An Introduction to Correspondence Analysis

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 gammaij . 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 tau 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 pii|j for Nominal Variables 78 3.3.1 The Generalised SVD of pii|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 gammaij . 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 pii|j for Ordinal and Nominal Variables 132 5.3.1 The Hybrid Decomposition of pii|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 gammaijk . 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 gammaijk . 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 gammaijk . 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 pii|j,k and the Partition of tauM 194 7.5.1 The Marcotrochino Residual 194 7.5.2 The Partition of tauM 196 7.5.3 Partition of tauM for the Alligator Data 197 7.6 Non-symmetrical Multi-way Correspondence Analysis 198 7.6.1 Tucker3 Decomposition of pii|j,k 198 7.6.2 Tucker3 Decomposition of pii|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