Permutation Techniques Based on Euclidean Analysis Spaces: A New and Powerful Statistical Method for Ecological Research

Abstract

The statistical procedures that are most widely used in ecological population and community research belong to the family of parametric methods. Embedded in these procedures are assumptions about the normal distribution of the underlying population, homogeneity of variances and linear response patterns. One of the problems encountered in ecological and vegetation studies, however, is that these assumptions are very difficult, if not impossible, to meet. In addition, a very serious shortcoming of the most widely used statistical methods is the lack of congruence between the geometry of the data space, which is for the most part Euclidean, and the analysis space, which in the standard parametric tests and most of the nonparametric tests, is not Euclidean. In ecological and vegetation studies, the combination of a failure to meet model assumptions and a lack of congruence between the geometries of the data space and the analysis space can lead, as shown in this paper, to gross errors in data interpretation and hypothesis testing.

A new and powerful statistical technique (MRPP) is presented in this paper which is free from assumptions about the underlying distribution model of the population under analysis, can easily handle nonlinear data structures and more importantly meets the congruence principle (a common geometry for both the data and analysis spaces). The theoretical formulations for MRPP and its randomized block design counterpart MRBP as well as their relationship to other statistical methods are outlined in the first part of a paper. This is followed by computer algorithms and programs needed for their implementation as well as a series of detailed examples which demonstrate major advantages of MRPP and MRBP over the currently most widely used statistical methods. Computer programs are available from the authors free of charge (send a blank diskette).