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Unconstrained models for the covariance structure of multivariate longitudinal data

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  • Kim, Chulmin
  • Zimmerman, Dale L.

Abstract

The constraint that a covariance matrix must be positive definite presents difficulties for modeling its structure. Pourahmadi (1999, 2000) [18,19] proposed a parameterization of the covariance matrix for univariate longitudinal data in which the parameters are unconstrained, which is based on the modified Cholesky decomposition of the covariance matrix. We extend this approach to multivariate longitudinal data by developing a modified Cholesky block decomposition that provides an alternative unconstrained parameterization for the covariance matrix, and we propose parsimonious models within this parameterization. A Fisher scoring algorithm is developed for obtaining maximum likelihood estimates of parameters, assuming that the observations are normally distributed. The asymptotic distribution of the maximum likelihood estimators is derived. The performance of the estimators for finite samples is investigated by simulation and compared with that of estimators obtained under a separable (Kronecker product) covariance model. Estimation and model selection are illustrated using bivariate longitudinal data from a study of poplar growth.

Suggested Citation

  • Kim, Chulmin & Zimmerman, Dale L., 2012. "Unconstrained models for the covariance structure of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 104-118.
    • Handle: RePEc:eee:jmvana:v:107:y:2012:i:c:p:104-118
    DOI: 10.1016/j.jmva.2012.01.004
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    References listed on IDEAS

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    1. Lu, Nelson & Zimmerman, Dale L., 2005. "The likelihood ratio test for a separable covariance matrix," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 449-457, July.
    2. Mathew, Thomas, 1989. "MANOVA in the multivariate components of variance model," Journal of Multivariate Analysis, Elsevier, vol. 29(1), pages 30-38, April.
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    4. Dayanand Naik & Shantha Rao, 2001. "Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix," Journal of Applied Statistics, Taylor & Francis Journals, vol. 28(1), pages 91-105.
    5. Zhao Wei & Hou Wei & Littell Ramon C. & Wu Rongling, 2005. "Structured Antedependence Models for Functional Mapping of Multiple Longitudinal Traits," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-28, November.
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    1. Kohli, Priya & Garcia, Tanya P. & Pourahmadi, Mohsen, 2016. "Modeling the Cholesky factors of covariance matrices of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 87-100.
    2. Rhee, Anbin & Kwak, Min-Sun & Lee, Keunbaik, 2022. "Robust modeling of multivariate longitudinal data using modified Cholesky and hypersphere decompositions," Computational Statistics & Data Analysis, Elsevier, vol. 170(C).
    3. Yicong Lin & Hanno Reuvers, 2019. "Efficient Estimation by Fully Modified GLS with an Application to the Environmental Kuznets Curve," Papers 1908.02552, arXiv.org, revised Aug 2020.
    4. Lee, Keunbaik & Lee, Chang-Hoon & Kwak, Min-Sun & Jang, Eun Jin, 2021. "Analysis of multivariate longitudinal data using ARMA Cholesky and hypersphere decompositions," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    5. Feng, Sanying & Lian, Heng & Xue, Liugen, 2016. "A new nested Cholesky decomposition and estimation for the covariance matrix of bivariate longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 98-109.

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