Leveraging molecular quantitative trait loci to understand the genetic architecture of diseases and complex traits

Overview of Methods

Our goal is to construct molecular QTL-based annotations that are maximally enriched for disease heritability. For a given molecular QTL data set, we construct a probabilistic (continuous-valued) annotation as follows. First, for each molecular phenotype (e.g. each gene) with at least one significant (FDR $<$ 5%) cis-QTL (e.g. 1Mb from TSS), we compute the causal posterior probability (CPP) of each cis SNP in the fine-mapped 95% credible set, using our CAVIAR fine-mapping method³⁵ (see URLs). Then, for each SNP in the genome, we assign an annotation value based on the maximum value of CPP across all molecular phenotypes; SNPs that do not belong to any 95% credible set are assigned an annotation value of 0. We refer to this annotation as MaxCPP. For comparison purposes, we also construct three other molecular QTL-based annotations. First, we construct a binary annotation containing all SNPs that are a significant (FDR $<$ 5%) cis-QTL for at least one molecular phenotype^19,20; we refer to this annotation as AllcisQTL. Second, we construct a binary annotation containing all SNPs that belong to the 95% credible set (see above) for at least one molecular phenotype; we refer to this annotation as 95%CredibleSet. Third, we construct a binary annotation containing the most significant SNP for each molecular phenotype with at least one significant (FDR $<$ 5%) QTL. We refer to this annotation as TopcisQTL (see Online Methods).

We apply a previously developed method, stratified LD score regression (S-LDSC)^32,33, to partition disease heritability using functional annotations. We use two metrics to quantify an annotation’s contribution to disease heritability: enrichment and standardized effect size ( $\tau \ast$). Enrichment is defined as the proportion of heritability explained by SNPs in an annotation divided by the proportion of SNPs in the annotation³²; here, we generalize this definition to probabilistic annotations such as MaxCPP. Standardized effect size ( $\tau \ast$) is defined as the proportionate change in per-SNP heritability associated to a one standard deviation increase in the value of the annotation, conditional on other annotations included in the model³³. Unlike enrichment, $\tau \ast$ quantifies effects that are unique to the focal annotation (see Online Methods).

We constructed MaxCPP and other annotations using eQTL data from the GTEx Consortium^12,14 and eQTL, hQTL, sQTL and meQTL data from the BLUEPRINT Consortium¹⁸ (Table 1; see URLs). We included a broad set of 75 functional annotations from the baselineLD model (Supplementary Table 1) in most analyses. We have made our annotations and partitioned LD scores freely publicly available (see URLs).

Simulations

We performed a comprehensive set of simulations to assess whether S-LDSC produces unbiased estimates of an annotation’s contribution to disease heritability for the AllcisQTL, 95%CredibleSet, TopcisQTL and MaxCPP annotations. We performed simulations using real genotypes from the UK Biobank, restricting to 749,024 SNPs on chromosome 1 (see Online Methods). In our main simulation, we simulated gene expression phenotypes for 500 individuals assuming that 10% of cis-variants for a gene are causal cis-eQTL (heritability=16%), simulated complex trait phenotypes for an independent set of 40,000 individuals assuming that the set of causal variants is exactly the set of causal eQTLs, with independent effect sizes (heritability=20%), and subsequently assumed that 10% of causal eQTL are missing from the data analyzed. We also performed secondary simulations under other genetic architectures and assumptions about missing data (see below). We estimated each annotation’s contribution to complex trait heritability using S-LDSC. We performed 400 independent simulations, and averaged results across simulations.

Enrichment estimates and true enrichments for each annotation are displayed in Figure 1 and Supplementary Table 2. We determined that S-LDSC estimates are severely upward biased for the TopcisQTL annotation On the other hand, S-LDSC estimates were slightly conservative for the AllcisQTL, 95%CredibleSet and MaxCPP annotations. Thus, we restrict our analyses to these three annotations in our analyses of real phenotypes below Of these three annotations, MaxCPP had the highest enrichment (Figure 1 and Supplementary Table 2). For comparison purposes, we also computed estimates using GCTA^27,28 (see URLs), a method that has previously been applied to assess eQTL enrichment for complex traits^19,20; we computed GCTA estimates for all annotations except MaxCPP, as GCTA is only applicable to binary annotations. We determined that GCTA estimates generally exhibited greater bias than S-LDSC estimates (Figure 1 and Supplementary Tables 2-3). We obtained similar results for both S-LDSC and GCTA at other simulation parameters (Supplementary Table 3). We also obtained similar results for both S-LDSC and GCTA using an alternative simulation framework, drawn from our previous work²⁵, that directly uses simulated gene expression to generate complex trait phenotypes (see Online Methods and Supplementary Table 4).

All functional enrichment methods (GCTA^27,28, BOLT-REML³⁰, S-LDSC^32,33, and LDAK³¹) that we are currently aware of assume that causal disease effect sizes are independent and identically distributed (i.i.d) conditional on MAF, LD and function annotation values. However, this assumption may be violated for molecular QTL-based annotations when causal variants are sparse (see Online Methods for an example); in particular, this is a limitation of our S-LDSC method. Indeed, our simulations confirm these biases (Figure 1 and Supplementary Tables 2-4). We note that other functional enrichment methods are also subject to this limitation. Specifically, GCTA^27,28 is shown by our simulations to exhibit greater biases than S-LDSC (Figure 1 and Supplementary Tables 2-4). BOLT-REML³⁰ is a computationally efficient method that produces the same results as GCTA^27,28, and LDAK³¹ has been shown in separate work to produce biased estimates in much simpler settings³⁸. S-LDSC produces slightly conservative estimates across a comprehensive set of simulations for the AllcisQTL, 95%CredibleSet and MaxCPP annotations that we consider in our analyses of real phenotypes below.

Fine-mapped eQTL are enriched for disease heritability

We used the GTEx eQTL data set (Table 1) to construct AllcisQTL, 95%CredibleSet and MaxCPP annotations. We constructed annotations using each of the 44 tissues (Supplementary Table 5). We applied S-LDSC to assess each annotation’s contribution to disease heritability for each of 41 independent diseases and complex trait data sets (average $N$=320K); for six traits we analyzed two different data sets, leading to a total of 47 data sets analyzed (see Supplementary Table 6). We meta-analyzed results across the 47 data sets, which were chosen to be independent (see Online Methods). We computed enrichment and $\tau \ast$ for each annotation, in analyses that included 75 functional annotations (Supplementary Table 1) from the baselineLD model³³. Results for whole blood, a widely studied tissue, are reported in Figure 2 and Supplementary Table 7. We determined that the MaxCPP annotation had far higher values of enrichment and $\tau \ast$ than AllcisQTL or 95%CredibleSet; the enrichment estimates remain much higher for MaxCPP even after accounting for the fact that S-LDSC generally produces more conservative estimates for AllcisQTL and 95%CredibleSet than for MaxCPP in our simulations (1.10-1.58x and 0.76-1.23x more conservative respectively, across all simulations; Figure 2 and Supplementary Table 7). The MaxCPP annotation also had higher values of enrichment and $\tau \ast$ in analyses that did not condition on the baselineLD model (Supplementary Figure 1 and Supplementary Table 8).

We investigated whether the $\tau \ast$ for MaxCPP in each respective tissue varied with sample size. We observed a correlation ( $R^2$ = 0.69, P = 1.36e-12) between sample size and $\tau \ast$ (Figure 3 and Supplementary Table 9). We observed a similar pattern in simulations (Supplementary Figure 2). This suggests that the correlation between sample size and $\tau \ast$ in GTEx data is related to statistical power and not because tissue-specific eQTL from tissues with larger sample size are more relevant for the 41 traits analyzed. Thus, annotations constructed from tissues with larger sample sizes are more informative for disease/trait architectures.

To maximize sample size, we performed a fixed-effect meta-analysis of eQTL effect sizes across the 44 tissues (FE-Meta-Tissue)(see Online Methods). We determined that FE-Meta-Tissue annotations had slightly larger enrichments and much larger $\tau \ast$ (due to larger annotation size) than annotations constructed from individual tissues (Figure 2, Supplementary Table 7 and Supplementary Table 10). As with individual tissues, the MaxCPP annotation had far higher values of enrichment (5.84x, s.e. = 0.40; P = 1.19e-31) and $\tau \ast$ ( $\tau \ast$ = 0.52, s.e. = 0.05; P = 2.73e-27) than AllCisQTL (enrichment = 1.80x and $\tau \ast$ = 0.03) or 95%CredibleSet (enrichment = 2.75x and $\tau \ast$ = 0.17). It is particularly notable that the $\tau \ast$ for GTEx FE-Meta-Tissue MaxCPP (conditional on the baselineLD model) is much larger than the $\tau \ast$ values of the 6 continuous annotations that we introduced in our previous work³³ (Supplementary Table 11). A histogram of MaxCPP annotation values for GTEx FE-Meta-Tissue is provided in Supplementary Figure 3, and correlations with baselineLD model annotations and their LD scores are provided in Supplementary Figures 4 and 5.

Tissue-specific fine-mapped eQTL enriched for heritability

Although the FE-Meta-Tissue MaxCPP annotation outperformed each of the 44 tissue-specific MaxCPP annotations in the meta-analysis across 41 traits (Figure 2 and Figure 3), this was not the case for every trait. We examined six autoimmune diseases, five blood cell traits, and eight brain-related diseases and traits in detail (see Online Methods). We first analyzed the six autoimmune diseases, analyzing MaxCPP annotations for blood and FE-Meta-Tissue separately (conditional on the baselineLD model) and meta-analyzing results across the six diseases. We obtained higher estimates of $\tau \ast$ (and higher or comparable estimates of enrichment) for blood than for FE-Meta-Tissue or any other individual tissue (Supplementary Table 12). We then analyzed MaxCPP annotations for blood and FE-Meta-Tissue jointly (conditional on the baselineLD model). We obtained a significantly positive $\tau \ast$ estimate for blood ( $\tau \ast$ = 0.91, s.e. = 0.34; P = 9.15e-03) (Figure 4 and Supplementary Table 13), implying that fine-mapped blood eQTL provides additional information about these six diseases conditional on fine-mapped FE-Meta-Tissue eQTL. We repeated these analyses for the five blood cell traits. When analyzing MaxCPP annotations for blood and FE-Meta-Tissue separately, we obtained higher estimates of $\tau \ast$ (and higher or comparable estimates of enrichment) for blood than for FE-Meta-Tissue or any other individual tissue (Supplementary Table 14). When analyzing MaxCPP annotations for blood and FE-Meta-Tissue jointly, we obtained a significantly positive $\tau \ast$ estimate for blood ( $\tau \ast$ = 1.17, s.e. = 0.24; P = 1.77e-06) (Figure 4 and Supplementary Table 13), implying that fine-mapped blood eQTL provides additional information about these five traits conditional on fine-mapped FE-Meta-Tissue eQTL. The correlations of blood MaxCPP annotation values with baselineLD model annotations and their LD scores are provided in Supplementary Figures 6 and 7.

We then analyzed the eight brain-related diseases and traits. We performed a fixed-effect meta-analysis of eQTL effect sizes across the 10 brain tissues and one nerve tissue (Brain+Nerve). When analyzing MaxCPP annotations for Brain+Nerve and FE-Meta-Tissue separately, we obtained higher or comparable estimates of enrichment and $\tau \ast$ for Brain+Nerve than for FE-Meta-Tissue or any individual tissue (Supplementary Table 15). When analyzing MaxCPP annotations for Brain+Nerve and FE-Meta-Tissue jointly, we obtained a significantly positive $\tau \ast$ estimate for Brain+Nerve ( $\tau \ast$ = 0.28, s.e. = 0.07; P = 9.81e-05) (Figure 4 and Supplementary Table 13), implying that fine-mapped Brain+Nerve eQTL provides additional information about these eight traits conditional on fine-mapped FE-Meta-Tissue eQTL. The correlations of Brain+Nerve MaxCPP annotation values with baselineLD model annotations and their LD scores are provided in Supplementary Figures 6 and 7. We repeated these analyses for each trait and each tissue separately, but determined that only three blood cell traits (white blood count, red blood cell distribution width, and eosinophil count traits), in conjunction with MaxCPP for blood, attained a significantly positive (FDR $<$5%) tissue-specific $\tau \ast$ (Supplementary Tables 16 and 17). Overall, these results demonstrate that tissue-specific eQTL effects on steady-state expression can be significant for diseases and complex traits, despite the well-documented high cis-genetic correlations of eQTL effect sizes across tissues^12,36.

MaxCPP signal is concentrated in disease-relevant gene sets

Recent studies have identified gene sets that are depleted for coding variants and enriched for de novo coding mutations impacting disease^37,39,40. To investigate the importance of non-coding common variants in these gene sets, for each gene set $S$ we used GTEx FE-Meta-Tissue to construct a new annotation MaxCPP( $S$), defined as the maximum CPP restricted to genes in $S$. For comparison purposes, we also constructed an annotation allSNP( $S$), defined as the set of all SNPs within 100Kb of genes in $S$.

We first analyzed the ExAC gene set, consisting of 3,230 genes that are strongly depleted for protein-truncating variants³⁷. We determined that MaxCPP(ExAC) was very strongly enriched in an analysis conditional on the baselineLD model, meta-analyzed across 41 independent traits (see Figure 5a and Supplementary Table 18). In particular, MaxCPP(ExAC) was much more strongly enriched (17.06x, s.e. = 1.28; P = 1.20e-35) than MaxCPP(All Genes) (5.84x) (P = 4.90e-17 for difference). This implies that eQTL for these 3,230 genes have a disproportionately strong impact on disease heritability, consistent with the fact that these genes are depleted for eQTL³⁷. We then analyzed MaxCPP(ExAC) and MaxCPP(All Genes) annotations jointly (conditional on the baselineLD model). We obtained a significantly positive $\tau \ast$ for MaxCPP(ExAC) ( $\tau \ast$ = 0.41, s.e. = 0.04; P = 1.40e-23; Figure 5b and Supplementary Table 19), implying that MaxCPP(ExAC) provides additional information about disease heritability conditional on MaxCPP (All Genes). We observed that the effect size ( $\tau \ast$) for MaxCPP(ExAC) conditional on MaxCPP (All Genes) and baselineLD is five times larger, and more statistically significant, than the $\tau \ast$ of allSNP (ExAC) conditional on the baselineLD model (Supplementary Table 20). Thus, MaxCPP can increase power to identify enriched gene sets.

We analyzed four additional gene sets $S$: a set of 1,003 genes that are strongly depleted for missense mutations⁴⁰ (Samocha); a set of 2,984 genes with strong selection against protein-truncating variants³⁹ (Cassa); a set of 1,878 genes predicted to be essential based on CRISPR experiments in a human cancer cell line⁴¹ (Wang); and a set of 11,983 genes with evidence of allelic heterogeneity in analyses of GTEx gene expression data using our previously developed methods (AH)⁴². For each of these gene sets, MaxCPP( $S$) was strongly enriched in analyses conditional on the baselineLD model, meta-analyzed across 41 independent traits (Figure 5a and Supplementary Table 18). In addition, for each gene set except the Wang gene set, we obtained a significantly positive $\tau \ast$ for MaxCPP( $S$) (after correcting for five gene sets tested) when analyzing MaxCPP( $S$) and MaxCPP(All Genes) jointly (conditional on the baselineLD model) (Figure 5b, Supplementary Table 19 and Supplementary Tables 21-25). As with the ExAC gene set, the $\tau \ast$ for MaxCPP( $S$) conditional on MaxCPP(All Genes) and the baselineLD model were substantially larger than the $\tau \ast$ of allSNPs (ExAC) conditional on the baselineLD model and were often more statistically significant (Supplementary Table 20), indicating that MaxCPP can increase power to identify enriched gene sets in which regulatory variants play an important role.

Fine-mapped molecular QTL are enriched for heritability

We analyzed five molecular QTL from the BLUEPRINT data set (Table 1), including eQTL, hQTL (H3K27ac and H3K4me1), sQTL and meQTL. In each case, we constructed AllcisQTL, 95%CredibleSet and MaxCPP annotations using each of the three immune cell types (CD14+ monocytes, CD16+ neutrophils, and naive CD4+ T cells; Supplementary Table 26) as well as a fixed-effect meta-analysis of molecular QTL effect sizes across the 3 cell types (FE-Meta-Tissue). We determined that for each QTL data set the MaxCPP annotation outperformed the AllcisQTL and 95%CredibleSet annotations (Supplementary Table 27). A histogram of MaxCPP annotation values for each QTL data set is provided in Supplementary Figure 8. MaxCPP for each molecular QTL was significantly enriched in an analysis conditional on the baselineLD model, meta-analyzed across the 41 traits: eQTL (5.44x, s.e. = 0.55; P = 3.26e-16), H3K27ac (4.28x, s.e. = 0.37; P = 2.59e-19), H3K4me1 (4.27x, s.e. = 0.36; P = 1.29e-20), sQTL (3.61x, s.e. = 0.40; P =1.39e-10), and meQTL (2.81x, s.e. = 0.19; P = 8.36e-22) (Figure 6a and Supplementary Table 28); the enrichment for BLUEPRINT eQTL was almost as large as the enrichment for GTEx eQTL (5.84x), despite the much smaller total sample size of FE-Meta-Tissue in BLUEPRINT. This implies that BLUEPRINT sample sizes, though small, are adequately powered for eQTL detection. Consistent with this finding, we observed a high replication rate between cis-QTL in GTEx and BLUEPRINT (see Supplementary Table 29), confirming that GTEx FE-Meta-Tissue provides increased power relative to GTEx blood (Supplementary Table 28). BLUEPRINT FE-Meta-Tissue generally attained higher enrichments and $\tau \ast$ than MaxCPP computed using each of the three immune cell types individually (Supplementary Table 30), similar to our GTEx results (Supplementary Tables 7 and 10). MaxCPP computed using FE-Meta-Tissue also generally outperformed each of the three cell types in a meta-analysis across the six autoimmune diseases (Supplementary Table 31) and a meta-analysis across the five blood cell traits (Supplementary Table 32), in contrast to the stronger enrichments for tissue-specific GTEx blood eQTL annotations for blood cell traits (Supplementary Tables 12 and 14). FE-Meta-Tissue generally attained higher enrichments and $\tau \ast$ than MaxCPP computed using each of the three immune cell types individually (Supplementary Table 30), similar to our GTEx results (Supplementary Tables 7 and 10). MaxCPP computed using FE-Meta-Tissue also generally outperformed each of the three cell types in a meta-analysis across the six autoimmune diseases (Supplementary Table 31) and a meta-analysis across the five blood cell traits (Supplementary Table 32), in contrast to tissue-specific results in GTEx (Supplementary Tables 12 and 14).

Finally, we jointly analyzed MaxCPP annotations for GTEx eQTL and each of the five BLUEPRINT molecular QTL (conditional on the baselineLD model). The purpose of this analysis was to determine whether each of these molecular QTL provides independent information about disease and complex trait architectures. We determined that $\tau \ast$ remained statistically significant for all molecular QTL except sQTL (Figure 6b and Supplementary Table 33); a joint analysis of just the five BLUEPRINT molecular QTL (conditional on the baselineLD model) produced similar findings (Supplementary Table 34). LD scores of the sQTL annotation had the highest correlation with LD scores of the GTEx eQTL and BLUEPRINT eQTL annotations ( $R=0.56-0.57$; see Supplementary Figure 5), implying that much of the informativeness of sQTL in this analysis is captured by eQTL. However, eQTL, hQTL (H3K27ac and H3K4me1) and meQTL are each uniquely informative for disease and complex trait architectures.

Molecular QTL-based annotations

We construct four annotations for any given QTL data set using the observed marginal association statistics. The four annotations are MaxCPP, AllcisQTL, 95%CredibleSet, and TopcisQTL. Each annotation is a vector that assigns a value to each SNP. Let $\mathrm{a}$ indicate our annotation for one QTL data set where ${\mathrm{a}}_{\mathrm{j}}$ indicates the value assigned to SNP $\mathrm{j}$. For binary annotations (AllcisQTL, 95%CredibleSet, and TopcisQTL) ${\mathrm{a}}_{\mathrm{j}}\in \{0,1\}$, and ${\mathrm{a}}_{\mathrm{j}}=0$ indicates that SNP $\mathrm{j}$ is not included in the annotation, while ${\mathrm{a}}_{\mathrm{j}}=1$ indicates that SNP $\mathrm{j}$ is included in the annotation. For continuous probabilistic annotations (MaxCPP), $0\le {\mathrm{a}}_{\mathrm{j}}\le 1$.

Let $\mathrm{S}=({\mathrm{s}}_1,{\mathrm{s}}_2,\cdots {\mathrm{s}}_{\mathrm{g}})$ indicate an ( $\mathrm{m}\times \mathrm{g}$) matrix of the observed marginal association statistics obtained for each QTL data set, where $\mathrm{m}$ is the number of SNPs and $\mathrm{g}$ is the number of eGenes (e.g., genes that have at least one significant cis-eQTL). Let ${\mathrm{s}}_{\mathrm{i}}$ be the vector of marginal association statistics of gene $\mathrm{i}$ for all the cis-variants. Utilizing ${\mathrm{s}}_{\mathrm{i}}$ and the LD structure, we can compute the causal posterior probability (CPP) for each variant. CPP is the probability that a variant is causal. Let ${\mathrm{\alpha }}_{\text{ji}}$ be the posterior probability that the SNP $\mathrm{j}$ is causal for the gene $\mathrm{i}$. We obtain the CPP values from CAVIAR³⁵. In addition to the CPP values, CAVIAR provides a 95%credible set that contains all of the causal variants with probability at least 95%. Let ${\mathrm{\theta }}_{\text{ji}}$ indicate whether SNP $\mathrm{j}$ is in the 95%credible set for the gene $\mathrm{i}$ (i.e., ${\mathrm{\theta }}_{\text{ji}}$=1 indicates that the SNP $\mathrm{j}$ is in the gene $\mathrm{i}$ 95%credible set and ${\mathrm{\theta }}_{\text{ji}}$=0 otherwise). We construct the MaxCPP annotation for SNP $\mathrm{j}$ by computing the maximum value of CPP over all genes where SNP $\mathrm{j}$ is in the 95%credible set of the gene $\mathrm{i}$. More formally, we have: ${\mathrm{a}}_{\mathrm{j}}={\max }_{\mathrm{i}}{\mathrm{\alpha }}_{\text{ji}}$ where the maximum is over genes $\mathrm{i}$ with ${\mathrm{\theta }}_{\text{ij}}=1$.

AllcisQTL annotation is a binary annotation, where any variant whose marginal association statistic for at least one gene passes the significance threshold (FDR $<$ 0.05) has annotation value 1, and each other variant has annotation value 0. Let ${\mathrm{t}}_{\text{ji}}$ indicate whether the SNP $\mathrm{j}$ is statistically significant for the gene $\mathrm{i}$ ( ${\mathrm{t}}_{\text{ji}}=1$ when FDR(j) $<$ 0.05 and ${\mathrm{t}}_{\text{ij}}=0$ otherwise). More formally, we have: ${\mathrm{a}}_{\mathrm{j}}={\max }_{\mathrm{i}}{\mathrm{t}}_{\text{ji}}$.

95%CredibleSet is a binary annotation, any variant that is in a 95%credible set of at least one gene has annotation value 1 and each other variant has annotation value 0. More formally, we have: ${\mathrm{a}}_{\mathrm{j}}={\max }_{\mathrm{i}}{\mathrm{\theta }}_{\text{ji}}$.

TopcisQTL is a binary annotation where any variant that is the most significant variant for at least one gene has annotation value 1 and each other variant has annotation value 0. Let ${\mathrm{\gamma }}_{\text{ji}}$ indicate whether the SNP $\mathrm{j}$ is the most significant SNP for the gene $\mathrm{i}$ (i.e., ${\mathrm{\gamma }}_{\text{ji}}$=1 if SNP $\mathrm{j}$ is the most significant SNP among all cis-variants for the gene $\mathrm{i}$ and ${\mathrm{\gamma }}_{\text{ji}}$=0 otherwise). More formally, we have: ${\mathrm{a}}_{\mathrm{j}}={\max }_{\mathrm{i}}{\mathrm{\gamma }}_{\text{ji}}$.

Enrichment and effect size ( $\mathrm{\tau }\ast$) metrics

We use two metrics to measure the importance of an annotation in the context of diseases and complex traits: Enrichment and standardized effect size ( $\mathrm{\tau }\ast$) of annotation. We use S-LDSC^32,33 to compute enrichment and standardized effect size ( $\mathrm{\tau }\ast$). Let ${\mathrm{a}}_{\text{cj}}$ indicate the annotation value of the SNP $\mathrm{j}$ for the annotation $\mathrm{c}$. S-LDSC^32,33 assumes that the variance of each SNP is a linear additive contribution to each annotation:

where ${\mathrm{\tau }}_{\mathrm{c}}$ is the contribution of the annotation $\mathrm{c}$ to per-SNP heritability. S-LDSC^32,33 estimates ${\mathrm{\tau }}_{\mathrm{c}}$ using the following equation:

where $\mathrm{N}$ is the GWAS sample size and $\mathrm{l}(\mathrm{j},\mathrm{c})$ is the LD score of the SNP $\mathrm{j}$ for the annotation $\mathrm{c}$. S-LDSC computes the LD scores as follow: $\mathrm{l}(\mathrm{j},\mathrm{c})={\displaystyle \sum }_{\mathrm{k}}{\mathrm{a}}_{\text{ck}}{\mathrm{r}}_{\text{jk}}^2$ where ${\mathrm{r}}_{\text{jk}}$ is the genetic correlation between the SNPs $\mathrm{j}$ and $\mathrm{k}$.

Since ${\mathrm{\tau }}_{\mathrm{c}}$ depends on the trait heritability and the size of the annotation, ref.³³ defined ${\mathrm{\tau }}_{\mathrm{c}}\ast$ for an annotation as the standardized annotation effect size:

where $\text{sd}(\mathrm{c})$ is the standard deviation of the annotation $\mathrm{c}$, ${\mathrm{h}}_{\mathrm{g}}^2$ is the SNP-heritability, and $\mathrm{M}$ is the number of variants used to compute ${\mathrm{h}}_{\mathrm{g}}^2$. In our experiments, $\mathrm{M}$ is equal to 5,961,159 (see below).

The enrichment of an annotation is defined as the fraction of heritability captured by the annotation divide by the fraction of SNPs in that annotation. We extend the definition of enrichment to continuous probabilistic annotations with values between 0 and 1:

where ${\mathrm{h}}_{\mathrm{g}}^2(\mathrm{c})$ is the heritability captured by the c-th annotation. We can compute this quantity as follows:

Although both enrichment and $\mathrm{\tau }\ast$ are computed using a model that includes all annotations, $\mathrm{\tau }\ast$ quantifies effects that are unique to the focal annotation (after conditioning on all other annotations in the model), whereas enrichment quantifies effects that are either unique and/or non-unique to the focal annotation. For example, consider a model that includes two annotations, where the first annotation is a highly disease-informative functional annotation and the second annotation is the first annotation plus a random set of SNPs. Only the first annotation will have significant $\mathrm{\tau }\ast$, but both annotations will be significantly enriched. We confirmed via simulation that, under a generative model in which only the baselineLD and GTEx FE-Meta-Tissue MaxCPP annotations directly influence trait heritability, $\mathrm{\tau }\ast$ estimates for the GTEx-Whole-Blood MaxCPP annotations are equal to 0 on average, with a correctly calibrated null distribution of P-values for nonzero $\mathrm{\tau }\ast$ (Supplementary Figure 11).

We computed the statistical significance level (p-value) of enrichment for each annotation via block-jackknife, as described in our previous studies^32,33,50. We computed the statistical significance (p-value) of standardized effect size ( $\mathrm{\tau }\ast$) for each annotation by assuming that $\frac{\mathrm{\tau }\ast }{\text{se}(\mathrm{\tau }\ast )}$ follows a normal distribution with mean 0 and variance of 1 ( $\frac{\mathrm{\tau }\ast }{\text{se}(\mathrm{\tau }\ast )}\sim \mathrm{N}(0,1)$)³³.

Simulation framework

Concrete example for S-LDSC bias estimates of TopcisQTL

Let x be a SNP in the TopcisQTL annotation, let y be a SNP not in the TopcisQTL annotation that is in LD with x, and let z be a random SNP not in the TopcisQTL annotation. Then, in expectation, y has a larger causal disease effect size than z (violating S-LDSC model assumptions), because it is possible that y is a causal molecular QTL (tagged by x, which may be more significant due to statistical chance), and that y is also causal for disease. The fact that SNPs in LD with the TopcisQTL annotation have larger causal disease effect sizes may cause TopcisQTL enrichment to be overestimated by S-LDSC, which attributes higher ${\mathrm{\chi }}^2$ statistics for such SNPs entirely to tagging of causal TopcisQTL enrichment. On the other hand, enrichments of more inclusive annotations may be underestimated by S-LDSC, because SNPs in the annotation with high LD to other SNPs in the annotation are expected to have lower causal disease effect sizes than random SNPs in the annotation (again violating S-LDSC model assumptions).

Set of 41 independent diseases and complex traits

We initially considered 34 GWAS summary association statistic data sets that are publicly available and 55 UK Biobank traits (see URLs) for which summary association statistics were computed using BOLT-LMM^30,54 (see URLs; up to N=459K European-ancestry samples). We restricted our analyses to 47 data sets with z-scores of total SNP heritability at least 6 (Supplementary Table 6). The 47 data sets included 6 traits that were duplicated in two different data sets (genetic correlation of at least 0.9). Thus, we analyzed 41 independent diseases and complex traits. We ran S-LDSC using the same procedures described in previous studies^32,33 (see above). All analyses that included the baselineLD model are based on baselineLD model v1.1, which is identical to the baselineLD model as previously described³³ except that we fixed an error in the promoter annotation (inherited from previous studies^32,33); we determined that fixing this error did not affect our results (see Supplementary Table 38). The z-score of total SNP heritability was computed using S-LDSC with the baselineLD model, and the genetic correlation between pairs of traits was computed using cross-trait LDSC⁵⁵. The meta-analyzed values of enrichment and $\mathrm{\tau }\ast$ across the 47 data sets were computed using a random-effect meta-analysis, as implemented in the $\text{rmeta}$ R package.

Fixed-effect meta-analysis of eQTL effect sizes (FE-Meta-Tissue)

Given a set of effect sizes for SNP $\mathrm{i}$ $({\mathrm{\beta }}_1,{\mathrm{\beta }}_2,\cdots {\mathrm{\beta }}_{\mathrm{t}})$ for $\mathrm{t}$ tissues, where ${\mathrm{\beta }}_{\mathrm{j}}$ is the eQTL effect size for tissue $\mathrm{j}$, we used fixed-effect meta-analysis (FE-Meta-Tissue) to compute inverse-variance weighted meta-analysis z-scores ${\mathrm{z}}_{\text{FE}}$ as follows:

where ${\mathrm{w}}_{\mathrm{j}}=\frac{1}{\text{se}{({\mathrm{\beta }}_{\mathrm{j}})}^2}$ and $\text{se}({\mathrm{\beta }}_{\mathrm{j}})$ is the standard error of ${\mathrm{\beta }}_{\mathrm{j}}$. We note that equation (7) is equivalent to computing a weighted average of z-scores.

Our use of FE-Meta-Tissue has two limitations. First, expression levels in two tissues in the same individual are not independent (sample overlap). Second, true effect sizes in two tissues may be different (heterogeneity). We discuss each limitation in turn.

Regarding the sample overlap limitation, we determined that noise is largely uncorrelated across tissues. For example, the correlation of normalized gene expression (read count) between whole blood and brain hippocampus is 0.11. Furthermore, the genetic correlation between these two tissues is $\sim$0.67³⁶ and the heritability explained by cis-eQTL is $\sim$0.12. This implies that the bulk of gene expression correlation is due to genetic correlation. Thus, the noise (environmental and measurement) in expression levels in two tissues in the same individual is close to independent.

Regarding the heterogeneity limitation, we determined that recently developed random-effect cross-tissue eQTL meta-analysis methods^48,49,56,57 are not applicable to our problem. The Meta-Tissue method⁴⁰ is computationally intractable for data sets as large as GTEx (number of tissues and sample size). Other existing methods^49,56,57, which are Bayesian methods, do not produce summary statistics (e.g. z-scores) that are required to compute the MaxCPP annotation. Thus, these methods are not applicable to our work. Previous studies^12,36 have shown that eQTL effects are highly correlated across tissues, suggesting that our fixed-effect meta-analysis approach is likely to be fairly close to optimal.

We note that the above limitations pertain only to power and not to false positives in our setting, which involves building eQTL annotations to apply to independent disease data. Our results (Figure 2 and Supplementary Tables 7 and 27) show that we have improved our results by utilizing FE-Meta-Tissue. Furthermore, utilizing FE-Meta-Tissue increases replication rates for both eQTL and hQTL (Supplementary Table 29).

Blood and brain related diseases and complex traits

We analyzed six autoimmune diseases: Crohn’s disease⁵⁸, Rheumatoid arthritis (ref.⁵⁹ and UK Biobank), Ulcerative colitis⁵⁸, Lupus⁶⁰, Celiac⁶¹, and all autoimmune and inflammatory diseases in UK Biobank).

We analyzed five blood cell traits: white blood cell count, red blood cell count, platelet count, eosinophil count, and red blood cell distribution width. All of these data sets were obtained from UK Biobank.

We analyzed eight independent brain-related diseases and complex traits: Age at menarche⁶², BMI (ref.⁶³ and UK Biobank), Bipolar Disorder⁶⁴, Depressive symptoms², Neuroticism (UK Biobank), Schizophrenia¹, Smoking Status (ref.⁶⁵ and UK Biobank), and Year of education (ref.⁶⁶ and UK Biobank). These traits are a subset of traits from Supplementary Table 6 that were reported to be brain-enriched^32,50.

Data availability

The S-LDSC software, baselineLD, and MaxCPP QTL-based annotations, and a tutorial on how to use S-LDSC with QTL-based annotations are available online (see URLs). A Life Sciences Reporting Summary is available.

URLs

• CAVIAR: http://genetics.cs.ucla.edu/caviar/

• GTEx (Release v6, dbGaP Accession phs000424.v6.p1): http://www.gtexportal.org.

• GCTA: cnsgenomics.com/software/gcta/

• BLUEPRINT: ftp://ftp.ebi.ac.uk/pub/databases/blueprint/blueprint_Epivar/qtl_as/

• baselineLD annotations: https://data.broadinstitute.org/alkesgroup/LDSCORE/

• MaxCPP QTL-based annotations and partitioned LD scores: https://data.broadinstitute.org/alkesgroup/LDSCORE/LDSC_QTL/

• 95%Causal Set QTL-based annotations and partitioned LD scores: https://data.broadinstitute.org/alkesgroup/LDSCORE/LDSC_QTL/

• 1000 Genomes Project Phase 3 data: ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/release/20130502

• PLINK software: https://www.cog-genomics.org/plink2

• BOLT-LMM software: https://data.broadinstitute.org/alkesgroup/BOLT-LMM

• BOLT-LMM summary statistics for UK Biobank traits: https://data.broadinstitute.org/alkesgroup/UKBB

• UK Biobank: http://www.ukbiobank.ac.uk/

• UK Biobank Genotyping and QC Documentation: http://www.ukbiobank.ac.uk/wp-content/uploads/2014/04/UKBiobank_genotyping_QC_documentation-web.pdf