinkage parameter of Vilhjálmsson et al.^{18} is \(\lambda _k = M{\mathrm{/}}h_{{\mathrm{SNP}}_k}^2\) as they assume that the error variance is 1 rather than \(1  h_{{\mathrm{SNP}}_k}^2\) in our implementation. The third difference is that in the LDpredInf model, Vilhjálmsson et al.^{18} calculate BLUP effects for blocks of a certain number of SNPs following a tiling window approach giving a block diagonal structure to L, whereas our implementation within the software GCTA (see URLs) follows a sliding window approach giving a banded diagonal to L. Assuming an error variance of \(1  h_{{\mathrm{SNP}}_k}^2\) is more appropriate because cumulatively the SNP markers explain \(h_{{\mathrm{SNP}}_k}^2\) of the phenotypic variance. In both implementations, a window is used to capture the LD around SNP markers in order to avoid the large computational costs of inverting a dense M dimensional SNP LD matrix, with only little loss of information (see below).
For multiple phenotypes, the elements of Eq. (11) become: \(\widehat {\bf{b}}_{{{{\rm OLS}{\prime}}}} = \left[ {\widehat {\bf{b}}_{{\mathrm{OLS}}_1{\prime} } \ldots \widehat {\bf{b}}_{{\mathrm{OLS}}_k{\prime} }} \right]\) and N = \(\left[ {\begin{array}{*{20}{c}} {{\bf{N}}_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\bf{N}}_k} \end{array}} \right]\), meaning that Eq. (11) can be extended as:
Equation (12) approximates Eq. (9) using only publically available GWAS summary statistic data and an independent genomic reference sample. However, there remains the large computational cost associated with the inversion of the nondiagonal matrix \(\left[ {{\bf{I}}_k \otimes {\bf{L}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{  1}{\bf{N}}^{  1} \otimes {\bf{I}}_M} \right]\).
Index weighted multitrait BLUP SNP effects
An alternative to Eq. (12), is to obtain k \(\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{SBLUP}}}\) effects by combining together k singletrait \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) estimates of Eq. (11), using an optimal index weighting for each trait. The index weighting to derive \(\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{SBLUP}}}\) from \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) estimates is identical to the index weighting to derive \(\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}\) from \(\widehat {\bf{b}}_{{\mathrm{BLUP}}}\) estimates.
For SNP j and focal trait f, we have \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) values for k traits, and we wish to obtain the index weights, w_{j,k}, for each \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{j,k}}\) effect as:
In animal and plant breeding, selection indices have been developed, which combine many singletrait BLUP predictors of an individual’s genetic value together in an index weighting to optimise the selection of individuals with the most favourable multitrait phenotype for breeding programs^{33,34,35,36}. Utilising a selection index approach, the solution for w_{SBLUP} of Eq. (13) can be obtained as:
where C_{SBLUP} a k × 1 column vector of the covariance of the \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}\) values of the k traits, with the true genetic effects of the SNPs for the focal trait, and V_{SBLUP} a k × k variance–covariance matrix of the \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) effects:
Therefore, if V_{SBLUP} and C_{SBLUP} can be approximated then \(\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{SBLUP}}}\) of Eq. (12) can be obtained from k singletrait \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) estimates from Eq. (11).
To derive the approximations, we first consider the diagonal elements of V_{SBLUP}, which comprise the variance of the SBLUP SNP solutions, \({\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)\). These can be approximated from theory under the assumption that \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}\) have BLUP properties \({\mathit{E}}\left[ {{\bf{b}}{\hat{\bf b}}} \right] = {\hat{\bf b}}\), which in turn implies that \({\mathrm{cov}}\left( {{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)\). Following Daetwyler et al.^{37} and Wray et al.^{38}, the squared correlation between a phenotype, y_{ k }, in an independent sample and a singletrait BLUP predictor of the phenotype, \(\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}\), is approximately:
where \(\widehat {\bf{g}}_{{\mathrm{BLUP}}_k} = {\bf{W}}\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}\) and \(h_k^2\) is the proportion of phenotypic variance attributable to additive genetic effects for trait k. Note that M_{eff} is the effective number of chromosome segments or the number of independent SNPs, which is a function of effective population size (N_{e}) and can be empirically obtained as an inverse of the variance of genomic relationships^{39,40}. Here we use an estimate of M_{eff} of 60,000, which is in line both with our estimates from the genomic relationships in our simulation data and with previously reported estimates^{41}. In Eq. (16), \(R_k^2\) occurs on both the left and righhand side. Solving for \(R_k^2\) gives \(R_k^2 = \frac{{\varphi + h^2  \sqrt {\left( {\varphi + h^2} \right)^2  4\varphi h^4} }}{{2\varphi }}\), where φ is \(\frac{{M_{{\mathrm{eff}}}}}{N}\).
With a phenotypic variance of 1 and individuallevel genetic effects g_{ k } = Wb_{ k }, then \(h_k^2 = \sigma _{g_k}^2 = M\sigma _{b_k}^2\) and the squared correlation between the true, g_{ k }, and estimated BLUP effects, \(\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}\), is:
Rearranging Eq. (17) gives \(R_k^2 = h_k^2R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = h_k^2\frac{{{\mathrm{cov}}\left( {{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)^2}}{{{\mathrm{var}}\left( {{\bf{g}}_k} \right){\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)}}\), which given the BLUP properties \({\mathrm{cov}}\left( {{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)\) and \(h_k^2 = \sigma _{g_k}^2\) with a phenotypic variance of 1, reduces to \(R_k^2 = {\mathrm{cov}}\left( {{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)\) = \({\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right) = M{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)\). Therefore:
Second, we consider the offdiagonal elements of V_{SBLUP}, which are comprised of the covariance of BLUP SNP solutions among the k traits. These can again be approximated from theory given the covariance of genetic effects among traits k and l is cov(b_{ k }, b_{ l }) = r_{G}h_{ k }h_{ l }/M, with r_{G} the genetic correlation, and given the squared correlation between the true genetic effects of the SNPs, b_{ k }, and \(\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}\) which is given by Eq. (17) as \(R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = \frac{{R_k^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_k^2{\mathrm{/}}h_k^2\). The covariance of BLUP SNP predictors is then:
Finally, we can consider the column vector C_{SBLUP}, which is composed of the covariance between the true genetic effects of the SNPs for the focal trait, b_{ f }, and \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}\) for all of the k traits. The first element of C_{SBLUP} is covariance between the true genetic effects of the SNPs for the focal trait b_{ f } and \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}_f}\) for the focal trait \({\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}} \right) = \frac{{R_f^2}}{M}\). The remaining elements of C_{SBLUP} are \({\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)\), which can be approximated from theory by considering a regression of b_{ f } on b_{ k } where the regression coefficient \(\beta _{f,k} = r_{\mathrm{G}}\sqrt {{\mathrm{var}}\left( {{\bf{b}}_f} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_k} \right)}\). The covariance of b_{ f } and \(\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}\) can then be written as:
If we consider a twotrait example where the focal trait that we want to predict is matched to the first of the two traits, Eqs. (18), (19) and (20) combine as:
giving the index for the focal trait as: \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}_f} = {\bf{w}}_1\widehat {\bf{b}}_{S{\mathrm{BLUP}}_f} + {\bf{w}}_2\widehat {\bf{b}}_{{\mathrm{SBLUP}}_2}\) with solutions for the index weights of:
For traits with low power, \(R_k^2\) is usually very small. In that case, V_{SBLUP} can be well approximated by a diagonal matrix with entries \(\frac{{R_k^2}}{M}\). w_{ f } will become 1 and w_{ k } for all other traits will be \(r_{{\mathrm{G}}_{f,k}}\frac{{h_f}}{{h_k}}\). It may appear surprising that traits with higher SNP heritability have smaller weights than traits with lower SNP heritability. This can be explained by the fact that the variance of each BLUP predictor \(\left( {R_k^2} \right)\) is approximately proportional to \(h_k^4N\) if M_{eff} is large, and thus a trait with higher SNP heritability will still have a larger contribution to the multitrait predictor than a trait with lower SNP heritability.
Equation (17) implies \(R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = R_k^2{\mathrm{/}}h_k^2\) and thus the index weights of Eq. (15) can be applied equally to BLUP solutions for the SNP effects or BLUP predictors for individuals of each trait as described in the main text in Eq. (1) through (3). Both \(r_{{\mathrm{G}}_{k,l}}\) and \(h_k^2\) of Eq. (15) can be obtained from summary statistic data using LD score regression^{22} and therefore \(\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}\) effects of Eq. (10), which would traditionally require individuallevel phenotype–genotype data for all traits, can be approximated accurately in a computationally efficient manner using only publically available GWAS summary statistic data and an independent genomic reference sample.
Index weighted multitrait OLS SNP effects
In the previous section, we have shown how \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) estimates for multiple traits can be combined to yield more accurate \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}\) SNP effects, which can be turned into \(\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}\) individual predictors that approach \(\widehat {\bf{g}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}\) accuracy. However, using a similar weighting we can also combine \(\widehat {\bf{b}}_{{\mathrm{OLS}}}\) estimates for multiple traits into \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{OLS}}}\).
For SNP j and focal trait f, we have \(\widehat {\bf{b}}_{{\mathrm{OLS}}}\) values for k traits, and we wish to obtain the index weights, w_{j,k}, for each \(\widehat {\bf{b}}_{{\mathrm{OLS}}_{j,k}}\) effect as:
Just like before, the optimal weights can be derived as:\({\bf{w}}_{{\mathrm{OLS}}} = {\bf{V}}_{{\mathrm{OLS}}}^{  1}{\bf{C}}_{{\mathrm{OLS}}}\), where C_{OLS} is now a k × 1 column vector of the covariances of the \(\widehat {\bf{b}}_{{\mathrm{OLS}}_k}\) values of the k traits with the true genetic effects of the SNPs for the focal trait, and V_{OLS} is a k × k variance–covariance matrix of the \(\widehat {\bf{b}}_{{\mathrm{OLS}}}\) effects:
The diagonal elements of V_{OLS} are:
The offdiagonal elements for trait k and l are
C_{OLS} now has elements
If we again consider a twotrait example, Eqs. (25), (26) and (27) combine as:
These weights are considerably different from the BLUP weights, which reflects the different variances of BLUP effects and OLS effects. Here we include this section for completeness but focus our analyses on multitrait BLUP effects, because they are more accurate in expectation than multitrait OLS effects.
Index weighted multitrait SNP effects using LDPred
For phenotypes with a genetic architecture characterised by a few loci of very large effect sizes, this approach may not be ideal. Models that assume a mixture distribution for SNP effects, such as LDpred or BayesR, can yield higher prediction accuracies in traits of noninfinitesimal genetic architecture^{18,42}. As outlined above, Eq. (17) implies \(R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = R_k^2{\mathrm{/}}h_k^2\) and thus the index weights of Eq. (15) can be applied equally to BLUP solutions for the SNP effects or BLUP predictors for individuals of each trait as described in the main text in Eq. (1) through (3). LDpred aims to estimate the posterior mean phenotype that provides best unbiased prediction. Therefore, singletrait individuallevel predictors obtained from LDPred can also be weighted together to create an approximate multitrait predictor.
Prediction accuracy of weighted multitrait BLUP predictors
The prediction accuracy of \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}}\) effects obtained from Eq. (15) can be derived by considering the correlation of b_{ f } and \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_k}\) as:
Equation (13) gives \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_f} = {{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}}\) and thus the covariance of b_{ f } and \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_f}\) is:
The variance of the \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}}\) effects obtained from Eq. (15) is:
Additionally, w = V^{−1}C and Vw = C, and thus w′C = w′Vw or written another way \({\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_f}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}}} \right)\) following BLUP properties. Substituting into Eq. (19), the correlation of b_{ f } and \(\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_k}\) can then be written as:
which gives the squared correlation as \(R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_f}}^2\) = \({\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}}} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_f} \right)\) = \(\frac{{R_f^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_f^2{\mathrm{/}}h_k^2\). Therefore, the squared correlation between a phenotype and a multiple trait index weighted BLUP predictor of the phenotype is approximately:
If we consider a twotrait example then prediction accuracy for a focal trait \(R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_k}}^2\) can be written as:
where V_{1,2} is the offdiagonal element of the matrix V of Eqs. (15) and (21). The value of \(R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}_f}}^2\) can then be compared to the prediction accuracy of the singletrait BLUP predictor of Eq. (16) and to the prediction accuracy of a crosstrait predictor^{43}, where a BLUP predictor of the second trait is used to predict the focal trait phenotype, which is given by: \(R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2 = R_{{\bf{y}}_2,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2r_{\mathrm{G}}\sqrt {\left( {h_2{\mathrm{/}}h_f} \right)}\). This comparison is of interest, because we expect the multitrait predictor to be more accurate than any available singletrait predictor, even if the most accurate singletrait predictor is across two different traits. Crosstrait prediction is equivalent to the proxyphenotype method, which has been used to predict cognitive performance from educational attainment GWAS data^{44}.
Loss of prediction accuracy from BLUP approximation
Equations (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33) and (34) assume that \({\mathrm{cov}}\left( {{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)\) = \({\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)\), or in other words that SBLUP SNP solutions have BLUP properties. The use of an independent LD reference sample to create an approximate singletrait BLUP predictor in Eq. (11) does not affect the covariance between the true SNP effect sizes and the approximate BLUP SNP solution, meaning that the approximate singletrait BLUP predictors have BLUP properties. However, the variance of \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) is likely affected, which may potentially result in a loss of prediction accuracy of a weighted multitrait BLUP predictor. The variance of \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) is:
The loss of information from using an independent data set as an LD reference to obtain L, rather than directly using the individuallevel data to calculate W′W, can be approximated by considering the scenario where SNP makers are unlinked, resulting in diag[L]. The diagonal elements of \(\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2\) for SNP j are then:
The diagonal elements of diag[(W′W)(W′W)] can be approximated as \({\mathrm{diag}}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)} \right]\) ≈ \(N^2\left( {1 + {\Bbb E}\left[ {r^2} \right]M} \right)\) + \(N^2\left( {1 + M{\mathrm{/}}N} \right)\), where the expectation of the LD correlation of the SNPs, \({\Bbb E}\left[ {r^2} \right]\), is 1/N as the SNP markers are unlinked. Equation (36) can then be written as:
From Eq. (37), the squared correlation between true SNP effects and SBLUP SNP effects can be written as:
This can be contrasted to Eq. (17), which gives the squared correlation between the true genetic effects of the SNPs, b_{ k }, and \(\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}\) as:
Equation (39) is similar to Eq. (38) apart from the factor \(1  R_k^2\). Therefore, the relative loss of prediction accuracy from using an SBLUP predictor is given as a ratio of Eqs. (39) and (38) as:
For a phenotype of SNP heritability 0.5, with effective number of independent markers (independent genomic segments), M_{eff}, of ~ 60,000 and sample size, N, of 500,000, \(R_{{\bf{b}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2\) from summary statistics in an independent reference sample will be 91% of the value of \(R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2\) if individuallevel data were available. Likewise, for a twotrait example where both traits have h^{2} = 0.5 and N = 500,000, the accuracy of the multitrait SBLUP predictor will also be 91% of the accuracy of the multitrait BLUP predictor.
It should be noted that here we assume L to be a a diagonal matrix, which will lead to a conservative estimate of the accuracy of SBLUP relative to the accuracy of BLUP, and that this estimate is in fact equivalent to the expected accuracy of a polygenic risk predictor based on marginal OLS effects^{28}. In practice, approximating L through an external reference data set leads to SBLUP predictors, which are more accurate than predictors based on marginal OLS effects but less accurate than predictors based on BLUP effects.
Computation time
Computing weights and combing up to 930,000 SNP effects of 34 traits takes <10 min on an Intel Xeon E78837 processor with 2.76 GHz. Memory requirements do not extend much beyond the amount necessary to read in the summary statistics. Calculation of singletrait SBLUP effects is more computationally demanding, so we split the data by chromosome. Runtime for chromosome one was <40 min, with memory usage just under 10 GB.
Simulation study
To compare the accuracy of singletrait and multitrait genetic predictors created from SNP effects obtained from both individuallevel and summary statistic data, we conducted a simulation study based on real genotypes from the Kaiser Permanente study (Genetic Epidemiology Research on Adult Health and Aging: GERA cohort) and simulated phenotypes.
From the GERA cohort, we selected 50,000 individuals of European ancestry (for definitions of European individuals and quality control (QC) of the genotypic data, see ref. ^{45}). SNP QC procedures on the initial data sets consisted of perSNP missing data rate of <0.01, minor allele frequency >0.01, perperson missing data rate <0.01 and Hardy–Weinberg disequilibrium pvalue < 1 × 10^{−6}. For the subsequent imputation, the data were first haplotyped using HAPIUR. After that, Impute2 was used to impute the haplotypes to the 1000 genomes reference panel (release 1, version 3). Bestguess genotypes at common SNPs included in the HapMap 3 European sample were then extracted and filtered for imputation info score >0.5, missing data rate of <0.01, minor allele frequency >0.01, perperson missing data rate <0.01 and Hardy–Weinberg disequilibrium pvalue < 1 × 10^{−6}. Next, we performed principal component analysis and removed individuals with principal eigenvector values that were >7 SD from the mean of the European cluster. Lastly, we identified pairs of individuals with genetic relatedness matrix >0.05 and removed one individual from each of these pairs.
The Atherosclerosis Risk in Communities study (ARIC data) was used as an independent LD reference when estimating SBLUP SNP effects of Eq. (11). Eight thousand seven hundred and fortyfour European individuals were selected and the data were imputed and QC conducted in the same way as described above for the GERA cohort. We then reduced the SNPs used in both the GERA and ARIC cohorts to overlapping HapMap3 SNPs, which gave 557,034 SNPs that were used in the simulation study.
We then randomly assigned 20,000, 20,000 and 10,000 individuals from the GERA cohort to create three data sets: training set one, training set two, and a testing set. We simulated two genetically correlated traits by randomly selecting 2000 causal SNPs. Effect sizes for the causal markers were simulated from a bivariate normal distribution with mean 0, variances of \(\frac{{h_1^2}}{M}\) and \(\frac{{h_2^2}}{M}\) and covariance of \(r_{\mathrm{G}}\sqrt {h_1^2h_2^2}\). These effect sizes were then multiplied with the standardised genotype dosages (mean 0 and variance 1) to create a genetic value for each individual. Normally distributed environmental effects e ~ N(0, 1 − h^{2}) were added to this genetic value for each individual to create phenotypes with mean 0 and variance of 1. To remove any effects of population stratification, the simulated phenotypes were then regressed against the first 20 genetic principal components, and the residuals from this regression were used in all subsequent analyses.
In training set 1, we simulated trait 1 and we then estimated: (i) OLS SNP effects using Eq. (5) \(\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}}} \right)\), (ii) BLUP SNP effects from the individuallevel data using Eq. (8) \(\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right)\), and (iii) approximate SBLUP effects using the OLS SNP effects from Eq. (5) and the ARIC data as a reference \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)\). In training set 2, we simulated trait 2 and estimated \(\widehat {\bf{b}}_{{\mathrm{OLS}}}\), \(\widehat {\bf{b}}_{{\mathrm{BLUP}}}\) and \(\widehat {\bf{b}}_{{\mathrm{SBLUP}}}\) in the same manner. We then estimated multitrait BLUP SNP effects using Eq. (9) \(\left( {\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}} \right)\) from individuallevel data by combining trait 1 from training set 1 and trait 2 from training set 2.
In the testing set, we then used the estimated SNP effects from the training sets to produce genetic predictors for both traits. Singletrait genetic predictors were created for both simulated traits from (i) the OLS SNP effects \(\left( {\widehat {\bf{g}}_{{\mathrm{OLS}}}} \right)\), (ii) the BLUP SNP effects \(\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}}} \right)\) and (iii) the SBLUP SNP effects \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)\). We then created multitrait predictors where trait 1 was the focal trait from: (i) individuallevel multitrait BLUP predictor \(\left( {\widehat {\bf{g}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}} \right)\), (ii) weighted multitrait SBLUP predictor \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}} \right)\), (iii) a weighted multitrait BLUP predictorbased individuallevel singletrait BLUP estimates \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}}} \right)\), and (iv) a weighted multitrait GWAS predictor based on GWAS OLS estimates \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{OLS}}}} \right)\). We simulated phenotypic values for both traits using the same effect sizes as those used to generate the phenotypes in the training sets and normally distributed environmental effects sampled independently for each trait as e ~ N(0, 1 − h^{2}). We also compared estimates obtained using LDPred with the proportion of SNPs in the model of 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1 or using the LDPredInf option.
We created two simulation scenarios. Heritability of the first and second trait and genetic correlations were \(h_1^2\) = 0.2, \(h_2^2\) = 0.8 and r_{G} = 0.8, respectively, in the first scenario and were \(h_1^2\) = 0.5, \(h_2^2\) = 0.5, and r_{G} = 0.5, respectively, in the second scenario. In each setup, six replicates were conducted, each with a different set of randomly selected causal markers. We then repeated all analyses on a permuted data set, where the values of the genotype matrix were permuted across all individuals, for each SNP. This creates a genotype matrix where the allele frequency distribution remains the same, but all LD structure is removed, allowing us to determine the degree to which differences between the simulations results are driven by the LD structure in the real genotype data. Finally, because prediction accuracy is expected to be reduced by the error introduced by using an external LD reference data set and a restricted LD window when implementing Eq. (8) (see above), we examined how changing the LD reference and restricting the LD window size influences to optimal value of shrinkage parameter λ when implementing Eq. (8) (see Supplementary Fig. 3).
Application to PGC schizophrenia and bipolar disorder
We then applied our approach to the schizophrenia (SCZ) and bipolar disorder (BIP) samples from both wave 1 and wave 2 data of the PGC (PGC1 and PGC2). A description of the data collection and imputation of the SNP genotype data can be found elsewhere^{25,26,46}.
We selected these two disorders because there is a high genetic correlation between them (estimate for r_{G} between schizophrenia and bipolar disorder using ldsc: 0.72, SE: 0.03; estimated using metaanalysis of all PGC2 schizophrenia and bipolar cohorts, excluding cohorts which were used as test set in the initial PGC1 analysis), and it enabled us to draw a direct comparison between the approach described here and a previous study, which estimated multitrait BLUP SNP effects \(\left( {\widehat {\bf{b}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}} \right)\) from individuallevel data in an approach equivalent to Eq. (9). The previous study used PGC1 data in the training set and selected four cohorts for schizophrenia and three cohorts for bipolar disorder as test sets. For schizophrenia, the training set comprised 17 cohorts (8826 cases, 6106 controls) and for bipolar disorder the training set comprised 11 cohorts (5867 cases, 3328 controls). The test set of 4 cohorts for schizophrenia contained 4068 cases and 5471 controls, and the test set of 3 cohorts for bipolar disorder contained 2029 cases and 5338 controls. The analyses on the PGC1 data were performed on 745,705 HapMap3 SNPs in common across all data sets. To have a direct comparison to our previous study, we began by reanalysing the same PGC1 training set data to estimate: (i) OLS SNP effects using Eq. (5) \(\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}}} \right)\), (ii) BLUP SNP effects from the individuallevel data using Eq. (8) \(\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right)\), and (iii) approximate SBLUP effects using the OLS SNP effects from Eq. (5) and the ARIC data as a reference \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)\) using Eq. (11). For the estimation of schizophrenia SBLUP effects, λ was set to 1,100,000, corresponding roughly to 1,000,000 markers and an observed scale SNP heritability estimate of 0.47, and for the estimation of bipolar disorder SBLUP effects, lambda was set to 1,200,000, corresponding roughly to 1,000,000 markers and an observed scale SNP heritability estimate of 0.45. For the four SCZ testing cohorts and the three BIP testing cohorts used in the previous study, we created: (i) weighted multitrait SBLUP predictors \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}} \right)\), (ii) weighted multitrait BLUP predictorbased individuallevel singletrait BLUP estimates \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{BLUP}}}} \right)\), and (iii) weighted multitrait GWAS predictor based on GWAS OLS estimates \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{OLS}}}} \right)\). We then compared the prediction accuracy we obtained using the weighted multitrait SBLUP predictors to the individuallevel multitrait BLUP predictor \(\left( {\widehat {\bf{g}}_{{\mathrm{MT}}  {\mathrm{BLUP}}}} \right)\) used in the previous study^{16}.
We then extended our analysis to the PGC2 data set. There were 36 cohorts for schizophrenia (26,412 cases and 32,440 controls in total) and 23 cohorts for bipolar disorder (18,865 cases and 30,460 controls in total) available to us. The number of SNPs used in the PGC2 analyses varied between cohorts. Summary statistics for each of the PGC2 cohorts was available to an imputed SNP set of >10,000,000 SNPs. After intersecting this set of SNPs with the HapMap3 SNPs and the ARIC SNPs, 932,344 SNPs remained that were used to create predictors.
We applied a crossvalidation approach as we observed that prediction accuracy as well as accuracy differences between predictors can be highly dependent on the choice of the test set in the extended PGC2 data set (Supplementary Figs. 5 and 6), which is supported by previous results showing highly variable prediction accuracy across cohorts in the PGC2 data set^{25}. A crossvalidation approach allowed us to get a more robust estimate of the increase of prediction accuracy achieved by our multitrait prediction method compared to a singletrait predictor. We employed a leaveoneout crossvalidation approach, where, for each test set cohort, all cohorts of the same disease without any highly related individuals were chosen to be in the training set for the singletrait predictor and all cohorts of both diseases without any highly related individuals were chosen to be in the training set for the multitrait predictor. To identify pairs of cohorts with highly related individuals, genetic relatedness for all pairs of individuals (across all pairs of cohorts) was calculated based on chromosome 22, and whenever at least one pair of individuals had relatedness >0.8, that pair of cohorts was not simultaneously used in the training set and the test set.
The full genotypes from the PGC2 cohorts that were used as test sets underwent stringent QC and only comprised 458,744–860,576 SNPs for schizophrenia and 556,278–859,034 SNPs for bipolar disorder. We refrained from using the intersection between all these cohorts to not reduce the number of SNPs used in prediction by too much. This meant that different iterations in the crossvalidations were based on predictions using a different number of SNPs. However, each comparison between a singletrait predictor and a multitrait predictor is based on the same number of SNPs.
In each iteration of the crossvalidation, a different cohort acts as the test set and a different set of cohorts comprises the training set. To create a predictor from a particular set of cohorts, we first had to obtain effect size estimates from this particular set of cohorts. This is achieved by performing a metaanalysis of the summary statistics of the cohorts that comprise the training set. The metaanalysed beta values b_{META} are calculated as:
where b_{ s } is the effect size in cohort s and SE_{ s } is the standard error in cohort s. Conversion between beta values and odds ratios (OR) simply follows the equality b = log(OR). The weights derived for each trait make assumptions about the variance of SNP effects. We found that, in the summary statistics we used, the observed variance across SNP effects often departed from the expected value. To correct for that, we scaled the SNP effect estimates for each trait to have a variance of one and multiplied the weights for the unscaled SNP effects by the expected standard deviation across all SNPs.
We created approximate SBLUP effects \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)\) using the OLS SNP effects from Eq. (5) and the ARIC data as an LD reference using Eq. (11) and set the shrinkage parameter, λ, to 1,300,000 for schizophrenia and to 2,000,000 for bipolar disorder, corresponding to observed scale SNP heritability estimates of 0.43 and 0.33 for schizophrenia and bipolar disorder, respectively. We then used the PLINK “score” function to turn SNP effects \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}},\widehat {\bf{b}}_{{\mathrm{GWAS}}}} \right)\) into individual predictors \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}},\widehat {\bf{g}}_{{\mathrm{GWAS}}}} \right)\) for each metaanalysed schizophrenia or bipolar disorder crossvalidation set. For the multitrait weighting, we estimated the heritability of schizophrenia and bipolar disorder and their genetic correlation using LD score regression from publicly available PGC2 schizophrenia summary statistics and the PGC1 bipolar disorder summary statistics. These estimates were then used to calculate the index weights of Eq. (15) for the weighted multitrait SBLUP predictors \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}},\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{GWAS}}}} \right)\) of SCZ and BIP, and these were not altered between different crossvalidation sets.
To test the degree to which the choice of weights affects the accuracy of the multitrait predictor, we compared the accuracy of multitrait predictors based on a spectrum of other weights (Supplementary Figs. 5 and 6). For this, we took advantage of two things: First, when individual predictors \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}},\widehat {\bf{g}}_{{\mathrm{GWAS}}}} \right)\) are weighted rather than SNP effects \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}},\widehat {\bf{b}}_{{\mathrm{GWAS}}}} \right)\), the conversion from SNP effects to individual effects does not have to be repeated for different weights. Second, the scaling of a predictor does not influence its accuracy in terms of correlation between prediction and outcome. Therefore, rather than testing each combination of weights of schizophrenia and bipolar disorder, it is sufficient to vary the relative weight of schizophrenia to bipolar disorder to explore the whole range of possible multitrait predictors for these two traits. For each test cohort, this enabled us to test whether the weights of our multitrait predictor derived from theory deviate from the weights that would result in the highest prediction accuracy for that data set.
Application to phenotypes in the UK Biobank study
We applied our approach to a large range of phenotypes for which GWAS summary statistics are publicly available. We started with GWAS summary statistics for 46 phenotypes. However, in some circumstance the same studies (i.e., based on the same individuals) had generated summary statistics for multiple similar phenotypes, so we chose only one phenotype per study, which left us with 34 phenotypes. For example, out of “Cigarettes per day” and “Smoking Ever” we only selected the latter to have only one trait for smoking. We used 112,338 unrelated individuals of European descent in the UK biobank data as the testing set. We paired 6 phenotypes out of the 34 summary statistic phenotypes to phenotypes in the UK Biobank: Height, BMI, fluid intelligence score, depression, angina, and diabetes. The first three are quantitative traits and the latter three are disease traits for which we could identify at least 1000 cases in the UK Biobank data. For details, see Supplementary Table 1.
For the disease traits, we used the selfreported diagnoses rather than ICD10 diagnoses, as they tend to have larger sample sizes. For depression, we used a more refined definition of cases and controls, where individuals were not counted as cases if they had any history of psychiatric symptoms or diagnoses other than depression or if they were prescribed drugs that are indicative of such diagnoses. Individuals were selected as controls only when there was an absence of any psychiatric symptoms or diagnoses and only when they were not prescribed any drugs that could be indicative of such diagnoses. All 6 traits in the UK Biobank were corrected for age, sex and the first 10 principal components by regressing the phenotype on these covariates and using the residuals from that regression for further analysis. For each trait, the SNPs that went into the analysis were based on the overlap between the GWAS summary statistics, the HapMap3 SNPs, the GERA data set, which was used as an LD reference in the SBLUP analysis, and the imputed SNPs from the UK Biobank. (For details on the QC process and imputation, see URLs.) Depending on the trait, the total number of SNPs ranged from around 660,000 to around 930,000.
We created singletrait \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)\) as well as multitrait \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}} \right)\) predictors for the six paired phenotypes. To create SBLUP SNP effects \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)\) from summary statistic trait, we used a λ value of \(M\left( {1  h_{{\mathrm{SNP}}_k}^2} \right){\mathrm{/}}h_{{\mathrm{SNP}}_k}^2\) for each trait k, where M is assumed to be 1,000,000. As LD reference set, we used a random subset of 10,000 people of European descent from the GERA data set, and we set the LD window size to 2000 kb. We then used the PLINK “–score” function to turn SNP effects \(\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)\) into individual predictors \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)\) for each trait. For the multitrait weighting, we used LD score regression to calculate SNP heritability and genetic correlation between all pairs of cohorts. For dichotomous disease traits, SNP heritability was calculated on the observed scale. For each phenotype for which a multitrait predictor was created, we selected all phenotypes that had a genetic correlation estimate significantly different from 0 at p = 0.05 with the focal trait, as well as the focal trait itself. The summary statistics based singletrait SBLUP predictors of the selected phenotypes were then combined into multitrait SBLUP \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}} \right)\) predictors. The weights for each phenotype were calculated according to Eq. (15). These weights require the singletrait predictors to have exactly the right variance. Since the summary statistics data slightly diverged from this expectation, we scaled each singletrait SBLUP predictor to have mean 0 and variance 1 and then multiplied it with its expected standard deviation, to ensure everything is on exactly the correct scale. We followed the same approach when using singletrait LDPred predictors.
We compared the performance of the multitrait predictors \(\left( {\widehat {\bf{g}}_{{\mathrm{wMT}}  {\mathrm{SBLUP}}}} \right)\) not only to the performance of the singletrait predictor \(\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)\) for the same trait but also to the performance of all other (crosstrait) singletrait predictors for the traits that exhibited significant r_{G} with the focal trait (Fig. 4). This is appropriate because in some traits the singletrait predictor from the same trait is not the most accurate singletrait predictor.
Code availability
Code reported in this manuscript is available from https://github.com/uqrmaie1/smtpred.
For GCTA, see http://cnsgenomics.com/software/gcta/
For LDSC, see https://github.com/bulik/ldsc
For MTG2, see https://sites.google.com/site/honglee0707/mtg2
For LDpred, see https://github.com/bvilhjal/ldpred/
For UK Biobank, see http://www.ukbiobank.ac.uk/
For PLINK2, see http://www.coggenomics.org/plink2
Data availability
PGC summary statistics data are available from http://www.med.unc.edu/pgc/resultsanddownloads
For UK Biobank data, see https://www.ukbiobank.ac.uk/
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
The University of Queensland group is supported by the Australian Research Council (Discovery Project 160103860 and 160102400), the Australian National Health and Medical Research Council (NHMRC grants 1087889, 1080157, 1048853, 1050218, 1078901, and 1078037) and the National Institute of Health (NIH grants R21ESO2505201 and PO1GMO99568). J.Y. is supported by a Charles and Sylvia Viertel Senior Medical Research Fellowship. M.R.R. is supported by the University of Lausanne. We thank all the participants and researchers of the many cohort studies that make this work possible, as well as our colleagues within The University of Queensland’s Program for Complex Trait Genomics and the Queensland Brain Institute IT team for comments and suggestions and technical support. The UK Biobank research was conducted using the UK Biobank Resource under project 12514. Statistical analyses of PGC data were carried out on the Genetic Cluster Computer (http://www.geneticcluster.org) hosted by SURFsara and financially supported by the Netherlands Scientific Organization (NWO 48005003) along with a supplement from the Dutch Brain Foundation and the VU University Amsterdam. Numerous (>100) grants from government agencies along with substantial private and foundation support worldwide enabled the collection of phenotype and genotype data, without which this research would not be possible; grant numbers are listed in primary PGC publications. This study makes use of data from dbGaP (Accession Numbers: phs000090.v3.p1, phs000674.v2.p2, phs000021.v2.p1, phs000167.v1.p1 and phs000017.v3.p1). A full list of acknowledgements to these data sets can be found in Supplementary Note 1.
Author information
Affiliations
Queensland Brain Institute, University of Queensland, Queensland, QLD, 4072, Australia
 Robert M. Maier
 , Sang Hong Lee
 , Naomi R. Wray
 , Jian Yang
 & Peter M. Visscher
Stanley Center for Psychiatric Research, Broad Institute, Cambridge, MA, 02142, USA
 Robert M. Maier
 & Stephan Ripke
Analytic and Translational Genetics Unit, Massachusetts General Hospital and Harvard Medical School, Boston, MA, 02114, USA
 Robert M. Maier
 & Stephan Ripke
Institute for Molecular Bioscience, University of Queensland, Queensland, QLD, 4072, Australia
 Zhihong Zhu
 , Maciej Trzaskowski
 , Naomi R. Wray
 , Jian Yang
 , Peter M. Visscher
 & Matthew R. Robinson
Centre for Population Health Research, School of Health Sciences and Sansom Institute of Health Research, University of South Australia, Adelaide, SA, 5000, Australia
 Sang Hong Lee
Division of Genetic Medicine, Department of Medicine, Psychiatry and Biomedical Informatics, Vanderbilt Genetics Institute, Vanderbilt University Medical Center, Nashville, TN, 37235, USA
 Douglas M. Ruderfer
Institute for Genomics and Multiscale Biology, Icahn School of Medicine at Mount Sinai, New York, NY, 10029, USA
 Eli A. Stahl
Department of Psychiatry and Psychotherapy, Charité, Campus Mitte, 10117, Berlin, Germany
 Stephan Ripke
Department of Computational Biology, University of Lausanne, 1015, Lausanne, Switzerland
 Matthew R. Robinson
Swiss Institute of Bioinformatics, CH1015, Lausanne, Switzerland
 Matthew R. Robinson
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Contributions
P.M.V., N.R.W. and M.R.R. conceived and designed the study. R.M.M. conducted all analyses and developed the software with assistance and guidance from J.Y., Z.Z. and S.H.L. provided statistical advice. M.T. conducted quality control on summary statistics data. R.M.M., N.R.W., P.M.V. and M.R.R. wrote the manuscript. D.M.R., E.A.S. and S.R. provided access to the PGC data. All authors reviewed and approved the final manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Robert M. Maier or Peter M. Visscher or Matthew R. Robinson.
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