An improved multiply robust estimator for the average treatment effect

Wang, Ce; Wei, Kecheng; Huang, Chen; Yu, Yongfu; Qin, Guoyou

doi:10.1186/s12874-023-02056-7

BMC Medical Research Methodology

Table 2 Simulation results with different sample sizes = 200, 500 or 2000 in the situation where the parametric models did not include the correct models and the neural network model included true covariates

From: An improved multiply robust estimator for the average treatment effect

Estimator	N = 200			N = 500			N = 2000
Estimator	Bias (%)	RMSE	CR (%)	Bias (%)	RMSE	CR (%)	Bias (%)	RMSE	CR (%)
IPW.model1	6.356	0.624	98.2	5.464	0.273	99.5	2.539	0.115	97.2
IPW.model2	46.619	0.567	69.6	44.788	0.491	42.6	45.272	0.465	1.2
IPW.model3	67.238	0.708	27.0	66.532	0.680	0.5	67.402	0.677	0
OR.model1	-0.983	0.197	97.6	-1.139	0.112	98.3	0.484	0.054	98.1
OR.model2	42.129	0.484	57.2	40.344	0.431	24.3	41.260	0.420	0.1
OR.model3	67.206	0.714	20.3	66.563	0.683	0.7	67.402	0.678	0
MR100000	6.289	0.372	98.1	4.006	0.210	97.4	1.397	0.100	95.5
MR010000	46.435	0.535	58.5	45.227	0.484	28.0	45.235	0.461	0
MR001000	67.323	0.708	15.9	66.650	0.681	0.4	67.405	0.677	0
MR000100	0.421	0.192	96.2	-0.975	0.112	97.6	0.455	0.055	97.8
MR000010	42.265	0.487	58.3	40.458	0.433	27.0	41.312	0.421	0.1
MR000001	67.395	0.715	19.9	66.618	0.683	0.7	67.425	0.678	0
MR100100	1.217	0.274	97.4	-0.427	0.154	98.0	0.085	0.075	95.8
MR010010	45.952	0.528	57.3	44.782	0.479	25.7	45.093	0.459	0
MR100010	4.864	0.355	98.4	2.818	0.202	97.8	1.105	0.096	95.6
MR010100	1.584	0.212	96.1	-0.283	0.133	95.5	-0.169	0.072	97.1
MR110000	5.109	0.380	98.1	4.001	0.210	97.3	1.501	0.100	95.3
MR000110	0.766	0.193	96.0	-0.648	0.116	96.6	0.328	0.057	97.4
MR000101	0.887	0.194	95.9	-0.722	0.111	97.3	0.498	0.055	97.8
MR000011	41.818	0.484	58.4	40.029	0.429	27.8	40.989	0.417	0.1
MR001100	0.484	0.181	97.8	-0.834	0.106	97.3	0.542	0.054	96.9
MR101000	5.245	0.389	98.5	4.602	0.219	97.9	1.414	0.098	94.6
MR001001	66.695	0.703	18.8	66.293	0.678	0.5	67.372	0.677	0
MR111000	4.091	0.389	98.8	4.438	0.218	97.9	1.583	0.099	94.7
MR000111	1.185	0.195	95.5	-0.481	0.116	96.1	0.426	0.057	97.6
MR001101	1.076	0.275	97.0	-0.302	0.154	98.3	0.172	0.074	96.1
MR110100	1.283	0.279	96.9	-0.446	0.153	98.1	0.075	0.075	95.8
MR100110	0.668	0.277	97.8	-0.484	0.155	97.6	0.159	0.074	95.9
MR101101	0.215	0.269	97.6	-0.354	0.151	98.1	0.242	0.071	95.8
MR110110	0.954	0.274	97.4	-0.483	0.154	98.0	0.165	0.073	96.1
MR011011	44.420	0.512	61.9	43.909	0.467	24.2	44.925	0.457	0
MR111011	5.377	0.356	98.3	4.277	0.203	97.8	1.426	0.091	95.5
MR011111	2.146	0.202	97.5	-0.015	0.123	96.9	0.070	0.063	96.6
MR111111	0.575	0.263	98.2	-0.327	0.151	98.0	0.254	0.070	96.0

Bias (%) mean relative bias, RMSE Root mean square error, CR Coverage rate, IPW Inverse probability weighting, OR Outcome regression, MR Multiply robust; MR estimators are denoted as “MR000000”, where each digit of the four numbers, from left to right, indicates if \({\pi }^{1}\left({\varvec{X}}\right)\), \({\pi }^{2}\left({\varvec{X}}\right)\),\({\pi }^{3}\left({\varvec{X}}\right),{m}^{1}\left({\varvec{X}},Z\right)\), \({m}^{2}\left({\varvec{X}},Z\right)\) or \({m}^{3}\left({\varvec{X}},Z\right)\) is included in the estimator (“1” means yes and “0” means no)

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ISSN: 1471-2288

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