. Author manuscript; available in PMC: 2009 May 2.

Published in final edited form as: Stat Med. 2008 Nov 10;27(25):5195–5216. doi: 10.1002/sim.3361

Table II.

Empirical means of β̂_X (with empirical standard deviations in parentheses) in 1000 simulations when the disease model is logistic, estimated by different methods: naïve logistic regression of Y on W, regression calibration for W in the main study (RC1), regression calibration for M̄ in calibration sub-study (RC2), efficient regression calibration (ERC), moment reconstruction (MR) and multiple imputation (IM).

β_X	Γ₀_y	γ_Xy	$σ_{δ y}^{2}$	ρ_XW^*	Naïve	RC1	RC2	ERC	MR	IM
Cohort, non-differential error (N =100000, n = 1000)
0.3	0	0.5	1	0.45	0.119 (0.010)	0.300 (0.034)	0.302 (0.131)	0.299 (0.034)	0.303 (0.132)	0.299 (0.128)
0.3	0	0.5	2	0.33	0.066 (0.007)	0.302 (0.050)		0.300 (0.047)	0.305 (0.133)	0.307 (0.135)
0.3	0	0.5	4	0.24	0.035 (0.005)	0.303 (0.066)		0.297 (0.059)	0.298 (0.136)	0.297 (0.138)
0.6	0	0.5	1	0.45	0.233 (0.009)	0.585 (0.053)	0.601 (0.130)	0.585 (0.050)	0.607 (0.134)	0.603 (0.129)
0.6	0	0.5	2	0.33	0.129 (0.007)	0.587 (0.077)		0.585 (0.066)	0.609 (0.133)	0.603 (0.135)
0.6	0	0.5	4	0.24	0.068 (0.005)	0.592 (0.108)		0.582 (0.078)	0.614 (0.137)	0.608 (0.143)
Case/control, non-differential error(N = 2000, n = 200)
0.3	0	0.5	1	0.45	0.121 (0.042)	0.324 (0.161)	0.317 (0.199)	0.298 (0.114)	0.310 (0.201)	0.310 (0.196)
0.3	0	0.5	2	0.33	0.065 (0.031)	0.381 (0.860)		0.282 (0.139)	0.301 (0.197)	0.301 (0.198)
0.3	0	0.5	4	0.24	0.035 (0.022)	0.419 (1.150)		0.276 (0.152)	0.305 (0.187)	0.303 (0.193)
0.6	0	0.5	1	0.45	0.231 (0.040)	0.683 (1.247)	0.629 (0.213)	0.590 (0.166)	0.618 (0.220)	0.608 (0.210)
0.6	0	0.5	2	0.33	0.131 (0.031)	0.666 (2.203)		0.586 (0.179)	0.623 (0.212)	0.618 (0.223)
0.6	0	0.5	4	0.24	0.068 (0.021)	0.814 (16.94)		0.577 (0.201)	0.622 (0.217)	0.612 (0.222)
Case/control, differential error (N= 2000, n= 200)
0.3	0/0.25	1	1		0.276 (0.033)	0.565 (0.108)	0.317 (0.199)	0.494 (0.095)	0.311 (0.192)	0.303 (0.165)
0.3	0	0.5/1	1		0.175 (0.035)	0.469 (0.178)	0.307 (0.188)	0.374 (0.114)	0.301 (0.182)	0.298 (0.166)
0.3	0	1	2/0.5		0.132 (0.031)	0.414 (0.140)	0.302 (0.192)	0.364 (0.110)	0.298 (0.188)	0.291 (0.165)

ρ_XW is the correlation of X and W.