Hostname: page-component-89b8bd64d-dvtzq Total loading time: 0 Render date: 2026-05-08T02:55:53.074Z Has data issue: false hasContentIssue false
  • English
  • Français

Improved Regression Calibration

Published online by Cambridge University Press:  01 January 2025

Anders Skrondal  and
Jouni Kuha
Anders Skrondal*
Affiliation:
Division of Epidemiology, Norwegian Institute of Public Health
Jouni Kuha
Affiliation:
Department of Statistics, London School of Economics
*
Requests for reprints should be sent to Anders Skrondal, Division of Epidemiology, Norwegian Institute of Public Health, P.O. Box 4404, Nydalen, 0403 Oslo, Norway. E-mail: anders.skrondal@fhi.no
Get access Rights & Permissions [Opens in a new window]

Abstract

The likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form, which makes maximum likelihood estimation taxing. A popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation. We propose an improved regression calibration approach, a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood. It is both consistent and computationally efficient, and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood. Simulations suggest that improved regression calibration, which is easy to implement in standard software, works well in a range of situations.

Keywords

covariate measurement errormeasurement modelgeneralized linear modelpseudo maximum likelihood estimationregression calibration

Information

Type
Original Paper
Information
Psychometrika , Volume 77 , Issue 4 , October 2012 , pp. 649 - 669
Copyright
Copyright © 2012 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Albert, P.S., & Follmann, D.A. (2000). Modeling repeated count data subject to informative dropout. Biometrics, 56, 667–677.CrossRefGoogle ScholarPubMed
Armstrong, B. (1985). Measurement error in generalized linear models. Communications in Statistics. Series B, 16, 529–544.CrossRefGoogle Scholar
Bentler, P.M. (1983). Some contributions to efficient statistics in structural models: specification and estimation of moment structures. Psychometrika, 48, 493–517.CrossRefGoogle Scholar
Blackburn, M., & Neumark, D. (1992). Unobserved ability, efficiency wages, and interindustry wage differentials. Quarterly Journal of Economics, 107, 1421–1436.CrossRefGoogle Scholar
Buonaccorsi, J., Demidenko, E., & Tosteson, T. (2000). Estimation in longitudinal random effects models with measurement error. Statistica Sinica, 10, 885–903.Google Scholar
Buonaccorsi, J. (2010). Measurement error: models, methods and applications. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
Burr, D. (1988). On errors-in-variables in binary regression—Berkson case. Journal of the American Statistical Association, 83, 739–743.Google Scholar
Buzas, J.S., & Stefanski, L.A. (1995). Instrumental variable estimation in generalized linear measurement error models. Journal of the American Statistical Association, 91, 999–1006.CrossRefGoogle Scholar
Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C.M. (2006). Measurement error in nonlinear models (2 ed.). Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
Carroll, R.J., Spiegelman, C.H., Lan, K.G., Bailey, K.T., & Abbott, R.D. (1984). On errors-in-variables for binary regression models. Biometrika, 71, 19–25.CrossRefGoogle Scholar
Carroll, R.J., & Stefanski, L.A. (1990). Approximate quasi-likelihood estimation in models with surrogate predictors. Journal of the American Statistical Association, 85, 652–663.CrossRefGoogle Scholar
Clayton, D.G. (1992). Models for the analysis of cohort and case-control studies with inaccurately measured exposures. In Dwyer, J.H., Feinlieb, M., Lippert, P., & Hoffmeister, H. (Eds.), Statistical models for longitudinal studies on health (pp. 301–331). New York: Oxford University Press.Google Scholar
Davis, P.J., & Rabinowitz, P. (1984). Methods of numerical integration (2 ed.). New York: Academic Press.Google Scholar
Gleser, L.J. (1990). Improvements of the naive approach to estimation in nonlinear errors-in-variables regression models. In Brown, P.J., & Fuller, W.A. (Eds.), Statistical analysis of measurement error models and applications (pp. 99–114). Providence: American Mathematical Society.CrossRefGoogle Scholar
Gong, G., & Samaniego, F.J. (1981). Pseudo maximum likelihood estimation: theory and applications. Annals of Statistics, 9, 861–869.CrossRefGoogle Scholar
Gourieroux, C., & Monfort, A. (1995). Statistics and econometric models. Cambridge: Cambridge University Press.Google Scholar
Griliches, Z. (1976). Wages of very young men. Journal of Political Economy, 85, S69–S86.CrossRefGoogle Scholar
Gustafson, P. (2004). Measurement error and misclassification in statistics and epidemiology: impacts and Bayesian adjustments. Boca Raton: Chapman & Hall/CRC.Google Scholar
Higdon, R., & Schafer, D.W. (2001). Maximum likelihood computations for regression with measurement error. Computational Statistics & Data Analysis, 35, 283–299.CrossRefGoogle Scholar
Jöreskog, K.G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36, 109–133.CrossRefGoogle Scholar
Jöreskog, K.G., & Goldberger, A.S. (1975). Estimation of a model with multiple indicators and multiple causes of a single latent variable. Journal of the American Statistical Association, 70, 631–639.Google Scholar
Kuha, J. (1997). Estimation by data augmentation in regression models with continuous and discrete covariates measured with error. Statistics in Medicine, 16, 189–202.3.0.CO;2-H>CrossRefGoogle ScholarPubMed
Lesaffre, E., & Spiessens, B. (2001). On the effect of the number of quadrature points in a logistic random-effects model: an example. Journal of the Royal Statistical Society. Series C, 50, 325–335.CrossRefGoogle Scholar
Liang, K.-Y., & Liu, X.-H. (1991). Estimating equations in generalized linear models with measurement error. In Godambe, V.P. (Eds.), Estimating functions (pp. 47–63). Oxford: Oxford University Press.CrossRefGoogle Scholar
Lütkepohl, H. (1996). Handbook of matrices. Chichester: Wiley.Google Scholar
McCullagh, P., & Nelder, J.A. (1989). Generalized linear models (2 ed.). London: Chapman & Hall.CrossRefGoogle Scholar
McDonald, R.P., (1967). Nonlinear factor analysis (Psychometric Monograph No. 15). Richmond: Psychometric Corporation. .Google Scholar
Parke, W.R. (1986). Pseudo maximum likelihood estimation: the asymptotic distribution. Annals of Statistics, 14, 355–357.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2003). Maximum likelihood estimation of generalized linear models with covariate measurement error. The Stata Journal, 3, 385–410.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized multilevel structural equation modeling. Psychometrika, 69, 167–190.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., & Pickles, A., (2004b). Gllamm manual (Technical report 160). U.C. Berkeley Division of Biostatistics. Downloadable from http://www.bepress.com/ucbbiostat/paper160/.Google Scholar
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128, 301–323.CrossRefGoogle Scholar
Rabe-Hesketh, S., & Skrondal, A. (2012). Multilevel and longitudinal modeling using Stata, vol. II: categorical responses, counts, and survival (3 ed.). College Station: Stata Press.Google Scholar
Richardson, S., & Gilks, W.S. (1993). Conditional independence models for epidemiological studies with covariate measurement error. Statistics in Medicine, 12, 1703–1722.CrossRefGoogle ScholarPubMed
Robinson, G.K. (1991). That BLUP is a good thing: the estimation of random effects. Statistical Science, 6, 15–51.Google Scholar
Robinson, P.M. (1974). Identification, estimation, and large sample theory for regressions containing unobservable variables. International Economic Review, 15, 680–692.CrossRefGoogle Scholar
Rosner, B., Spiegelman, D., & Willett, W.C. (1990). Correction of logistic regression relative risk estimates and confidence intervals for measurement error: the case of multiple covariates measured with error. American Journal of Epidemiology, 132, 734–745.CrossRefGoogle ScholarPubMed
Rosner, B., Willett, W.C., & Spiegelman, D. (1989). Correction of logistic regression relative risk estimates and confidence intervals for systematic within-person measurement error. Statistics in Medicine, 8, 1031–1040.CrossRefGoogle ScholarPubMed
Rubin, D.B. (1976). Inference and missing data. Biometrika, 63, 581–592.CrossRefGoogle Scholar
Schafer, D.W. (1987). Covariate measurement error in generalized linear models. Biometrika, 74, 385–391.CrossRefGoogle Scholar
Schafer, D.W. (1993). Likelihood analysis for probit regression with measurement error. Biometrika, 80, 899–904.CrossRefGoogle Scholar
Schafer, D.W., & Purdy, K.G. (1986). Likelihood analysis for errors-in-variables regression with replicate measurements. Biometrika, 83, 813–824.CrossRefGoogle Scholar
Shapiro, A. (2007). Statistical inference of moment structures. In Lee, S.Y. (Eds.), Handbook of latent variable and related models (pp. 229–259). Amsterdam: Elsevier.Google Scholar
Skrondal, A., & Laake, P. (2001). Regression among factor scores. Psychometrika, 66, 563–575.CrossRefGoogle Scholar
Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
Skrondal, A., & Rabe-Hesketh, S. (2007). Latent variable modelling: a survey. Scandinavian Journal of Statistics, 34, 712–745.CrossRefGoogle Scholar
Skrondal, A., & Rabe-Hesketh, S. (2009). Prediction in multilevel generalized linear mixed models. Journal of the Royal Statistical Society. Series A, 172, 659–687.CrossRefGoogle Scholar
Stephens, D.A., & Dellaportas, P. (1992). Bayesian analysis of generalised linear models with covariate measurement error. In Bernardo, J.M., Berger, J.O., Dawid, A.P., & Smith, A.F.M. (Eds.), Bayesian statistics (Vol. 4, pp. 813–820). Oxford: Oxford University Press.Google Scholar
Thisted, R.A. (1988). Elements of statistical computing. London: Chapman & Hall.Google Scholar