Multivariate Statistics for Wildlife and Ecology Research

1 Introduction and Overview.- 1.1 Objectives.- 1.2 Multivariate Statistics: An Ecological Perspective.- 1.3 Multivariate Description and Inference.- 1.4 Multivariate Confusion!.- 1.5 Types of Multivariate Techniques.- 1.5.1 Ordination.- 1.5.2 Cluster Analysis.- 1.5.3 Discriminant Analysis.- 1.5.4 Canonical Correlation Analysis.- 2 Ordination: Principal Components Analysis.- 2.1 Objectives.- 2.2 Conceptual Overview.- 2.2.1 Ordination.- 2.2.2 Principal Components Analysis (PCA).- 2.3 Geometric Overview.- 2.4 The Data Set.- 2.5 Assumptions.- 2.5.1 Multivariate Normality.- 2.5.2 Independent Random Sample and the Effects of Outliers.- 2.5.3 Linearity.- 2.6 Sample Size Requirements.- 2.6.1 General Rules.- 2.6.2 Specific Rules.- 2.7 Deriving the Principal Components.- 2.7.1 The Use of Correlation and Covariance Matrices.- 2.7.2 Eigenvalues and Associated Statistics.- 2.7.3 Eigenvectors and Scoring Coefficients.- 2.8 Assessing the Importance of the Principal Components.- 2.8.1 Latent Root Criterion.- 2.8.2 Scree Plot Criterion.- 2.8.3 Broken Stick Criterion.- 2.8.4 Relative Percent Variance Criterion.- 2.8.5 Significance Tests.- 2.9 Interpreting the Principal Components.- 2.9.1 Principal Component Structure.- 2.9.2 Significance of Principal Component Loadings.- 2.9.3 Interpreting the Principal Component Structure.- 2.9.4 Communality.- 2.9.5 Principal Component Scores and Associated Plots.- 2.10 Rotating the Principal Components.- 2.11 Limitations of Principal Components Analysis.- 2.12 R-Factor Versus Q-Factor Ordination.- 2.13 Other Ordination Techniques.- 2.13.1 Polar Ordination.- 2.13.2 Factor Analysis.- 2.13.3 Nonmetric Multidimensional Scaling.- 2.13.4 Reciprocal Averaging.- 2.13.5 Detrended Correspondence Analysis.- 2.13.6 Canonical Correspondence Analysis.- Appendix 2.1.- 3 Cluster Analysis.- 3.1 Objectives.- 3.2 Conceptual Overview.- 3.3 The Definition of Cluster.- 3.4 The Data Set.- 3.5 Clustering Techniques.- 3.6 Nonhierarchical Clustering.- 3.6.1 Polythetic Agglomerative Nonhierarchical Clustering.- 3.6.2 Polythetic Divisive Nonhierarchical Clustering.- 3.7 Hierarchical Clustering.- 3.7.1 Polythetic Agglomerative Hierarchical Clustering.- 3.7.2 Polythetic Divisive Hierarchical Clustering.- 3.8 Evaluating the Stability of the Cluster Solution.- 3.9 Complementary Use of Ordination and Cluster Analysis.- 3.10 Limitations of Cluster Analysis.- Appendix 3.1.- 4 Discriminant Analysis.- 4.1 Objectives.- 4.2 Conceptual Overview.- 4.2.1 Overview of Canonical Analysis of Discriminance.- 4.2.2 Overview of Classification.- 4.2.3 Analogy with Multiple Regression Analysis and Multivariate Analysis of Variance.- 4.3 Geometric Overview.- 4.4 The Data Set.- 4.5 Assumptions.- 4.5.1 Equality of Variance-Covariance Matrices.- 4.5.2 Multivariate Normality.- 4.5.3 Singularities and Multicollinearity.- 4.5.4 Independent Random Sample and the Effects of Outliers.- 4.5.5 Prior Probabilities Are Identifiable.- 4.5.6 Linearity 153.- 4.6 Sample Size Requirements.- 4.6.1 General Rules.- 4.6.2 Specific Rules.- 4.7 Deriving the Canonical Functions.- 4.7.1 Stepwise Selection of Variables.- 4.7.2 Eigenvalues and Associated Statistics.- 4.7.3 Eigenvectors and Canonical Coefficients.- 4.8 Assessing the Importance of the Canonical Functions.- 4.8.1 Relative Percent Variance Criterion.- 4.8.2 Canonical Correlation Criterion.- 4.8.3 Classification Accuracy.- 4.8.4 Significance Tests.- 4.8.5 Canonical Scores and Associated Plots.- 4.9 Interpreting the Canonical Functions.- 4.9.1 Standardized Canonical Coefficients.- 4.9.2 Total Structure Coefficients.- 4.9.3 Covariance-Controlled Partial F-Ratios.- 4.9.4 Significance Tests Based on Resampling Procedures.- 4.9.5 Potency Index.- 4.10 Validating the Canonical Functions.- 4.10.1 Split-Sample Validation.- 4.10.2 Validation Using Resampling Procedures.- 4.11 Limitations of Discriminant Analysis.- Appendix 4.1.- 5 Canonical Correlation Analysis.- 5.1 Objectives.- 5.2 Conceptual Overview.- 5.3 Geometric Overview.- 5