Analysis of multivariate longitudinal data using ARMA Cholesky and hypersphere decompositions

Abstract

In longitudinal data with many replications, the high-order autoregressive (AR) structure of covariance matrix is required to capture the serial correlations between repeated outcomes. Thus, the high-order AR structure requires many parameters underlying the dynamic data dependence. In this paper, we proposed an autoregressive moving-average (ARMA) structure of covariance matrix involving multivariate linear models instead of the high-order AR structure of covariance matrix. We decomposed the covariance matrix using autoregressive moving-average Cholesky decomposition (ARMACD) to explain the correlations between responses at each time point, the correlation within separate responses over time, and the cross-correlation between different responses at different times. The ARMACD facilitates nonstationarity and heteroscedasticity of the covariance matrix, and the estimated covariance matrix is guaranteed to be positive definite. We illustrated the proposed methods using data derived from a study of nonalcoholic fatty liver disease.