An empirical evaluation of approximate and exact regression-based causal mediation approaches for a binary outcome and a continuous or a binary mediator for case-control study designs

Table 8 Comparison of approaches for the estimation of natural effects on the odds ratio scale for Scenario 3 with a binary mediator (based on 1000 data sets of size \(n=1000\))

Effect	Approach	True value	Mean	Bias	Relative bias (%)	SD	RMSE	CP (%) delta	CP (%) boot
NDE	Approx_Naive	1.100	1.112	0.013	1.138	0.151	0.152	94.7	94.5
	Approx_C	1.100	1.120	0.020	1.844	0.152	0.153	94.8	94.2
	Approx_IPW	1.100	1.117	0.018	1.634	0.152	0.153	95.2	94.5
	Exact_Naive	1.100	1.092	-0.007	-0.638	0.144	0.144	94.8	94.3
	Exact_IPW	1.100	1.113	0.013	1.221	0.150	0.151	95.1	95.1
	Unified	1.100	1.119	0.019	1.736	0.152	0.153	94.9	-
NIE	Approx_Naive	0.939	0.964	0.025	2.704	0.016	0.030	61.1	64.6
	Approx_C	0.939	0.932	-0.007	-0.750	0.033	0.034	88.1	95.2
	Approx_IPW	0.939	0.942	0.003	0.303	0.028	0.028	87.1	94.4
	Exact_Naive	0.939	0.954	0.015	1.609	0.023	0.028	81.5	87.5
	Exact_IPW	0.939	0.937	-0.001	-0.156	0.031	0.031	88.9	95.3
	Unified	0.939	0.932	-0.007	-0.722	0.033	0.034	94.5	-
TE	Approx_Naive	1.032	1.072	0.040	3.830	0.143	0.149	93.9	94.4
	Approx_C	1.032	1.042	0.010	0.949	0.136	0.136	95.1	94.5
	Approx_IPW	1.032	1.051	0.019	1.842	0.139	0.140	95.1	94.4
	Exact_Naive	1.032	1.042	0.009	0.911	0.136	0.136	94.6	93.9
	Exact_IPW	1.032	1.042	0.010	0.956	0.136	0.137	95.4	94.1
	Unified	1.032	1.041	0.009	0.870	0.136	0.136	94.6	-

boot Bootstrap, CP Coverage probability, NDE Natural direct effect, NIE Natural indirect effect, RMSE Root mean squared error, SD Standard deviation, TE Total effect

ISSN: 1471-2288