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 9 Comparison of approaches for the estimation of natural effects on the odds ratio scale for Scenario 4 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	1.042	1.214	0.172	16.484	0.191	0.257	85.0	83.6
	Approx_C	1.042	1.151	0.109	10.480	0.196	0.224	93.0	91.9
	Approx_IPW	1.042	1.183	0.141	13.532	0.193	0.239	89.6	88.1
	Exact_Naive	1.042	0.965	-0.077	-7.398	0.213	0.227	90.0	91.0
	Exact_IPW	1.042	1.052	0.010	0.983	0.197	0.197	94.1	93.9
	Unified	1.042	1.153	0.111	10.699	0.196	0.226	92.9	-
NIE	Approx_Naive	1.927	1.701	-0.226	-11.736	0.154	0.274	72.1	66.3
	Approx_C	1.927	1.778	-0.149	-7.735	0.179	0.233	92.0	86.5
	Approx_IPW	1.927	1.745	-0.182	-9.452	0.166	0.246	86.2	79.0
	Exact_Naive	1.927	2.165	0.238	12.341	0.399	0.465	87.9	88.4
	Exact_IPW	1.927	1.958	0.031	1.592	0.265	0.266	94.3	93.0
	Unified	1.927	1.772	-0.155	-8.065	0.178	0.236	87.1	-
TE	Approx_Naive	2.008	2.052	0.044	2.174	0.292	0.295	94.7	94.2
	Approx_C	2.008	2.027	0.019	0.926	0.277	0.278	95.7	94.8
	Approx_IPW	2.008	2.048	0.040	1.999	0.287	0.290	95.0	94.3
	Exact_Naive	2.008	2.026	0.018	0.891	0.277	0.278	95.1	94.0
	Exact_IPW	2.008	2.026	0.018	0.892	0.277	0.278	95.6	95.0
	Unified	2.008	2.024	0.015	0.762	0.277	0.277	94.9	-

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