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 4 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 all 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	4.087	0.756	99.3	4.534	0.327	99.6	2.474	0.116	98.2
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	-3.473	0.236	97.7	-1.343	0.130	98.6	0.430	0.059	98.7
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	5.769	0.393	98.1	4.767	0.220	98.6	1.275	0.092	96.1
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	1.296	0.221	97.5	-0.800	0.126	98.4	0.383	0.060	98.5
MR000010	42.265	0.487	58.3	40.458	0.433	27	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	2.534	0.288	98.2	-0.128	0.164	98.4	0.135	0.075	97.4
MR010010	45.952	0.528	57.3	44.782	0.479	25.7	45.093	0.459	0
MR100010	4.220	0.397	98.4	3.676	0.215	98.7	0.926	0.089	96.8
MR010100	1.865	0.229	96.5	-0.101	0.141	97.4	-0.359	0.075	97.2
MR110000	4.939	0.404	98.6	5.023	0.219	98.6	1.464	0.092	96.2
MR000110	1.280	0.223	96.6	-0.359	0.130	97.6	0.218	0.063	98.1
MR000101	1.986	0.219	97.3	-0.578	0.126	98.3	0.440	0.060	98.5
MR000011	41.818	0.484	58.4	40.029	0.429	27.8	40.989	0.417	0.1
MR001100	1.435	0.205	97.6	-0.645	0.121	98.0	0.408	0.058	98.4
MR101000	4.584	0.403	98.8	5.113	0.211	98.7	1.261	0.089	97.0
MR001001	66.695	0.703	18.8	66.293	0.678	0.5	67.372	0.677	0
MR111000	3.688	0.408	98.8	5.317	0.214	98.7	1.400	0.089	96.3
MR000111	1.888	0.221	96.9	-0.163	0.128	97.6	0.293	0.062	98.0
MR001101	3.140	0.291	98.3	0.087	0.162	98.8	0.137	0.074	97.3
MR110100	2.706	0.293	98.3	-0.023	0.165	98.6	0.097	0.076	97.6
MR100110	2.341	0.290	97.9	0.091	0.163	98.5	0.149	0.073	97.6
MR101101	2.972	0.293	98.3	0.054	0.159	99.0	0.087	0.072	97.6
MR110110	2.789	0.292	97.9	0.116	0.164	98.5	0.130	0.074	97.4
MR011011	44.420	0.512	61.9	43.909	0.467	24.2	44.925	0.457	0
MR111011	5.161	0.395	98.7	5.598	0.208	98.8	1.538	0.087	96.4
MR011111	2.330	0.218	97.6	0.091	0.130	98.1	0.055	0.067	97.4
MR111111	2.725	0.289	98.3	0.402	0.157	99.0	0.107	0.071	97.3

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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