The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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Survey of Models for Random Effects Covariance Matrix in Generalized Linear Mixed Model

일반화 선형혼합모형의 임의효과 공분산행렬을 위한 모형들의 조사 및 고찰

  • Kim, Jiyeong (Department of Statistics, Sungkyunkwan University) ;
  • Lee, Keunbaik (Department of Statistics, Sungkyunkwan University)
  • Received : 2015.03.13
  • Accepted : 2015.03.30
  • Published : 2015.04.30
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Abstract

Generalized linear mixed models are used to analyze longitudinal categorical data. Random effects specify the serial dependence of repeated outcomes in these models; however, the estimation of a random effects covariance matrix is challenging because of many parameters in the matrix and the estimated covariance matrix should satisfy positive definiteness. Several approaches to model the random effects covariance matrix are proposed to overcome these restrictions: modified Cholesky decomposition, moving average Cholesky decomposition, and partial autocorrelation approaches. We review several approaches and present potential future work.

일반화 선형혼합모델은 일반적으로 경시적 범주형 자료를 분석하는데 사용된다. 이 모델에서 임의효과는 반복 측정치들의 시간에 따른 의존성을 설명한다. 임의효과 공분산행렬의 추정은 여러가지 제약조건들 때문에 쉽지 않은 문제이다. 제약조건으로는 행렬의 모수들의 수가 많으며, 또한 추정된 공분산행렬은 양정치성을 만족하여야 한다. 이러한 제한을 극복하기 위해, 임의효과 공분산행렬의 모형화를 위한 여러가지 방법이 제안되었다: 수정 단냠레스키분해, 이동평균 단냠레스키분해와 부분 자기상관행렬을 이용한 방법이 있다. 이 논문에서 위의 제안된 방법들을 소개한다.

Keywords

References

  1. Agresti, A. (2013). Categorical Data Analysis, 3rd Edition, Wiley.
  2. Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models, Journal of the American Statistical Association, 88, 125-134.
  3. Daniels, J. M. and Zhao, Y. D. (2003). Modeling the random effects covariance matrix in longitudinal data, Statistics in Medicine, 22, 1631-1647. https://doi.org/10.1002/sim.1470
  4. Daniels, M. J. and Pourahmadi, M. (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
  5. Daniels, M. J. and Pourahmadi, M. (2009). Modeling covariance matrices via partial autocorrelations, Journal of Multivariate Analysis, 100, 2352-2363. https://doi.org/10.1016/j.jmva.2009.04.015
  6. Diggle, P. J., Heagerty, P., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data, 2nd Edition, Analysis of Longitudinal Data.
  7. Heagerty, P. J. (1999). Marginally specified logistic-normal models for longitudinal binary data, Biometrics, 55, 688-698. https://doi.org/10.1111/j.0006-341X.1999.00688.x
  8. Heagerty, P. J. (2002). Marginalized transition models and likelihood inference for longitudinal categorical data, Biometrics, 58, 342-351. https://doi.org/10.1111/j.0006-341X.2002.00342.x
  9. Heagerty, P. J. and Kurland, B. F. (2001). Misspecified maximum likelihood estimates and generalized linear mixed models, Biometrika, 88, 973-985. https://doi.org/10.1093/biomet/88.4.973
  10. 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
  11. Lee, K., Daniels, M. and Joo, Y. (2013). Flexible marginalized models for bivariate longitudinal ordinal data, Biostatistics, 14, 462-476. https://doi.org/10.1093/biostatistics/kxs058
  12. Lee, K. and Yoo, J. K. (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
  13. Lee, K., Yoo, J. K., 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. https://doi.org/10.1016/j.csda.2011.09.011
  14. Liang, K. Y. and Zeger, S. L. (1986). Longitudinal analysis using generalized linear models, Biometrika, 73, 13-22. https://doi.org/10.1093/biomet/73.1.13
  15. Pan, J. and Mackenzie, G. (2003). On modeling mean-covariance structure in longitudinal studies, Biometrika, 90, 239-244. https://doi.org/10.1093/biomet/90.1.239
  16. Pan, J. and Mackenzie, G. (2006). Regression models for covariance structures in longitudinal studies, Statistical Modeling, 6, 43-57. https://doi.org/10.1191/1471082X06st105oa
  17. Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation, Biometrika, 86, 677-690. https://doi.org/10.1093/biomet/86.3.677
  18. Pourahmadi, M. (2000). Maximum likelihood estimation of generalized linear models for multivariate normal covariance matrix, Biometrika, 87, 425-435. https://doi.org/10.1093/biomet/87.2.425
  19. Pourahmadi, M. (2007). Cholesky decompositions and estimation of a covariance matrix: Orthogonality of variance-correlation parameters, Biometrika, 94, 1006-1013. https://doi.org/10.1093/biomet/asm073
  20. Pourahmadi, M. and Daniels, M. J. (2002). Dynamic conditionally linear mixed models for longitudinal data, Biometrics, 58, 225-231. https://doi.org/10.1111/j.0006-341X.2002.00225.x
  21. Zhang, W. and Leng, C. (2012). A moving average Cholesky factor model in covariance modeling for longitudinal data, Biometrika, 99, 141-150. https://doi.org/10.1093/biomet/asr068

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