Small-sample adjustments to tests with unbalanced repeated measures assuming several covariance structures

Recent advances in methods for analysis of longitudinal data arid incomplete repeated measures have been in the area of maximum likelihood (ML) and restricted maximum likelihood (REML) methods (e.g., Laird and Ware, 1982 Biometrics, Jennrich and Schluchter, 1986 Biometrics). This paper outlines the ML and REML approaches to the analysis of incomplete repeated measures data and growth curves, and then examines methods for small-sample adjustment of asymptotic Wald-type chi-square tests constructed from ML and REML estimates under four different assumed covariance structures. These adjustments involve transformation of the Wald Ghi-square statistic to an approximate F-statistic. In certain cases when data are complete and balanced, the transformed test statistics have exact F-distribution under the null hypothesis. The first three covariance structures: (1) Compound Symmetry, (2) First-Order Autoregressive, and (3) Multivariate (unstructured), are examined in the context of the analysis of a repeated measures d sign having a single between-subjects factor and a single within-subjects factor. The fourth model, which implies a special type of covariance structure, is a Linear Random Effects Growth Curve Model. For each covariance structure model, we review known exact results both for the case of balanced and unbalanced data. We then examine the type I error rates of the various tests via a small simulation study. It is shown that the most appropriate type of small-sample correction depends upon the form of the assumed covariance structure, whether ML or REML procedures are used, and whether the test is a 'between groups' or 'within-subject' test. The results emphasize the importance of applying a correction to the asymptotic tests in small samples.