. 2024 Jul 24;111(8):1736–1749. doi: 10.1016/j.ajhg.2024.06.012

Table 1.

Simulation results evaluating the performance of mintMR and competing methods when the number of IVs is limited

	Equal probability of non-zero effects across all tissues			Higher probability of non-zero effects in one tissue and lower in others
Variance of outcome explained by UHP effects

	0.05	0.10	0.15	0.05	0.10	0.15

Power

mintMR	0.859	0.786	0.657	0.841	0.794	0.677
mintMR_oracle	0.903	0.842	0.773	0.898	0.830	0.775
mintMR_single-gene	0.718	0.629^∗	0.567^∗	0.691	0.610^∗	0.582^∗
IVW+metaIV	0.351^∗	0.317^∗	0.323^∗	0.362^∗	0.341^∗	0.342^∗
Egger	0.308^∗	0.275^∗	0.262^∗	0.305^∗	0.270^∗	0.273^∗
MVMR-IVW	0.663^∗	0.534^∗	0.473^∗	0.652^∗	0.523^∗	0.461^∗
MVMR-Egger	0.573^∗	0.518^∗	0.443^∗	0.510^∗	0.441^∗	0.384^∗
MVMR-Lasso	0.770^∗	0.730^∗	0.704^∗	0.764^∗	0.710^∗	0.681^∗
MVMR-Median	0.641	0.572^∗	0.519^∗	0.677	0.578^∗	0.500^∗
MVMR-Robust	0.455	0.374	0.315	0.444	0.365	0.295

Type I error rate

mintMR	0.050	0.049	0.048	0.051	0.050	0.046
mintMR_oracle	0.048	0.050	0.048	0.050	0.048	0.048
mintMR_single-gene	0.072	0.120^∗	0.158^∗	0.074	0.116^∗	0.155^∗
IVW+metaIV	0.149^∗	0.160^∗	0.162^∗	0.146^∗	0.157^∗	0.158^∗
Egger	0.131^∗	0.138^∗	0.141^∗	0.131^∗	0.135^∗	0.139^∗
MVMR-IVW	0.121^∗	0.128^∗	0.134^∗	0.122^∗	0.130^∗	0.134^∗
MVMR-Egger	0.133^∗	0.138^∗	0.138^∗	0.121^∗	0.136^∗	0.141^∗
MVMR-Lasso	0.159^∗	0.214^∗	0.259^∗	0.158^∗	0.210^∗	0.257^∗
MVMR-Median	0.089	0.117^∗	0.132^∗	0.087	0.115^∗	0.127^∗
MVMR-Robust	0.062	0.076	0.080	0.062	0.077	0.080

Two types of causal effects of genes on outcomes are simulated. For the first type, genes affect outcomes in multiple tissues, with each gene having an equal probability (5%) of having non-zero effects in any tissue. For the second type, in one tissue, 15% of the genes have non-zero effects on outcome. In each of the rest of the tissues, 3% of the genes have non-zero effects. The proportion of variation in outcome explained by UHP effects varies from 0.05 to 0.15. The sample size of the outcome is 50,000 and 500 for exposure. The number of IVs is 15. Two exposures are generated and each exposure has 5 tissues. The causal effects are generated with $N (0, 0.015)$ . The type I error rate and power are calculated based on the p value cutoff of 0.05. Methods with type I error rates between 0.05 and 0.1 are considered to have borderline but tolerable control, and we still compare their power without highlighting their mildly inflated type I error rates. Methods with inflated type I error rates (≥0.1) are indicated with an asterisk (^∗) to ensure a fair power comparison.