Table Of ContentBiomathematics
Volume 12
Managing Editor
S.A. Levin
Editorial Board
M. Arbib H.J. Bremermann J. Cowan W.M. Hirsch
S. Karlin J. Keller K. Krickeberg R.C. Lewontin
R.M. May J.D. Murray L.A. Segel
R Gittins
Canonical Analysis
A Review with Applications in Ecology
With 16 Figures
Springer-Verlag
Berlin Heidelberg NewY ork Tokyo
Robert Gittins
Department of Plant Sciences
University of Western Ontario
London, Ontario, Canada N6A 5B7
AMS-MOS Classification (1980): 62-02, 62H20, 62H 17,
62J05, 92-02, 92A17
ISBN -13 :978-3-642-69880-4 e-ISBN -13 :978-3-642-69878-1
DOl: 10.1007/978-3-642-69878-1
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use of illustra
tions, broadcasting, reproduction by photocopying machine or similar means, and stor
age in data banks. Under § 54 of the German Copyright Law where copies are made
for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1985
Softcover reprint of the hardcover 1st edition 1985
2141/3020-543210
For my wife Ong
and daughter Genevieve
Preface
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. The applicability of canonical analysis is of
course not confined to plant ecology. Indeed, a surprisingly wide class of
problems that arise in such allied disciplines as animal ecology, biogeography,
climatology, epidemiology, palaeoclimatology, geology, psychology, taxonomy,
agriculture, forestry and fisheries can be formulated and analyzed in terms
of relationships between two or more sets of variables. Potential applications
in these and other fields no doubt will occur to readers with interests in
particular areas.
The book is intended primarily for ecologists and others interested in rela
tionships between variables of different kinds. More generally, the aim is to
raise issues encountered in analyzing systems of this sort so as to encourage
interaction between statisticians and ecologists. The work may appeal to statisti
cians by exposing areas where further theoretical work is needed and by provid
ing a source of worked numerical examples. The book may also serve a useful
purpose by drawing the attention of ecologists to opportunities afforded by
canonical analysis not only as a means of addressing a varied assortment of
ecological problems, but also as a unifying structure in terms of which many
statistical methods can be compared and related.
VIII Preface
The book has been aimed at the graduate or research level. Suitable prerequi
sites would be a course in linear algebra and one in multivariate analysis. Nahik
ian's A Modern Algebra for Biologists or G. Strang's Linear Algebra and its
Applications would provide much of the necessary mathematical background.
With respect to statistical requirements, P.E. Green's Analyzing Multivariate
Data covers the relevant material at a level appropriate to understanding the
book. The same author's Mathematical Tools for Applied Multivariate Analysis
provides a useful exposition of the same ground from a geometric and more
intuitively appealing viewpoint. The text is in three parts. Part I reviews the
theoretical foundations of the subject, Part II consists of accounts of applications
of canonical analysis to real data, and Part III is devoted to assessing the
applications and to a consideration of the prospects for canonical analysis in
ecology and more generally. Chapter 1 provides a general introduction to the
study of relationships between two sets of variables. Chapter 2 deals with the
formulation and derivation of canonical analysis before proceeding to review
properties of the solution and the computation of canonical correlations and
variates. In Chap. 3 several numerical indices which are helpful in interpreting
the results of analysis are introduced and a number of extensions and generaliza
tions of canonical analysis described. The chapter closes with an account of
hypothesis testing in canonical analysis. Two special cases of canonical analysis
important in applications form the subject matter of Chaps. 4 and 5, namely
canonical variate analysis and dual scaling, respectively. The second of these
chapters concludes the first part of the book.
The applications of Part II cover field surveys and designed experiments
though the balance between the two is heavily in favor of the former. Chapter
6 describes a canonical analysis of spatial variation in the representation of
three plant species. Soil/vegetation relationships are the focus of attention in
Chaps. 7 and 8. In Chap. 9 canonical analysis is applied in a study of the
temporal stability of vegetation. Chapter 10 deals with the analysis of vegetation
structure. Chapter 11 reports a comparative study of the multivariate responses
of several grass species to an imposed treatment regime in a controlled experi
ment. In Chap. 12, the final chapter of Part II, canonical analysis is used to
explore relationships between the occurrence of several species of large herbi
vores and environmental conditions in an East African rangeland. Part III con
sists of Chaps. 13 and 14. In Chap. 13 the results of the seven applications
are reviewed and the worth of the analyses in ecological terms assessed. Chapter
14 is to some extent speculative. In recent years statisticians have devoted a
great deal of attention to two topics which have a strong bearing on canonical
analysis - statistical data analysis and linear regression methodology. The spirit of
this work is well-conveyed by Gnanadesikan's text Methods for Statistical Data
Analysis of Multivariate Observations. In Chap. 14 developments in statistical
data analysis and modern regression methodology are reviewed and an attempt
is made to embed canonical analysis in the context provided by them. In effect,
the chapter seeks to anticipate and delineate changes in the way in which canoni
cal analysis will come to be used during the course of the next decade. The
text closes with Appendices devoted to multivariate regression and to tabulations
of the data and other information pertaining to the worked examples.
Preface IX
Part of the book was written while I was visiting the Department of Plant
Sciences at the University of Western Ontario. My debt to Professor L. Orl6ci
for making my visit to Canada possible and for his support and encouragement
during this period is considerable. I am also much indebted to Dr. N.A. Camp
bell of the Division of Mathematics and Statistics, CSIRO, Australia who read
the theoretical chapters in manuscript and made valuable suggestions for their
improvement. Portions of the manuscript were also read by Professor S.W.
Nash of the Department of Mathematics, University of British Columbia and
by Mr. J.C. Gower of the Statistics Department, Rothamsted Experimental
Station. Both suggested improvements for which I am most grateful. Naturally,
responsibility for all errors and obscurities which remain is mine. Dr. J. Ogden
kindly placed the rain forest data used in the analyses of Chaps. 8 and 9 at
my disposal and helped in interpreting the results obtained. Finally, it is a
pleasure to thank Mrs. Eleanor Lowther for her skill and patience in so expertly
preparing the typescript for publication.
October 1984 Robert Gittins
Contents
1. Introduction . . . . . . . . 1
1.1 The study of relationships 1
1.2 Objectives ..... . 3
1.3 Canonical analysis: overview 4
PART I. THEORY 11
2. Canonical correlations and canonical variates 13
2.1 Introduction 13
2.2 Formulation 13
2.3 Derivation of canonical correlation coefficients and canonical
variates ........... . 15
2.3.1 Eigenanalysis . . . . . . . 15
2.3.2 Singular value decompositon 17
2.3.3 Other derivations .... . 18
2.3.4 Concluding remarks ... . 20
2.4 Properties of canonical correlation coefficients, weights and
variates ................... 22
2.4.1 Properties of canonical correlation coefficients 22
2..4.2 Properties of canonical weights 25
2.4.3 Properties of canonical variates 27
2.5 Computation . . . . . 31
2.5.1 Numerical methods 31
2.5.2 Further remarks . 35
3. Extensions and generalizations 37
3.1 Introduction ..... 37
3.2 Further interpretive devices 37
3.2.1 Correlations between canonical variates and the original
variables . . . . . . . . . . . . . . 38
3.2.2 Variance extracted by a canonical variate 40
3.2.3 Redundancy 40
3.2.4 Total redundancy .......... 41
XII Contents
3.2.5 Variable communalities 42
3.2.6 Concluding remarks 43
3.3 Extensions and generalizations 45
3.3.1 Redundancy analysis: an alternative to canonical
analysis . 45
3.3.2 Improving the interpretability of canonical weights 45
3.3.3 Rotation of canonical variates 50
3.3.4 Validation 50
3.3.5 Predicting a criterion of maximum utility 52
3.3.6 Generalizations of canonical analysis 53
3.3.7 Concluding remarks 55
3.4 Hypothesis testing 56
3.4.1 Independence 56
3.4.2 Dimensionality 59
3.4.3 The contribution of particular variables 62
3.4.4 Hypothesis tests for nonnormal data 63
3.4.5 Residuals from a fitted model 63
4. Canonical variate analysis 67
4.1 Introduction 67
4.2 Binary-valued dummy variables 68
4.3 Formulation and derivation 69
4.3.1 Point conceptualizations of and 70
NXp NZq
4.3.2 Derivation 73
4.4 Further aspects of canonical variate analysis 75
4.5 Hypothesis testing 80
4.5.1 Equality of g vector-means 81
4.5.2 Dimensionality 81
4.6 Affinities with other methods 82
4.6.1 Canonical variate analysis, multivariate analysis
of variance and multiple discriminant analysis 82
4.6.2 Canonical variate analysis and principal component
analysis . 87
4.7 Imposition of structure 88
4.7.1 Designed comparisons 89
4.7.2 Separating the sources of variation 90
4.7.3 Further comments 91
4.8 Concluding remarks 92
5. Dual scaling 96
5.1 Introduction 96
5.2 Formulation and derivation 98
5.2.1 Maximizing the correlation between rows and columns 99
5.2.2 Maximizing the separation between rows and columns 103
