Introduction to Correspondence Analysis

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 published by 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.

• Preface xiii1 Introduction 11.1 Data Visualisation 11.2 Correspondence Analysis in a “Nutshell” 31.3 Data Sets 41.3.1 Traditional European Food Data 41.3.2 Temperature Data 61.3.3 Shoplifting Data 61.3.4 Alligator Data 71.4 Symmetrical Versus Asymmetrical Association 81.5 Notation 101.5.1 The Two-way Contingency Table 101.5.2 The Three-way Contingency Table 111.6 Formal Test of Symmetrical Association 121.6.1 Test of Independence for Two-way Contingency Tables 121.6.2 The Chi-squared Statistic for a Two-way Table 131.6.3 Analysis of the Traditional European Food Data 131.6.4 The Chi-squared Statistic for a Three-way Table 151.6.5 Analysis of the Alligator Data 161.7 Formal Test of Asymmetrical Association 171.7.1 Test of Predictability for Two-way Contingency Tables 171.7.2 The Goodman–Kruskal tau Index 171.7.3 Analysis of the Traditional European Food Data 181.7.4 Test of Predictability for Three-way Contingency Tables 191.7.5 Marcotorchino’s Index 191.7.6 Analysis of the Alligator Data 201.7.7 The Gray–Williams Index and Delta Index 211.8 Correspondence Analysis and R 221.9 Overview of the Book 25Part I Classical Analysis of Two Categorical Variables 292 Simple Correspondence Analysis 312.1 Introduction 312.2 Reducing Multi-dimensional Space 322.2.1 Profiles Cloud of Points 322.2.2 Profiles for the Traditional European Food Data 332.2.3 Weighted Centred Profiles 332.3 Measuring Symmetric Association 392.3.1 The Pearson Ratio 392.3.2 Analysis of the Traditional European Food Data 402.4 Decomposing the Pearson Residual for Nominal Variables 412.4.1 The Generalised SVD of 𝛾ij − 1 412.4.2 SVD of the Pearson Ratio’s 442.4.3 GSVD and the Traditional European Food Data 442.5 Constructing a Low-Dimensional Display 462.5.1 Standard Coordinates 462.5.2 Principal Coordinates 472.6 Practicalities of the Low-Dimensional Plot 502.6.1 The Two-Dimensional Correspondence Plot 502.6.2 What is NOT Being Shown in a Two-Dimensional Correspondence Plot? 542.6.3 The Three-Dimensional Correspondence Plot 572.7 The Biplot Display 592.7.1 Definition 592.7.2 Isometric Biplots of the Traditional European Food Data 602.7.3 What is NOT Being Shown in a Two-Dimensional Biplot? 632.8 The Case for No Visual Display 632.9 Detecting Statistically Significant Points 642.9.1 Confidence Circles and Ellipses 642.9.2 Confidence Ellipses for the Traditional European Food Data 652.10 Approximate p-values 692.10.1 The Hypothesis Test and its p-value 692.10.2 P-values and the Traditional European Food Data 702.11 Final Comments 703 Non-Symmetrical Correspondence Analysis 713.1 Introduction 713.2 Quantifying Asymmetric Association 723.2.1 The Goodman–Kruskal tau Index 723.2.2 The 𝜏 Index and the Traditional European Food Data 723.2.3 Weighted Centred Column Profile 733.2.4 Profiles of the Traditional European Food Data 733.3 Decomposing 𝜋i|j for Nominal Variables 763.3.1 The Generalised SVD of 𝜋i|j 763.3.2 GSVD and the Traditional Food Data 773.4 Constructing a Low-Dimensional Display 793.4.1 Standard Coordinates 793.4.2 Principal Coordinates 793.5 Practicalities of the Low-Dimensional Plot 823.5.1 The Two-Dimensional Correspondence Plot 823.5.2 The Three-Dimensional Correspondence Plot 853.6 The Biplot Display 893.6.1 Definition 893.6.2 The Column Isometric Biplot for the Traditional Food Data 903.6.3 The Three-Dimensional Biplot 913.7 Detecting Statistically Significant Points 923.7.1 Confidence Circles and Ellipses 923.7.2 Confidence Ellipses for the Traditional Food Data 933.8 Final Comments 96Part II Ordinal Analysis of Two Categorical Variables 994 Simple Ordinal Correspondence Analysis 1014.1 Introduction 1014.2 A Simple Correspondence Analysis of the Temperature Data 1024.3 On the Mean and Variation of Profiles with Ordered Categories 1044.3.1 Profiles of the Temperature Data 1044.3.2 Defining Scores 1054.3.3 On the Mean of the Profiles 1074.3.4 On the Variation of the Profiles 1084.3.5 Mean and Variation of Profiles for the Temperature Data 1084.4 Decomposing the Pearson Residual for Ordinal Variables 1114.4.1 The Bivariate Moment Decomposition of 𝛾ij − 1 1114.4.2 BMD and the Temperature Data 1134.5 Constructing a Low-Dimensional Display 1154.5.1 Standard Coordinates 1154.5.2 Principal Coordinates 1164.5.3 Practicalities of the Ordered Principal Coordinates 1194.6 The Biplot Display 1204.6.1 Definition 1204.6.2 Ordered Column Isometric Biplot 1204.6.3 Ordered Row Isometric Biplot 1204.6.4 Ordered Isometric Biplots for the Temperature Data 1214.7 Final Comments 1245 Ordered Non-symmetrical Correspondence Analysis 1255.1 Introduction 1255.2 The Goodman–Kruskal tau Index Revisited 1265.3 Decomposing 𝜋i|j for Ordinal and Nominal Variables 1285.3.1 The Hybrid Decomposition of 𝜋i|j 1285.3.2 Hybrid Decomposition and the Shoplifting Data 1315.4 Constructing a Low-Dimensional Display 1335.4.1 Standard Coordinates 1335.4.2 Principal Coordinates 1345.5 The Biplot 1355.5.1 An Overview 1355.5.2 Column Isometric Biplot 1355.5.3 Column Isometric Biplot of the Shoplifting Data 1355.5.4 Row Isometric Biplot 1375.5.5 Row Isometric Biplot of the Shoplifting Data 1375.5.6 Distance Measures and the Row Isometric Biplots 1405.6 Some FinalWords 141Part III Analysis of Multiple Categorical Variables 1436 Multiple Correspondence Analysis 1456.1 Introduction 1456.2 Crisp Coding and the Indicator Matrix 1466.2.1 Crisp Coding 1466.2.2 The Indicator Matrix 1466.2.3 Crisp Coding and the Alligator Data 1476.2.4 Application of Multiple Correspondence Analysis using the Indicator Matrix 1486.3 The Burt Matrix 1526.4 Stacking 1566.4.1 A Definition 1566.4.2 Stacking and the Alligator Data – Lake(Size)× Food 1566.4.3 Stacking and the Alligator Data – Food(Size)× Lake 1596.5 Final Comments 1617 Multi-way Correspondence Analysis 1637.1 An Introduction 1637.2 Pearson’s Residual 𝛾ijk − 1 and the Partition of X2 1647.2.1 The Pearson Residual 1647.2.2 The Partition of X2 1657.2.3 Partition of X2 for theAlligator Data 1657.3 Symmetric Multi-way Correspondence Analysis 1677.3.1 Tucker3 Decomposition of 𝛾ijk − 1 1677.3.2 T3D and the Analysis of Two Variables 1707.3.3 On the Choice of the Number of Components 1717.3.4 Tucker3 Decomposition of 𝛾ijk − 1 and the Alligator Data 1717.4 Constructing a Low-Dimensional Display 1757.4.1 Principal Coordinates 1757.4.2 The Interactive Biplot 1767.4.3 Column-Tube Interactive Biplot for the Alligator Data 1817.4.4 Row Interactive Biplot for the Alligator Data 1857.5 The Marcotorchino Residual 𝜋i|j,k and the Partition of 𝜏M 1887.5.1 The Marcotrochino Residual 1887.5.2 The Partition of 𝜏M 1897.5.3 Partition of 𝜏M for the Alligator Data 1907.6 Non-symmetrical Multi-way Correspondence Analysis 1917.6.1 Tucker3 Decomposition of 𝜋i|j,k 1917.6.2 Tucker3 Decomposition of 𝜋i|j,k and the Alligator Data 1937.7 Constructing a Low-Dimensional Display 1947.7.1 On the Choice of Coordinates 1947.7.2 Column–Tube Interactive Biplot for the Alligator Data 1957.8 Final Comments 199References 201Author Index 213Subject Index 217