  Compositional data arise when count observations are normalised into
proportions adding up to unity. To allow use of standard statistical methods,
compositional proportions can be mapped from the simplex into the Euclidean
space through the isometric log-ratio (ilr) transformation. When the counts
follow a multinomial distribution with fixed class-specific probabilities, the
distribution of the ensuing ilr coordinates has been shown to be asymptotically
multivariate normal. We here derive an asymptotic normal approximation to the
distribution of the ilr coordinates when the counts show overdispersion under
the Dirichlet-multinomial mixture model. Using a simulation study, we then
investigate the practical applicability of the approximation against the
empirical distribution of the ilr coordinates under varying levels of
extra-multinomial variation and the total count. The approximation works well,
except with a small total count or high amount of overdispersion. These
empirical results remain even under population-level heterogeneity in the total
count. Our work is motivated by microbiome data, which often exhibit
considerable extra-multinomial variation and are increasingly treated as
compositional through scaling taxon-specific counts into proportions. We
conclude that if the analysis of empirical data relies on normality of the ilr
coordinates, it may be advisable to choose a taxonomic level where counts are
less sparse so that the distribution of taxon-specific class probabilities
remains unimodal.
