Table 4.

Results of the simulation study with n₁ = 500 cases and n₀ = 500 controls, disease rate of approximately 4.5%, with heteroscedastic errors. Here “Normal” means that ε = Normal(0, 1), while “Gamma” means that ε is a centered and scale Gamma random variable with shape 0.4, mean zero and variance one. The analyses performed were using controls only (“Controls”), the semiparametric efficient method that assumes normality and homoscedasticity (“Param”), the method of Wei, et al. (2012), (“Robust”), and our method (“Semi”). Over 1,000 simulations, we computed the mean estimated β (“Mean”), its standard deviation (“s.d.”), the mean estimated standard deviation (“Est. sd”), the coverage for a nominal 90% confidence interval (“90%”), the coverage for a nominal 95% confidence interval (“95%”), and the mean squared error efficiency compared to using only the controls (“MSE Eff”).

	Normal				Gamma
α2 = 0.00	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.996	0.996	1.000	1.005	0.992	0.994	1.000	1.002
s.d.	0.099	0.071	0.071	0.076	0.099	0.070	0.073	0.077
Est. sd	0.096	0.070	0.072	0.082	0.096	0.070	0.071	0.078
90%	0.887	0.892	0.895	0.898	0.887	0.903	0.893	0.898
95%	0.932	0.953	0.949	0.950	0.944	0.946	0.947	0.951
MSE Eff		1.948	1.961	1.692		1.971	1.847	1.663
α2 = 0.25	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.986	1.044	0.973	0.997	0.983	1.063	0.964	0.995
s.d.	0.100	0.072	0.066	0.077	0.094	0.082	0.069	0.071
Est. sd	0.096	0.071	0.070	0.081	0.094	0.083	0.072	0.074
90%	0.880	0.838	0.907	0.912	0.894	0.825	0.863	0.904
95%	0.936	0.907	0.953	0.959	0.946	0.900	0.934	0.950
MSE Eff		1.415	1.984	1.717		0.852	1.516	1.801
α2 = 0.50	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.972	1.088	0.949	0.991	0.962	1.145	0.906	0.993
s.d.	0.099	0.072	0.068	0.083	0.095	0.096	0.076	0.082
Est. sd	0.096	0.072	0.071	0.102	0.090	0.100	0.076	0.105
90%	0.877	0.664	0.842	0.897	0.857	0.591	0.655	0.900
95%	0.936	0.789	0.914	0.946	0.909	0.714	0.756	0.935
MSE Eff		0.816	1.479	1.519		0.343	0.717	1.546