Document Type : Original Paper
Authors
1 1 Department of Statistics, Tarbiat Modares University, Tehran, Islamic Republic of Iran
2 2 School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Islamic Republic of Iran
Abstract
Ignoring measurement errors in generalized linear mixed models (GLMMs) as well as other regression models will inevitably lead to significant biases, deviations, and incorrect inferences in the estimation of the parameters. In the presence of measurement error, numerous approaches have been proposed to rectify this issue. Furthermore, the application of the frequentist approach of GLMMs is intricate due to the emergence of an intractable numerical integration process. In this study, the complete likelihood inference of a multinomial logit random effects model with covariate measurement error in conjunction with replicate measures is suggested via the multivariate Gauss-Hermite quadrature approximation. To achieve this objective, the likelihood function will be evaluated and approximated for two distinct scenarios; the proposed method which incorporates the classical additive structural measurement error model for the error-prone covariate and the naive method. We will describe and compare different upshots of parameter estimation in these two different situations. The results of performing the proposed method, assessed through simulation, show that the proposed method performs well when correcting for measurement error in terms of bias, empirical standard error, root of mean squared error and coverage ratio. The application of the proposed method is further highlighted with real-world data based on a multilevel study concerning the prevalence of contraceptive methods used by women in Bangladesh.
Keywords
- Mixed Models
- Nominal Response
- Multivariate Gauss-Hermite Quadrature
- Multinomial Logit Model
- Measurement Error
Main Subjects
- Goldstein H. Multilevel Statistical Models. 4th ed. London: John Wiley & Sons, 2011.
- Searle SR, Casella G and McCullach CE. Variance Components. New Jersey: John Wiley & Sons, 1992.
- McCullach CE, Searle SR and Neuhaus JM. Generalized, Linear, and Mixed Models. 2nd ed. London: John Wiley & Sons, 2008.
- McCulloch CE. Maximum likelihood variance components estimation for binary data. Journal of the American Statistical Association. 1994 Mar 1;89(425):330-335.
- Tanner MA. Tools for Statistical inference: Observed Data and Data Augmentation. 2nd ed. New York: Springer Science and Business Media, 1993.
- Diggle PJ, Liang KY and Zeger SL. Analysis of Longitudinal Data. Oxford: Oxford University Press, 1994.
- Skrondal, A., and Rabe-Hesketh, S. Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models. Boca Raton: CRC Press, 2004.
- Wu L. Mixed Effects Models for Complex Data. Boca Raton: CRC Press, 2009.
- Stroup WW. Generalized Linear Mixed Models: Modern Concepts, Methods and Applications. Boca Raton: CRC Press, 2012.
- Carroll RJ, Ruppert D, Stefanski LA and Crainiceanu CM. Measurement Error in Nonlinear Models: A Modern Perspective. Boca Raton: CRC Press, 2006.
- Wang N, Lin X, Guttierrez RG. A bias correction regression calibration approach in generalized linear mixed measurement error models. Communications in Statistics-Theory and Methods. 1999 Jan 1;28(1):217-32.
- Wang N, Lin X, Gutierrez RG, Carroll RJ. Bias analysis and SIMEX approach in generalized linear mixed measurement error models. Journal of the American Statistical Association. 1998 Mar 1;93(441):249-61.
- Buonaccorsi JP, Romeo G, Thoresen M. Model-based bootstrapping when correcting for measurement error with application to logistic regression. Biometrics. 2018 Mar;74(1):135-44.
- Lele SR, Dennis B, Lutscher F. Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods. Ecology letters. 2007 Jul;10(7):551-63.
- Lele SR, Nadeem K, Schmuland B. Estimability and likelihood inference for generalized linear mixed models using data cloning. Journal of the American Statistical Association. 2010 Dec 1;105(492):1617-25.
- Torabi M. Likelihood inference in generalized linear mixed measurement error models. Computational statistics & data analysis. 2013 Jan 1;57(1):549-57.
- Zhang H, Wu L. An approximate method for generalized linear and nonlinear mixed effects models with a mechanistic nonlinear covariate measurement error model. Metrika. 2019 May 1;82(4):471-99.
- Xie X, Xue X, Strickler HD. Generalized linear mixed model for binary outcomes when covariates are subject to measurement errors and detection limits. Statistics in medicine. 2018 Jan 15;37(1):119-36.
- Hoque ME, Torabi M. Modeling the random effects covariance matrix for longitudinal data with covariates measurement error. Statistics in Medicine. 2018 Dec 10;37(28):4167-84.
- Lipovetsky, S. Handbook of Measurement Error Models. Boca Raton: Chapman & Hall, 2023.
- Hartzel J, Agresti A, Caffo B. Multinomial logit random effects models. Statistical Modelling. 2001 Jul;1(2):81-102.
- Jaeckel P. A Note on Multivariate Gauss-Hermite Quadrature. London: ABN Amro, 2005. Retrieved from:
23.http://www.pjaeckel.webspace.virginmedia.com/ANoteOnMultivariateGaussHermiteQuadrature.pdf
- 23. Agresti A. Categorical Data Analysis. New York: John Wiley & Sons, 2002.
- Skrondal A and Rabe-Hesketh S. Generalized Latent Variable Modeling. Boca Raton: CRC Press, 2004.
- Molenberghs G and Verbeke G. Models for Discrete Longitudinal Data. New York: Springer Science & Business Media, 2006.
- Pan JX, Thompson R. Generalized linear mixed models: An improved estimating procedure. InCOMPSTAT: Proceedings in Computational Statistics 14th Symposium held in Utrecht, The Netherlands, 2000 2000 (pp. 373-378). Physica-Verlag HD.
- Nelder JA and Mead RA. A simplex algorithm for function minimization. Comput J. 1965;7:308-13.
- Steele F, Diamond I. Contraceptive switching in Bangladesh. Studies in Family Planning. 1999 Dec;30(4):315-28.
- Huq NM and Cleland J. Bangladesh fertility survey, National Institute of Population Research and Training (NIPORT), 1990.
- Rasbash J, Steele F, Browne WJ, Goldstein H. A user’s guide to MLwiN, v2. 10. Centre for multilevel modelling, University of Bristol. 2009;192.
- Hossain MB. Analysing the relationship between family planning workers' contact and contraceptive switching in rural Bangladesh using multilevel modelling. Journal of biosocial science. 2005 Sep 1;37(5):529.
|
|
- DeGraff DS. Increasing contraceptive use in Bangladesh: The role of demand and supply factors. Demography. 1991 Feb;28:65-81.
- Yi G and Cook RJ. Errors in the Measurement of Covariates. The Encyclopedia of Biostatistics. 2005; 3:1741-1748.
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