Table 1.

Biases and root mean squared errors (RMSEs) for ordinary logistic regression, retrospective maximum likelihood, and our approach, where disease status (D), the genetic variant (G), and the environmental covariate (X) are binary. The environmental variable is measured with error, with misclassification probabilities being 0.20 for exposed and 0.10 for nonexposed subjects. The results are based on a simulation study with 500 replications for 1000 cases and 1000 controls. Results are given when pr(D = 1) is known and when it is unknown.

			Logistic		Retrospective		Pseudo-likelihood
pr(D = 1)	Parameter	True value	Bias	RMSE	Bias	RMSE	Bias	RMSE
Known	β0	−5.000	4.294	4.295	−0.006	0.108	−0.006	0.108
	βg	0.693	0.239	0.323	−0.005	0.305	−0.004	0.305
	βx	1.099	−0.327	0.344	0.005	0.155	0.005	0.155
	βxg	0.693	−0.284	0.395	0.001	0.327	0.001	0.327
	pr(X = 1)	0.100			0.002	0.021	0.002	0.022
	pr(G = 1)	0.100			0.000	0.009	0.000	0.008
Unknown	β0	−5.000	4.294	4.295	−1.016	2.042	−1.016	2.042
	βg	0.693	0.239	0.323	−0.009	0.306	−0.009	0.306
	βx	1.099	−0.327	0.344	0.004	0.155	0.004	0.155
	βxg	0.693	−0.284	0.395	0.013	0.333	0.013	0.333
	pr(X = 1)	0.100			0.023	0.022	0.002	0.022
	pr(G = 1)	0.100			0.000	0.009	0.000	0.009
	pr(D = 1)	0.016			0.002	0.019	0.002	0.019