A Bayes Linear Analysis of Multilevel Models

Abstract

In this thesis, Bayes Linear methods for modeling multilevel data are presented
and discussed. Second-order exchangeability judgements are exploited to formulate
subjectivist versions of multilevel models. Bayes linear methods are applied to
estimate model parameters and for diagnostic checks. Closed-form expressions of
estimators are derived, allowing insight into relationships between the quantities
thereof. The canonical analysis and resolution transforms are used to guide sample
design and sample size determination under cost constraints. A finite version of a
multilevel model is formulated, analysed and compared to infinite versions, giving
further insight into sample design issues via the finite resolution transform.
A new Bayes Linear Minimum Variance Estimation (BLIMVE) approach is de-
veloped to estimate variances. Estimated variances are used to perform two-stage
Bayes linear analysis of more complex multilevel models. The methods developed
are shown to be applicable in cases of small level-2 samples. The Bayes linear analy-
ses of multilevel models are applied to an educational data set using special-purpose
codes written in the R Statistical Language.