Table.

Results from 80,000 Monte Carlo simulations with N = 10,000 and N^* = 100. The ${\overline{R}}_W^2$ and $\overline{\text{Var}}V\overline{a}r(W)$ are the out-of-sample prediction $R_W^2$ and variance of predicted exposures, respectively, averaged over 80,000 Monte Carlo simulations. The standard deviation and fraction of Monte Carlo runs statistically significant are given for estimates of α̂₃ in the correctly specified exposure model only, because this parameter is not included in the misspecified model. The bias, standard deviation, root mean squared error (RMSE), and 95% confidence interval (CI) coverage are given for estimates of the health effect parameter β̂_X. We also report the average estimated standard error (SE) for β̂_X. The Monte Carlo standard error in estimating the bias of β̂_X in all models is less than 0.001. (R code to produce these results is provided in an electronic appendix, http://links.lww.com.)

	σ2 = 1		σ2 = 0.1
	Correct model	Misspecified model	Correct model	Misspecified model
Exposure predictions
${\overline{R}}_W^2$	0.74	0.49	0.73	0.50
$\overline{\text{Var}}V\overline{a}r(W)$	48.5	32.7	50.2	32.3
Exposure model parameter estimate α̂3
Standard deviation	0.41	-	1.37	-
Statistically significant (p < 0.05)	100%	-	83%	-
Health effect parameter estimate β̂x
Bias	−0.007	−0.001	−0.035	0.001
Standard deviation	0.12	0.21	0.23	0.16
RMSE	0.12	0.21	0.23	0.16
E(SE)	0.038	0.049	0.038	0.049
95% CI coverage	45%	35%	26%	46%