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 3 Comparison of approaches for the estimation of natural effects on the odds ratio scale for Scenario 3 with a continuous 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	1.646	1.669	0.024	1.443	0.236	0.238	96.3	95.9
	Approx_C	1.646	1.675	0.029	1.772	0.241	0.243	96.6	95.7
	Approx_IPW	1.646	1.673	0.028	1.686	0.242	0.243	96.4	95.8
	Exact_Naive	1.646	1.662	0.017	1.018	0.245	0.245	95.6	95.3
	Exact_IPW	1.646	1.670	0.024	1.457	0.244	0.245	96.4	95.9
	Unified	1.646	1.676	0.030	1.844	0.241	0.243	96.3	-
NIE	Approx_Naive	1.150	1.157	0.007	0.606	0.094	0.095	94.4	94.0
	Approx_C	1.150	1.144	-0.005	-0.473	0.079	0.079	97.1	94.3
	Approx_IPW	1.150	1.151	0.001	0.100	0.087	0.087	96.8	93.9
	Exact_Naive	1.150	1.154	0.004	0.383	0.091	0.091	94.0	94.0
	Exact_IPW	1.150	1.149	0.000	-0.031	0.085	0.085	96.9	93.9
	Unified	1.150	1.143	-0.006	-0.548	0.078	0.078	95.0	-
TE	Approx_Naive	1.892	1.922	0.031	1.615	0.255	0.257	95.7	95.3
	Approx_C	1.892	1.908	0.016	0.850	0.245	0.245	95.8	95.0
	Approx_IPW	1.892	1.917	0.025	1.321	0.249	0.250	95.6	95.1
	Exact_Naive	1.892	1.908	0.016	0.843	0.244	0.245	95.3	95.5
	Exact_IPW	1.892	1.909	0.018	0.929	0.246	0.246	95.6	95.0
	Unified	1.892	1.908	0.016	0.851	0.244	0.245	95.3	-

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