Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Bayesian modeling of random effects precision/covariance matrix in cumulative logit random effects models

  • Kim, Jiyeong (Department of Statistics, Sungkyunkwan University) ;
  • Sohn, Insuk (Center for Biostatistics and Clinical Epidemiology, Samsung Medical Center) ;
  • Lee, Keunbaik (Department of Statistics, Sungkyunkwan University)
  • Received : 2016.10.18
  • Accepted : 2016.12.05
  • Published : 2017.01.31
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Abstract

Cumulative logit random effects models are typically used to analyze longitudinal ordinal data. The random effects covariance matrix is used in the models to demonstrate both subject-specific and time variations. The covariance matrix may also be homogeneous; however, the structure of the covariance matrix is assumed to be homoscedastic and restricted because the matrix is high-dimensional and should be positive definite. To satisfy these restrictions two Cholesky decomposition methods were proposed in linear (mixed) models for the random effects precision matrix and the random effects covariance matrix, respectively: modified Cholesky and moving average Cholesky decompositions. In this paper, we use these two methods to model the random effects precision matrix and the random effects covariance matrix in cumulative logit random effects models for longitudinal ordinal data. The methods are illustrated by a lung cancer data set.

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References

  1. Breslow NE and Clayton DG (1993). Approximate inference in generalized linear mixed models, Journal of the American Statistical Association, 88, 9-25.
  2. Chen MH and Shao QM (2001). Propriety of posterior distribution for dichotomous quantal response models, Proceedings of the American Mathematical Society, 129, 293-302.
  3. Daniels JM and PourahmadiM(2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data, Biometrika, 89, 553-566. https://doi.org/10.1093/biomet/89.3.553
  4. Daniels JM and Zhao YD (2003). Modelling the random effects covariance matrix in longitudinal data, Statistics in Medicine, 22, 1631-1647. https://doi.org/10.1002/sim.1470
  5. Daniels MJ and Hogan JW (2008). Missing Data in Longitudinal Studies:Strategies for Bayesian Modeling and Sensitivity Analysis, Chapman& Hall/CRC, Boca Raton, FL.
  6. Diggle PJ, Heagerty P, Liang KY, and Zeger SL (2002). Analysis of Longitudinal Data(2nd ed), Oxford University Press, New York.
  7. Galecki AT (1994) General class of covariance structures for two or more repeated factors in longitudinal data analysis, Communications in Statistics-Theory and Method, 23, 3105-3119. https://doi.org/10.1080/03610929408831436
  8. Gelfand AE and Ghosh SK (1998). Model choice: a minimum posterior predictive loss approach, Biometrika, 85, 1-11. https://doi.org/10.1093/biomet/85.1.1
  9. Gelman A and Rubin D (1992). Inference from iterative simulation using multiple sequences, Statistical Science, 7, 457-511. https://doi.org/10.1214/ss/1177011136
  10. Gibbons RD and Hedeker D (1997). Random effects probit and logistic regression models for three-level data, Biometrics, 53, 1527-1537. https://doi.org/10.2307/2533520
  11. Heagerty PJ and Kurland BF (2001). Misspecified maximum likelihood estimates and generalised linear mixed models, Biometrika, 88, 973-985. https://doi.org/10.1093/biomet/88.4.973
  12. Ibrahim JG and Laud PW (1991). On Bayesian analysis of generalized linear models using Jeffreys's prior, Journal of the American Statistical Association, 86, 981-986. https://doi.org/10.1080/01621459.1991.10475141
  13. Kim J and Lee K (2015). Survey of models for random effects covariance matrix in generalized linear mixed model, Korean Journal of Applied Statistics, 28, 211-219. https://doi.org/10.5351/KJAS.2015.28.2.211
  14. Kim ST, Uhm JE, Lee J, Sun J, Sohn I, Kim SW, Jung S, Park YH, Ahn JS, Park K, and Ahn MJ (2012). Randomized phase II study of gefitinib versus erlotinib in patients with advanced non-small cell lung cancer who failed previous chemotherapy, Lung Cancer, 75, 82-88. https://doi.org/10.1016/j.lungcan.2011.05.022
  15. Lee K (2013). Bayesian modeling of random effects covariance matrix for generalized linear mixed models, Communication for Statistical Applications and Methods, 20, 235-240. https://doi.org/10.5351/CSAM.2013.20.3.235
  16. Lee K and Daniels MJ (2008). Marginalized models for longitudinal ordinal data with application to quality of life studies, Statistics in Medicine, 27, 4359-4380. https://doi.org/10.1002/sim.3352
  17. Lee K, Daniels MJ, and Joo Y (2013). Flexible marginalized models for bivariate longitudinal ordinal data, Biostatistics, 14, 462-476. https://doi.org/10.1093/biostatistics/kxs058
  18. Lee K, Kang S, Liu X, and Seo D (2011). Likelihood-based approach for analysis of longitudinal nominal data using marginalized random effects models, Journal of Applied Statistics, 38, 1577-1590. https://doi.org/10.1080/02664763.2010.515675
  19. Lee K and Yoo JK (2014). Bayesian Cholesky factor models in random effects covariance matrix for generalized linear mixed models, Computational Statistics & Data Analysis, 80, 111-116. https://doi.org/10.1016/j.csda.2014.06.016
  20. Lee K, Lee J, Hagan J, and Yoo JK (2012). Modeling the random effects covariance matrix for the generalized linear mixed models, Computational Statistics & Data Analysis, 56, 1545-1551. https://doi.org/10.1016/j.csda.2011.09.011
  21. Liu I and Agresti A (2005). The analysis of ordered categorical data: an overview and a survey of recent developments, Test, 14, 1-73. https://doi.org/10.1007/BF02595397
  22. Litiere S, Alonso A, and Molenberghs G (2007). Type I and type II error under random-effects mis-specifiecation in generalized linear mixed models, Biometrics, 63, 1038-1944. https://doi.org/10.1111/j.1541-0420.2007.00782.x
  23. Litiere S, Alonso A, and Molenberghs G (2008). The impact of a misspecified random-effects distribution on the estimation and the performance of inferential procedures in generalized linear mixed models, Statistics in Medicine, 27, 3125-3144.
  24. McCullagh P (1980). Regression models for ordinal data, Journal of the Royal Statistical Society Series B (Methodological), 42, 109-142. https://doi.org/10.1111/j.2517-6161.1980.tb01109.x
  25. Nooraee N, Molenberghs G, and van den Heuvel ER (2014). GEE for longitudinal ordinal data: comparing R-geepack, R-multgee, R-repolr, SAS-GENMOD, SPSS-GENLIN, Computational Statistics & Data Analysis, 77, 70-83. https://doi.org/10.1016/j.csda.2014.03.009
  26. Pan J and MacKenzie G (2003). On modelling mean-covariance structures in longitudinal studies, Biometrika, 90, 239-244. https://doi.org/10.1093/biomet/90.1.239
  27. Pan J and MacKenzie G (2006). Regression models for covariance structures in longitudinal studies, Statistical Modelling, 6, 43-57. https://doi.org/10.1191/1471082X06st105oa
  28. Pourahmadi M (1999). Joint mean-covariance models with applications to longitudinal data: uncon- strained parameterisation, Biometrika, 86, 677-690. https://doi.org/10.1093/biomet/86.3.677
  29. Pourahmadi M (2000). Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix, Biometrika, 87, 425-435 https://doi.org/10.1093/biomet/87.2.425
  30. Pourahmadi M and Daniels MJ (2002). Dynamic conditionally linear mixed models for longitudinal data, Biometrics, 58, 225-231. https://doi.org/10.1111/j.0006-341X.2002.00225.x
  31. Rothman AJ, Levina E, and Zhu J (2010). A new approach to Cholesky-based covariance regularization in high dimensions, Biometrika, 97, 539-550. https://doi.org/10.1093/biomet/asq022
  32. Spiegelhalter D, Best N, Carlin B, and van der Linde A (2002). Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society Series B (Statistical Methodology), 64, 583-639. https://doi.org/10.1111/1467-9868.00353
  33. Spiegelhalter D, Thomas A, Best N, and Lunn D (2003). WinBUGS Version 1.4 User Manual, Medical Research Council Biostatistics Unit, Cambridge, UK.
  34. Sturtz S, Ligges U, and Gelman A (2005). R2WinBUGS: a package for running WinBUGS from R Journal of Statistical Software, 12, 1-16.
  35. Zhang W and Leng C (2012). A moving average Cholesky factor model in covariance modelling for longitudinal data, Biometrika, 99, 141-150. https://doi.org/10.1093/biomet/asr068