Comparison of two independent populations of compositional data with positive correlations among components using a nested dirichlet distribution.

Compositional data are multivariate data made up of components that sum to a fixed value. Often the data are presented as proportions of a whole, where the value of each component is constrained to be between 0 and 1 and the sum of the components is 1. There are many applications in psychology and other disciplines that yield compositional data sets including Morris water maze experiments, psychological well-being scores, analysis of daily physical activity times, and components of household expenditures. Statistical methods exist for compositional data and typically consist of two approaches. The first is to use transformation strategies, such as log ratios, which can lead to results that are challenging to interpret. The second involves using an appropriate distribution, such as the Dirichlet distribution, that captures the key characteristics of compositional data, and allows for ready interpretation of downstream analysis. Unfortunately, the Dirichlet distribution has constraints on variance and correlation that render it inappropriate for some applications. As a result, practicing researchers will often resort to standard two-sample t test or analysis of variance models for each variable in the composition to detect differences in means. We show that a recently published method using the Dirichlet distribution can drastically inflate Type I error rates, and we introduce a global two-sample test to detect differences in mean proportion of components for two independent groups where both groups are from either a Dirichlet or a more flexible nested Dirichlet distribution. We also derive confidence interval formulas for individual components for post hoc testing and further interpretation of results. We illustrate the utility of our methods using a recent Morris water maze experiment and human activity data. (PsycInfo Database Record (c) 2025 APA, all rights reserved).