Overlapping-sample Mendelian randomisation with multiple exposures: a Bayesian approach

Zou, Linyi; Guo, Hui; Berzuini, Carlo

doi:10.1186/s12874-020-01170-0

BMC Medical Research Methodology

Table 1 Causal effects estimated from simulated data using our Bayesian method and classic methods (2SLS when sample overlap = 100% and IVW otherwise) when β=(β₁,β₂,β₃)=0.3

From: Overlapping-sample Mendelian randomisation with multiple exposures: a Bayesian approach

			\(\hat {\beta }_{1}\)								\(\hat {\beta }_{2}\)								\(\hat {\beta }_{3}\)
overlap	α	δ	Bayesian				Classic				Bayesian				Classic				Bayesian				Classic
			mean	sd	coverage	power	mean	sd	coverage	power	mean	sd	coverage	power	mean	sd	coverage	power	mean	sd	coverage	power	mean	sd	coverage	power
100%	0.5	1	0.300	0.001	0.96	1	0.301	0.020	0.96	1	0.300	0.002	0.97	1	0.298	0.025	0.99	1	0.300	0.001	0.99	1	0.301	0.019	0.99	1
		0.5	0.301	0.006	0.96	1	0.300	0.017	0.98	1	0.301	0.004	0.99	1	0.302	0.024	0.97	1	0.300	0.002	0.99	1	0.299	0.018	0.97	1
		0.1	0.301	0.006	0.96	1	0.300	0.015	0.95	1	0.301	0.009	0.98	1	0.300	0.022	0.97	1	0.300	0.006	0.95	1	0.300	0.016	0.96	1
	0.1	1	0.302	0.006	0.97	1	0.286	0.100	0.98	0.72	0.303	0.010	0.97	1	0.309	0.141	0.98	0.47	0.300	0.006	0.98	1	0.292	0.106	0.97	0.76
		0.5	0.306	0.009	0.95	1	0.302	0.087	0.96	0.92	0.309	0.015	0.98	1	0.305	0.130	0.96	0.65	0.301	0.009	0.99	1	0.294	0.092	0.98	0.86
		0.1	0.319	0.029	0.95	1	0.301	0.071	0.97	0.97	0.324	0.050	0.94	1	0.304	0.104	0.97	0.79	0.300	0.029	0.98	1	0.296	0.079	0.95	0.96
80%	0.5	1	0.304	0.020	0.93	1	0.297	0.108	0.95	0.78	0.304	0.022	0.95	1	0.311	0.117	0.99	0.66	0.300	0.013	0.97	1	0.294	0.100	0.96	0.83
		0.5	0.302	0.010	0.96	1	0.302	0.099	0.97	0.87	0.302	0.015	0.96	1	0.300	0.101	1	0.71	0.299	0.011	0.96	1	0.306	0.098	0.97	0.89
		0.1	0.302	0.011	0.95	1	0.307	0.098	0.92	0.91	0.300	0.016	0.97	1	0.311	0.092	0.97	0.85	0.301	0.011	0.96	1	0.298	0.093	0.95	0.90
	0.1	1	0.306	0.048	0.98	1	0.319	0.393	0.96	0.11	0.307	0.072	0.98	0.98	0.317	0.577	0.96	0.11	0.302	0.060	0.97	1	0.284	0.406	0.94	0.12
		0.5	0.320	0.041	0.98	1	0.276	0.335	0.95	0.17	0.320	0.060	0.98	1	0.301	0.461	0.97	0.12	0.303	0.047	1	1	0.320	0.355	0.96	0.15
		0.1	0.320	0.050	0.97	1	0.296	0.296	0.94	0.22	0.320	0.075	0.97	1	0.310	0.437	0.93	0.14	0.301	0.052	0.98	1	0.295	0.301	0.94	0.18
60%	0.5	1	0.302	0.020	0.95	1	0.295	0.078	0.94	0.97	0.299	0.027	0.95	1	0.310	0.081	0.99	0.93	0.302	0.018	0.94	1	0.295	0.073	0.94	0.99
		0.5	0.299	0.014	0.97	1	0.305	0.068	0.95	1	0.301	0.018	0.96	1	0.303	0.070	1	0.99	0.301	0.013	0.96	1	0.298	0.069	0.96	1
		0.1	0.300	0.012	0.98	1	0.304	0.060	0.96	1	0.303	0.018	0.96	1	0.301	0.070	0.98	0.98	0.298	0.012	0.95	1	0.298	0.065	0.96	1
	0.1	1	0.283	0.066	0.96	0.98	0.304	0.283	0.94	0.22	0.285	0.106	0.95	0.77	0.336	0.376	0.96	0.14	0.307	0.086	0.95	0.97	0.299	0.298	0.95	0.20
		0.5	0.300	0.057	0.97	1	0.304	0.236	0.92	0.27	0.307	0.085	0.97	0.93	0.324	0.319	0.95	0.17	0.304	0.066	0.98	1	0.287	0.229	0.96	0.26
		0.1	0.301	0.061	0.97	1	0.306	0.203	0.94	0.41	0.303	0.085	0.97	0.95	0.318	0.246	0.97	0.20	0.300	0.059	0.96	1	0.280	0.183	0.94	0.32
40%	0.5	1	0.300	0.025	0.91	1	0.299	0.060	0.96	1	0.298	0.033	0.91	1	0.304	0.059	1	1	0.301	0.021	0.93	1	0.298	0.055	0.96	1
		0.5	0.301	0.017	0.93	1	0.297	0.058	0.94	1	0.300	0.023	0.95	1	0.303	0.055	0.99	1	0.298	0.015	0.94	1	0.297	0.057	0.94	1
		0.1	0.300	0.013	0.97	1	0.300	0.049	0.97	1	0.300	0.019	0.95	1	0.308	0.055	0.99	1	0.300	0.014	0.95	1	0.296	0.058	0.93	1
	0.1	1	0.270	0.085	0.94	0.90	0.265	0.200	0.97	0.24	0.277	0.143	0.97	0.57	0.298	0.323	0.94	0.18	0.301	0.111	0.92	0.82	0.291	0.237	0.95	0.29
		0.5	0.292	0.073	0.95	0.99	0.299	0.174	0.96	0.38	0.296	0.112	0.94	0.74	0.321	0.276	0.95	0.27	0.298	0.085	0.94	0.95	0.273	0.192	0.95	0.37
		0.1	0.301	0.063	0.98	1	0.313	0.154	0.97	0.55	0.296	0.097	0.98	0.84	0.302	0.213	0.94	0.27	0.301	0.067	0.97	1	0.307	0.155	0.96	0.51
20%	0.5	1	0.306	0.028	0.86	1	0.304	0.045	0.99	1	0.297	0.040	0.86	1	0.301	0.058	1	1	0.301	0.025	0.91	1	0.301	0.051	0.95	1
		0.5	0.301	0.018	0.96	1	0.298	0.050	0.95	1	0.301	0.029	0.93	1	0.304	0.050	0.99	1	0.299	0.021	0.93	1	0.299	0.044	0.97	1
		0.1	0.302	0.015	0.96	1	0.304	0.041	0.96	1	0.300	0.021	0.94	1	0.298	0.047	0.99	1	0.299	0.016	0.92	1	0.304	0.047	0.95	1
	0.1	1	0.281	0.097	0.93	0.82	0.267	0.171	0.96	0.28	0.273	0.147	0.95	0.49	0.298	0.235	0.98	0.21	0.286	0.124	0.93	0.74	0.308	0.177	0.97	0.41
		0.5	0.283	0.092	0.94	0.86	0.294	0.153	0.95	0.47	0.291	0.127	0.97	0.63	0.276	0.201	0.97	0.24	0.292	0.097	0.92	0.89	0.306	0.151	0.95	0.49
		0.1	0.316	0.073	0.95	0.99	0.304	0.132	0.97	0.64	0.308	0.099	0.99	0.82	0.307	0.178	0.96	0.38	0.308	0.071	0.97	0.99	0.290	0.130	0.97	0.57
0%	0.5	1	0.303	0.028	0.88	1	0.299	0.049	0.95	1	0.295	0.037	0.91	1	0.296	0.055	0.97	1	0.302	0.028	0.87	1	0.303	0.050	0.93	1
		0.5	0.304	0.021	0.93	1	0.301	0.042	0.97	1	0.302	0.030	0.91	1	0.300	0.047	0.98	1	0.300	0.020	0.94	1	0.308	0.043	0.95	1
		0.1	0.302	0.015	0.97	1	0.301	0.039	0.97	1	0.301	0.025	0.92	1	0.299	0.041	1	1	0.300	0.015	0.96	1	0.308	0.041	0.95	1
	0.1	1	0.323	0.135	0.93	0.74	0.289	0.158	0.98	0.41	0.289	0.188	0.91	0.43	0.294	0.280	0.89	0.31	0.309	0.140	0.90	0.74	0.311	0.194	0.91	0.46
		0.5	0.319	0.110	0.90	0.91	0.290	0.135	0.96	0.58	0.303	0.153	0.91	0.62	0.293	0.229	0.89	0.35	0.308	0.110	0.90	0.85	0.309	0.162	0.91	0.62
		0.1	0.307	0.081	0.95	0.97	0.316	0.120	0.95	0.74	0.286	0.108	0.93	0.75	0.312	0.184	0.92	0.45	0.307	0.082	0.94	0.95	0.299	0.135	0.89	0.73

Mean, standard deviation (sd), coverage and power are displayed for the estimated causal effects of the exposures X₁,X₂ and X₃ on the outcome Y, denoted as \(\hat {\beta }_{1}, \hat {\beta }_{2}\) and \(\hat {\beta }_{3}\) respectively. There are 36 configurations containing six sample overlapping rates (100%, 80%, 60%, 40%, 20%, 0%), two levels of IV strengths (α=(α₁,α₂,α₃)=0.5 and 0.1) and three levels of effects of the confounder U on the exposures (δ=(δ₁,δ₂,δ₃)=1,0.5 and 0.1). The effect of U on Y (δ₄) is set to 1

Back to article page

ISSN: 1471-2288

Contact us

General enquiries: journalsubmissions@springernature.com