Table 5.
Misspecification of the measurement error model in logistic regression: E(Y|X) = {1 + exp(−β0 − β1)X)}−1, where X = L + Ub; W = L + Uc; β = (−1, ln(5))′; M = 1 − 2L + V; Q = −1 + 2X + ∊; θ = 0.7; L ~ N(0.5,1); Ub - N(0, ) is the Berkson error; Uc ~ N(0, ) is the classical error; V ~ N(0,1); ∊ ~ N(0,1); n = 500; Naive, “naive” regression replacing X by W; RC1, Regression calibration replacing X by its conditional expectation given W; RC2, Regression calibration replacing X by its conditional expectation given W and Q, M in the calibration sample; SIMEX, simulation extrapolation procedure; EEE, Expected estimating equation method.
= 0.3 |
= 0.5 |
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β | Naive | RC1 | RC2 | SIMEX | EEE | Naive | RC1 | RC2 | SIMEX | EEE | ||
Incorrectly assuming a classical error model | ||||||||||||
0.2 | β 0 | Bias | 0.315 | 0.159 | 0.068 0.148 0.009 | 0.414 0.194 | 0.078 | 0.225 | 0.005 | |||
SD | 0.115 | 0.125 | 0.140 | 0.133 | 0.158 | 0.108 | 0.125 | 0.143 | 0.126 | 0.165 | ||
ASE | 0.119 | 0.129 | 0.143 | 0.137 | 0.161 | 0.114 | 0.128 | 0.148 | 0.133 | 0.171 | ||
CP | 0.249 | 0.772 | 0.916 | 0.802 | 0.949 | 0.067 | 0.646 | 0.931 | 0.582 | 0.959 | ||
β 1 | Bias | −0.564 | −0.254 | −0.096 | −0.251 | −0.001 | −0.750 | −0.315 | −0.117 | −0.399 | 0.001 | |
SD | 0.110 | 0.146 | 0.172 | 0.152 | 0.206 | 0.095 | 0.141 | 0.170 | 0.135 | 0.208 | ||
ASE | 0.112 | 0.143 | 0.165 | 0.151 | 0.197 | 0.098 | 0.140 | 0.173 | 0.135 | 0.212 | ||
CP | 0.008 | 0.551 | 0.898 | 0.581 | 0.957 | 0.000 | 0.403 | 0.874 | 0.200 | 0.946 | ||
0.4 | β 0 | Bias | 0.370 | 0.221 | 0.096 | 0.217 | 0.039 | 0.449 | 0.245 | 0.119 | 0.210 | −0.006 |
SD | 0.113 | 0.128 | 0.143 | 0.131 | 0.161 | 0.117 | 0.131 | 0.154 | 0.154 | 0.200 | ||
ASE | 0.117 | 0.126 | 0.144 | 0.133 | 0.162 | 0.112 | 0.126 | 0.150 | 0.146 | 0.196 | ||
CP | 0.113 | 0.562 | 0.902 | 0.613 | 0.938 | 0.049 | 0.491 | 0.859 | 0.678 | 0.962 | ||
β 1 | Bias | −0.634 | −0.340 | −0.119 | −0.346 | −0.026 | −0.805 | −0.398 | −0.139 | −0.481 | −0.022 | |
SD | 0.105 | 0.138 | 0.162 | 0.139 | 0.194 | 0.097 | 0.137 | 0.169 | 0.133 | 0.207 | ||
ASE | 0.109 | 0.139 | 0.166 | 0.144 | 0.200 | 0.096 | 0.138 | 0.176 | 0.132 | 0.219 | ||
CP | 0.000 | 0.340 | 0.884 | 0.358 | 0.946 | 0.000 | 0.207 | 0.870 | 0.105 | 0.957 | ||
| ||||||||||||
Incorrectly assuming a Berkson error model | ||||||||||||
0.2 | β 0 | Bias | 0.315 | 0.315 | 0.200 | 0.282 | 0.161 | 0.414 | 0.414 | 0.278 | 0.395 | 0.248 |
SD | 0.115 | 0.115 | 0.127 | 0.122 | 0.139 | 0.108 | 0.108 | 0.119 | 0.112 | 0.129 | ||
ASE | 0.119 | 0.119 | 0.130 | 0.126 | 0.142 | 0.114 | 0.114 | 0.129 | 0.117 | 0.138 | ||
CP | 0.249 | 0.249 | 0.670 | 0.398 | 0.777 | 0.067 | 0.067 | 0.415 | 0.113 | 0.574 | ||
β 1 | Bias | −0.564 | −0.564 | −0.355 | −0.514 | −0.293 | −0.750 | −0.750 | −0.502 | −0.722 | −0.455 | |
SD | 0.110 | 0.110 | 0.132 | 0.118 | 0.152 | 0.095 | 0.095 | 0.122 | 0.099 | 0.140 | ||
ASE | 0.112 | 0.112 | 0.131 | 0.120 | 0.151 | 0.098 | 0.098 | 0.127 | 0.101 | 0.145 | ||
CP | 0.008 | 0.008 | 0.259 | 0.030 | 0.497 | 0.000 | 0.000 | 0.041 | 0.000 | 0.159 | ||
0.4 | β 0 | Bias | 0.370 | 0.370 | 0.208 | 0.315 | 0.136 | 0.449 | 0.449 | 0.256 | 0.415 | 0.195 |
SD | 0.113 | 0.113 | 0.128 | 0.124 | 0.146 | 0.117 | 0.117 | 0.136 | 0.127 | 0.153 | ||
ASE | 0.117 | 0.117 | 0.133 | 0.129 | 0.152 | 0.112 | 0.112 | 0.131 | 0.120 | 0.147 | ||
CP | 0.113 | 0.113 | 0.642 | 0.325 | 0.853 | 0.049 | 0.049 | 0.501 | 0.118 | 0.673 | ||
β 1 | Bias | −0.634 | −0.634 | −0.341 | −0.548 | −0.225 | −0.805 | −0.805 | −0.455 | −0.755 | −0.357 | |
SD | 0.105 | 0.105 | 0.127 | 0.122 | 0.155 | 0.097 | 0.097 | 0.126 | 0.109 | 0.153 | ||
ASE | 0.109 | 0.109 | 0.133 | 0.126 | 0.165 | 0.096 | 0.096 | 0.125 | 0.105 | 0.153 | ||
CP | 0.000 | 0.000 | 0.289 | 0.018 | 0.711 | 0.000 | 0.000 | 0.084 | 0.000 | 0.353 |
Note: SD denotes the sample standard deviation of the estimates; ASE is the average of the estimated standard errors; CP represents the coverage probability of the 95% confidence intervals.