  A first step when fitting multilevel models to continuous responses is to
explore the degree of clustering in the data. Researchers fit
variance-component models and then report the proportion of variation in the
response that is due to systematic differences between clusters. Equally they
report the response correlation between units within a cluster. These
statistics are popularly referred to as variance partition coefficients (VPCs)
and intraclass correlation coefficients (ICCs). When fitting multilevel models
to categorical (binary, ordinal, or nominal) and count responses, these
statistics prove more challenging to calculate. For categorical response
models, researchers appeal to their latent response formulations and report
VPCs/ICCs in terms of latent continuous responses envisaged to underly the
observed categorical responses. For standard count response models, however,
there are no corresponding latent response formulations. More generally, there
is a paucity of guidance on how to partition the variation. As a result,
applied researchers are likely to avoid or inadequately report and discuss the
substantive importance of clustering and cluster effects in their studies. A
recent article drew attention to a little-known exact algebraic expression for
the VPC/ICC for the special case of the two-level random-intercept Poisson
model. In this article, we make a substantial new contribution. First, we
derive exact VPC/ICC expressions for more flexible negative binomial models
that allows for overdispersion, a phenomenon which often occurs in practice.
Then we derive exact VPC/ICC expressions for three-level and random-coefficient
extensions to these models. We illustrate our work with an application to
student absenteeism.
