In order to investigate potential limitations of our approach, we performed null and causal simulations under genetic architectures that violate LCV model assumptions. These simulations and their results are described in detail in the Supplementary Note. We simulated four types of LCV model violations: (1) null simulations with a bivariate Gaussian mixture model, where one of the mixture components generates imperfectly correlated effect sizes on the two traits; (2) null simulations with two latent causal variables; (3) causal simulations with a bivariate Gaussian mixture model; and (4) causal simulations with an additional latent confounder. LCV produced well-calibrated p-values under models of type (1) (Supplementary Figure 1a-c); in addition, these simulations recapitulated the limitations of existing methods (Figure 2). Models of type (2) sometimes caused LCV (and existing Methods) to produce false positives (Supplementary Figure 1d-e); however, extreme values of the simulation parameters were required in order for LCV to produce high gcp estimates, implying that results with high gcp estimates are extremely unlikely to be false positives (Supplementary Figure 2). Causal models of type (3-4) lead to reduced power for LCV (and other Methods) (Supplementary Figure 1f-g), as well as downwardly biased gcp estimates for LCV (Supplementary Tables 4-5).
