The sample datasets, Canoco and CanoDraw project files, and the files with results of the analyses discussed in this book are available from our web site: http://regent.bf.jcu.cz/maed/

You can download the sample files from there and find additional information, including errata and additional notes about the topics discussed in this book.

The sample files are available in compressed ZIP files, either separately for each chapter or in one large ZIP file, containing all the files.

While we can think of potential applications for linear discriminant analysis (LDA) in ecology (such as finding differences in habitat conditions among a priori recognized community types), its real use is quite infrequent. This is probably caused by the fact that many problems that seem to be a good fit for discriminant analysis produce weak rather than convincing results. A good example of this might be an attempt to use discriminant analysis for the selection of plant species indicating a particular community type. The discriminant analysis then attempts to separate the community types using a linear combination of species abundances. However, because the species occurrences are often strongly correlated and each of them alone provides little information, it is usually impossible to single out few such diagnostic species, even if the species composition differs strongly among the distinguished types. We will therefore use an example from a numerical taxonomy here, in the hope that the topic will be easily understood by ecologists and that they can apply the hints provided to their own problems.

Data

The sample data were taken from a taxonomic study of several species of Melampyrum, hemiparasitic plants from the Scrophulariaceae family. Eighty plant specimens were measured, selected from four taxonomic groups: M. bohemicum from Czech localities, M. bohemicum from Austrian localities, hybrid populations of M. bohemicum × M. nemorosum, and M. nemorosum. Each group was represented by 20 specimens, originating usually from several local populations. You will be looking for the plant vegetative and flower-related characteristics, which would allow the four taxonomic groups to be distinguished.

Multivariate methods are no longer restricted to the exploration of data and to the generation of new hypotheses. In particular, constrained ordination is a powerful tool for analysing data from manipulative experiments. In this chapter, we review the basic types of experimental design, with an emphasis on manipulative field experiments. Generally, we expect that the aim of the experiment is to compare the response of studied objects (e.g. an ecological community) to several treatments (treatment levels). Note that one of the treatment levels is usually a control treatment (although in real ecological studies, it might be difficult to decide what is the control; for example, when we compare several types of grassland management, which of the management types is the control one?). Detailed treatment of the topics handled in this chapter can be found for example in Underwood (1997).

If the response is univariate (e.g. number of species, total biomass), then the most common analytical tools are ANOVA, general linear models (which include both ANOVA, linear regression and their combinations), or generalized linear models. Generalized linear models are an extension of general linear models for the cases where the distribution of the response variable cannot be approximated by the normal distribution.

Completely randomized design

The simplest design is the completely randomized one (Figure 2–1). We first select the plots, and then randomly assign treatment levels to individual plots. This design is correct, but not always the best, as it does not control for environmental heterogeneity. This heterogeneity is always present as an unexplained variability. If the heterogeneity is large, use of this design might decrease the power of the tests.