Table 2.

Results of treatment effect estimates from propensity matching and propensity weighting when assuming X₂ is an effect modifier and no unmeasured confounder exists. Here, multiple imputation is done in two ways; commonly used method (no interaction term) and elaborated method (interaction terms included)

		Heterogeneous treatment effect
		Propensity matching				Propensity weighting
		Coefficient		Bias	MSE	Coefficient		MSE
		Mean	SD			Mean	SD
No missing	No adjustment	1.736	0.156	1.409	2.011	1.736	0.156	3.040
	After adjustment	0.327	0.093	0.000	0.009	− 0.003	0.152	0.023
MCAR	Complete case analysis	0.300	0.133	− 0.027	0.018	− 0.003	0.219	0.048
	Missing indicator	0.574	0.120	0.247	0.075	0.305	0.162	0.119
	No interaction term
	Multiple imputation
	With Y	0.315	0.103	− 0.012	0.011	− 0.021	0.168	0.029
	Without Y	0.542	0.108	0.215	0.058	0.297	0.158	0.113
	Combined method
	With Y	0.315	0.102	− 0.012	0.011	− 0.021	0.169	0.029
	Without Y	0.541	0.110	0.214	0.058	0.297	0.158	0.113
	Interaction terms
	Multiple imputation	0.316	0.103	− 0.011	0.011	− 0.002	0.166	0.028
	Combined method	0.316	0.104	− 0.011	0.011	− 0.003	0.166	0.028
MAR	Complete case analysis	0.129	0.147	− 0.198	0.061	− 0.200	0.241	0.098
	Missing indicator	0.620	0.122	0.293	0.101	0.272	0.179	0.106
	No interaction term
	Multiple imputation
	With Y	0.251	0.107	− 0.076	0.017	− 0.093	0.181	0.042
	Without Y	0.579	0.112	0.252	0.076	0.286	0.173	0.111
	Combined method
	With Y	0.250	0.108	− 0.077	0.017	− 0.092	0.182	0.042
	Without Y	0.580	0.113	0.253	0.077	0.285	0.173	0.111
	Interaction terms
	Multiple imputation	0.330	0.116	0.003	0.013	0.010	0.185	0.034
	Combined method	0.330	0.116	0.003	0.013	0.010	0.185	0.034
MNAR	Complete case analysis	− 0.111	0.141	− 0.438	0.211	− 0.411	0.224	0.219
	Missing indicator	0.588	0.121	0.261	0.082	0.230	0.171	0.082
	No interaction term
	Multiple imputation
	With Y	0.151	0.114	− 0.176	0.044	− 0.238	0.207	0.100
	Without Y	0.586	0.112	0.259	0.080	0.350	0.165	0.150
	Combined method
	With Y	0.140	0.111	− 0.187	0.047	− 0.248	0.206	0.104
	Without Y	0.546	0.108	0.219	0.060	0.248	0.165	0.089
	Interaction terms
	Multiple imputation	0.182	0.117	− 0.145	0.035	− 0.192	0.208	0.080
	Combined method	0.170	0.114	− 0.157	0.038	− 0.205	0.264	0.112