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 1 Simulation results with different sample sizes = 200, 500 or 2000 in the situation where the parametric models included 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	7.234	0.472	98.8	1.497	0.306	96.2	0.702	0.166	95.5
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	0.713	0.157	94.4	-0.544	0.095	95.1	0.245	0.046	95.6
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	3.434	0.292	93.7	1.196	0.194	92.3	0.604	0.100	93.4
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	0.737	0.159	93.7	-0.441	0.097	95.2	0.253	0.046	95.3
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	1.053	0.197	94.3	-0.201	0.127	93.2	0.131	0.061	93.7
MR100010	1.058	0.244	97.0	0.077	0.135	97.3	0.206	0.061	93.8
MR010100	0.860	0.241	95.3	-0.223	0.147	95.1	0.048	0.075	95.9
MR110000	4.005	0.366	98.1	2.045	0.204	97.2	1.331	0.100	95.2
MR000110	-2.888	0.169	95.1	-1.898	0.099	96.1	-0.039	0.046	96.0
MR000101	0.887	0.194	95.9	-0.722	0.111	97.3	0.498	0.055	97.8
MR000011	1.155	0.159	94.2	-0.212	0.098	94.9	0.307	0.046	95.5
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	1.872	0.374	98.5	2.412	0.212	97.7	1.324	0.098	95.1
MR000111	-2.446	0.168	94.9	-1.639	0.100	95.8	0.011	0.047	96.0
MR001101	1.076	0.275	97.0	-0.302	0.154	98.3	0.172	0.074	96.1
MR110100	1.500	0.270	96.9	-0.513	0.156	97.6	0.100	0.076	95.2
MR100110	0.677	0.247	97.1	-0.167	0.136	97.3	0.220	0.061	94.6
MR101101	0.215	0.269	97.6	-0.354	0.151	98.1	0.242	0.071	95.8
MR110110	1.180	0.244	96.8	-0.225	0.137	96.7	0.225	0.062	94.0
MR011011	1.383	0.204	94.5	0.060	0.131	94.5	0.209	0.062	94.2
MR111011	0.871	0.256	96.9	-0.110	0.139	97.5	0.225	0.063	94.2
MR011111	1.104	0.204	95.5	0.120	0.131	94.4	0.177	0.062	94.3
MR111111	0.800	0.249	96.7	-0.225	0.141	97.2	0.231	0.063	94.5

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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