  Many scientific and industrial processes produce data that is best analysed
as vectors of relative values, often called compositions or proportions. The
Dirichlet distribution is a natural distribution to use for composition or
proportion data. It has the advantage of a low number of parameters, making it
the parsimonious choice in many cases. In this paper we consider the case where
the outcome of a process is Dirichlet, dependent on one or more explanatory
variables in a regression setting. We explore some existing approaches to this
problem, and then introduce a new simulation approach to fitting such models,
based on the Bayesian framework. We illustrate the advantages of the new
approach through simulated examples and an application in sport science. These
advantages include: increased accuracy of fit, increased power for inference,
and the ability to introduce random effects without additional complexity in
the analysis.
