Table 2 Comparison of methods with some invalid instruments.

From: A robust and efficient method for Mendelian randomization with hundreds of genetic variants

	20 invalid variants			40 invalid variants			60 invalid variants
Method	Mean	SD	Power	Mean	SD	Power	Mean	SD	Power
Null causal effect: θ = 0
Scenario 2: balanced pleiotropy, InSIDE satisfied
Inverse-variance weighted	0.000	0.043	5.1	−0.001	0.057	5.5	0.000	0.068	5.4
MR-Egger	0.002	0.120	5.4	0.000	0.155	5.0	0.001	0.186	5.0
Weighted median	0.000	0.034	4.3	−0.000	0.043	7.5	0.001	0.057	12.4
MR-PRESSO	−0.000	0.030	7.8	−0.000	0.043	12.4	0.003	0.060	9.5
Weighted MBE	−0.004	0.118	0.3	0.002	0.066	0.9	−0.001	0.161	1.8
Contamination mixture	0.000	0.033	6.6	0.000	0.043	9.1	0.000	0.066	15.1
Scenario 3: directional pleiotropy, InSIDE satisfied
Inverse-variance weighted	0.133	0.031	96.6	0.266	0.038	100.0	0.400	0.044	100.0
MR-Egger	0.005	0.111	5.3	0.010	0.135	5.4	0.015	0.148	5.9
Weighted median	0.050	0.033	26.6	0.129	0.044	88.3	0.274	0.071	99.9
MR-PRESSO	0.056	0.030	50.9	0.168	0.042	99.6	0.330	0.056	100.0
Weighted MBE	0.008	0.068	0.3	0.028	0.109	2.1	0.086	0.156	22.3
Contamination mixture	0.016	0.034	9.7	0.042	0.044	25.4	0.144	0.173	60.9
Scenario 4: pleiotropy via confounder, InSIDE violated
Inverse-variance weighted	0.118	0.040	85.6	0.213	0.041	99.6	0.289	0.040	100.0
MR-Egger	0.280	0.121	81.1	0.400	0.110	96.1	0.457	0.100	99.2
Weighted median	0.080	0.044	56.6	0.212	0.072	98.1	0.339	0.062	100.0
MR-PRESSO	0.051	0.038	44.8	0.155	0.051	96.2	0.260	0.051	99.9
Weighted MBE	0.028	0.300	0.7	0.137	0.403	15.9	0.259	1.589	33.6
Contamination mixture	0.010	0.035	9.1	0.034	0.059	20.2	0.159	0.159	52.3
Positive causal effect: θ = +0.1
Scenario 2: balanced pleiotropy, InSIDE satisfied
Inverse-variance weighted	0.094	0.044	58.2	0.095	0.058	38.4	0.095	0.069	28.2
MR-Egger	0.063	0.120	8.7	0.060	0.155	6.3	0.060	0.188	6.3
Weighted median	0.091	0.036	69.1	0.092	0.045	59.9	0.092	0.060	51.5
MR-PRESSO	0.095	0.033	89.0	0.095	0.044	73.7	0.098	0.066	58.2
Weighted MBE	0.088	0.254	15.7	0.084	0.112	16.2	0.090	0.315	17.3
Contamination mixture	0.097	0.036	82.7	0.098	0.047	70.8	0.101	0.074	56.9
Scenario 3: directional pleiotropy, InSIDE satisfied
Inverse-variance weighted	0.228	0.032	100.0	0.362	0.039	100.0	0.495	0.045	100.0
MR-Egger	0.069	0.115	9.7	0.075	0.137	9.0	0.078	0.150	9.0
Weighted median	0.144	0.035	98.1	0.227	0.046	100.0	0.372	0.071	100.0
MR-PRESSO	0.156	0.032	100.0	0.278	0.046	100.0	0.440	0.058	100.0
Weighted MBE	0.090	0.096	25.0	0.132	0.233	44.3	0.189	0.247	69.1
Contamination mixture	0.114	0.036	92.5	0.146	0.050	94.4	0.305	0.228	98.1
Scenario 4: pleiotropy via confounder, InSIDE violated
Inverse-variance weighted	0.214	0.041	100.0	0.307	0.042	100.0	0.384	0.041	100.0
MR-Egger	0.360	0.123	91.8	0.482	0.111	98.9	0.550	0.101	99.8
Weighted median	0.176	0.047	99.1	0.309	0.073	100.0	0.434	0.063	100.0
MR-PRESSO	0.149	0.042	99.0	0.256	0.054	100.0	0.361	0.051	100.0
Weighted MBE	0.102	0.487	8.4	0.225	0.401	22.5	0.453	1.416	37.8
Contamination mixture	0.109	0.040	88.4	0.140	0.070	86.4	0.301	0.165	91.5

No one method outperformed all others in every scenario, but the contamination mixture method had good overall performance across scenarios with up to 40 invalid instruments out of 100. Performance with 60 invalid instruments was generally poor for all methods. Mean, standard deviation (SD) of estimates and empirical power (%) in simulation study for Scenarios 2, 3 and 4. MR-PRESSO Mendelian Randomization Pleiotropy RESidual Sum and Outlier method of Verbanck et al.¹⁷; MBE mode-based estimate of Hartwig et al.¹⁶

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Table 2 Comparison of methods with some invalid instruments.

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