Multilevel covariance component models

Goldstein (1986) describes the analysis of the multilevel mixed effects linear model with random coefficients, where the variance and covariance components have a nested structure across levels. The purpose of the present note is to show how a simple extension to the formulae in that paper can accommodate cross-classifications of the components within any level of the nesting, thus enabling quite general covariance component models to be specified and efficient parameter estimates obtained. For simplicity the 3-level model is used, with the extension to 4 or more levels being straightforward. We write the random part of the 3-level model as