Table 1 Simulation results comparing the ATT estimation under the alternative hypothesis.

From: G-computation, propensity score-based methods, and targeted maximum likelihood estimator for causal inference with different covariates sets: a comparative simulation study

n	method	selection strategy	mean bias				log OR
n	method	selection strategy	π₀	π₁	Δπ	log OR	MSE	MSE*	VEB (%)	coverage (%)	power (%)
100	GC	outcome	0.000	−0.001	−0.001	0.012	0.526	0.716	−6.2	94.1	17.7
		treatment	0.002	−0.001	−0.003	0.006	0.580	0.786	−5.7	94.1	14.0
		common	0.002	−0.001	−0.003	0.006	0.552	0.735	−4.2	94.8	15.1
		entire	−0.001	−0.001	−0.001	0.013	0.558	0.768	−8.8	93.3	16.9
	IPTW	outcome	0.000	−0.001	−0.001	0.008	0.578	0.727	10.8	97.3	7.8
		treatment	−0.000	−0.001	−0.001	0.000	0.716	0.837	−1.2	95.1	9.8
		common	0.002	−0.001	−0.003	0.003	0.587	0.743	6.6	96.8	8.8
		entire	−0.003	−0.001	0.002	0.005	0.741	0.838	−1.5	95.2	9.6
	TMLE	outcome	−0.001	−0.001	0.000	0.002	0.694	0.794	30.0	95.7	5.8
		treatment	0.000	−0.001	−0.001	−0.020	0.876	0.955	183.3	98.8	1.0
		common	−0.000	−0.001	−0.001	−0.001	0.702	0.794	10.4	95.3	7.3
		entire	−0.003	−0.001	0.001	−0.013	0.886	0.953	412.2	98.8	0.5
	FM	outcome	−0.004	−0.001	0.003	0.022	0.665	0.787	−16.7	90.1	18.9
		treatment	−0.006	−0.001	0.004	0.017	0.822	0.911	−32.3	81.3	25.2
		common	−0.001	−0.001	−0.000	0.010	0.653	0.795	−15.3	91.0	17.5
		entire	−0.008	−0.001	0.006	0.022	0.842	0.921	−33.8	80.3	26.7
300	GC	outcome	0.001	−0.001	−0.002	−0.021	0.283	0.555	−1.6	94.5	43.6
		treatment	0.002	−0.001	−0.003	−0.024	0.319	0.606	−2.3	94.3	35.2
		common	0.002	−0.001	−0.003	−0.023	0.304	0.561	−1.5	94.8	38.5
		entire	0.001	−0.001	−0.002	−0.022	0.297	0.600	−2.6	94.0	39.9
	IPTW	outcome	0.002	−0.001	−0.003	−0.027	0.301	0.556	16.4	97.9	24.0
		treatment	0.001	−0.001	−0.002	−0.026	0.372	0.628	6.6	96.2	21.4
		common	0.003	−0.001	−0.004	−0.028	0.318	0.563	9.1	96.8	26.1
		entire	0.001	−0.001	−0.002	−0.025	0.361	0.622	11.7	97.2	20.0
	TMLE	outcome	0.000	−0.001	−0.001	−0.023	0.358	0.577	−2.3	93.6	29.0
		treatment	0.002	−0.001	−0.003	−0.035	0.454	0.683	51.2	99.1	6.8
		common	0.001	−0.001	−0.002	−0.023	0.378	0.582	−3.5	93.0	26.5
		entire	0.002	−0.001	−0.003	−0.035	0.432	0.674	81.8	99.3	4.4
	FM	outcome	−0.000	−0.001	−0.001	−0.020	0.351	0.579	−11.7	91.9	37.2
		treatment	−0.001	−0.001	−0.000	−0.022	0.444	0.656	−30.2	82.7	38.9
		common	0.001	−0.001	−0.002	−0.024	0.363	0.587	−14.6	90.4	36.9
		entire	−0.001	−0.001	0.000	−0.020	0.439	0.662	−29.3	83.2	39.1
500	GC	outcome	0.001	−0.001	−0.002	−0.014	0.217	0.509	−1.1	94.7	64.5
		treatment	0.001	−0.001	−0.002	−0.014	0.245	0.556	−1.5	94.4	53.6
		common	0.001	−0.001	−0.002	−0.015	0.233	0.618	−0.8	94.8	57.6
		entire	0.001	−0.001	−0.002	−0.014	0.228	0.552	−2.0	94.2	60.5
	IPTW	outcome	0.002	−0.001	−0.003	−0.019	0.230	0.509	16.5	97.9	43.3
		treatment	0.000	−0.001	−0.001	−0.013	0.285	0.574	6.8	96.6	35.4
		common	0.002	−0.001	−0.003	−0.018	0.244	0.514	9.2	96.8	43.7
		entire	0.000	−0.001	−0.001	−0.014	0.274	0.571	12.3	97.2	33.9
	TMLE	outcome	0.001	−0.001	−0.002	−0.015	0.272	0.521	−4.7	93.4	48.5
		treatment	0.001	−0.001	−0.002	−0.018	0.347	0.618	35.0	99.1	15.9
		common	0.000	−0.001	−0.001	−0.013	0.289	0.527	−4.8	93.1	43.7
		entire	0.001	−0.001	−0.002	−0.019	0.328	0.611	51.1	99.3	12.9
	FM	outcome	0.001	−0.001	−0.002	−0.015	0.265	0.525	−9.9	92.4	53.0
		treatment	−0.001	−0.001	−0.000	−0.011	0.346	0.597	−31.0	82.7	51.7
		common	0.001	−0.001	−0.001	−0.014	0.283	0.530	−15.8	90.1	52.3
		entire	−0.002	−0.001	0.001	−0.008	0.340	0.596	−29.8	83.2	52.6
2000	GC	outcome	0.000	0.000	−0.000	−0.002	0.108	0.479	−1.7	94.7	99.6
		treatment	0.001	0.000	−0.000	−0.003	0.122	0.524	−1.2	94.8	98.6
		common	0.001	0.000	−0.000	−0.003	0.116	0.480	−0.9	94.7	99.1
		entire	0.000	0.000	−0.000	−0.002	0.113	0.523	−1.8	94.5	99.4
	IPTW	outcome	0.002	0.000	−0.001	−0.006	0.113	0.478	16.3	97.6	98.1
		treatment	0.000	0.000	−0.000	−0.002	0.138	0.539	7.9	96.4	93.0
		common	0.002	0.000	−0.001	−0.006	0.120	0.480	9.4	97.0	97.7
		entire	0.000	0.000	−0.000	−0.002	0.131	0.537	13.9	97.4	93.6
2000	TMLE	outcome	0.001	0.000	−0.000	−0.002	0.132	0.483	−5.9	93.3	97.5
		treatment	0.000	0.000	0.000	−0.002	0.169	0.568	18.2	98.2	71.8
		common	−0.000	0.000	0.000	−0.000	0.142	0.486	−5.6	93.6	95.5
		entire	0.001	0.000	−0.000	−0.004	0.158	0.565	23.5	98.6	75.3
	FM	outcome	0.000	0.000	−0.000	−0.002	0.134	0.484	−12.0	91.6	97.7
		treatment	0.001	0.000	−0.000	−0.005	0.203	0.548	−41.6	74.6	89.9
		common	0.001	0.000	−0.000	−0.003	0.149	0.485	−20.5	88.5	96.7
		entire	0.000	0.000	0.000	−0.002	0.162	0.543	−26.9	84.5	94.8

*MSE in the presence of an unmeasured confounder. Theoretical values: \({\pi }_{1}=0.701,{\pi }_{0}=0.589,\theta =0.492\).

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Table 1 Simulation results comparing the ATT estimation under the alternative hypothesis.

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