Bayesian baseline-category logit random effects models for longitudinal nominal data

  • Kim, Jiyeong (Laboratory of Low Dose Risk Assessment, National Radiation Emergency Medical Center, Korea Institute of Radiological and Medical Sciences) ;
  • Lee, Keunbaik (Department of Statistics, Sungkyunkwan University)
  • Received : 2019.09.19
  • Accepted : 2019.12.24
  • Published : 2020.03.31


Baseline-category logit random effects models have been used to analyze longitudinal nominal data. The models account for subject-specific variations using random effects. However, the random effects covariance matrix in the models needs to explain subject-specific variations as well as serial correlations for nominal outcomes. In order to satisfy them, the covariance matrix must be heterogeneous and high-dimensional. However, it is difficult to estimate the random effects covariance matrix due to its high dimensionality and positive-definiteness. In this paper, we exploit the modified Cholesky decomposition to estimate the high-dimensional heterogeneous random effects covariance matrix. Bayesian methodology is proposed to estimate parameters of interest. The proposed methods are illustrated with real data from the McKinney Homeless Research Project.


Supported by : National Research Foundation of Korea (KRF)


  1. Breslow NE and Clayton DG (1993). Approximate inference in generalized linear mixed models, Journal of the American Statistical Association, 88, 125-134.
  2. Chen B, Yi GY, and Cook RJ (2009). Likelihood analysis of joint marginal and conditional models for longitudinal categorical data, Canadian Journal of Statistics, 37, 182-205.
  3. Daniels MJ and Gatsonis C (1997). Hierarchical polytomous regression models with applications to health services research, Statistics in Medicine, 16, 2311-2325.<2311::AID-SIM654>3.0.CO;2-E
  4. Daniels JM and Zhao YD (2003). Modeling the random effects covariance matrix in longitudinal data, Statistics in Medicine, 22, 1631-1647.
  5. Hartzel J, Agresti A, and Caffo B (2001). Multinomial logit random effects models, Statistical Modelling, 1, 81-102.
  6. Hedeker D (2003). A mixed-effects multinomial logistic regression model, Statistics in Medicine, 22, 1433-1446.
  7. Hedeker D and Gibbons RD (2006). Longitudinal Data Analysis, Wiley, Hoboken, New Jersey.
  8. Hurlburt MS, Wood PA, and Hough RL (1996). Providing independent housing for the homeless mentally ill: A novel approach to evaluating long-term longitudinal housing patterns, Journal of Community Psychology, 24, 291-310.
  9. Lee K (2013). Bayesian modeling of random effects covariance matrix for generalized linear mixed models, Communication for Statistical Applications and Methods, 20, 235-240.
  10. Lee K, Cho H, Kwak MS,and Jang EJ (2020). Estimation of covariance matrix of multivariate longitudinal data using modified Choleksky and hypersphere decompositions, Biometrics, 76, 75-86.
  11. 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.
  12. Lee K and Mercante D (2010). Longitudinal nominal data analysis using marginalized models, Computational Statistics & Data Analysis, 54, 208-218.
  13. Lee K, Yoo JK, Lee J, and Hagan J (2012). Modeling the random effects covariance matrix for the generalized linear mixed models, Computational Statistics & Data Analysis, 56, 1545-1551.
  14. Pan J and Mackenzie G (2006). Regression models for covariance structures in longitudinal studies, Statistical Modelling, 6, 43-57.
  15. Pourahmadi M (1999). Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation, Biometrika, 86, 677-690.
  16. Pourahmadi M (2000). Maximum likelihood estimation of generalized linear models for multivariate normal covariance matrix, Biometrika, 87, 425-435.
  17. Revelt D and Train K (1998). Mixed logit with repeated choices: households' choices of appliance efficiency level, Review of Economics and Statistics, 80, 647-657.
  18. Theil H (1969). A multinomial extension of the linear logit model, International Economic Review, 10, 251-259.
  19. Theil H (1970). On the estimation of relationships involving qualitative variables, American Journal of Sociology, 76, 103-154.