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Relationships between sets of variables of different kinds are of interest in many branches of science. The question of the analysis of relationships of this sort has nevertheless rather surprisingly received less attention from statisticians and others than it would seem to deserve. Of the available methods, that address­ ing the question most directly is canonical correlation analysis, here referred to for convenience as canonical analysis. Yet canonical analysis is often coolly received despite a lack of suitable alternatives. The purpose of this book is to clarify just what may and what may not be accomplished by means of canoni­ cal analysis in one field of scientific endeavor. Canonical analysis is concerned with reducing the correlation structure be­ tween two (or more) sets of variables to its simplest possible form. After a review of the nature and properties of canonical analysis, an assessment of the method as an exploratory tool of use in ecological investigations is made. Applications of canonical analysis to several sets of ecological data are described and discussed with this objective in mind. The examples are drawn largely from plant ecology. The position is adopted that canonical analysis exists primarily to be used; the examples are accordingly worked through in some detail with the aim of showing how canonical analysis can contribute towards the attainment of ecological goals, as well as to indicate the kind and extent of the insight afforded.

Inhaltsverzeichnis

Frontmatter

Introduction

1. Introduction

Abstract
Ecology deals with relationships between plants and animals and between them and the places where they live. Consequently, many questions of interest to ecologists call for the investigation of relationships between variables of two distinct but associated kinds. Such relationships may involve those, for example, between the plant and animal constituents of a biotic community. They might also involve, as in plant ecology, connections between plant communities and their component species, on the one hand, and characteristics of their physical environment on the other. As another example, comparative relationships among a number of affiliated species or populations with respect to a particular treatment regime in a designed experiment might be studied. In more general terms, the question which arises calls for the exploration of relationships between any two or more sets of variables of ecological interest.
Robert Gittins

Theory

Frontmatter

2. Canonical correlations and canonical variates

Abstract
In this chapter we shall summarize the essential elements of the theory of canonical correlations and variates. We shall begin by formulating the problem. The derivation of canonical correlations and canonical variates will then be taken up. Canonical analysis can be derived in several ways. Two derivations will be described rather fully — those leading to eigenanalysis and singular value decomposition, respectively. Several other derivations are of interest for the light which they shed on canonical analysis and its relations with other methods, and these will be described more briefly. We shall then consider the properties of canonical correlations, weights and variates. Finally, computational aspects of canonical analysis will be taken up.
Robert Gittins

3. Extensions and generalizations

Abstract
In the preceding chapter we introduced the essential ingredients of the theory of canonical correlation. The theory stems from the work of Hotelling in the 1930s. Since Hotelling’s pioneering studies the theory has been added to in a number of ways. The present chapter deals with developments of this kind. We shall begin by considering several numerical indices which have been devised in order to further the interpretation of results in addition to the canonical correlations, weights and variates of the classical solution. Other developments are directed towards widening the applicability of the original theory and several generalizations of this kind will be mentioned. Tests of statistical hypotheses are sometimes of interest in canonical analysis and the chapter closes with an account of hypothesis testing.
Robert Gittins

4. Canonical variate analysis

Abstract
In studies of dependency it is often useful to distinguish between studies of relationship and studies of structure. The term structure is used here to refer to the way in which the sample is organized or constructed. In studies of relationship it is the connections between two sets of observed or measured variables which are of interest. In structural investigations, on the other hand, it is affinities among subsamples making up a sample with respect to p response variables considered simultaneously which are of interest. For this purpose a set of q artificial variables may be created to account for the design characteristics of the survey or experiment which gave rise to the data, or to embody hypotheses concerning structural relationships among subsamples. In this chapter we shall be concerned with the use of canonical analysis in structural investigations of these kinds. The topic is a large one and we cannot hope to do more than touch upon a few selected aspects. Many of the theoretical results presented are illustrated by worked, numerical examples in Chaps. 10 and 11.
Robert Gittins

5. Dual scaling

Abstract
In the previous chapter we saw that canonical analysis can be applied to a sample of p-variate observations partitioned into g≥2 classes for the purpose of obtaining insight into relationships among the classes. Canonical variate analysis was introduced as a special case of canonical analysis in which the data matrix for one of the two sets of variables is specialized so as to account for the partition of a sample. Suppose now that we have a sample of individuals which are simultaneously classified with respect to two categorical variables. Dual scaling may be set up as a double canonical variate analysis in which both data matrices are specialized so that each corresponds to one of the two variables of classification for the analysis of data of this kind. Doubly-classified observations arise frequently. In palynology, for example, the data generally consist of counts of fossil pollen taxa at a number of stratigraphic levels. In a similar way, studies of natural communities often lead to estimates of the density of different animal species in a variety of plant communities. The data in such cases are conveniently organized in the form of a 2-way array or table. We may then enquire as to the nature of the relationships within and between the row and column categories of the table. This is the question addressed by dual scaling. Dual scaling is in fact applicable to an m-way classification of individuals (m≥2), though for simplicity we shall confine our attention to the case where m is strictly equal to two.
Robert Gittins

Applications

Frontmatter

General introduction

6. Experiment 1: an investigation of spatial variation

Abstract
The experiment concerns the joint pattern of spatial variation common to three plant species belonging to a limestone grassland community in Anglesey, North Wales. The species were Phleum bertolonii DC., Dactylis glomerata L. and Galium verum L. From previous work, the species were known to share the same relatively simple generalized distribution pattern, which was centered on the southwestern extremity of the study area. Furthermore, the direction and extent of the departure of the distribution of each species from the common trend was also known. P. bertolonii was the most circumscribed of the species, being strictly confined to the southwestern corner. G. verum, on the other hand, was the most widespread species, occurring throughout the study area although its maximum representation occurred in the southwest. D. glomerata was less extreme with a distribution intermediate between those of the other species, but resembling that of P. bertolonii more closely than that of G. verum.
Robert Gittins

7. Experiment 2: soil-species relationships in a limestone grassland community

Abstract
This experiment concerns relationships between the abundances of several plant species and associated soil characteristics in a limestone grassland community in Anglesey, North Wales. Previous work (Gittins, 1969) suggested that the factor-complexes of soil moisture and soil fertility are influential in determining the representation of many of the species present. Knowledge of the salient soil-species relationships provides a basis against which the ability of canonical analysis to recover such relationships can be assessed.
Robert Gittins

8. Soil-vegetation relationships in a lowland tropical rain forest

Abstract
The aim of the following application of canonical analysis is to clarify relationships between soils and vegetation in a 1 km2 area of rain forest in the Bartica Triangle region of Guyana, South America. The available field data consist of estimates of the composition of twenty-five 100 m × 100 m stands of forest vegetation together with determinations of selected soil characteristics of the sites. Stands were arranged on a square lattice at intervals of 200 m, and, in recording their vegetation, attention was confined to the predominant, woody component of the vegetation. One hundred and seventy species of woody plant were encountered in all, the representation of these being estimated in terms of basal area. Ten soil samples were collected in each site and pooled to form a single, composite sample on which determinations of twenty physical and chemical properties were later made in the laboratory. A comprehensive account of the field sampling procedures and of methods of soil analysis has been given by Ogden (1966).
Robert Gittins

9. Dynamic status of a lowland tropical rain forest

Abstract
The stability of vegetation over time is frequently of interest to plant ecologists. Stability, however, is a property of vegetation which is not readily investigated because its definitive study calls for sequential observation over time. This is especially true of forest vegetation where the time-scales involved are often too long to permit direct observation. One means of circumventing the difficulty in the case of forest communities is to regard size class measurements as if they constituted observations at different times. Differences in size correspond to differences in age which, cautiously interpreted, may be substituted for observations at particular points in time or of particular seral stages. In this way it may be possible to gain insight into the dynamic status of forest vegetation over the short term, at least. The present chapter concerns a study of this kind.
Robert Gittins

10. The structure of grassland vegetation in Anglesey, North Wales

Abstract
Many ecological endeavors lead naturally to comparisons between plant communities with respect to their overall species’ composition. The object of comparative studies of this kind is to obtain through a description of community differences insight into the processes responsible for the differences and an indication of the avenues along which more intensive study might be rewarding. In this chapter the use of canonical analysis in a comparative study is illustrated. Previous work on the composition and structure of grassland vegetation overlying a small (5 ha) area of Carboniferous Limestone in Anglesey, North Wales, showed the vegetation to be composed of three comparatively distinct communities (Gittins, 1965). The communities recognized were described as limestone, neutral and fertile grassland. With the aim of obtaining further insight into the structure of the vegetation a clearer understanding of interrelationships among the communities was sought. Examination of the sample in a geometric space chosen to emphasize both the distinctiveness of the communities and their mutual relationships would be appropriate for this purpose. Such a mapping could be achieved by a canonical analysis in which one set of variables consisted of estimates of species’ representation in the stands surveyed while the variables of the second set consisted of dummy variables specifying the community affiliation of stands — that is, in other words, by a canonical variate analysis.
Robert Gittins

11. The nitrogen nutrition of eight grass species

Abstract
Identification of factors which control the occurrence and representation of plant species in the field is a prime objective of much ecological work. Preliminary indications of the identity of operative factors may be obtained either by direct observation in the field or by use of exploratory multivariate techniques in the analysis of survey data. The next step towards identification involves attempts to substantiate provisional insights of these kinds. The most direct and convincing means of substantiation is provided by examining the response of species to treatments of one or more of the indicated factors in a designed experiment. In this chapter the use of canonical analysis in the analysis of experimental data is described and illustrated. The study concerns the comparative response of eight grass species to five treatment levels of nitrate nitrogen. General field observation had indicated that the species in question differed in their response to variation in soil fertility. The experiment was designed to enable evidence for the differential response of the species to one component of the factor-complex of soil fertility, namely, nitrate nitrogen, to be assessed.
Robert Gittins

12. Herbivore-environment relationships in the Rwenzori National Park, Uganda

Abstract
The Rwenzori National Park lies astride the equator in the Western Rift Valley of Uganda. The Park contains a variety of plant communities, including forests, swamps and several kinds of grassland. The grasslands are noteworthy in that they support the highest recorded large mammal biomass of any natural area of the world (Bourlière, 1965; Coe, Cumming & Phillipson, 1976). An estimate of 294.9 kg ha−1 for the average year-round standing-crop herbivore biomass is given by Field and Laws (1970). Interrelationships between the herbivore species, in particular, and their environment — vegetation, climate, soils and other animal species — have been studied by Field and Laws (1970). Herbivore distribution was found to show considerable heterogeneity in relation to the occurrence of different plant communities, the presence of standing water and the incidence of fire. Table 2 of Field and Laws’ (1970) paper gives estimates of the mean density km−2 of nine herbivore species in ten sites or study areas. These data provided the starting point for the present investigation. Proceeding along different lines from Field and Laws, the data were analyzed in an attempt to clarify relationships within and between herbivores and study areas. Two methods were employed for the purpose — dual scaling and the contingency table analysis of E.J. Williams (1952). At a more fundamental level, the aims of the study were to illustrate the use of canonical analysis in an ecological context involving two categorial variables and to verify connections between dual scaling and contingency table analysis described in Sect. 5.5.1.
Robert Gittins

Appraisal and prospect

Frontmatter

13. Applications: assessment and conclusions

Abstract
The analyses of Chaps. 6–12 were intended to illustrate how and to what extent canonical analysis can contribute towards the attainment of ecological goals. More specifically, the analyses were directed towards revealing something of the varied opportunities offered by canonical analysis in analyzing ecological data; the flexibility of canonical analysis which results under specialization of the variables; and the existence of connections between canonical analysis and other statistical methods widely used in ecology. The present chapter addresses the question of the worth of the analyses in ecological terms. In evaluating the analyses, appeal will be made to information outside the analyses themselves as well as to internal, data-based criteria. Scaling methods generally, of which canonical analysis is one, are difficult to evaluate except by reference to existing substantive knowledge — in other words, that is, by reference to information external to a particular analysis. This process is sometimes known as validation and has the general goal of adding conviction to the results of an analysis. The most convincing evidence for the validity of a result is that it leads to further insight. External assessment will be guided and informed by the use of internal, data-based indices. In particular, reliance will be placed on indices of variance accounted for. However, neither external knowledge nor explained variance offer entirely satisfactory means of evaluating an analysis. As the true relationships of interest are rarely known with certainty, in seeking external support reliance has inevitably to be placed on preconceived ideas as to what the true relationships might be.
Robert Gittins

14. Research issues and future developments

Abstract
We have endeavored in the preceding chapters to draw attention to the richness and flexibility of canonical analysis. The flexibility of the underlying model, in particular, is expressed in part by the number of apparently distinct methods which prove on examination to be special cases of canonical correlation analysis. Canonical variate analysis and dual scaling were both shown abstractly to be equivalent to canonical correlation analysis. The model’s flexibility is complemented by a correspondingly diverse range of application. While, in relation to the worked examples of Chaps. 6–12, it may be that the ability of canonical analysis to have parsimoniously summarized the dependence structures in question is to be attributed to the small scale of these investigations, within the class of small-scale investigations the applicability of canonical analysis was shown to be very wide. Altogether, it seems justifiable to conclude that canonical analysis offers an unusually versatile structure by means of which statistical theory, ecological theory and empirical data can be harnessed to provide insight into the organization of ecological systems.
Robert Gittins

Backmatter

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