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Abstract
In this thesis we use discrete copulas to develop novel methods to model multivariate abundance data in ecology. These data, which consist of measures of abundance for many species at a set of sites, occur naturally in ecological sampling. Multivariate abundance data are therefore very common, but also challenging to analyse. The responses are discrete and sparse, with many variables relative to sample size. We propose the use of Gaussian copulas, combined with covariance modelling, to create flexible models for multivariate abundance data that take the aforementioned properties into account. Copulas are not commonly used in ecology, but are well suited to modelling multivariate abundance data due to their flexibility. The modelling framework we propose extends the flexibility of copulas further, by combining any set of discrete or continuous response distributions, with any covariance modelling algorithm designed for Gaussian data. We first propose a novel estimation method for such models, and explore the use of these models to study patterns in correlation between species, using covariance modelling techniques. Then we introduce a tool to visualise species interactions using copula Gaussian graphical models. We demonstrate this on a large dataset of New Zealand native forest species, where we are able to uncover known species relationships as well as generate new hypotheses for how species interact. We then use Gaussian copula models to carry out marginal inference. In particular, when it comes to marginal hypothesis testing and model selection, the likelihood based inference implemented with Gaussian copulas has several advantages over the commonly used approach based on generalised estimating equations.