Freedman Laurence S, Fainberg Vitaly, Kipnis Victor, Midthune Douglas, Carroll Raymond J
Department of Mathematics and Statistics, Bar Ilan University, Ramat Gan 52900, Israel.
Biometrics. 2004 Mar;60(1):172-81. doi: 10.1111/j.0006-341X.2004.00164.x.
We introduce a new method, moment reconstruction, of correcting for measurement error in covariates in regression models. The central idea is similar to regression calibration in that the values of the covariates that are measured with error are replaced by "adjusted" values. In regression calibration the adjusted value is the expectation of the true value conditional on the measured value. In moment reconstruction the adjusted value is the variance-preserving empirical Bayes estimate of the true value conditional on the outcome variable. The adjusted values thereby have the same first two moments and the same covariance with the outcome variable as the unobserved "true" covariate values. We show that moment reconstruction is equivalent to regression calibration in the case of linear regression, but leads to different results for logistic regression. For case-control studies with logistic regression and covariates that are normally distributed within cases and controls, we show that the resulting estimates of the regression coefficients are consistent. In simulations we demonstrate that for logistic regression, moment reconstruction carries less bias than regression calibration, and for case-control studies is superior in mean-square error to the standard regression calibration approach. Finally, we give an example of the use of moment reconstruction in linear discriminant analysis and a nonstandard problem where we wish to adjust a classification tree for measurement error in the explanatory variables.
我们介绍了一种新的方法——矩重构，用于校正回归模型中协变量的测量误差。其核心思想与回归校准类似，即有测量误差的协变量值被“调整后”的值所取代。在回归校准中，调整后的值是在测量值条件下真实值的期望。在矩重构中，调整后的值是在结果变量条件下真实值的方差保持经验贝叶斯估计。因此，调整后的值与未观测到的“真实”协变量值具有相同的前两个矩以及与结果变量相同的协方差。我们表明，在线性回归情况下，矩重构等同于回归校准，但对于逻辑回归会得出不同的结果。对于采用逻辑回归且协变量在病例组和对照组内呈正态分布的病例对照研究，我们表明回归系数的所得估计是一致的。在模拟中我们证明，对于逻辑回归，矩重构的偏差小于回归校准，并且对于病例对照研究，其均方误差优于标准回归校准方法。最后，我们给出了矩重构在线性判别分析中的应用示例以及一个非标准问题，即我们希望针对解释变量中的测量误差调整分类树。
