Covariate measurement error in generalized linear models
Covariate measurement error in generalized linear models
复制标题
DOI：
10.1093/biomet/74.2.385
复制
发表时间：
1987-06
期刊：
影响因子：
2.7
通讯作者：
D. W. Schafer
中科院分区：
文献类型：
--
作者：
D. W. Schafer
摘要当正态分布的协变量被正态分布的测量误差掩盖时，EM算法被用来获得具有典型联系的广义线性模型的回归系数的估计。通过将真实协变量转换为“缺失数据”，EM程序提出了一种迭代方案，其中每个循环由E步骤组成，需要计算真实协变量的近似第一和第二条件矩，然后是M步骤，其中回归参数根据这些近似值通过迭代重新加权最小二乘进行更新。所提出的程序进行了数值比较与精确的最大似然解，通过使用高斯积分，而不是在EM算法的E-步骤的近似值，并与其他估计简单的logistic回归测量误差。拟议程序的结果令人鼓舞。本文提出了一种估计广义线性模型中一个或多个协变量有误差测量时回归系数的方法。这项工作与卡罗尔等人（1984）的论文有关，其中考虑了结构性逻辑回归模型的最大似然估计。这些作者提供了计算简单的概率回归模型的估计，并建议结构逻辑回归可以在原则上进行。这篇论文继续了这一建议，但在细节上与Armstrong（1985）的类似工作有所不同。Stefanski &卡罗尔（1985）、Wolter & Fuller（1982）和普伦蒂斯（1982）提出了其他具有相关目的的程序。为了快速介绍所提出的方法，它是方便地显示在其标准的计算形式，忽略测量误差的朴素估计。如果yi是观测到的响应变量，xi是观测到的协变量，那么对于将yi的分布的典型参数等同于xtf 3的模型，f3的最大似然估计可以通过迭代重新加权最小二乘获得。在（s +1）个周期之后的f3的估计由下式给出：
SUMMARY The EM algorithm is used to obtain estimators of regression coefficients for generalized linear models with canonical link when normally distributed covariates are masked by normally distributed measurement errors. By casting the true covariates as 'missing data', the EM procedure suggests an iterative scheme in which each cycle consists of an E-step, requiring the computation of approximate first and second conditional moments of the true covariates given the observed data, followed by an M-step in which regression parameters are updated by iteratively reweighted least squares based on these approximations. The proposed procedure is compared numerically with the exact maximum likelihood solution, obtained by using Gaussian quadrature instead of the approximations in the E-step of the EM algorithm, and with alternative estimators for simple logistic regression with measurement error. The results for the proposed procedure are encouraging. A procedure is proposed in this paper for estimating regression coefficients in generalized linear models when one or more of the covariates is measured with error. This work is related to the paper by Carroll et al. (1984) in which maximum likelihood estimates are considered for the structural logistic regression model. These authors provide estimates for the computationally simpler probit regression model and suggest that the structural logistic regression can be done in principle. This paper follows up on that suggestion but differs in detail from similar work by Armstrong (1985). Other procedures with related aims have been suggested by Stefanski & Carroll (1985), Wolter & Fuller (1982) and Prentice (1982). For a quick introduction to the proposed method it is convenient to display the naive estimator which ignores measurement error in its standard computational form. If yi is an observed response variable and xi is the observed covariate, then, for the model which equates the canonical parameter of the distribution of yi to xtf3, the maximum likelihood estimator of f3 can be obtained by iteratively reweighted least-squares. The estimate of f3 after (s +1) cycles is given by