The Effect of Covariance Structure on Variance Estimation in Balanced Growth-Curve Models with Random Parameters

Abstract

Intuition suggests that altering the covariance structure of a parametric model for repeated-measures data alters the variances of the model's estimated mean parameters. The purpose of this article is to sharpen such intuition for a family of growth-curve models with differing numbers of random effects for the individual sampling units and with a fixed structure on the mean. For every member of this family, the maximum likelihood (ML) estimator of the fixed effects is identical to the ordinary least squares (OLS) estimator. In addition, simple closed-form ML and restricted maximum likelihood estimators for the variance and covariance parameters exist for every member. As a consequence, closed-form expressions for the estimated variance-covariance matrix of the OLS estimator of the fixed effects also exist for the entire family. We derive explicit relationships between the variance and covariance parameter estimators from different members of the family and thereby extend some familiar results. For example, it is well known that for balanced and complete longitudinal designs the compound symmetry assumption for the covariance structure of the serial observations (i.e., assuming one random effect for each sampling unit) yields a more precise estimate of the slope of the population growth curve than of its intercept. It is also well known that for such designs the diagonal covariance structure assumption of OLS regression (i.e., no random effects for the units) yields a more precise estimate of the intercept than does the compound symmetry assumption, and a less precise estimate of the slope. We extend such relationships as these to growth-curve models whose covariance structures are of increasing linear complexity (i.e., assuming two or more random effects for each of the sampling units). We find that, in general, the variance of the OLS estimator depends strongly on assumed covariance structure. We also find that, specifically, the linear growth-curve assumption (i.e., two random effects for each sampling unit) is a conservative assumption, in that such will not give misleadingly small variance estimates for both intercept and slope even if a more complex covariance structure actually holds. We illustrate these points with a data example. In the conclusion, we extend our results by showing how they may apply to more general families of balanced growth-curve models that allow wider ranges of covariance structures and designs for the random effects, such as when different population subgroups require different covariance structures. We also address several computational issues involved in transformations of growth-curve models to canonical form.

Key Words:

• Covariance components
• Design of experiments
• Longitudinal data analysis
• Random-effects models
• Repeated measures
• Restricted maximum likelihood estimation
• Variance components