Unimodal models to relate species to environment

To assess the impact of environmental change on biological communities knowledge about species-environment relationships is indispensable. Ecologists attempt to uncover the relationships between species and environment from data obtained from field surveys. In the survey, species are scored on their presence or their abundance at each of several sampling sites and environmental variables that ecologists believe to be important are measured. The research that led to this thesis aimed to unravel the assumptions required for the application of statistical methods that are popular among ecologists to analyse such data. From a statistical point of view, species data are difficult to analyse: - there are many species involved (10 - 500), - many species occur at a few sites only. So the data contain numerous zeroes, - relations between species and environmental variables are not linear, but unimodal: a plant, for example, preferably grows under for that species optimal moisture conditions and is encountered less frequently at drier or wetter sites. A mathematical model for a unimodal relationship is the Gaussian response model. Standard statistical methods such as linear regression, principal components analysis and canonical correlation analysis are often inappropriate for analysing species data because they are based on linear relationships. One of the methods that ecologists use instead is correspondence analysis. This thesis contributes to the understanding of the underlying response model. With correspondence analysis, species and sites are arranged to discover the structure in the data (ordination) and the arrangement is subsequently related to environmental variables. It is an indirect method to detect relations between species and environment, hence R.H. Whittaker's term "indirect gradient analysis". Correspondence analysis has been invented around 1935 but did not receive interest from ecologists before 1973 when M.O. Hill derived the technique once more as the repeated application of "weighted averaging" - a method that was familiar to ecologists ever since 1930. Weighted averaging has the advantage of being simple to apply. The method can be used for two different aims: (1) to estimate the optimum of a species for an environmental variable and (2) to estimate the value of an environmental variable at a site from known optima of the species present (calibration). In chapter 2, estimating optima by weighted averaging is compared with the results of non-linear regression on the basis of the Gaussian response model. Under particular conditions, both methods agree precisely. In other cases, weighted averaging gives a biased estimate of the optimum and non-linear regression is the method to be preferred. An additional advantage of non-linear regression is that it can also be used to fit response models with more than one environmental variable. In chapter 3, weighted averaging to estimate the value of an environmental variable is compared with calibration on the basis of the Gaussian response model. Also in this context the techniques are sometimes equivalent. Chapter 4 deals with correspondence analysis. It is shown that, under particular conditions, correspondence analysis approximates ordination on the basis of the Gaussian response model, which is computationally much more complicated. To detect relations, indirect methods have an important disadvantage. The impact of some environmental variables on the species composition can be so large that the impact of other interesting environmental variables may fail to be detected. This problem can be overcome by using non-linear regression, but with many species and environmental variables this is laborious. In chapter 5, a simpler "direct" method is proposed, canonical correspondence analysis. In chapter 6, canonical correspondence analysis turns out to be a multivariate extension of weighted averaging. The results can be displayed graphically. In chapter 7, an extension with "covariables" is discussed, which leads to partial canonical correspondence analysis. Chapter 7 also shows that Gaussian models and, hence, canonical correspondence analysis are relevant to the analysis of contingency tables. Chapter 8 describes a study to estimate ecological amplitudes of plant species with respect to Ellenberg's moisture scale from species data alone. The question that is addressed as well, is how consequent Ellenberg's moisture indicator values are. Finally, chapter 9 cross-tabulates various gradient-analysis techniques by the type of problem (regression, calibration, ordination, etc.) and the response model (linear or unimodal). Furthermore, improvements are proposed for detrended correspondence analysis. A computer program, named MOM is written which can perform most of the methods discussed.