An Introduction to Correspondence Analysis

Master the fundamentals of correspondence analysis with this illuminating resource

An Introduction to Correspondence Analysis assists researchers in improving their familiarity with the concepts, terminology, and application of several variants of correspondence analysis. The accomplished academics and authors deliver a comprehensive and insightful treatment of the fundamentals of correspondence analysis, including the statistical and visual aspects of the subject.

Written in three parts, the book begins by offering readers a description of two variants of correspondence analysis that can be applied to two-way contingency tables for nominal categories of variables. Part Two shifts the discussion to categories of ordinal variables and demonstrates how the ordered structure of these variables can be incorporated into a correspondence analysis. Part Three describes the analysis of multiple nominal categorical variables, including both multiple correspondence analysis and multi-way correspondence analysis.

Readers will benefit from explanations of a wide variety of specific topics, for example:

• Simple correspondence analysis, including how to reduce multidimensional space, measuring symmetric associations with the Pearson Ratio, constructing low-dimensional displays, and detecting statistically significant points
• Non-symmetrical correspondence analysis, including quantifying asymmetric associations
• Simple ordinal correspondence analysis, including how to decompose the Pearson Residual for ordinal variables
• Multiple correspondence analysis, including crisp coding and the indicator matrix, the Burt Matrix, and stacking
• Multi-way correspondence analysis, including symmetric multi-way analysis

Perfect for researchers who seek to improve their understanding of key concepts in the graphical analysis of categorical data, An Introduction to Correspondence Analysis will also assist readers already familiar with correspondence analysis who wish to review the theoretical and foundational underpinnings of crucial concepts.

Eric J. Beh is Professor of Statistics at the School of Mathematical & Physical Sciences at the University of Newcastle, Australia. He has been actively researching in many areas of categorical data analysis including ecological inference, measures of association and categorical models. For the past 25 years his research has focused primarily on the technical, computational and practical development of correspondence analysis. He has over 100 publications and, with Rosaria Lombardo, has authored Correspondence Analysis: Theory, Methods and New Strategies with Wiley. Together, they have given short courses and workshops around the world on this topic

 Rosaria Lombardo is Associate Professor of Statistics at the Department of Economics of the University of Campania “L. Vanvitelli”, Italy. Her research interests include non-linear multivariate data analysis, quantification theory and, in particular, correspondence analysis and data visualization. Since receiving her PhD in Computational Statistics and Applications at the University of Naples “Federico II”, she has authored over 100 publications including those in Statistical Science, Psychometrika, Computational Statistics & Data Analysis, and the Journal of Statistical Planning and Inference.

Dedication iii

Preface xi

1 Introduction 1

1.1 Data Visualisation 1

1.2 Correspondence Analysis in a “Nutshell” 3

1.3 Data Sets 4

1.3.1 Traditional European Food Data 4

1.3.2 Temperature Data 4

1.3.3 Shoplifting Data 5

1.3.4 Alligator Data 6

1.4 Symmetrical vs Asymmetrical Association 7

1.5 Notation 9

1.5.1 The Two-way Contingency Table 9

1.5.2 The Three-way Contingency Table 10

1.6 Formal Test of Symmetrical Association 11

1.6.1 Test of Independence for Two-way Contingency Tables 11

1.6.2 The Chi-squared Statistic for a Two-way Table 12

1.6.3 Analysis of the Traditional European Food Data 12

1.6.4 The Chi-squared Statistic for a Three-way Table 14

1.6.5 Analysis of the Alligator Data 15

1.7 Formal Test of Asymmetrical Association 15

1.7.1 Test of Predictability for Two-way Contingency Tables 15

1.7.2 The Goodman-Kruskal tau Index 16

1.7.3 Analysis of the Traditional European Food Data 17

1.7.4 Test of Predictability for Three-way Contingency Tables 17

1.7.5 Marcotorchino’s Index 18

1.7.6 Analysis of the Alligator Data 19

1.7.7 The Gray-Williams Index & Delta Index 19

1.8 Correspondence Analysis and R 20

1.9 Overview of the Book 24

Part One Classical Analysis of Two Categorical Variables 27

2 Simple Correspondence Analysis 29

2.1 Introduction 29

2.2 Reducing Multidimensional Space 30

2.2.1 Profiles Cloud of Points 30

2.2.2 Profiles for the Traditional European Food Data 31

2.2.3 Weighted Centred Profiles 34

2.3 Measuring Symmetric Association 39

2.3.1 The Pearson Ratio 39

2.3.2 Analysis of the Traditional European Food Data 40

2.4 Decomposing the Pearson Residual for Nominal Variables 42

2.4.1 The Generalised SVD of γij − 1 42

2.4.2 SVD of the Pearson Ratio’s 44

2.4.3 GSVD and the Traditional European Food Data 45

2.5 Constructing a Low-Dimensional Display 45

2.5.1 Standard Coordinates 45

2.5.2 Principal Coordinates 47

2.6 Practicalities of the Low-Dimensional Plot 51

2.6.1 The Two-Dimensional Correspondence Plot 51

2.6.2 What is NOT Being Shown in a Two-Dimensional Correspondence

Plot? 54

2.6.3 The Three-Dimensional Correspondence Plot 58

2.7 The Biplot Display 58

2.7.1 Definition 58

2.7.2 Isometric Biplots of the Traditional European Food Data 61

2.7.3 What is NOT Being Shown in a Two-Dimensional Biplot? 64

2.8 The Case for No Visual Display 64

2.9 Detecting Statistically Significant Points 65

2.9.1 Confidence Circles and Ellipses 65

2.9.2 Confidence Ellipses for the Traditional European Food Data 66

2.10 Approximate P-values 70

2.10.1 The Hypothesis Test and its P-value 70

2.10.2 P-values and the Traditional European Food Data 71

2.11 Final Comments 71

3 Non-Symmetrical Correspondence Analysis 73

3.1 Introduction 73

3.2 Quantifying Asymmetric Association 74

3.2.1 The Goodman-Kruskal tau Index 74

3.2.2 The τ Index and the Traditional European Food Data 74

3.2.3 Weighted Centred Column Profile 75

3.2.4 Profiles of the Traditional European Food Data 75

3.3 Decomposing πi|j

for Nominal Variables 78

3.3.1 The Generalised SVD of πi|j 78

3.3.2 GSVD and the Traditional Food Data 79

3.4 Constructing a Low-Dimensional Display 81

3.4.1 Standard Coordinates 81

3.4.2 Principal Coordinates 82

3.5 Practicalities of the Low-Dimensional Plot 85

3.5.1 The Two-Dimensional Correspondence Plot 85

3.5.2 The Three-Dimensional Correspondence Plot 87

3.6 The Biplot Display 91

3.6.1 Definition 91

3.6.2 The Column Isometric Biplot for the Traditional Food Data 92

3.6.3 The Three-Dimensional Biplot 95

3.7 Detecting Statistically Significant Points 95

3.7.1 Confidence Circles and Ellipses 95

3.7.2 Confidence Ellipses for the Traditional Food Data 96

3.8 Final Comments 98

Part Two Ordinal Analysis of Two Categorical Variables 101

4 Simple Ordinal Correspondence Analysis 103

4.1 Introduction 103

4.2 A Simple Correspondence Analysis of the Temperature Data 104

4.3 On the Mean and Variation of Profiles with Ordered Categories 106

4.3.1 Profiles of the Temperature Data 106

4.3.2 Defining Scores 107

4.3.3 On the Mean of the Profiles 110

4.3.4 On the Variation of the Profiles 111

4.3.5 Mean & Variation of Profiles for the Temperature Data 112

4.4 Decomposing the Pearson Residual for Ordinal Variables 114

4.4.1 The Bivariate Moment Decomposition of γij − 1 114

4.4.2 BMD and the Temperature Data 116

4.5 Constructed a Low-Dimensional Display 119

4.5.1 Standard Coordinates 119

4.5.2 Principal Coordinates 119

4.5.3 Practicalities of the Ordered Principal Coordinates 123

4.6 The Biplot Display 123

4.6.1 Definition 123

4.6.2 Ordered Column Isometric Biplot 123

4.6.3 Ordered Row Isometric Biplot 124

4.6.4 Ordered Isometric Biplots for the Temperature Data 124

4.7 Final Comments 127

5 Ordered Non-symmetrical Correspondence Analysis 129

5.1 Introduction 129

5.2 The Goodman-Kruskal tau Index Revisited 130

5.3 Decomposing πi|j

for Ordinal and Nominal Variables 132

5.3.1 The Hybrid Decomposition of πi|j 132

5.3.2 Hybrid decomposition and the Shoplifting Data 135

5.4 Constructing a Low-Dimensional Display 138

5.4.1 Standard Coordinates 138

5.4.2 Principal Coordinates 138

5.5 The Biplot 139

5.5.1 An Overview 139

5.5.2 Column Isometric Biplot 139

5.5.3 Column Isometric Biplot of the Shoplifting Data 140

5.5.4 Row Isometric Biplot 141

5.5.5 Row Isometric Biplot of the Shoplifting Data 142

5.5.6 Distance Measures and the Row Isometric Biplots 145

5.6 Some Final Words 146

Part Three Analysis of Multiple Categorical Variables 147

6 Multiple Correspondence Analysis 149

6.1 Introduction 149

6.2 Crisp Coding and the Indicator Matrix 150

6.2.1 Crisp Coding 150

6.2.2 The Indicator Matrix 150

6.2.3 Crisp Coding and the Alligator data 151

6.2.4 Application of Multiple Correspondence Analysis using the Indicator

Matrix 151

6.3 The Burt Matrix 157

6.4 Stacking 162

6.4.1 A Definition 162

6.4.2 Stacking and the Alligator Data - Lake(Size)×Food 162

6.4.3 Stacking and the Alligator Data - Food(Size)×Lake 166

6.5 Final Comments 167

7 Multi-way Correspondence Analysis 169

7.1 An Introduction 169

7.2 Pearson’s Residual γijk − 1 and the Partition of X2 170

7.2.1 The Pearson Residual 170

7.2.2 The Partition of X2 171

7.2.3 Partition of X2

for the Alligator Data 171

7.3 Symmetric Multi-way Correspondence Analysis 173

7.3.1 Tucker3 Decomposition of γijk − 1 173

7.3.2 T3D and the Analysis of Two Variables 176

7.3.3 On the Choice of the Number of Components 177

7.3.4 Tucker3 Decomposition of γijk − 1 and the Alligator Data 178

7.4 Constructing a Low-Dimensional Display 182

7.4.1 Principal Coordinates 182

7.4.2 The Interactive Biplot 182

7.4.3 Column-Tube Interactive Biplot for the Alligator Data 188

7.4.4 Row Interactive Biplot for the Alligator Data 192

7.5 The Marcotorchino Residual πi|j,k and the Partition of τM 194

7.5.1 The Marcotrochino Residual 194

7.5.2 The Partition of τM 196

7.5.3 Partition of τM for the Alligator Data 197

7.6 Non-symmetrical Multi-way Correspondence Analysis 198

7.6.1 Tucker3 Decomposition of πi|j,k 198

7.6.2 Tucker3 Decomposition of πi|j,k and the Alligator Data 200

7.7 Constructing a Low-Dimensional Display 201

7.7.1 On the Choice of Coordinates 201

7.7.2 Column-Tube Interactive Biplot for the Alligator Data 202

7.8 Final Comments 206

References 208

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