#### Abstract

Compositional data is a type of multivariate data where each component of a vector is sandwiched between 0 and 1 and the sum of the components is 1. For example, the proportion of time that each of 7 mice spend in one of four quadrants of a circular water maze is between 0 and 1, and the total proportion of time spent in the maze is 1. If there are two sets of mice, one set of normal mice and one set of cognitively impaired mice, the experiment has a two-sample design. Such data is frequently analyzed incorrectly by comparing the two samples via a t-test (or ANOVA for multiple samples) on one component of the vector at a time.

This problem is corrected by analyzing compositional datasets using nested Dirichlet distributions, generalized versions of Dirichlet distributions that allow for positive correlations among components. Specifically, we extend a previous result of two-sample comparisons using Dirichlet distributions and nested Dirichlet distributions to multi-sample comparisons. The performance of the new test in terms of type I error rates and power is established using simulation studies. In addition, to use a nested model, an appropriate tree which describes the relationship between components must first be found. An existing data driven tree finding algorithm is improved upon by including an extra step that prunes unnecessary nodes using confidence intervals for the differences between parameters at each level of the tree. The tree finding algorithm and multi-sample test are demonstrated on two datasets.

#### Degree Date

Winter 2022

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Statistical Science

#### Advisor

Dr. Monnie McGee

#### Subject Area

Statistics

#### Number of Pages

149

#### Format

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

#### Recommended Citation

Luedeker, Bianca, "Compositional Datasets and the Nested Dirichlet Distribution" (2022). *Statistical Science Theses and Dissertations*. 30.

https://scholar.smu.edu/hum_sci_statisticalscience_etds/30