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Structural Modeling of Measurement Error in Generalized Linear Models with Rasch Measures as Covariates


This paper proposes a structural analysis for generalized linear models when some explanatory variables are measured with error and the measurement error variance is a function of the true variables. The focus is on latent variables investigated on the basis of questionnaires and estimated using item response theory models. Latent variable estimates are then treated as observed measures of the true variables. This leads to a two-stage estimation procedure which constitutes an alternative to a joint model for the outcome variable and the responses given to the questionnaire. Simulation studies explore the effect of ignoring the true error structure and the performance of the proposed method. Two illustrative examples concern achievement data of university students. Particular attention is given to the Rasch model.

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  • Andersen, E.B. (1970). Asymptotic properties of conditional maximum-likelihood estimators. Journal of the Royal Statistical Society, Series B, 32, 283–301.

    Google Scholar 

  • Andrich, D. (1988). Rasch models for measurement. Newbury Park: Sage Publications.

    Google Scholar 

  • Bartholomew, D.J., & Knott, M. (1999). Latent variable models and factor analysis. London: Arnold Publishers.

    Google Scholar 

  • Battauz, M., Bellio, R., & Gori, E. (2008). Reducing measurement error in student achievement estimation. Psychometrika, 73, 289–302.

    Article  Google Scholar 

  • Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: an application of an EM-algorithm. Psychometrika, 46, 443–459.

    Article  Google Scholar 

  • Carroll, R.J., & Wang, Y. (2008). Nonparametric variance estimation in the analysis of microarray data: A measurement error approach. Biometrika, 95, 437–449.

    PubMed  Article  Google Scholar 

  • Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C.M. (2006). Measurement error in nonlinear models: a modern perspective (2nd ed.). London: Chapman and Hall.

    Book  Google Scholar 

  • Casella, G., & Berger, R.L. (2002). Statistical inference (2nd ed.). North Scituate: Duxbury Press.

    Google Scholar 

  • Cheng, C.L., & Van Ness, J. (1999). Statistical regression with measurement error. London: Arnold Publishers.

    Google Scholar 

  • Christensen, K.B. (2007). Latent covariates in generalized linear models: a Rasch model approach. In J.-L. Auget, N. Balakrishnan, M. Mesbah, & G. Molenberghs (Eds.), Advances in statistical methods for the health sciences (pp. 95–108). Boston: Birkhäuser.

    Chapter  Google Scholar 

  • Davison, A.C. (2003). Statistical models. Cambridge: Cambridge University Press.

    Google Scholar 

  • Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.

    Google Scholar 

  • Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80, 27–38.

    Article  Google Scholar 

  • Fox, J.P., & Glas, A.W. (2003). Bayesian modeling of measurement error in predictor variables using item response theory. Psychometrika, 68, 169–191.

    Article  Google Scholar 

  • Higdon, R., & Schafer, D.W. (2001). Maximum likelihood computations for regression with measurement error. Computational Statistics & Data Analysis, 35, 283–299.

    Article  Google Scholar 

  • Hoijtink, H., & Boomsma, A. (1995). On person parameter estimation in the dichotomous Rasch model. In G.H. Fischer & I.W. Molenaar (Eds.), Rasch models: foundations, recent developments, and applications (pp. 53–68). New York: Springer.

    Google Scholar 

  • Kosmidis, I., & Firth, D. (2009). Bias reduction in exponential family nonlinear models. Biometrika, 96, 793–804.

    Article  Google Scholar 

  • McCullagh, P., & Nelder, J.A. (1989). Generalized linear models (2nd ed.). London: Chapman and Hall.

    Google Scholar 

  • Mislevy, R.T. (1985). Estimation of latent group effects. Journal of the American Statistical Association, 80, 993–997.

    Article  Google Scholar 

  • Monahan, J.F. (2001). Numerical methods of statistics. Cambridge: Cambridge University Press.

    Google Scholar 

  • Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized multilevel structural equation modeling. Psychometrika, 69, 167–190.

    Article  Google Scholar 

  • R Development Core Team (2010). R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

  • Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

    Google Scholar 

  • Schafer, D.W. (1987). Covariate measurement error in generalized linear models. Biometrika, 74, 385–391.

    Article  Google Scholar 

  • Schafer, D.W., & Purdy, K.G. (1996). Likelihood analysis for errors-in-variables regression with replicate measurements. Biometrika, 83, 813–824.

    Article  Google Scholar 

  • van der Linden, W.J., & Hambleton, R.K. (1997). Handbook of modern item response theory. New York: Springer.

    Google Scholar 

  • Wang, Y., Ma, Y., & Carroll, R.J. (2009). Variance estimation in the analysis of microarray data. Journal of the Royal Statistical Society, Series B, 71, 425–445.

    Article  Google Scholar 

  • Warm, T.A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427–450.

    Article  Google Scholar 

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Correspondence to Michela Battauz.

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Battauz, M., Bellio, R. Structural Modeling of Measurement Error in Generalized Linear Models with Rasch Measures as Covariates. Psychometrika 76, 40–56 (2011).

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  • heteroscedastic error
  • IRT
  • maximum likelihood estimation
  • measurement error
  • Rasch model
  • structural model