The Analysis of Very Small Samples of Repeated Measurements

Abstract

The statistical analysis of repeated measures or longitudinal data always requires the accommodation of the covariance structure of the repeated measurements at some stage in the analysis. The general linear mixed model is often used for such analyses, and allows for the specification of both a mean model and a covariance structure. Often the covariance structure itself is not of direct interest, but only a means to producing valid inferences about the response. This thesis considers methods for the analysis of repeated measurements which arise from very small samples. In Part 1, existing methods of analysis are shown to be inadequate for very small samples. More precisely, statistical measures of goodness of fit are not necessarily the right measure of the appropriateness of a covariance structure and inferences based on conventional Wald type procedures (with small sample adjustments) do not approximate sufficiently well their nominal properties when data are unbalanced or incomplete. In Part 2, adaptive-estimation techniques are considered for the sample covariance matrix which smooth between unstructured and structured forms; 'direct' smoothing, a weighted average of the unstructured and structured estimates, and an estimate chosen via penalised likelihood. Whilst attractive in principle, these approaches are shown to have little success in practice, being critically dependent on the 'correct' choice of smoothing structure. Part 3 considers methods which are less dependent on the covariance structure. A generalisation of a small sample adjustment to the empirical sandwich estimator is developed which accounts for its inherent bias and increased variance. This has nominal properties but lacks power. Also, a modification to Box's correction, an ANOVA F-statistic which accounts for departures from independence, is given which has both nominal properties and acceptable power. Finally, Part 4 recommends the adoption of the modified Box statistic for repeated measurements data where the sample size is very small.