Table 5.

Results of the simulation study with n₁ = 500 cases and n₀ = 500 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 10%
α2 = 0.50	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.905	0.897	1.078	0.990	0.950	0.931	1.065	1.001
s.d.	0.083	0.073	0.091	0.117	0.101	0.073	0.071	0.108
Est. sd	0.083	0.072	0.089	0.115	0.100	0.072	0.072	0.111
90%	0.676	0.600	0.770	0.895	0.847	0.765	0.781	0.895
95%	0.766	0.698	0.850	0.947	0.914	0.851	0.859	0.955
MSE Eff		0.998	1.107	1.154		1.258	1.370	1.089
	disease rate 0.5%
α2 = 0.50	Controls	Param	Robust	Semi	Controls	Param	Robust	Semi
Mean	0.997	1.113	0.973	1.007	0.991	1.296	0.890	0.995
s.d.	0.098	0.073	0.067	0.082	0.102	0.113	0.087	0.072
Est. sd	0.101	0.072	0.070	0.088	0.098	0.112	0.084	0.071
90%	0.906	0.541	0.892	0.897	0.895	0.145	0.630	0.907
95%	0.951	0.663	0.957	0.942	0.937	0.231	0.745	0.941
MSE Eff		0.531	1.842	1.419		0.104	0.533	2.013