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 6 Comparison of approaches for the estimation of natural effects on the odds ratio scale for Scenario 1 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	2.152	2.042	-0.110	-5.131	0.306	0.325	92.1	91.5
	Approx_C	2.152	2.179	0.027	1.233	0.306	0.307	94.5	93.3
	Approx_IPW	2.152	2.166	0.014	0.637	0.312	0.312	94.4	93.4
	Exact_Naive	2.152	2.193	0.041	1.923	0.304	0.307	93.9	93.4
	Exact_IPW	2.152	2.181	0.029	1.365	0.312	0.313	94.2	93.3
	Unified	2.152	2.177	0.024	1.137	0.305	0.306	94.0	-
NIE	Approx_Naive	1.047	1.044	-0.003	-0.274	0.032	0.032	89.6	93.7
	Approx_C	1.047	1.046	-0.001	-0.141	0.025	0.025	98.9	95.6
	Approx_IPW	1.047	1.047	0.000	0.008	0.027	0.027	98.8	96.2
	Exact_Naive	1.047	1.037	-0.010	-0.996	0.024	0.026	86.5	89.4
	Exact_IPW	1.047	1.046	-0.001	-0.136	0.025	0.025	98.9	95.4
	Unified	1.047	1.046	-0.001	-0.119	0.025	0.025	97.1	-
TE	Approx_Naive	2.254	2.133	-0.121	-5.378	0.328	0.349	92.0	91.5
	Approx_C	2.254	2.277	0.023	1.035	0.315	0.316	94.5	93.5
	Approx_IPW	2.254	2.267	0.013	0.594	0.323	0.323	94.6	93.4
	Exact_Naive	2.254	2.274	0.020	0.875	0.314	0.314	93.5	93.3
	Exact_IPW	2.254	2.280	0.026	1.174	0.322	0.323	94.5	93.4
	Unified	2.254	2.276	0.022	0.960	0.314	0.315	93.4	-

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