Nonetheless, existing methods have important limitations. First, they make restrictive assumptions about the correlations of non-zero effects among conditions. For example, Flutre et al.5 assume these correlations are non-negative and equal. Correlations may be negative in some applications; e.g., genetic variants that increase one trait may decrease another. And some conditions may be more correlated than others; for example, in our eQTL application (below), brain tissues are strongly correlated with one another. Second, the most flexible methods are computationally intractable for moderate numbers of conditions (e.g., 44 tissues in our eQTL application). Existing solutions to this problem substantially reduce flexibility. For example, Flutre et al.5 solve the computational problem by restricting effects to be shared in all conditions, or to be specific to a single condition. Alternatively, Wei et al.12 allow for all possible patterns of sharing, but only under the restrictive assumption that non-zero effects are uncorrelated among conditions. Third, existing methods typically focus only on testing for significant effects in each condition, and not on estimating effect sizes which, as we illustrate, is important for assessing heterogeneity among conditions.
