Table 3.

Simulation results from 1000 simulated case-control samples taken from a population with a disease rate of approximately 4.5%, and independent genetic and environmental variables, under the logistic model with gene–environment interaction. The results for G ∼ $\mathcal{B}$ (0.6) and X ∼ $\mathcal{G}$ (20, 1) is displayed on the left whereas the results for G ~ $\mathcal{N}$ N(0, 1) and X ~ $\mathcal{G}$ (20, 1) is on the right. Each replicate contains N₁ = 1000 cases and N₀ = 1000 controls, and is analyzed through two approaches, (1) “Logistic” is ordinary logistic regression, and (2) “Semi” is our semiparametric efficient estimator. Here, we list the sample mean (“mean”), the sample standard error (“se”), the mean estimated standard error (“est se”) and the coverage for the nominal 95% confidence intervals (“95%”) for both methods. In addition, we computed the mean squared error efficiency of the “Semi” method compared to the “Logistic” approach.

		Binary G, Gamma X			Normal G, Gamma X
	β	3.577	0.080	−0.141	3.577	0.080	−0.141
Logistic	Mean	3.599	0.081	−0.142	3.592	0.080	−0.141
	se	0.456	0.018	0.022	0.269	0.012	0.012
	est se	0.462	0.018	0.022	0.259	0.012	0.012
	95%	0.957	0.953	0.949	0.937	0.950	0.942
Semi	Mean	3.586	0.080	−0.141	3.569	0.080	−0.140
	se	0.375	0.016	0.018	0.230	0.011	0.010
	est se	0.369	0.016	0.017	0.202	0.011	0.009
	95%	0.950	0.949	0.942	0.914	0.940	0.919
	MSE Eff	1.484	1.305	1.559	1.372	1.059	1.437