Our tests relate to previously defined estimators of genetic correlation and covariance between traits. We consider two definitions of genetic covariance at a locus: 1) the covariance between the genetic component of expression and the genetic component of trait; 2) the covariance between the causal effect sizes for expression and the causal effect-sizes for trait. Under assumptions of independent effect-sizes, these definitions yield asymptotically identical quantities37. Assuming a substantially large training set where the genetic component of expression can be perfectly predicted, the direct TWAS tests for a significant association between the genetic component of expression and the trait; equivalent to testing definition #1 for a polygenic trait. Likewise, the summary-based TWAS tests for a significant sum of products of the causal expression effect sizes and the causal trait effect sizes; equivalent to definition #2 up to a scaling factor. The TWAS approach therefore fits naturally with the broader study of shared genetic etiology of multiple phenotypes. At the sample sizes evaluated in this study, the TWAS approach is substantially better powered than an LD-based estimate of local genetic correlation (Supplementary Note).
