A cautionary note on generalized linear models for covariance of unbalanced longitudinal data

Abstract

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations.

Nonparametric covariance estimation with shrinkage toward stationary models

“…In the case that there is no fixed number of measurements and set of associated observation times for each subject, it is unclear how to define the discrete lag as in the usual formulation of autoregressive models. The ambiguity surrounding the definition of a discrete lag with unbalanced data gives rise to incoherence in the autoregressive parameters and innovation variances as noted by Huang, Chen, Maadooliat, and Pourahmadi (2012). This makes treatment of individual subdiagonals of the Cholesky factor or the covariance matrix itself infeasible.…”