Correspondence Analysis

Eric J. BehSchool of Mathematics & Physical Sciences, University of Newcastle, AustraliaRosaria LombardoDepartment of Economics, Second University of Naples, Italy

"the book is outstandingly comprehensive and informative, well written, and clear. If the book is adopted for courses in Statistics for not only students in applied fields, but also for students in Statistics, it will provide them with an excellent up-to-date knowledge of the entire spectrum of correspondence analysis. I would also like to recommend the book very strongly to most researchers including seasoned researchers in data analysis, for the book will undoubtedly fill in the gap of their knowledge about versatile correspondence analysis. I learned a lot, reading the book." (Psychometrika 2016)

• Foreword xvPreface xviiPart One Introduction 11 Data Visualisation 31.1 A Very Brief Introduction to Data Visualisation 31.1.1 A Very Brief History 31.1.2 Introduction to Visualisation Tools for Numerical Data 41.1.3 Introduction to Visualisation Tools for Univariate Categorical Data 61.2 Data Visualisation for Contingency Tables 101.2.1 Fourfold Displays 111.3 Other Plots 121.4 Studying Exposure to Asbestos 131.4.1 Asbestos and Irving J. Selikoff 131.4.2 Selikoff’s Data 171.4.3 Numerical Analysis of Selikoff’s Data 171.4.4 A Graphical Analysis of Selikoff’s Data 181.4.5 Classical Correspondence Analysis of Selikoff’s Data 201.4.6 Other Methods of Graphical Analysis 221.5 Happiness Data 251.6 Correspondence Analysis Now 291.6.1 A Bibliographic Taste 291.6.2 The Increasing Popularity of Correspondence Analysis 291.6.3 The Growth of the Correspondence Analysis Family Tree 321.7 Overview of the Book 341.8 R Code 35References 362 Pearson’s Chi-Squared Statistic 442.1 Introduction 442.2 Pearson’s Chi-Squared Statistic 442.2.1 Notation 442.2.2 Measuring the Departure from Independence 452.2.3 Pearson’s Chi-Squared Statistic 472.2.4 Other 2 Measures of Association 482.2.5 The Power Divergence Statistic 492.2.6 Dealing with the Sample Size 502.3 The Goodman--Kruskal Tau Index 512.3.1 Other Measures and Issues 522.4 The 2 × 2 Contingency Table 522.4.1 Yates’ Continuity Correction 532.5 Early Contingency Tables 542.5.1 The Impact of Adolph Quetelet 552.5.2 Gavarret’s (1840) Legitimate Children Data 582.5.3 Finley’s (1884) Tornado Data 582.5.4 Galton’s (1892) Fingerprint Data 592.5.5 Final Comments 612.6 R Code 612.6.1 Expectation and Variance of the Pearson Chi-Squared Statistic 612.6.2 Pearson’s Chi-Squared Test of Independence 622.6.3 The Cressie--Read Statistic 64References 67Part Two Correspondence Analysis of Two-Way Contingency Tables 713 Methods of Decomposition 733.1 Introduction 733.2 Reducing Multidimensional Space 733.3 Profiles and Cloud of Points 743.4 Property of Distributional Equivalence 793.5 The Triplet and Classical Reciprocal Averaging 793.5.1 One-Dimensional Reciprocal Averaging 803.5.2 Matrix Form of One-Dimensional Reciprocal Averaging 813.5.3 -Dimensional Reciprocal Averaging 833.5.4 Some Historical Comments 833.6 Solving the Triplet Using Eigen-Decomposition 843.6.1 The Decomposition 843.6.2 Example 853.7 Solving the Triplet Using Singular Value Decomposition 863.7.1 The Standard Decomposition 863.7.2 The Generalised Decomposition 883.8 The Generalised Triplet and Reciprocal Averaging 893.9 Solving the Generalised Triplet Using Gram--Schmidt Process 913.9.1 Ordered Categorical Variables and a priori Scores 913.9.2 On Finding Orthogonalised Vectors 923.9.3 A Recurrence Formulae Approach 943.9.4 Changing the Basis Vector 963.9.5 Generalised Correlations 973.10 Bivariate Moment Decomposition 1003.11 Hybrid Decomposition 1003.11.1 An Alternative Singly Ordered Approach 1023.12 R Code 1033.12.1 Eigen-Decomposition in R 1033.12.2 Singular Value Decomposition in R 1033.12.3 Singular Value Decomposition for Matrix Approximation 1043.12.4 Generating Emerson’s Polynomials 1063.13 A Preliminary Graphical Summary 1093.14 Analysis of Analgesic Drugs 112References 1154 Simple Correspondence Analysis 1204.1 Introduction 1204.2 Notation 1214.3 Measuring Departures from Complete Independence 1224.3.1 The ‘Duplication Constant’ 1234.3.2 Pearson Ratios 1234.4 Decomposing the Pearson Ratio 1244.5 Coordinate Systems 1264.5.1 Standard Coordinates 1264.5.2 Principal Coordinates 1274.5.3 Biplot Coordinates 1324.6 Distances 1364.6.1 Distance from the Origin 1364.6.2 Intra-Variable Distances and the Metric 1374.6.3 Inter-Variable Distances 1384.7 Transition Formulae 1404.8 Moments of the Principal Coordinates 1414.8.1 The Mean of 1424.8.2 The Variance of 1424.8.3 The Skewness of 1434.8.4 The Kurtosis of 1434.8.5 Moments of the Asbestos Data 1444.9 How Many Dimensions to Use? 1454.10 R Code 1474.11 Other Theoretical Issues 1544.12 Some Applications of Correspondence Analysis 1564.13 Analysis of a Mother’s Attachment to Her Child 158References 1655 Non-Symmetrical Correspondence Analysis 1775.1 Introduction 1775.2 The Goodman--Kruskal Tau Index 1805.2.1 The Tau Index as a Measure of the Increase in Predictability 1805.2.2 The Tau Index in the Context of ANOVA 1825.2.3 The Sensitivity of 1825.2.4 A Demonstration: Revisiting Selikoff’s Asbestos Data 1855.3 Non-Symmetrical Correspondence Analysis 1865.3.1 The Centred Column Profile Matrix 1865.3.2 Decomposition of 1875.4 The Coordinate Systems 1885.4.1 Standard Coordinates 1885.4.2 Principal Coordinates 1895.4.3 Biplot Coordinates 1935.5 Transition Formulae 1975.5.1 Supplementary Points 1985.5.2 Reconstruction Formulae 1985.6 Moments of the Principal Coordinates 1995.6.1 The Mean of 1995.6.2 The Variance of 2005.6.3 The Skewness of 2015.6.4 The Kurtosis of 2015.7 The Distances 2015.7.1 Column Distances 2015.7.2 Row Distances 2035.8 Comparison with Simple Correspondence Analysis 2045.9 R Code 2045.10 Analysis of a Mother’s Attachment to Her Child 209References 2126 Ordered Correspondence Analysis 2166.1 Introduction 2166.2 Pearson’s Ratio and Bivariate Moment Decomposition 2216.3 Coordinate Systems 2226.3.1 Standard Coordinates 2226.3.2 The Generalised Correlations 2236.3.3 Principal Coordinates 2256.3.4 Location, Dispersion and Higher Order Components 2296.3.5 The Correspondence Plot and Generalised Correlations 2306.3.6 Impact on the Choice of Scores 2326.4 Artificial Data Revisited 2336.4.1 On the Structure of the Association 2336.4.2 A Graphical Summary of the Association 2336.4.3 An Interpretation of the Axes and Components 2346.4.4 The Impact of the Choice of Scores 2356.5 Transition Formulae 2366.6 Distance Measures 2386.6.1 Distance from the Origin 2386.6.2 Intra-Variable Distances 2396.7 Singly Ordered Analysis 2396.8 R Code 2416.8.1 Generalised Correlations and Principal Inertias 2416.8.2 Doubly Ordered Correspondence Analysis 245References 2487 Ordered Non-Symmetrical Correspondence Analysis 2517.1 Introduction 2517.2 General Considerations 2527.2.1 Orthogonal Polynomials Instead of Singular Vectors 2537.3 Doubly Ordered Non-Symmetrical Correspondence Analysis 2547.3.1 Bivariate Moment Decomposition 2547.3.2 Generalised Correlations in Bivariate Moment Decomposition 2557.4 Singly Ordered Non-Symmetrical Correspondence Analysis 2577.4.1 Hybrid Decomposition for an Ordered Predictor Variable 2577.4.2 Hybrid Decomposition in the Case of Ordered Response Variables 2587.4.3 Generalised Correlations in Hybrid Decomposition 2587.5 Coordinate Systems for Ordered Non-Symmetrical Correspondence Analysis 2597.5.1 Polynomial Plots for Doubly Ordered Non-Symmetrical Correspondence Analysis 2607.5.2 Polynomial Biplot for Doubly Ordered Non-Symmetrical Correspondence Analysis 2627.5.3 Polynomial Plot for Singly Ordered Non-Symmetrical Correspondence Analysis with an Ordered Predictor Variable 2627.5.4 Polynomial Biplot for Singly Ordered Non-Symmetrical Correspondence Analysis with an Ordered Predictor Variable 2637.5.5 Polynomial Plot for Singly Ordered Non-Symmetrical Correspondence Analysis with an Ordered Response Variable 2647.5.6 Polynomial Biplot for Singly Ordered Non-Symmetrical Correspondence Analysis with an Ordered Response Variable 2657.6 Tests of Asymmetric Association 2657.7 Distances in Ordered Non-Symmetrical Correspondence Analysis 2667.7.1 Distances in Doubly Ordered Non-Symmetrical Correspondence Analysis 2677.7.2 Distances in Singly Ordered Non-Symmetrical Correspondence Analysis 2697.8 Doubly Ordered Non-Symmetrical Correspondence of Asbestos Data 2697.8.1 Trends 2707.9 Singly Ordered Non-Symmetrical Correspondence Analysis of Drug Data 2777.9.1 Predictability of Ordered Rows Given Columns 2787.10 R Code for Ordered Non-Symmetrical Correspondence Analysis 283References 3008 External Stability and Confidence Regions 3028.1 Introduction 3028.2 On the Statistical Significance of a Point 3038.3 Circular Confidence Regions for Classical Correspondence Analysis 3048.4 Elliptical Confidence Regions for Classical Correspondence Analysis 3068.4.1 The Information in the Optimal Correspondence Plot 3068.4.2 The Information in the First Two Dimensions 3088.4.3 Eccentricity of Elliptical Regions 3098.4.4 Comparison of Confidence Regions 3098.5 Confidence Regions for Non-Symmetrical Correspondence Analysis 3118.5.1 Circular Regions in Non-Symmetrical Correspondence Analysis 3128.5.2 Elliptical Regions in Non-Symmetrical Correspondence Analysis 3128.6 Approximate -values and Classical Correspondence Analysis 3138.6.1 Approximate -values Based on Confidence Circles 3138.6.2 Approximate -values Based on Confidence Ellipses 3148.7 Approximate -values and Non-Symmetrical Correspondence Analysis 3158.8 Bootstrap Elliptical Confidence Regions 3158.9 Ringrose’s Bootstrap Confidence Regions 3168.9.1 Confidence Ellipses and Covariance Matrix 3178.10 Confidence Regions and Selikoff’s Asbestos Data 3188.11 Confidence Regions and Mother--Child Attachment Data 3228.12 R Code 3258.12.1 Calculating the Path of a Confidence Ellipse 3268.12.2 Constructing Elliptical Regions in a Correspondence Plot 327References 3359 Variants of Correspondence Analysis 3379.1 Introduction 3379.2 Correspondence Analysis Using Adjusted Standardised Residuals 3379.3 Correspondence Analysis Using the Freeman--Tukey Statistic 3409.4 Correspondence Analysis of Ranked Data 3429.5 R Code 3439.5.1 Adjusted Standardised Residuals 3439.5.2 Freeman--Tukey Statistic 3499.6 The Correspondence Analysis Family 3539.6.1 Detrended Correspondence Analysis 3539.6.2 Canonical Correspondence Analysis 3549.6.3 Inverse Correspondence Analysis 3559.6.4 Ordered Correspondence Analysis 3559.6.5 Grade Correspondence Analysis 3559.6.6 Symbolic Correspondence Analysis 3569.6.7 Correspondence Analysis of Proximity Data 3569.6.8 Residual (Scaling) Correspondence Analysis 3609.6.9 Log-Ratio Correspondence Analysis 3629.6.10 Parametric Correspondence Analysis 3649.6.11 Subset Correspondence Analysis 3649.6.12 Foucart’s Correspondence Analysis 3659.7 Other Techniques 365References 366Part Three Correspondence Analysis of Multi-Way Contingency Tables 37310 Coding and Multiple Correspondence Analysis 37510.1 Introduction to Coding 37510.2 Coding Data 37710.2.1 B-Splines 37710.2.2 Crisp Coding 38010.2.3 Fuzzy Coding 38210.3 Coding Ordered Categorical Variables by Orthogonal Polynomials 38210.4 Burt Matrix 38410.5 An Introduction to Multiple Correspondence Analysis 38610.6 Multiple Correspondence Analysis 38810.6.1 Notation 38810.6.2 Decomposition Methods 38910.6.3 Coordinates, Transition Formulae and Adjusted Inertia 39310.7 Variants of Multiple Correspondence Analysis 39510.7.1 Joint Correspondence Analysis 39610.7.2 Stacking and Concatenation 39710.8 Ordered Multiple Correspondence Analysis 39810.8.1 Orthogonal Polynomials in Multiple Correspondence Analysis 39810.8.2 Hybrid Decomposition of Multiple Indicator Tables 39910.8.3 Two Ordered Variables and Their Contingency Table 40010.8.4 Test of Statistical Significance 40110.8.5 Properties of Ordered Multiple Correspondence Analysis 40310.8.6 Graphical Displays in Ordered Multiple Correspondence Analysis 40410.9 Applications 40510.9.1 Customer Satisfaction in Health Care Services 40610.9.2 Two Quality Aspects 41110.10 R Code 41710.10.1 B-Spline Function 41710.10.2 Crisp and Fuzzy Coding Using B-Splines in R 42110.10.3 Crisp Coding and the Burt Table by Indicator Functions in R 42510.10.4 Classical and Multiple Correspondence Analysis in R 428References 44411 Symmetrical and Non-Symmetrical Three-Way Correspondence Analysis 45111.1 Introduction 45111.2 Notation 45311.3 Symmetric and Asymmetric Association in Three-Way Contingency Tables 45411.4 Partitioning Three-Way Measures of Association 45511.4.1 Partitioning Pearson’s Three-Way Statistic 45711.4.2 Partitioning Marcotorchino’s and Gray--William’s Three-Way Indices 45811.4.3 Marcotorchino’s Index 46011.4.4 Partitioning the Three-Way Delta Index 46111.4.5 Three-Way Delta Index 46311.5 Formal Tests of Predictability 46311.5.1 Testing Pearson’s Statistic 46411.5.2 Testing the Marcotorchino’s Index 46411.5.3 Testing the Delta Index 46511.5.4 Discussion 46511.6 Tucker3 Decomposition for Three-Way Tables 46611.7 Correspondence Analysis of Three-Way Contingency Tables 46711.7.1 Symmetrically Associated Variables 46711.7.2 Asymmetrically Associated Variables 46811.7.3 Additional Property 46911.8 Modelling of Partial and Marginal Dependence 47011.9 Graphical Representation 47111.9.1 Interactive Plot 47111.9.2 Interactive Biplot 47211.9.3 Category Contribution 47411.10 On the Application of Partitions 47411.10.1 Olive Data: Partitioning the Asymmetric Association 47411.10.2 Job Satisfaction Data: Partitioning the Asymmetric Association 47611.11 On the Application of Three-Way Correspondence Analysis 47711.11.1 Job Satisfaction and Three-Way Symmetrical Correspondence Analysis 47711.11.2 Job Satisfaction and Three-Way Non-Symmetrical Correspondence Analysis 48311.12 R Code 490References 511Part Four The Computation of Correspondence Analysis 51712 Computing and Correspondence Analysis 51912.1 Introduction 51912.2 A Look Through Time 51912.2.1 Pre-1990 51912.2.2 From 1990 to 2000 52012.2.3 The Early 2000s 52212.3 The Impact of R 52312.3.1 Overview of Correspondence Analysis in R 52312.3.2 MASS 52412.3.3 Nenadi´c and Greenacre’s (2007) ca 52512.3.4 Murtagh (2005) 52712.3.5 ade4 53012.4 Some Stand-Alone Programs 53312.4.1 JMP 53312.4.2 SPSS 53312.4.3 PAST 53412.4.4 DtmVic5.6+ 535References 540Index 545