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Some Recent Advances in Measurement Error Models and Methods

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Abstract

A measurement error model is a regression model with (substantial) measurement errors in the variables. Disregarding these easurement errors in estimating the regression parameters results in asymptotically biased estimators. Several methods have been roposed to eliminate, or at least to reduce, this bias, and the relative efficiency and robustness of these methods have been compared. The aper gives an account of these endeavors. In another context, when data are of a categorical nature, classification errors play a similar role as easurement errors in continuous data. The paper also reviews some recent advances in this field.

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This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the frame of the Sonderforschungsbereich SFB 386. We thank two anonymous eferees for their helpful comments.

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References

  • Amemiya, Y., Fuller, W. (1988). Estimation for the nonlinear functional relationship. Annals of Statistics16 147–160.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Augustin, T. (2002). Survival Analysis under Measurement Error. Habilitationsschrift (post-doctoral thesis). University of Munich.

    Google Scholar 

  • Augustin, T. (2004). An exact corrected log-likelihood function for Cox’s proportional hazards model under measurement error and some extensions. Scandinavian Journal of Statistics31 43–50.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Augustin, T., Schwarz, R. (2002). Cox’s proportional hazards model under co-variate measurement error — A review and comparison of methods. In Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications (S. Van Huffel, P. Lemmerling, eds.), 175–184. Kluwer, Dordrecht.

    Google Scholar 

  • Augustin, T., Wolff, J. (2004). A bias analysis of Weibull models under heaped data. Statistical Papers45 211–229.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Bender, R., Augustin, T., Bletter, M. (2005). Simulating survival times for Cox regression models. Statistics in Medicine24 1713–1723.

    CrossRef  PubMed  MathSciNet  Google Scholar 

  • Brown, P. J. (1982). Multivariate calibration. Journal of the Royal Statistical Society, Series B44 287–321.

    MATH  Google Scholar 

  • Brand, R. (2002). Microdata protection through noise addition. In Inference Control in Statistical Databases — From Theory to Practice. (J. Doningo-Ferrer ed.), Lecture Notes in Computer Science 2316. Springer, Berlin.

    Google Scholar 

  • Buzas, J. S. (1998). Unbiased scores in proportional hazards regression with covariate measurement error. Journal of Statistical Planning and Inference67 247–257.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Card, D. (2001). Estimating the return to schooling: Progress on some persistent econometric problems. Econometrica69 1127–1160.

    CrossRef  Google Scholar 

  • Carroll, R. J., Ruppert, D., Stefanski, L. A. (1995). Measurement Error in Nonlinear Models. Chapman and Hall, London.

    MATH  Google Scholar 

  • Carroll, R. J., Küchenhoff, H., Lombard, F., Stefanski, L. A. (1996). Asymptotics for the Simex estimator in structural measurement error models. Journal of the American Statistical Association91 242–250.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Cheng, C.-L., Schneeweiss, H. (1998). Polynomial regression with errors in the variables. Journal of the Royal Statistical Society, Series B60 189–199.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Cheng, C.-L., Schneeweiss, H., Thamerus, M. (2000). A small sample estimator for a polynomial regression with errors in the variables. Journal of the Royal Statistical Society, Series B62 699–709.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Cheng, C.-L., Schneeweiss, H. (2002). On the polynomial measurement error model. In Total Least Squares and Errors-in-Variables Modeling (S. van Htuffel, P. Lemmerling, eds.), 131–143. Kluwer, Dordrecht.

    Google Scholar 

  • Cheng, C.-L., Van Ness, J. W. (1999). Statistical Regression with Measurement Error. Arnold, London.

    MATH  Google Scholar 

  • Cook, J., Stefanski, L. A. (1994). Simulation-extrapolation estimation for parametric measurement error models. Journal of the American Statistical Association89 1314–1328.

    CrossRef  MATH  Google Scholar 

  • Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of the Royal Statistical Society, Series B34 187–220.

    MATH  Google Scholar 

  • Flinn, C. J., Heckman, J. J. (1982). Models for the analysis of labor force dynamics. Advances in Econometrics1 35–95.

    Google Scholar 

  • Friedman, M. (1957). A Theory of the Consumption Function. Princeton University Press, Princeton.

    Google Scholar 

  • Fuller, W. A. (1987). Measurement Error Models. Wiley, New York.

    CrossRef  MATH  Google Scholar 

  • Gimenez, P., Bolfarine, H., Colosimo, E. A. (1999). Estimation in Weibull regression model with measurement error. Communications in Statistics — Theory and Methods28 495–510.

    CrossRef  MATH  Google Scholar 

  • Gleser, L. J. (1990). Improvement of the naive estimation in nonlinear errors-in-variables regression. In Statistical Analysis of Measurement Error Models and Application (P. J. Brown, W. A. Fuller, eds.), Contemporary Mathematics112 99–114.

    Google Scholar 

  • Hausman, J. A., Abrevaya, J., Scott-Morton, F. M. (1998). Misclassiflcation of the dependent variable in a discrete-response setting. Journal of Econometrics87 239–269.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Heid, I., Küchenhoff, H., Wellmann, J., Gerken, M., Kreienbrock, L. (2002). On the potential of measurement error to induce differential bias on odds ratio estimates: An example from radon epidemiology. Statistics in Medicine21 3261–3278.

    CrossRef  PubMed  CAS  Google Scholar 

  • Hsiao, C, Wang, Q.K. (2000). Estimation of structural nonlinear errors-in-variables models by simulated least-squares method. International Economic Review41 523–542.

    CrossRef  MathSciNet  Google Scholar 

  • Hu, P., Tsiatis A. A., Davidian M. (1998). Estimating the parameters in the Cox model when covariate variables are measured with error. Biometrics54 1407–1419.

    CrossRef  MATH  PubMed  CAS  MathSciNet  Google Scholar 

  • Hu, Y., Ridder, G. (2005). Estimating a nonlinear model with measurement error using marginal information. http://www-rcf.usc.edu/~ridder/Wpapers/EIV-marg-final.pdf.

    Google Scholar 

  • Huang, H. S., Huwang, L. (2001). On the polynomial structural relationship. The Canadian Journal of Statistics29 493–511.

    CrossRef  MathSciNet  Google Scholar 

  • Huang, Y., Wang, C. Y. (2000). Cox regression with accurate covariates unascertainable: A nonparametric-correction approach. Journal of the American Statistical Association95 1209–1219 (Correction: 98 779).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Hughes, M. D. (1993). Regression dilution in the proportional hazards model. Biometrics49 1056–1066.

    CrossRef  MATH  PubMed  CAS  MathSciNet  Google Scholar 

  • Kong, F. H., Gu, M. (1999). Consistent estimation in Cox proportional hazards model with covariate measurement errors. Statistica Sinica9 953–969.

    MATH  MathSciNet  Google Scholar 

  • Kong, F. H., Huang, W., Li, X. (1998). Estimating survival curves under proportional hazards model with covariate measurement errors. Scandinavian Journal of Statistics25 573–587.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Küchenhoff, H., (1992). Estimation in generalized linear models with covariate measurement error using the theory of misspecified models. In Statistical Modelling (P. van der Heijden, W. Jansen, B. Francis, G. Seeber, eds.), 185–193. Elsevier, Amsterdam.

    Google Scholar 

  • Küchenhoff, H. (1995). The identification of logistic regression models with errors in the variables. Statistical Papers36 41–48.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Küchenhoff, H., Bender, R., Langer, I., Lenz-Tönjes, R. (2003). Effect of Berkson measurement error on parameter estimates in Cox regression models. Discission Paper 346, Sonderforschungsbereich 386, University of Munich.

    Google Scholar 

  • Küchenhoff, H., Carroll, R. J. (1997). Segmented regression with errors in predictors: semiparametric and parametric methods. Statistics in Medicine16 169–188.

    CrossRef  PubMed  Google Scholar 

  • Küchenhoff, H., Mwalili, S., Lesaffre, E. (2005). A general method for dealing with misclassification in regression: The misclassification SIMEX. Biometrics (to appear).

    Google Scholar 

  • Kuha, J.T., Temple, J. (2003). Covariate measurement error in quadratic regression. International Statistical Review71 131–150.

    CrossRef  MATH  Google Scholar 

  • Kukush, A., Markovsky, I., Van Huffel, S. (2004). Consistent estimation in an implicit quadratic measurement error model. Computational Statistics & Data Analysis47 123–147.

    CrossRef  MathSciNet  Google Scholar 

  • Kukush, A., Schneeweiss, H., Wolf, R. (2001). Comparison of three estimators in a polynomial regression with measurement errors. Discussion Paper 233, Sonderforschungsbereich 386, University of Munich.

    Google Scholar 

  • Kukush, A., Schneeweiss, H. (2005). Comparing different estimators in a nonlinear measurement error model. I and II. Mathematical Methods of Statistics14 53–79 and 203–223.

    MathSciNet  Google Scholar 

  • Li, T. (2000). Estimation of nonlinear errors-in-variables models: A simulated minimum distance estimator. Statistics and Probability Letters47 243–248.

    CrossRef  MATH  ADS  MathSciNet  Google Scholar 

  • Nakamura, T. (1990). Corrected score functions for errors-in-variables models: Methodology and application to generalized linear models. Biometrika77 127–137.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Nakamura, T. (1992). Proportional hazards model with covariates subject to measurement error. Biometrics48 829–838.

    CrossRef  PubMed  CAS  MathSciNet  Google Scholar 

  • Nowak, E. (1993). The identification of multivariate linear dynamic error-in-variables models. Journal of Econometrics59 213–227.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Pepe, M. S., Self, M. S., Prentice, R. L. (1989). Further results in covariate measurement errors in cohort studies with time to response data. Statistics in Medicine8 1167–1178.

    CrossRef  PubMed  CAS  Google Scholar 

  • Prentice, R. L. (1982). Covariate measurement errors and parameter estimation in a failure time regression model. Biometrika69 331–342.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Ronning, G. (2005). Randomized response and the binary probit model. Economics Letters86 221–228.

    CrossRef  MathSciNet  Google Scholar 

  • Schmid, M., Schneeweiss, H., Küchenhoff, H. (2005a). Consistent estimation of a simple linear model under microaggregation. Discussion Paper 415, Sonderforschungsbereich 386, University of Munich.

    Google Scholar 

  • Schmid, M., Schneeweiss, H., Küchenhoff, H. (2005b). Statistical inference in a simple linear model under microaggregation. Discussion Paper 416, Sonderforschungsbereich 386, University of Munich.

    Google Scholar 

  • Schneeweiss, H., Cheng, C.-L. (2006). Bias of the structural quasi-score estimator of a measurement error model under misspecification of the regressor distribution. Journal of Multivariate Analysis97 455–473.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Schneeweiss, H., Mittag, H. J. (1986). Lineare Modelle mit fehlerbehafteten Daten. Physica, Heidelberg.

    Google Scholar 

  • Schuster, G. (1998). ML estimation from binomial data with misclassifications-a comparison: Internal validation versus repeated measurements. In Econometrics in Theory and Practice (R. Galata, H. Küchenhoff, eds.), 45–58. Physika, Heidelberg.

    Google Scholar 

  • Shklyar, S., Schneeweiss, H. (2005). A comparison of asymptotic covariance matrices of three consistent estimators in the Poisson regression model with measurement errors. Journal of Multivariate Analysis94 250–270.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Shklyar, S., Schneeweiss, H., Kukush, A. (2005). Quasi score is more efficient than corrected score in a polynomial measurement error model. Discussion Paper 445, Sonderforschungsbereich 386, University of Munich.

    Google Scholar 

  • Skinner, C. J., Humphreys, K. (1999). Weibull regression for lifetimes measured with error. Lifetime Data Analysis5 23–37.

    CrossRef  MATH  PubMed  CAS  MathSciNet  Google Scholar 

  • Stefanski, L. A. (1989). Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models. Communications in Statistics — Theory and Methods18 4335–4358.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Thamerus, M. (2003). Fitting a mixture distribution to a variable subject to heteroscedastic measurement errors. Computational Statistics18 1–17.

    MATH  MathSciNet  Google Scholar 

  • Tsiatis A., Davidian, M. (2004). Joint modeling of longitudinal and time-to-event data: An overview. Statistica Sinica14 809–834.

    MATH  MathSciNet  Google Scholar 

  • Wansbeek, T., Meijer, E. (2000). Measurement Error and Latent Variables in Econometrics. Elsevier, Amsterdam.

    MATH  Google Scholar 

  • Wolf, R. (2004). Vergleich von funktionalen und strukturellen Messfehlerverfahren. Logos Verlag, Berlin.

    Google Scholar 

  • Wolff, J., Augustin, T. (2003). Heaping and its consequences for duration analysis — a simulation study. Allgemeines Statistisches Archiv — Journal of the German Statistical Society87 1–28.

    MathSciNet  Google Scholar 

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Schneeweiß, H., Augustin, T. (2006). Some Recent Advances in Measurement Error Models and Methods. In: Hübler, O., Frohn, J. (eds) Modern Econometric Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-32693-6_13

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