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 3 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 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	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	-3.473	0.236	97.7	-1.343	0.130	98.6	0.430	0.059	98.7
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	5.769	0.393	98.1	4.767	0.220	98.6	1.275	0.092	96.1
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	1.296	0.221	97.5	-0.800	0.126	98.4	0.383	0.060	98.5
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	2.534	0.288	98.2	-0.128	0.164	98.4	0.135	0.075	97.4
MR010010	1.053	0.197	94.3	-0.201	0.127	93.2	0.131	0.061	93.7
MR100010	1.737	0.250	97.9	-0.037	0.142	97.7	0.156	0.063	94.6
MR010100	1.401	0.242	95.7	-0.059	0.149	96.2	-0.101	0.074	96.9
MR110000	4.103	0.398	98.8	1.673	0.210	98.2	1.090	0.091	96.1
MR000110	-3.647	0.178	96.7	-2.681	0.100	96.9	-0.272	0.046	96.7
MR000101	1.986	0.219	97.3	-0.578	0.126	98.3	0.440	0.060	98.5
MR000011	1.155	0.159	94.2	-0.212	0.098	94.9	0.307	0.046	95.5
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	2.864	0.398	99.1	2.326	0.205	98.6	1.060	0.088	96.7
MR000111	-3.181	0.177	96.9	-2.429	0.101	97.2	-0.215	0.046	96.8
MR001101	3.140	0.291	98.3	0.087	0.162	98.8	0.137	0.074	97.3
MR110100	2.774	0.288	97.8	-0.107	0.164	98.2	0.112	0.075	97.1
MR100110	2.179	0.257	97.1	-0.374	0.141	97.8	0.235	0.063	95.8
MR101101	2.972	0.293	98.3	0.054	0.159	99.0	0.087	0.072	97.6
MR110110	2.384	0.262	97.8	-0.257	0.143	98.0	0.207	0.064	95.7
MR011011	1.383	0.204	94.5	0.060	0.131	94.5	0.209	0.062	94.2
MR111011	1.983	0.275	97.7	0.168	0.148	97.7	0.120	0.064	94.9
MR011111	1.017	0.203	95.6	0.011	0.130	95.2	0.254	0.061	95.4
MR111111	2.366	0.268	97.5	-0.039	0.149	98.0	0.177	0.064	95.6

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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