A Markov Chain Monte Carlo Method for Approximating 2-Way Contingency Tables with Applications in the Stability Analysis of Ecological Ordination

Donald Ylvisaker
Ph.D., 1999
Advisor: Donald Ylvisaker

This dissertation develops a Markov Chain Monte Carlo method for approximating 2-way contingency tables with an eye toward assessing the stability of ecological ordination. Ecology is a part of biology that deals with the interrelationships between populations, communities and ecosystems and their environment. It draws on knowledge from many other disciplines such as climatology, physical geography, agronomy, and pedology [52]. Odum. [75] prefers the definition “Ecology is the study of structure and function of nature,” and stresses the role of ecosystem research in relation to the use of nature by man. Krebs [58] prefers to think of Ecology as the scientific study of the interactions that determine the distribution and abundance of organisms in nature. Ordination attempts to order environmental entities such as species (or sites) in such a way that similar entities are placed close to one another and dissimilar entities farther apart. The proximity of these entities to one another allows ecologists to draw conclusions concerning environmental factors that characterize the community. Although interpretations may be somewhat subjective, one can be confident in his results if the techniques utilized have the following qualities: The results are stable within technique : Similar results are obtained for a given data set under various sampling assumptions. Conversely, departures from these assumptions do not result in major differences in the results and interpretations; The results are stable between techniques: Similar results are obtained for a given data set utilizing various techniques. In other words, there exists a unique solution; The results make ecological sense: Ecological similarity is related to proximity in ordination space and clusters existing in nature should be obvious from the ordination. Conversely, the ordination does not produce clusters which do not exist. In order to examine the stability of methods, it is first necessary to produce data matrices that are consistent with our understanding of the underlying processes in nature. This is done by means of the Meteropolis-Hastings algorithm, a Markov Chain Monte Carlo technique. Results are then displayed using position plots, and stability is gauged visually.