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Estimation strategy of multilevel model for ordinal longitudinal data. (English) Zbl 1436.62200

Summary: This paper considers the shrinkage estimation of multilevel models that are appropriate for ordinal longitudinal data. These models can accommodate multiple random effects and, additionally, allow for a general form of model covariates that are related to the overall level of the responses and changes to the response over time. The likelihood inference for multilevel models is computationally burdensome due to intractable integrals. A maximum marginal likelihood (MML) method with Fisher’s scoring procedure is therefore followed to estimate the random and fixed effects parameters. In real life data, researchers may have collected many covariates for the response. Some of these covariates may satisfy certain constraints which can be used to produce a restricted estimate from the unrestricted likelihood function. The unrestricted and restricted MMLs can then be combined optimally to form the pretest and shrinkage estimators. Asymptotic properties of these estimators including biases and risks will be discussed. A simulation study is conducted to assess the performance of the estimators with respect to the unrestricted MML estimator. Finally, the relevance of the proposed estimators will be illustrated with a real data set.

MSC:

62H12 Estimation in multivariate analysis
62H11 Directional data; spatial statistics
62G07 Density estimation
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[1] Agresti, A. (2010). Analysis of Ordinal Categorical Data. Hoboken, New Jersey: Wiley. · Zbl 1263.62007 · doi:10.1002/9780470594001
[2] Ahmed, S. E., Doksum, K., Hossain, S., & You, J. (2007). Shrinkage, pretest and LASSO estimators in partially linear models. Australian and New Zealand Journal of Statistics, 49(4), 461-471. · Zbl 1158.62029 · doi:10.1111/j.1467-842X.2007.00493.x
[3] Albert, P. S., Hunsberger, S. A., & Biro, F. M. (1997). Modeling repeated measures with monotonic ordinal responses and misclassification, with applications to studying maturation. Journal of the American Statistical Association, 92, 1304-1211. · Zbl 0913.62101 · doi:10.1080/01621459.1997.10473651
[4] Bellamy, N., Buchanan, W. W., Goldsmith, C. H., Campbell, J., & Stitt, L. (1988). Validation study of womac: A health status instrument for measuring clinically-important patient-relevant outcomes following total hip or knee arthroplasty in osteoarthritis. Journal of Orthopedics and Rheumatology, 1, 95-108.
[5] Goldstein, H. (2010). Multilevel Statistical Models. Chichester: Wiley. · doi:10.1002/9780470973394
[6] Hedeker, D., & Gibbons, R. (1994). A random-effects ordinal regression model for multilevel analysis. Biometrics, 50(4), 933-944. · Zbl 0826.62049 · doi:10.2307/2533433
[7] Hedeker, D., & Gibbons, R. D. (2006). Longitudinal data analysis. New York: Wiley. · Zbl 1136.62075
[8] Hossain, S., Ahmed, S. E., & Doksum, K. A. (2015). Shrinkage, pretest, and penalty estimators in generalized linear models. Statistical Methodology, 24, 52-68. · Zbl 1486.62215 · doi:10.1016/j.stamet.2014.11.003
[9] Hossain, S., Ahmed, S. E., Yi, Y., & Chen, B. (2016). Shrinkage and pretest estimators for longitudinal data analysis under partially linear models. Jounal of Nonparametric Statistics, 28(3), 531-549. · Zbl 1407.62136 · doi:10.1080/10485252.2016.1190358
[10] Hox, J., Moerbeek, M., & van de Schoot, R. (2018). Multilevel Analysis: Techniques and Applications, Third Edition (Quantitative Methodology Series). New York: Routledge.
[11] Hox, J. J., Moerbeek, M., & Schoot, R. (2017). Multilevel analysis: Techniques and applications. New York: Routledge. · doi:10.4324/9781315650982
[12] Lee, K., & Daniels, M. J. (2008). Marginalized models for longitudinal ordinal data with application to quality of life studies. Statistics in Medicine, 27(21), 4359-4380. · doi:10.1002/sim.3352
[13] Lee, Y., & Nelder, J. A. (2004). Conditional and marginal models: Another view. Statistical Science, 19(2), 219-238. · Zbl 1100.62591 · doi:10.1214/088342304000000305
[14] Lian, H. (2012). Shrinkage estimation for identification of linear components in additive models. Statistics and Probability Letters, 82, 225-231. · Zbl 1237.62048 · doi:10.1016/j.spl.2011.10.009
[15] Liu, Q., & Pierce, D. A. (1994). A note on gauss-hermite quadrature. Biometrika, 81(3), 624-629. · Zbl 0813.65053
[16] Magnus, J. R. (1988). Linear Structures. Oxford, London: Charles Griffin. · Zbl 0667.15010
[17] Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical Linear Models: Applications and data analysis methods. Thousand Oaks: Sage Publications Ltd. · Zbl 1001.62004
[18] Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. London: Chapman and Hall/CRC. · Zbl 1097.62001 · doi:10.1201/9780203489437
[19] Snijders, T. A. B., & Bosker, R. J. (2012). Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling. London: SAGE Publications Ltd. · Zbl 1296.62008
[20] Stroud, A. H., & Sechrest, D. (1966). Gaussian quadrature formulas. Upper Saddle River: Prentice-Hall. · Zbl 0156.17002
[21] Thomson, T., & Hossain, S. (2018). Efficient shrinkage for generalized linear mixed models under linear restrictions. Sankhya A: The Indian Journal of Statistics, 80, 1-26. · doi:10.1007/s13171-017-0122-6
[22] Thomson, T., Hossain, S., & Ghahramani, M. (2016). Efficient estimation for time series following generalized linear models. Australian & New Zealand Journal of Statistics, 58, 493-513. · Zbl 1373.62457 · doi:10.1111/anzs.12169
[23] van der Vaart, A. W. (1998). Asymptotic Statistics. New York: Cambridge University Press. · Zbl 0943.62002 · doi:10.1017/CBO9780511802256
[24] Wu, C. O., & Chiang, C. T. (2000). Kernel smoothing on varying coefficient models with longitudinal dependent variable. Statistica Sinica, 10, 433-456. · Zbl 0945.62047
[25] Zeng, T., & Hill, R. C. (2016). Shrinkage estimation in the random parameters logit model. Open Journal of Statistics, 6, 667-674. · doi:10.4236/ojs.2016.64056
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