Abstract
| Compositional data are vectors of proportions defined on the unit simplex and
this type of constrained data occur frequently in applications. It is also
possible for the compositional data to be correlated due to the clustering or
grouping of the observations. We propose a new class of mixed model for
compositional data based on the Kent distribution for directional data, where
the random effects also have Kent distributions. The advantage of this approach
is that it handles zero components directly and the new model has a fully
flexible underlying covariance structure. One useful property of the new
directional mixed model is that the marginal mean direction has a closed form
and is interpretable. The random effects enter the model in a multiplicative way
via the product of a set of rotation matrices and the conditional mean direction
is a random rotation of the marginal mean direction. For estimation we apply a
quasi-likelihood method which results in solving a new set of generalised
estimating equations and these are shown to have low bias in typical situations.
For inference we use a nonparametric bootstrap method for clustered data which
does not rely on estimates of the shape parameters (shape parameters are
difficult to estimate in Kent models). The new approach is shown to be more
tractable than the traditional approach based on the logratio transformation. |