Flexible statistical methods for estimating and testing effects in genomic studies with multiple conditions.

Overview of fitting procedure in mash, which estimates the multivariate distribution of effects present in the data.The data consist of a matrix of summary data (e.g., Z scores) for a large number of units (e.g., gene-SNP pairs) in multiple conditions (e.g., tissues), and, optionally, their standard errors (not shown). Color indicates the sign (positive, negative) of an effect (blue, yellow) or covariance (blue, red), with shading intensity indicating size. After selecting rows containing the strongest signals (1)—in this example, the top 6 rows—we apply covariance estimation techniques to estimate candidate “data-driven” covariance matrices Uk (2). To these, we add “canonical” covariance matrices Uk , including the identity matrix, and matrices representing condition-specific effects. Each covariance matrix represents a pattern of effects that may occur in the data. We scale each covariance matrix by a grid of scaling factors, ωl, varying from “very small” to “very large”, which allows for a priori effect sizes to range from very small to very large. Using the entire data set, we compute maximum-likelihood estimates of the weights (relative frequencies) πk,l for each (Uk,ωl) combination (3), thereby learning how commonly each pattern-effect size combination occurs in the data. Finally, we compute posterior statistics using the fitted model (4); the posterior mean estimates shown in the bottom-right illustrate that effect estimates are “shrunk” adaptively using the fitted mash model.

Comparison of methods on simulated data.Results are shown for two simulation scenarios: “shared, structured effects”, in which the non-zero effects are shared among conditions in complex, structured ways similar to patterns of eQTL sharing in the GTEx data; and “shared, unstructured effects”, in which the non-zero effects are shared among conditions but independent. Each simulation result involves n = 20,000 independent units observed at R = 44 conditions, with 400 non-null units. Panels a–b show ROC curves for detecting significant units (n = 20,000 discoveries), based on unit-specific measures of significance (as in traditional meta-analyses). Panels c–d show ROC curves for detecting significant effects (n × R = 44 × 20,000 = 880,000 discoveries), which requires effect-specific measures of significance. In c–d, we also require the estimated sign (+/–) of each significant effect to be correct to be considered a “true positive”. Panels e and f summarize the error in the estimated effects relative to the error from a simple condition-by-condition analysis (Relative Root Mean Squared Error, or RRMSE for short). Our new method (mash) outperformed other methods, particularly in the “shared, structured effects” scenario.

Summary of primary patterns identified by mash in GTEx data.Shown are the heatmap of the correlation matrix (a) and bar plots of the first three eigenvectors (b, c, d) of the covariance matrix Uk corresponding to the dominant mixture component identified by mash (n = 16,069 independent gene-SNP pairs). This component accounts for 34% of all weight in the GTEx data. Tissues are color-coded as indicated by the tissue labels in the heatmap. The first eigenvector (b) reflects broad sharing among all tissues, with all effects in the same direction; the second eigenvector (c) captures differences between brain (and, to a lesser extent, testis and pituitary) and other tissues; the third eigenvector (d) primarily captures effects that are stronger in whole blood.

Examples illustrating that mash uses learned patterns of sharing to inform effect estimates in the GTEx data.In panel a, each colored dot shows the original (“raw”) effect estimate for a single tissue (color-coded as in Fig. 3), with grey bars indicating ±2 standard errors. These are the data provided to mash. Panel b shows the corresponding mash estimates. In each case, mash combines information across all tissues, using the background information (patterns of sharing) learned from data on all eQTLs to produce more precise estimates. Panel c shows, for contrast, the corresponding estimates from mash-bmalite, which, due to its more restricted model, fails to capture features clearly apparent in the original data, such as strong brain effects in MCPH1. In b and c, colored dots are posterior means, and error bars depict ±2 posterior standard deviations. For all estimates, n = 83–430 individuals, depending on the tissue (Supplementary Table 3).

Number of tissues shared by sign and magnitude.Histograms show estimated number of tissues in which top eQTLs are “shared,” considering all tissues (n = 12,171 gene-SNP pairs with a significant eQTL in at least one tissue), non-brain tissues (n = 12,117), and brain tissues only (n = 8,474), and using two different sharing definitions, by sign (a) and by magnitude (b). Sharing by sign means that the eQTLs have the same sign in the estimated effect; sharing by magnitude means that they also have similar effect sizes (within a factor of 2).

Pairwise sharing by magnitude of eQTLs among tissues.For each pair of tissues, we considered the top eQTLs that were significant (lfsr < 0.05) in at least one of the two tissues, and plotted the proportion of these that are “shared in magnitude”—that is, have effect estimates that are the same sign and within a factor of 2 in size of one another (n = 5,605–9,811 gene-SNP pairs, depending on pair of tissues compared). Brackets surrounding tissue labels highlight groups of biologically related tissues mentioned in the text as showing particularly high levels of sharing.