Electronic Journal of Statistics

Estimation via corrected scores in general semiparametric regression models with error-prone covariates

Arnab Maity and Tatiyana V. Apanasovich

Full-text: Open access


This paper considers the problem of estimation in a general semiparametric regression model when error-prone covariates are modeled parametrically while covariates measured without error are modeled nonparametrically. To account for the effects of measurement error, we apply a correction to a criterion function. The specific form of the correction proposed allows Monte Carlo simulations in problems for which the direct calculation of a corrected criterion is difficult. Therefore, in contrast to methods that require solving integral equations of possibly multiple dimensions, as in the case of multiple error-prone covariates, we propose methodology which offers a simple implementation. The resulting methods are functional, they make no assumptions about the distribution of the mismeasured covariates. We utilize profile kernel and backfitting estimation methods and derive the asymptotic distribution of the resulting estimators. Through numerical studies we demonstrate the applicability of proposed methods to Poisson, logistic and multivariate Gaussian partially linear models. We show that the performance of our methods is similar to a computationally demanding alternative. Finally, we demonstrate the practical value of our methods when applied to Nevada Test Site (NTS) Thyroid Disease Study data.

Article information

Electron. J. Statist., Volume 5 (2011), 1424-1449.

First available in Project Euclid: 4 November 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Generalized estimating equations generalized linear mixed models kernel method measurement error Monte Carlo Corrected Score semiparametric regression


Maity, Arnab; Apanasovich, Tatiyana V. Estimation via corrected scores in general semiparametric regression models with error-prone covariates. Electron. J. Statist. 5 (2011), 1424--1449. doi:10.1214/11-EJS647. https://projecteuclid.org/euclid.ejs/1320416980

Export citation


  • [1] Al-Abood, A. M. and Young, D. H. (1986). The power of approximate tests for the regression coefficients in a gamma regression model., IEEE Transactions On Reliability, R-35, 216-220.
  • [2] Apanasovich, T.V., Carroll, R. J., Maity, A. (2009). SIMEX and standard error estimation in semiparametric measurement error models., Electronic Journal of Statistics, 3, 318-348.
  • [3] Cameron, A.C. and Trivedi, P.K. (1998). Regression analysis of count data, Cambridge:Cambridge University, Press.
  • [4] Carroll, R. J., Ruppert, D., Crainiceanu, C. and Stefanski, L. A. (2006)., Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition. London: CRC Press.
  • [5] Cook, J. R. and Stefanski, L. A. (1994). Simulation-extrapolation estimation in parametric measurement error models., Journal of the American Statistical Association, 89, 1314-1328.
  • [6] Eckert, R.S., Carroll, R.J. and Wang, N. (1997). Transformations to additivity in measurement error models., Biometrics. 53, 262-272.
  • [7] Kerber, R. L., Till, J. E., Simon, S. L., Lyon, J. L. Thomas, D. C., Preston-Martin, S., Rollison, M. L., Lloyd, R. D. and Stevens, W. (1993). A cohort study of thyroid disease in relation to fallout from nuclear weapons testing., Journal of the American Medical Association, 270, 2076-2083.
  • [8] Li, Y., Guolo, A., Owen Hoffman, F., and Carroll, R. J. (2007). Shared Uncertainty in Measurement Error Problems, with Application to Nevada Test Site Fallout Data., Biometrics, 63, 1226-36.
  • [9] Liang, H., Haerdle, W. and Carroll, R.J. (1999). Estimation in a partially linear error-in-variables model., Annals of Statistics, 27, 1519-1535.
  • [10] Liang, H. and Ren, H.B. (2005). Generalized partially linear measurement error models., Journal of Computational and Graphical Statistics, 14, 237-250.
  • [11] Lin, X., Wang, N., Welsh, A. and Carroll, R. J. (2004). Equivalent kernels of smoothing splines in nonparametric regression for clustered data., Biometrika, 91, 177–193.
  • [12] Lin, X. and Carroll, R.J. (2006). Semiparametric estimation in general repeated measures problems., Journal of the Royal Statistical Society, Series B, 68, 69-88.
  • [13] Lubin, J. H., Schafer, D. W. Ron, E., Stovall, M. and Carroll, R. J. (2004). A reanalysis of thyroid neoplasms in the Israeli tinea capitis study accounting for dose uncertainties., Radiation Research, 161, 359-368.
  • [14] Lyon, J. L., Alder, S. C., Stone, M. B., Scholl, A., Reading, J. C. Holubkov, R., Sheng, X. White, G. L., Hegmann, K. T., Anspaugh, L., Hoffman, F. O., Simon, S. L., Thomas, B., Carroll, R. J. & Meikle, A. W. (2006). Thyroid disease associated with exposure to the Nevada Test Site radiation: a reevaluation based on corrected dosimetry and examination data., Epidemiology, 17, 604-614.
  • [15] Ma, Y. and Carroll, R.J. (2006). Locally efficient estimators for semiparametric models with measurement error., Journal of the American Statistical Association, 101, 1465-1474.
  • [16] Mallick, B., Hoffman, F. O. and Carroll, R. J. (2002). Semiparametric regression modeling with mixtures of Berkson and classical error, with application to fallout from the Nevada Test Site., Biometrics, 58, 13-20.
  • [17] Nakamura, T. (1990). Corrected score functions for error-in-variable models: methodology and application to generalized linear models, Biometrika, 77, 127-137.
  • [18] Novick, J.S. and Stefanski, L.A. (2002). Corrected score estimation via complex variable simulation extrapolation., Journal of the American Statistical Association, 97, 472-481.
  • [19] Pierce, D. A. and Kellerer, A. (2004). Adjusting for covariate errors with nonparametric assessment of the true covariate distribution., Biometrika, 91, 863-876.
  • [20] Reeves, G. K., Cox, D. R., Darby, S. C. and Whitley, E. (1998). Some aspects of measurement error in explanatory variables for continuous and binary regression models., Statistics in Medicine, 17, 2157-2177.
  • [21] Singpurwalla, N. D. (1971). A problem in accelerated Life testing., Journal of the American Statistical Association, 66, 841-845.
  • [22] Schafer, D. W. and Gilbert, E. S. (2006). Some statistical implications of does uncertainty in radiation dose-response analyses., Radiation Research, 166, 303-312.
  • [23] Schafer, D. W., Lubin, J. H., Ron, E., Stovall, M. and Carroll, R. J. (2001). Thyroid cancer following scalp irradiation: a reanalysis accounting for uncertainty in dosimetry., Biometrics, 57, 689-697.
  • [24] Simon, S. L., Till, J. E., Lloyd, R. D., Kerber, R. L., Thomas, D. C., Preston–Martin, S., Lyon, J. L. and Stevens, W. (1995). The Utah Leukemia case–control study: dosimetry methodology and results., Health Physics, 68, 460–471.
  • [25] Simon, S. L., Anspaugh, L. R., Hoffman, F. O., et al. (2006). 2004 update of dosimetry for the Utah Thyroid Cohort Study., Radiation Research, 165, 208-222.
  • [26] Stefanski, L.A. and Cook, J.R. (1995). Simulation-Extrapolation: the measurement error jackknife., Journal of the American Statistical Association, 90, 1247-1256.
  • [27] Stevens, W., Till, J. E., Thomas, D. C., et al. (1992). Assessment of leukemia and thyroid disease in relation to fallout in Utah: report of a cohort study of thyroid disease and radioactive fallout from the Nevada test site. University of, Utah.
  • [28] Stram, D. O. and Kopecky, K. J. (2003). Power and uncertainty analysis of epidemiological studies of radiation-related disease risk in which dose estimates are based on a complex dosimetry system: some observations., Radiation Research, 160, 408-417.
  • [29] Tsiatis, A. A. and Ma, Y. (2004). Locally efficient semiparametric estimators for functional measurement error models., Biometrika, 91, 835-848.
  • [30] Zhu, L. and Cui, H. (2003). A semiparametric regression model with errors in variables., Scan. J. Statist. 30, 429-442.