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 4 Comparison of approaches for the estimation of natural effects on the odds ratio scale for Scenario 4 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	1.516	1.551	0.034	2.270	0.246	0.249	95.5	95.3
	Approx_C	1.516	1.554	0.037	2.468	0.250	0.252	95.6	94.8
	Approx_IPW	1.516	1.552	0.036	2.393	0.249	0.251	95.5	94.6
	Exact_Naive	1.516	1.543	0.027	1.761	0.253	0.254	95.4	95.0
	Exact_IPW	1.516	1.548	0.031	2.077	0.252	0.254	95.6	94.8
	Unified	1.516	1.554	0.038	2.497	0.249	0.252	95.4	-
NIE	Approx_Naive	1.105	1.111	0.006	0.564	0.112	0.113	94.6	94.4
	Approx_C	1.105	1.101	-0.004	-0.350	0.101	0.101	95.8	94.3
	Approx_IPW	1.105	1.107	0.002	0.204	0.108	0.108	95.2	94.2
	Exact_Naive	1.105	1.110	0.005	0.450	0.110	0.110	94.4	94.3
	Exact_IPW	1.105	1.106	0.001	0.111	0.106	0.106	95.2	93.9
	Unified	1.105	1.100	-0.005	-0.410	0.099	0.100	94.9	-
TE	Approx_Naive	1.675	1.707	0.032	1.914	0.222	0.224	95.1	94.3
	Approx_C	1.675	1.696	0.021	1.225	0.219	0.220	95.2	95.0
	Approx_IPW	1.675	1.703	0.028	1.688	0.222	0.223	95.0	94.4
	Exact_Naive	1.675	1.696	0.020	1.212	0.219	0.220	95.1	94.9
	Exact_IPW	1.675	1.696	0.021	1.242	0.220	0.221	95.2	95.2
	Unified	1.675	1.696	0.020	1.216	0.219	0.220	95.1	-

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