The bivariate analyses used a bivariate extension of equation (1) (ref. 20). The two traits were measured in different individuals, but the equations were related through the genome-wide similarities estimated from SNPs. Genetic and residual variances for the traits were estimated as well as the genetic covariance σg12. The genetic correlation coefficient (rg) was calculated by (σg12/(σg1σg2)) and is approximately the same on the observed case-control scale as on the liability scale20 and so does not depend on specifications of K. The covariance σg12 can be transformed to the liability scale, accounting for assumed disorder risks and proportions of cases and controls in the samples of each disorder20, and it equals the coheritability52 rgh1h2. We used the approximated χ2 test statistic (estimate/s.e.)2 to test whether estimates were significantly different from zero. We checked that this simple approximation agreed well with the more formal and computer-intensive likelihood ratio test for several examples. Heterogeneity of SNP-based heritabilities was tested using Cochran’s Q (ref. 63) and Higgins’ I2 (ref. 64) values, acknowledging potential non-independence of the six estimates (three subsets plus three subset pairs).
