Bayesian analysis of covariance matrices and dynamic models for longitudinal data

• Published in: Biometrika Vol. 89; no. 3; pp. 553 - 566
• Main Authors: Daniels, Michael J., Pourahmadi, Mohsen
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.08.2002; Biometrika Trust; Oxford University Press for Biometrika Trust; Oxford Publishing Limited (England)
• Series: Biometrika
• Subjects: Analysis of covariance; Antedependence and autoregressive models; Autoregressive models; Bayes estimate; Bayesian analysis; Covariance; Covariance matrices; Decomposition; Dynamic modeling; Estimation methods; Estimators; Exact sciences and technology; Hierarchical model; Inference from stochastic processes; time series analysis; Linear inference, regression; Longitudinal data; Markov chain Monte Carlo; Mathematical vectors; Mathematics; Mixed model; Multivariate analysis; Parametric inference; Probability and statistics; Random variables; Sciences and techniques of general use; Shrinkage estimator; Statistical discrepancies; Statistical variance; Statistics; Time series; Time series model; Unconstrained parameterisation; Wishart distribution; Autoregression; Decomposition method; Shrinkage estimator; Autoregressive model; Covariate; Parameterization; Statistical simulation; Markov chain; Covariance; Cholesky method; Distribution function; Wishart matrix; Dynamic model; Data structure; Bayes estimation; Monte Carlo method; Data analysis; Covariance analysis; Prior distribution; Time series; Wishart distribution; Covariance matrix; Shrinkage; Gibbs sampling; Variance analysis; Statistical regression; Simulation; Hierarchical model
• ISSN: 0006-3444; 1464-3510
• DOI: 10.1093/biomet/89.3.553

Parsimonious modelling of the within‐subject covariance structure while heeding its positive‐definiteness is of great importance in the analysis of longitudinal data. Using the Cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation, we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse Wishart priors. Our priors offer many advantages with regard to elicitation, positive definiteness, computations using Gibbs sampling, shrinking covariances toward a particular structure with considerable flexibility, and modelling covariances using covariates. Bayesian estimation methods are developed and the results are compared using two simulation studies. These simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data.