Introduction

Randomized complete blocks design is probably the most popular experimental design in ecological studies, because it controls in a powerful way the environmental heterogeneity. For a univariate response (e.g. number of species, total biomass) the results of experiments set in randomized complete blocks are evaluated using a two-way ANOVA without interactions. The interaction mean square is used as the error term – the denominator in the calculation of F -statistics. In the following tutorial, you will use the program CANOCO in a similar way to evaluate the community response (i.e. a multivariate response of the species composition of the vegetation). The example is based on an experiment studying the effect of dominant species, plant litter and moss on the composition of a community of vascular plants, with special attention paid to seedling recruitment. In this way, some of the aspects of the importance of regeneration niche for species coexistence were tested. The experiment was established in four randomized complete blocks, the treatment had four levels and the response was measured once. The experiment is described in full by Špa čková et al. (1998). Here is a simplified description of the experiment.

The experiment was established in March 1994, shortly after snowmelt, in four randomized complete blocks. Each block contained four plots, each with a different treatment: (1) a control plot where the vegetation remained undisturbed; (2) a plot with the removal of litter; (3) a plot with removal of the dominant species Nardus stricta; and (4) a plot with removal of litter and mosses. Each plot was 2 m × 2 m square.

This chapter introduces four specialized or more advanced techniques, which build on the foundations of ordination methods and can be used with the Canoco for Windows package. All four methods are illustrated in one of the case studies presented in Chapters 11–17.

Testing the significance of individual constrained ordination axes

If you use several (partially) independent environmental variables in a constrained ordination, the analysis results in several constrained (canonical) ordination axes. In CANOCO, it is easy to test the effect of the first (most important) constrained axis or the effect of the whole set of constrained axes.

But you may be interested in reducing the set of constrained axes (the dimensionality of the canonical ordination space), and to do so you need to find how many canonical axes effectively contribute to the explanation of the response variables (to the explanation of community variation, typically). To do so, you must test the effects of individual constrained axes. Their independent (marginal) effects do not differ from their additional (conditional) effects, of course, because they are mutually linearly independent by their definition.

Let us start with the simplest situation, when you have a constrained ordination (RDA or CCA) with no covariables. If you need to test the significance of a second (or higher) canonical axis for such constrained analysis, you should clone the corresponding CANOCO project, i.e. create a new project, similar to the original one but containing, in addition, covariable data. You will use the scores of the samples on the constrained axes, which were calculated in the original project.

The methods for analysing species composition are usually divided into gradient analysis and classification. The term gradient analysis is used here in the broad sense, for any method attempting to relate species composition to the (measured or hypothetical) environmental gradients.

Traditionally, the classification methods, when used in plant community ecology, were connected with the discontinuum approach (or vegetation unit approach) or sometimes even with the Clementsian superorganismal approach, whereas the methods of gradient analysis were connected with the continuum concept or with the Gleasonian individualistic concept of (plant) communities (Whittaker 1975). While this might reflect the history of the methods, this distinction is no longer valid. The methods are complementary and their choice depends mainly on the purpose of a study. For example, in vegetation mapping some classification is necessary. Even if there are no distinct boundaries between adjacent vegetation types, we have to cut the continuum and create distinct vegetation units for mapping purposes. Ordination methods can help find repeatable vegetation patterns and discontinuities in species composition, and show any transitional types, etc. These methods are now accepted even in phytosociology. Also, the methods are no longer restricted to plant community ecology. They became widespread in most studies of ecological communities with major emphasis on species composition and its relationship with the underlying factors. In fact, it seems to us that the advanced applications of gradient analysis are nowadays found outside the vegetation sciences, for example in hydrobiological studies (see the bibliographies by Birks et al. 1996, 1998).

Why ordination?

When we investigate variation of plant or animal communities across a range of different environmental conditions, we usually find not only large differences in species composition of the studied communities, but also a certain consistency or predictability of this variation. For example, if we look at the variation of grassland vegetation in a landscape and describe the plant community composition using vegetation samples, then the individual samples can be usually ordered along one, two or three imaginary axes. The change in the vegetation composition is often small as we move our focus from one sample to those nearby on such a hypothetical axis.

This gradual change in the community composition can often be related to differing, but partially overlapping demands of individual species for environmental factors such as the average soil moisture, its fluctuations throughout the season, the ability of species to compete with other ones for the available nutrients and light, etc. If the axes along which we originally ordered the samples can be identified with a particular environmental factor (such as moisture or richness of soil nutrients), we can call them a soil moisture gradient, a nutrient availability gradient, etc. Occasionally, such gradients can be identified in a real landscape, e.g. as a spatial gradient along a slope from a riverbank, with gradually decreasing soil moisture. But more often we can identify such axes along which the plant or animal communities vary in a more or less smooth, predictable way, yet we cannot find them in nature as a visible spatial gradient and neither can we identify them uniquely with a particular measurable environmental factor. In such cases, we speak about gradients of species composition change.

The primary device for presenting the results of an ordination model is the ordination diagram. The contents of an ordination diagram can be used to approximate the species data table, the matrix of distances between individual samples, or the matrix of correlations or dissimilarities between individual species. In ordination analysis including environmental variables, we can use the ordination diagram to approximate the contents of the environmental data table, the relationship between the species and the environmental variables, the correlations among environmental variables, etc. The following two sections summarize what we can deduce from ordination diagrams that result from linear and unimodal ordination methods.

Before we discuss rules for interpreting ordination diagrams, we must stress that the absolute values of coordinates of objects (samples, species, explanatory variables) in ordination space do not have, in general, any meaning. When interpreting ordination diagrams, we use relative distances, relative directions, or relative ordering of projection points.

What we can infer from ordination diagrams: linear methods

An ordination diagram based on a linear ordination method (PCA or RDA) can display scores for samples (represented by symbols), species (represented by arrows), quantitative environmental variables (represented by arrows) and nominal dummy variables (represented by points – centroids – corresponding to the individual levels of a factor variable). Table 10–1 (after Ter Braak 1994) summarizes what can be deduced from ordination diagrams based on these scores. Additionally, the meaning of the length of species arrows in the two types of ordination scaling is discussed, and information about calculating the radius of the equilibrium contribution circle is given.

In this case study, we will demonstrate a common application of multivariate analysis: the search for a pattern in a set of vegetation samples with available measurements of environmental variables. The species data comprise classical vegetation relevés (records of all vascular plants and bryophytes, with estimates of their abundance using the Braun–Blanquet scale) in spring fen meadows in the westernmost Carpathian mountain ranges. The data represent an ordinal transformation of the Braun–Blanquet scale, i.e. the r, +, and 1 to 5 values of the Braun–Blanquet scale are replaced by values 1 to 7. The relevés are complemented with environmental data – chemical analyses of the spring water, soil organic carbon content and slope of the locality. The ion concentrations represent their molarity: note that the environmental data are standardized in all CANOCO procedures and, consequently, the choice of units does not play any role (as long as the various units are linearly related).+ The data are courtesy of Michal Hájek and their analysis is presented in Hájek et al. (2002). The aim of this case study is to describe basic vegetation patterns and their relationship with available environmental data, mainly to the chemical properties of the spring water.

The data are stored in the file meadows.xls, where one sheet represents the species data, and the other one the environmental data (in comparison with the original, the data are slightly simplified). The data were imported to CANOCO format, using the WCanoImp program. Two files were created, meadows.spe and meadows.env. Note that in the Excel file, the species data are transposed (i.e. species as rows, samples as columns).

In many multivariate methods, one of the first steps is to calculate a matrix of similarities (resemblance measures) either between the samples (relevés) or between the species. Although this step is not explicitly done in all the methods, in fact each of the multivariate methods works (even if implicitly) with some resemblance measure. The linear ordination methods can be related to several variants of Euclidean distance, while the weighted-averaging ordination methods can be related to chi-square distances. There semblance functions are reviewed in many texts (e.g. Gower & Legendre 1986; Legendre & Legendre 1998; Ludwig & Reynolds 1988; Orloci 1978), so here we will introduce only the most important ones.

In this chapter, we will use the following notation: we have n samples (relevés), containing m species. Yik represents the abundance of the kth species (k = 1, 2, …,m) in the ith sample (i = 1, 2, …, n).

It is technically possible to transpose the data matrix (or exchange i and k in the formulae), thus any of the resemblance functions can be calculated equally well for rows or columns (i.e. for samples or species). Nevertheless, there are generally different kinds of resemblance functions suitable for expressing the (dis)similarity among samples, and those suitable for describing similarity among the species. This is because the species set is usually a fixed entity: if we study vascular plants, then all the vascular plants in any plot are recorded – or at least we believe we have recorded them. But the sample set is usually just a (random) selection, something which is not fixed.

The use of any multivariate statistical method even for small datasets requires a computer program to perform the analysis. Most of the known statistical methods are implemented in several statistical packages. In this book, we demonstrate how to use the ordination methods with possibly the most widely employed package, Canoco for Windows. We also show how to use the methods not available in CANOCO (clustering, NMDS, ANOVA) with the general package Statistica for Windows. In the next paragraph, we provide information on obtaining a trial version of the CANOCO program, which you can use to work through the tutorials provided in this book, using the sample datasets (see Appendix A for information on how to obtain the datasets). We also provide an overview of other available software in tabular form and show both the freely available as well as the commercial software. The attention is focused on the specialized software packages, targeting ecologists (or biologists), so we do not cover general statistical packages such as S-Plus, SAS or GENSTAT.

The Canoco for Windows program is commercial software requiring a valid licence for its use. But we reached agreement with its distributor (Microcomputer Power, Ithaca, NY, USA), who will provide you on request with a trial version of the software, which will be functional for a minimum of one month. You can use it to try the sample analyses discussed in this book, using the data and CANOCO projects provided on our web site (see Appendix A). To contact Dr Richard Furnas from Microcomputer Power, write to the following E-mail address: trial@microcomputerpower.com.

In this chapter, we discuss constrained ordination and related approaches: stepwise selection of a set of explanatory variables, Monte Carlo permutation tests and variance partitioning procedure.

Linear multiple regression model

We start this chapter with a summary of the classical linear regression model, because it is very important for understanding the meaning of almost all aspects of direct gradient analysis.

Figure 5–1 presents a simple linear regression used to model the dependence of the values of a variable Y on the values of a variable X. The figure shows the fitted regression line and also the difference between the true (observed) values of the response variable Y and the fitted values (Ŷ; on the line). The difference between these two values is called the regression residual and is labelled as e in Figure 5–1.

An important feature of all statistical models (including regression models) is that they have two main components. The systematic component describes the variability in the values of the response variable(s) that can be explained by one or more explanatory variables (predictors), using a parameterized function. The simplest function is a linear combination of the explanatory variables, which is the function used by (general) linear regression models. The stochastic component is a term referring to the remaining variability in the values of the response variable, not described by the systematic part of the model. The stochastic component is usually defined using its probabilistic, distributional properties.

We judge the quality of the fitted model by the amount of the response variable's variance that can be described by the systematic component.

Overview of the package

The Canoco for Windows package is composed of several programs. In this section we summarize the role of each program in data analysis and in the interpretation of results. The following sections then deal with some typical aspects of this software use. This chapter is not meant to replace the documentation distributed with the Canoco for Windows package, but provides a starting point for efficient use of this software.

Canoco for Windows 4.5

This is the central piece of the package. Here you specify the data you want to use, the ordination model to apply, and the analysis options. You can also select subsets of the explained and explanatory variables to use in the analysis or change the weights for the individual samples. All these choices are collected in a CANOCO project.

Canoco for Windows allows one to use quite a wide range of ordination methods. The central ones are the linear methods (principal components analysis, PCA, and redundancy analysis, RDA) and the unimodal methods (correspondence analysis, CA, detrended correspondence analysis, DCA, and canonical correspondence analysis, CCA), but based on them you can use CANOCO to apply other methods such as multiple discriminant analysis (canonical variate analysis, CVA) or metric multidimensional scaling (principal coordinates analysis, PCoA) to your data set. From the widely used ordination methods, only non-metric multidimensional scaling (NMDS) is missing.

CANOCO 4.5

This program can be used as a less user-friendly, but slightly more powerful alternative to the Canoco for Windows program.

Introduction

In many cases, the effects of several explanatory variables need to be separated, even when the explanatory variables are correlated. The example below comes from a field fertilization experiment (Pyšek and Lepš 1991). A barley field was fertilized with three nitrogen fertilizers (ammonium sulphate, calcium-ammonium nitrate and liquid urea) and two different total nitrogen doses. For practical reasons, the experiment was not established in a correct experimental design; i.e. plots are pseudo-replications. The experiment was designed by hydrologists to assess nutrient runoff and, consequently, smaller plots were not practical. In 122 plots, the species composition of weeds was characterized by classical Braun–Blanquet relevés (for the calculations, the ordinal transformation was used, i.e. numbers 1–7 were used for grades of the Braun–Blanquet scale: r,+, 1,. …, 5). The percentage cover of barley was estimated in all relevés.

The authors expected the weed community to be influenced both directly by fertilizers and indirectly through the effect of crop competition. Based on the experimental manipulations, the overall fertilizer effect can be assessed. However, barley cover is highly correlated with fertilizer dose. As the cover of barley was not manipulated, there is no direct evidence of the effect of barley cover on the weed assemblages. However, the data enable us to partially separate the direct effects of fertilization from indirect effects of barley competition. This is done in a similar way to the separation of the effects of correlated predictors on the univariate response in multiple regression. The separation can be done using the variable of interest as an explanatory (environmental) variable and the other ones as covariables.

Introduction

Repeated observations of experimental units are typical in many areas of ecological research. In experiments, the units (plots) are sampled first before the experimental treatment is imposed on some of them. In this way, you obtain ‘baseline’ data, i.e. data where differences between the sampling units are caused solely by random variability. After the treatment is imposed, the units are sampled once or several times to reveal the difference between the development (dynamics) of experimental and control units. This design is sometimes called replicated BACI – before after control impact. To analyse the univariate response (e.g. number of species, or total biomass) in this design, you can usually apply the repeated measurements model of ANOVA.

There are two possibilities for analysing such data. You can use the split-plot ANOVA with time, i.e. the repeated measures factor, being the ‘within plot’ factor, or you can analyse the data using MANOVA. Although the theoretical distinction between those two is complicated, the first option is usually used because it provides a stronger test. Nevertheless, it also has stronger assumptions for its validity (see e.g. Lindsey 1993;Von Ende 1993). The interaction between time and the treatment reflects the difference in the development of the units between treatments. CANOCO can analyse the repeated observations of the species composition in a way equivalent to the univariate repeated measurements ANOVA. Whereas all the model terms are tested simultaneously in ANOVA, in CANOCO you must run a separate analysis to test each of the terms.

The aim of classification is to obtain groups of objects (samples, species) that are internally homogeneous and distinct from the other groups. When the species are classified, the homogeneity can be interpreted as their similar ecological behaviour, as reflected by the similarity of their distributions. The classification methods are usually categorized as in Figure 7–1.

Historically, numerical classifications were considered an objective alternative to subjective classifications (such as the Zürich-Montpellier phytosociological system, Mueller-Dombois & Ellenberg 1974). They are indeed ‘objective’ by their reproducibility, i.e. getting identical results with identical inputs (in classical phytosociology, the classification methods obviously provide varying results). But also in numerical classifications, the methodological choices are highly subjective and affect the final result.

Sample data set

The various possibilities of data classification will be demonstrated using vegetation data of 14 relevés from an altitudinal transect in Nízké Tatry Mts, Slovakia. Relevé 1 was recorded at an altitude of 1200 metres above sea level (m a.s.l.), relevé 14 at 1830ma.s.l. Relevés were recorded using the Braun–Blanquet scale (r, +, 1–5, see Mueller-Dombois & Ellenberg 1974). For calculations, the scale was converted into numbers 1 to 7 (ordinal transformation, Van der Maarel 1979). Data were entered as a classical vegetation data table (file tatry.xls) and further imported (using the WCanoImp program) into the condensed Cornell (CANOCO) format (file tatry.spe), to enable use of the CANOCO and TWINSPAN programs. The data were also imported into a Statistica file (file tatry.sta).

In this chapter, we will study the hierarchical components of variation of the crayfish community in the drainage of Spring River, north central Arkansas and south central Missouri, USA. The data were collected by Dr Camille Flinders (Flinders & Magoulick 2002, unpublished results).The statistical approach used in this study is described in Section 9.2.

Data and design

The species data consist of 567 samples of the crayfish community composition. There are 10 ‘species’, which actually represent only five cray-fish species, with each species divided into two size categories, depending on carapace length (above or below 15 mm). Note that the data matrix is quite sparse. In the 5670 data cells, there are only 834 non-zero values. Therefore, 85% of the data cells are empty! This would suggest high beta diversity and, consequently, use of a unimodal ordination method, such as CCA. There is a problem with that, however. There are 133 samples without any crayfish specimen present and such empty samples cannot be compared with the others using the chi-square distance, which is implied by unimodal ordination methods. You must, therefore, use a linear ordination method.

The sampling used for collecting data has a perfectly regular (balanced) design. The data were collected from seven different watersheds (WS). In each, three different streams (ST) were selected, and within each stream, three reaches (RE) were sampled. Each reach is (within these data) represented by three different runs (RU) and, finally, each run is represented by three different samples. This leads to the total of 567 samples = 7 WS · 3 ST · 3 RE · 3 RU · 3 replicates. These acronyms will be used, when needed, throughout this chapter.

The multidimensional data on community composition, properties of individual populations, or properties of environment are the bread and butter of an ecologist's life. They need to be analysed with taking their multidimensionality into account. A reductionist approach of looking at the properties of each variable separately does not work in most cases. The methods for statistical analysis of such data sets fit under the umbrella of ‘multivariate statistical methods’.

In this book, we present a hopefully consistent set of approaches to answering many of the questions that an ecologist might have about the studied systems. Nevertheless, we admit that our views are biased to some extent, and we pay limited attention to other less parametric methods, such as the family of non-metric multidimensional scaling (NMDS) algorithms or the group of methods similar to the Mantel test or the ANOSIM method. We do not want to fuel the controversy between proponents of various approaches to analysing multivariate data. We simply claim that the solutions presented are not the only ones possible, but they work for us, as well as many others.

We also give greater emphasis to ordination methods compared to classification approaches, but we do not imply that the classification methods are not useful. Our description of multivariate methods is extended by a short overview of regression analysis, including some of the more recent developments such as generalized additive models.

Our intention is to provide the reader with both the basic understanding of principles of multivariate methods and the skills needed to use those methods in his/her own work.

The primary goal of the analyses demonstrated in this case study is to describe the variability of bird communities, related to the differences in their habitat.

The data set originates from a field study by Mirek E. Šálek et al. (unpublished data) in Velka Fatra Mts. (Slovak Republic) where the bird assemblages were studied using a grid of equidistant points placed over the selected area of montane forest, representing a mix of spruce-dominated stands and beech-dominated stands. There was a varying cover of deforested area (primarily pastures) and the individual quadrats differed in their altitude, slope, forest density, cover and nature of shrub layer, etc. (see the next section for description of environmental variables). The primary data (species data) consist of the number of nesting pairs of individual bird species, estimated by listening to singing males at the centre points of each quadrat. The data values represent an average of four observations (performed twice in each of two consecutive seasons).

Data manipulation

The data are contained in the Excel spreadsheet file named birds.xls. This file has two sheets. We will use the data contained in the first sheet (named birds), where the bird species observed in only one of the quadrats were omitted (the full data are available in the other sheet, labelled birds-full). Note that the birds sheet contains both the species data (average abundance of individual bird species) and the environmental data (description of habitat characteristics for the particular quadrat).