在迴歸分析中共變數測量誤差的偏差效應

Bias Effect of Covariate Measurement Errors in Regression Analysis

10.29973/JCSA.200509.0002

黃麗惠(Li-Hui Huang)；王清雲(C. Y. Wang)；李燊銘(Shen-Ming Lee)

線性迴歸 ； 測量誤差 ； 動差校正法 ； 迴歸插補校正法 ； 估計函數 ； Estimating equation ； measurement error ； regression calibration ； Method of moments

中國統計學報

43卷3期（2005 / 09 / 01）

259 - 280

繁體中文

本文討論在迴歸分析中，共變數測量誤差所造成的偏差效應。我們證明測量誤差可造成各種不同的偏差效應。因此自然估計式(naive estimator)可能低估，亦可能高估參數，此現象並不同於單變數線性迴歸的衰減效應(effect of attenuation)。在線性及logistic迴歸上我們作偏差分析。我們證明了若測量誤差在若干參數下的線性迴歸造成低估效應時，則其在一樣參數下的logistic迴歸上之低估效應將更強。本文也證明在測量誤差項的共變異數矩陣之估計是相同下，線性迴歸中某動差校正估計與迴歸插補校正估計法是相同的。我們做了模擬探討且看出了不同的效應，也以實例來探討測量誤差對參數估計的影響。

We study the effect of measurement error in regression analysis, in which multiple covariate variables are not precisely measured. When covariate variables are measured with errors, we show that measurement errors may lead to various effects and hence the naive estimator may either under- or over-estimate the regression coefficients. This is different from the effect of attenuation under simple linear regression. Bias analyses are conducted for linear and logistic regression. We show that if in linear regression with certain parameters the measurement error effect is under-estimation, then the underestimation is stronger in logistic regression under the same parameters. We study a method of moments and the regression calibration estimator and show that these two consistent estimators are asymptotically equivalent in linear regression. Intensive simulation studies are presented and we show situations for various effects. We demonstrate the methods by data from an epidemiologic study.

1. Carroll, R. J.,Ruppert, D.,Stefanski, L. A.(1995).Nonlinear Measurement Error Models.Chapman and Hall.
2. Chen, C. L.,Van Ness, J. W.(1998).Statistical Regression with Measurement Error.Arnold, London:Oxford University Press.
3. Cook, J.,Stefanski, L. A.(1995).A simulation extrapolation method for parametric measurement error models.Journal of the American Statistical Association,89,1314-1328.
4. Fuller, W. A.(1987).Measurement Error Model.New York:John Willey and Sons.
5. Huang, Y.,Wang, C. Y.(2001).Consistent functional methods for logistic regression with error in covariates.Journal of the American Statistical Association,96,1469-1482.
6. Hughes, M. D.(1993).Regression dilution in the proportional hazards model.Biometrics,49,1056-1066.
7. Liang, K. Y.,Liu, X. H.,V. P. Godambe(ed.)(1991).Estimating Functions.Oxford:Clarendon Press.
8. Prentice, R. L.(1982).Covariate measurement errors and parameter estimation in a failure time regression model.Biometrika,69,331-342.
9. Stefanski, L. A.,Carroll, R. J.(1987).Conditional score and optimal scores for generalized linear measurement-error models.Biometrika,74,703-716.
10. Wang, C. Y.,Pepe, M.S.(2000).Expected estimating equations to accommodate coveriate measurement error.Journal of the Royal Statistical Society, Ser. B.,62,509-524.
11. 王清雲、陳瑞照(2000)。迴歸分析中的測量誤差模型淺介。統計薪傳，1，101-107。
12. 王清雲、黃以簡、陳瑞照(2001)。羅吉斯迴歸共變數含測量誤差之估計方法與數值計算比較。中國統計學報，39，125-138。
13. 黃登源(1998)。應用迴歸分析。台北:華泰書局。