Canonical Analysis

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.

1. Introduction.- 1.1 The study of relationships.- 1.2 Objectives.- 1.3 Canonical analysis: overview.- I. Theory.- 2. Canonical correlations and canonical variates.- 2.1 Introduction.- 2.2 Formulation.- 2.3 Derivation of canonical correlation coefficients and canonical variates.- 2.3.1 Eigenanalysis.- 2.3.2 Singular value decompositon.- 2.3.3 Other derivations.- 2.3.4 Concluding remarks.- 2.4 Properties of canonical correlation coefficients, weights and variates.- 2.4.1 Properties of canonical correlation coefficients.- 2.4.2 Properties of canonical weights.- 2.4.3 Properties of canonical variates.- 2.5 Computation.- 2.5.1 Numerical methods.- 2.5.2 Further remarks.- 3. Extensions and generalizations.- 3.1 Introduction.- 3.2 Further interpretive devices.- 3.2.1 Correlations between canonical variates and the original variables.- 3.2.2 Variance extracted by a canonical variate.- 3.2.3 Redundancy.- 3.2.4 Total redundancy.- 3.2.5 Variable communalities.- 3.2.6 Concluding remarks.- 3.3 Extensions and generalizations.- 3.3.1 Redundancy analysis: an alternative to canonical analysis.- 3.3.2 Improving the interpretability of canonical weights.- 3.3.3 Rotation of canonical variates.- 3.3.4 Validation.- 3.3.5 Predicting a criterion of maximum utility.- 3.3.6 Generalizations of canonical analysis.- 3.3.7 Concluding remarks.- 3.4 Hypothesis testing.- 3.4.1 Independence.- 3.4.2 Dimensionality.- 3.4.3 The contribution of particular variables.- 3.4.4 Hypothesis tests for nonnormal data.- 3.4.5 Residuals from a fitted model.- 4. Canonical variate analysis.- 4.1 Introduction.- 4.2 Binary-valued dummy variables.- 4.3 Formulation and derivation.- 4.3.1 Point conceptualizations of NXp and NZq.- 4.3.2 Derivation.- 4.4 Further aspects of canonical variate analysis.- 4.5 Hypothesis testing.- 4.5.1 Equality of g vector-means.- 4.5.2 Dimensionality.- 4.6 Affinities with other methods.- 4.6.1 Canonical variate analysis, multivariate analysis of variance and multiple discriminant analysis.- 4.6.2 Canonical variate analysis and principal component analysis.- 4.7 Imposition of structure.- 4.7.1 Designed comparisons.- 4.7.2 Separating the sources of variation.- 4.7.3 Further comments.- 4.8 Concluding remarks.- 5. Dual scaling.- 5.1 Introduction.- 5.2 Formulation and derivation.- 5.2.1 Maximizing the correlation between rows and columns.- 5.2.2 Maximizing the separation between rows and columns.- 5.3 Further aspects of dual scaling.- 5.4 Hypothesis testing.- 5.4.1 Independence.- 5.4.2 Dimensionality.- 5.5 Affinities with other methods.- 5.5.1 Dual scaling and the analysis of contingency tables.- 5.5.2 Dual scaling, correspondence analysis and principal component analysis.- 5.6 Relationships among statistical methods.- 5.7 Concluding remarks.- II. Applications.- General introduction.- 6. Experiment 1: an investigation of spatial variation.- 6.1 Introduction.- 6.2 Results.- 6.2.1 The canonical correlation coefficients.- 6.2.2 Independence.- 6.2.3 Dimensionality.- 6.2.4 The canonical variates.- 6.2.5 Variable communalities.- 6.3 Conclusions.- 7. Experiment 2: soil-species relationships in a limestone grassland community.- 7.1 Introduction.- 7.2 Results.- 7.2.1 The canonical correlation coefficients.- 7.2.2 Independence.- 7.2.3 Dimensionality.- 7.2.4 The canonical variates.- 7.2.5 Variable communalities.- 7.3 Conclusions.- 8. Soil-vegetation relationships in a lowland tropical rain forest.- 8.1 Introduction.- 8.2 Results.- 8.2.1 The canonical correlation coefficients.- 8.2.2 Independence.- 8.2.3 Dimensionality.- 8.2.4 The canonical variates.- 8.2.5 Variable communalities.- 8.3 Ecological assessment of the results.- 8.4 Conclusions.- 9. Dynamic status of a lowland tropical rain forest.- 9.1 Introduction.- 9.2 Results.- 9.2.1 The canonical correlation coefficients.- 9.2.2 Independence.- 9.2.3 Dimensionality.- 9.2.4 The canonical variates.- 9.2.5 Variable communalities.- 9.3 Ecological assessment of the results.- 9.4 Conclusions.- 10. The structure of grassland vegetation in Anglesey, North Wales.- 10.1 Introduction.- 10.2 Results.- 10.2.1 The canonical correlation coefficients.- 10.2.2 Equality of community centroids.- 10.2.3 Collinearity.- 10.2.4 The canonical variates.- 10.2.5 Variable communalities.- 10.3 Ecological assessment of the results.- 10.4 Conclusions.- 11. The nitrogen nutrition of eight grass species.- 11.1 Introduction.- 11.2 Multivariate analysis of variance.- 11.2.1 Results.- 11.2.2 Designed comparisons.- 11.3 Canonical variate analysis.- 11.3.1 Results.- 11.4 Relationships between multivariate analysis of variance, discriminant analysis and canonical variate analysis.- 11.5 Ecological assessment of the results.- 11.6 Conclusions.- 12. Herbivore-environment relationships in the Rwenzori National Park, Uganda.- 12.1 Introduction.- 12.2 Contingency table analysis.- 12.2.1 Results.- 12.3 Dual scaling.- 12.3.1 Results.- 12.4 Relationships between contingency table analysis and dual scaling.- 12.5 Ecological assessment of the results.- 12.6 Conclusions.- III. Appraisal and Prospect.- 13. Applications: assessment and conclusions.- 13.1 Introduction.- 13.2 Assessment.- 13.3 Conclusions.- 14. Research issues and future developments.- 14.1 Introduction.- 14.2 Data collection.- 14.2.1 Choice of variables.- 14.2.2 Experimental design.- 14.3 Initial data exploration.- 14.3.1 Homogeneity.- 14.3.2 Assessing joint distribution.- 14.3.3 Re-expressing variables.- 14.4 Potential data problems.- 14.4.1 Outlying or influential observations.- 14.4.2 Long-tailed distributions.- 14.4.3 Collinearity.- 14.5 Statistical assessment.- 14.5.1 Residuals in canonical analysis.- 14.5.2 Does the model fit?.- 14.5.3 Stability of results.- 14.5.4 Miscellanea.- 14.6 Concluding remarks.- Appendices.- A.1 Multivariate regression.- A.2 Data sets used in worked applications.- A.3 Species composition of a limestone grassland community.- References.- Species’ index.- Author index.