On the distribution of isometric log-ratio coordinates under extra-multinomial count data

Compositional data can be mapped from the simplex to the Euclidean space through the isometric log-ratio (ilr) transformation. When the underlying counts follow a multinomial distribution, the distribution of the ensuing ilr coordinates has been shown to be asymptotically multivariate normal. We derive conditions under which the asymptotic normality of the ilr coordinates holds under a compound multinomial distribution inducing overdispersion in the counts. We derive a normal approximation and investigate its practical applicability under extra-multinomial variation using a simulation study under the Dirichlet-multinomial distribution. The approximation works well, except with a small total count or high amount of overdispersion. Our work is motivated by microbiome data, which exhibit extra-multinomial variation and are increasingly treated as compositions. We conclude that if empirical data analysis relies on the normality of ilr coordinates, it may be advisable to choose a taxonomic level with less sparsity so that the distribution of taxon-specific class probabilities remains unimodal.