  Maximum likelihood estimates (MLEs) are asymptotically normally distributed,
and this property is used in meta-analyses to test the heterogeneity of
estimates, either for a single cluster or for several sub-groups. More
recently, MLEs for associations between risk factors and diseases have been
hierarchically clustered to search for diseases with shared underlying causes,
but an objective statistical criterion is needed to determine the number and
composition of clusters. To tackle this problem, conventional statistical tests
are briefly reviewed, before considering the posterior distribution for a
partition of data into clusters. The posterior distribution is calculated by
marginalising out the unknown cluster centres, and is different to the
likelihood associated with mixture models. The calculation is equivalent to
that used to obtain the Bayesian Information Criterion (BIC), but is exact,
without a Laplace approximation. The result includes a sum of squares term, and
terms that depend on the number and composition of clusters, that penalise the
number of free parameters in the model. The usual BIC is shown to be unsuitable
for clustering applications unless the number of items in each individual
cluster is sufficiently large.
