. Author manuscript; available in PMC: 2016 Jan 28.

Published in final edited form as: J R Stat Soc Series B Stat Methodol. 2015 Feb 15;78(1):127–151. doi: 10.1111/rssb.12107

Table 6.

Results of the simulation study with n₁ = 334 cases and n₀ = 666 controls, α₂ = 0.5, 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
	disease rate 4.5%
α₂ = 0.50	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.977	1.052	0.961	0.996	0.961	1.085	0.926	0.994
s.d.	0.084	0.070	0.063	0.077	0.083	0.087	0.066	0.082
Est. sd	0.087	0.072	0.064	0.083	0.08	0.087	0.067	0.090
90%	0.883	0.825	0.859	0.913	0.827	0.735	0.702	0.918
95%	0.939	0.892	0.930	0.952	0.905	0.831	0.806	0.954
MSE Eff		0.998	1.382	1.276		0.568	0.855	1.244
	disease rate 10%
α₂ = 0.50	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.911	0.909	1.021	1.001	0.956	0.937	1.027	1.000
s.d.	0.072	0.064	0.080	0.079	0.084	0.066	0.070	0.087
Est. sd	0.072	0.065	0.080	0.076	0.087	0.065	0.072	0.094
90%	0.654	0.595	0.895	0.901	0.867	0.772	0.877	0.906
95%	0.749	0.700	0.949	0.951	0.927	0.851	0.933	0.952
MSE Eff		1.058	1.915	2.099		1.080	1.597	1.188
	disease rate 0.5%
α₂ = 0.50	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.997	1.073	0.979	1.007	0.994	1.189	0.920	0.997
s.d.	0.088	0.073	0.066	0.078	0.084	0.100	0.071	0.070
Est. sd	0.087	0.072	0.063	0.086	0.085	0.099	0.073	0.069
90%	0.891	0.728	0.871	0.901	0.899	0.384	0.725	0.911
95%	0.950	0.820	0.929	0.952	0.953	0.539	0.829	0.960
MSE Eff		0.727	1.616	1.264		0.155	0.620	1.445