We introduce more flexible statistical methods that combine the most attractive features of existing approaches, while overcoming their major limitations. The methods, which we refer to as “multivariate adaptive shrinkage” (mash), build on recent approaches16 for testing and estimating effects in a single condition, extending these approaches to multiple conditions. Key features of mash include: (i) it is flexible, allowing for both shared and condition-specific effects, and arbitrary patterns of correlation among conditions; (ii) it is computationally tractable for hundreds of thousands of tests in dozens of conditions, or more; (iii) it provides not only measures of significance, but also estimates of effect sizes, together with measures of uncertainty; (iv) it is adaptive, meaning that its behavior adapts to the patterns present in the data; and (v) it is generic, requiring only a matrix containing the observed effects in each condition and a matrix of the corresponding standard errors. (Alternatively, mash can be supplied with just matrix of Z scores, although this reduces the ability to estimate effect sizes.) Together, these features make mash the most flexible and widely applicable method available for estimating and testing multiple effects in multiple conditions.
