Table 1.

Simulation results for the under-specification setting, in which the full model of interest has the form h⁻¹ {pr(D = 1)} = β₀ + β_XX + β_ZZ + β_XZXZ, where h⁻¹ denotes the inverse link function corresponding to a logistic model, and where the external model is under-specified but involves both covariates with the form h⁻¹ {pr(D = 1)} = θ₀ + θ_XX + θ_Z. Results multiplied by 10³ are presented, and the coverage probabilities (CP) are reported as percents.

	Int	β0 GR/ mGR	CML	Int	βX GR/ mGR	CML	Int	βZ GR/ mGR	CML	Int	βXZ GR/ mGR	CML
simple random; N = 1000
Bias	−8.94	−0.79	−1.12	2.42	2.16	2.46	1.29	1.29	1.65	1.39	1.30	1.87
SE	91.4	24.4	24.4	96.8	20.1	19.9	94.3	20.2	19.8	89.4	89.6	89.3
ESE	91.8	22.9	23.5	92.3	19.8	19.7	92.4	19.9	19.8	85.8	85.3	86.9
MSE	8.42	0.59	0.59	9.38	0.41	0.40	8.89	0.41	0.39	7.98	8.02	7.97
CP	95.4	88.6	90.8	94.3	94.6	95.2	94.6	94.4	95.5	93.6	93.4	93.8
case-control; N = 1000
Bias	-	-	1.70	2.40	28.4	2.23	5.06	27.6	1.73	−1.51	−1.20	−2.30
SE	-	-	17.4	75.7	11.3	16.7	72.2	11.4	16.6	72.9	72.8	71.7
ESE	-	-	16.6	73.3	11.6	16.5	73.1	11.6	16.5	71.4	71.3	70.1
MSE	-	-	0.30	5.73	0.93	0.28	5.24	0.89	0.28	5.31	5.29	5.15
CP	-	-	90.7	94.2	31.4	94.5	95.4	33.1	94.8	94.7	94.6	93.7

Int: internal-data only method

GR: generalized regression, mGR: modified GR for case-control sampling

CML: constrained maximum likelihood

ESE: estimated standard error

MSE: mean squared error

CP: coverage probability of a 95% confidence interval interval