Table 2 Simulation results comparing the ATE 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	set	mean bias				log OR
n	method	set	π₀	π₁	Δπ	log OR	MSE	MSE*	VEB (%)	coverage (%)	power (%)
100	GC	outcome	−0.001	−0.002	−0.001	−0.003	0.404	0.634	−7.3	93.2	24.7
		treatment	−0.002	−0.001	0.000	0.004	0.477	0.727	−9.5	92.4	19.9
		common	−0.001	−0.002	−0.001	−0.002	0.434	0.650	−6.6	93.5	22.1
		entire	−0.002	−0.001	0.001	0.003	0.450	0.714	−11.4	91.8	22.6
	IPTW	outcome	−0.003	−0.001	0.001	0.011	0.464	0.646	12.1	97.4	12.1
		treatment	−0.006	0.002	0.008	0.046	0.633	0.769	−7.6	93.8	16.7
		common	−0.002	−0.001	0.001	0.010	0.480	0.657	6.3	96.3	13.5
		entire	−0.006	0.003	0.009	0.053	0.647	0.773	−7.2	94.7	16.4
	TMLE	outcome	−0.001	−0.002	−0.000	0.003	0.438	0.642	−14.3	89.5	26.9
		treatment	−0.004	0.002	0.006	0.039	0.572	0.757	−24.9	84.3	27.5
		common	−0.001	−0.002	−0.001	0.002	0.469	0.657	−10.7	90.9	21.2
		entire	−0.005	0.003	0.007	0.043	0.544	0.748	−30.7	80.9	34.3
	FM	outcome	−0.005	0.002	0.006	0.039	0.549	0.710	−24.3	87.1	28.5
		treatment	−0.009	0.005	0.014	0.082	0.677	0.832	−37.7	78.0	35.1
		common	−0.005	0.001	0.006	0.038	0.563	0.713	−26.3	85.8	29.1
		entire	−0.007	0.006	0.014	0.082	0.674	0.830	−37.3	78.1	34.8
300	GC	outcome	−0.000	−0.000	0.000	0.001	0.221	0.532	−1.9	94.5	59.8
		treatment	−0.000	−0.000	0.000	0.001	0.259	0.608	−2.8	94.3	47.4
		common	−0.000	−0.000	0.000	0.001	0.237	0.539	−1.2	94.8	53.5
		entire	−0.000	−0.000	0.000	0.001	0.241	0.600	−3.4	94.0	53.0
	IPTW	outcome	−0.001	−0.000	0.001	0.006	0.239	0.533	20.2	98.0	34.7
		treatment	−0.002	0.000	0.003	0.014	0.330	0.615	4.6	96.0	29.5
		common	−0.001	−0.000	0.001	0.006	0.252	0.541	13.3	97.4	36.5
		entire	−0.002	0.000	0.002	0.013	0.326	0.607	7.9	96.6	28.5
	TMLE	outcome	−0.000	−0.001	−0.000	0.000	0.233	0.532	−3.0	93.9	54.2
		treatment	−0.001	0.000	0.002	0.009	0.310	0.612	−10.4	90.6	40.2
		common	−0.001	−0.001	0.000	0.001	0.249	0.540	−1.5	94.6	48.1
		entire	−0.001	0.000	0.001	0.008	0.290	0.603	−13.2	89.6	46.1
	FM	outcome	−0.002	0.000	0.002	0.010	0.294	0.552	−20.2	88.7	51.6
		treatment	−0.003	0.003	0.006	0.032	0.389	0.652	−39.3	77.0	53.3
		common	−0.001	−0.000	0.001	0.008	0.315	0.588	−25.5	86.2	51.3
		entire	−0.003	0.003	0.006	0.032	0.377	0.644	−37.4	77.8	52.2
500	GC	outcome	−0.000	0.000	0.001	0.003	0.168	0.501	−0.4	94.8	81.1
		treatment	−0.000	0.000	0.001	0.002	0.198	0.573	−1.0	94.8	69.0
		common	−0.000	0.000	0.000	0.002	0.183	0.505	−0.7	94.9	75.0
		entire	−0.000	0.000	0.001	0.004	0.183	0.569	−1.0	94.8	75.3
	IPTW	outcome	−0.001	0.000	0.001	0.005	0.180	0.501	22.2	98.3	58.5
		treatment	−0.001	0.001	0.001	0.007	0.248	0.573	8.1	96.5	42.3
		common	−0.001	0.000	0.001	0.005	0.193	0.505	13.8	97.3	58.6
		entire	−0.001	0.000	0.001	0.006	0.239	0.569	13.1	97.2	41.3
	TMLE	outcome	−0.000	0.000	0.000	0.002	0.177	0.501	−0.8	94.7	76.8
		treatment	−0.000	0.000	0.000	0.003	0.234	0.571	−5.9	92.7	56.1
		common	−0.000	0.000	0.000	0.002	0.190	0.505	−0.5	94.7	69.7
		entire	−0.000	0.000	0.000	0.003	0.218	0.566	−7.5	91.8	63.1
	FM	outcome	−0.001	0.000	0.001	0.005	0.219	0.518	−17.5	89.8	70.1
		treatment	−0.002	0.002	0.003	0.018	0.302	0.598	−39.8	76.2	65.5
		common	−0.001	−0.000	0.001	0.005	0.266	0.555	−31.8	82.3	66.4
		entire	−0.002	0.002	0.004	0.019	0.289	0.592	−37.1	78.3	66.2
2000	GC	outcome	−0.000	−0.000	−0.000	−0.001	0.085	0.482	−0.6	94.6	100.0
		treatment	0.000	−0.001	−0.001	−0.003	0.099	0.550	−0.6	94.7	99.8
		common	0.000	−0.001	−0.001	−0.003	0.092	0.483	−0.8	94.7	99.9
		entire	−0.000	−0.000	−0.000	−0.001	0.091	0.550	−0.6	94.7	99.9
	IPTW	outcome	−0.000	−0.000	0.000	0.002	0.090	0.482	21.2	98.2	99.8
		treatment	0.000	−0.001	−0.001	−0.002	0.122	0.547	9.3	96.7	95.1
		common	−0.000	−0.000	0.000	0.001	0.096	0.483	13.5	97.3	99.7
		entire	0.000	−0.000	−0.001	−0.002	0.117	0.546	14.3	97.5	95.6
2000	TMLE	outcome	−0.000	−0.000	−0.000	−0.001	0.088	0.482	−0.6	94.8	100.0
		treatment	0.000	−0.001	−0.001	−0.003	0.116	0.545	−2.2	94.4	98.7
		common	0.000	−0.000	−0.001	−0.002	0.095	0.483	−0.3	94.8	99.9
		entire	0.000	−0.000	−0.001	−0.002	0.108	0.544	−2.6	94.1	99.4
	FM	outcome	−0.000	−0.000	−0.000	0.000	0.129	0.497	−29.9	82.9	99.0
		treatment	−0.001	−0.000	0.000	0.003	0.169	0.569	−46.6	70.6	96.2
		common	0.000	−0.000	−0.001	−0.001	0.205	0.534	−55.9	61.1	92.7
		entire	−0.000	−0.000	0.000	0.002	0.145	0.549	−37.7	77.9	98.2

*MSE in the presence of an unmeasured confounder. Theoretical values: \({\pi }_{1}=0.557,{\pi }_{0}=0.441,\theta =0.466\).

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Table 2 Simulation results comparing the ATE estimation under the alternative hypothesis.

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