Analysis of Ecological Communities Using Mixture Models

In this dissertation we develop methodology to analyse the structure of ecological communities. Distributions of species within and between communities are characterized by highly skewed and heavy tailed distributions which often causes sparsity in an observed sample. We develop methodologies to account for these factors, focusing on three problems of interest. The first is estimating total species in a community. This quantity is an important measure of community health and is often used to assess sample quality, but due to sparsity when sampling rare species it is not readily determined. Two approaches are proposed, the first models the underlying rate distribution with a mixture distribution and estimates the expected number of zeros when mixed with a Poisson. The mixture distribution is estimated by minimizing a least squares objective function of observed and expected counts or by sampling a Multinomial-Dirichlet mixture. The second approach estimates total species by modelling the detection rate of new species with a parametric function. Methods of model selection and estimates of variance are presented for both approaches. For the second problem, we look at modelling the distribution of a species across multiple sites. Firstly, we differentiate structural zeros from sparse sampling zeros, an important distinction when modelling the underlying rates or fitting a presence-absence model. We also model the outcome as a mixture distribution to stratify the data, as observations from the low rates and heavy tail cannot be analysed on the same scale. The model is estimated using either the least squares or Multinomial-Dirichlet approach, adjusted for varying numbers of observations between sites. A modification of the expectation-maximization algorithm is also proposed to optimize the fit. For the third part, we focus on identifying factors that significantly affect community composition. We use the mixture distribution approach to estimate site specific distributions, then compute pairwise distances between sites to obtain the test statistic for a permutation test. Each method is tested via simulation, compared against current methods and illustrated with an applied example. The methods are motivated by analysis of microbiome data, but are designed to be generally applicable.Ph.D