A comprehensive overview of the internationalisation of correspondence analysis
Correspondence Analysis: Theory, Practice and New Strategies examines the key issues of correspondence analysis, and discusses the new advances that have been made over the last 20 years.
The main focus of this book is to provide a comprehensive discussion of some of the key technical and practical aspects of correspondence analysis, and to demonstrate how they may be put to use. Particular attention is given to the history and mathematical links of the developments made. These links include not just those major contributions made by researchers in Europe (which is where much of the attention surrounding correspondence analysis has focused) but also the important contributions made by researchers in other parts of the world.
Key features include:
* A comprehensive international perspective on the key developments of correspondence analysis.
* Discussion of correspondence analysis for nominal and ordinal categorical data.
* Discussion of correspondence analysis of contingency tables with varying association structures (symmetric and non-symmetric relationship between two or more categorical variables).
* Extensive treatment of many of the members of the correspondence analysis family for two-way, three-way and multiple contingency tables.
Correspondence Analysis offers a comprehensive and detailed overview of this topic which will be of value to academics, postgraduate students and researchers wanting a better understanding of correspondence analysis. Readers interested in the historical development, internationalisation and diverse applicability of correspondence analysis will also find much to enjoy in this book.
Autorentext
Eric J. Beh
School of Mathematics & Physical Sciences, University of Newcastle, Australia
Rosaria Lombardo
Department of Economics, Second University of Naples, Italy
Inhalt
Foreword xv
Preface xvii
Part One Introduction 1
1 Data Visualisation 3
1.1 A Very Brief Introduction to Data Visualisation 3
1.1.1 A Very Brief History 3
1.1.2 Introduction to Visualisation Tools for Numerical Data 4
1.1.3 Introduction to Visualisation Tools for Univariate Categorical Data 6
1.2 Data Visualisation for Contingency Tables 10
1.2.1 Fourfold Displays 11
1.3 Other Plots 12
1.4 Studying Exposure to Asbestos 13
1.4.1 Asbestos and Irving J. Selikoff 13
1.4.2 Selikoff's Data 17
1.4.3 Numerical Analysis of Selikoff's Data 17
1.4.4 A Graphical Analysis of Selikoff's Data 18
1.4.5 Classical Correspondence Analysis of Selikoff's Data 20
1.4.6 Other Methods of Graphical Analysis 22
1.5 Happiness Data 25
1.6 Correspondence Analysis Now 29
1.6.1 A Bibliographic Taste 29
1.6.2 The Increasing Popularity of Correspondence Analysis 29
1.6.3 The Growth of the Correspondence Analysis Family Tree 32
1.7 Overview of the Book 34
1.8 R Code 35
References 36
2 Pearson's Chi-Squared Statistic 44
2.1 Introduction 44
2.2 Pearson's Chi-Squared Statistic 44
2.2.1 Notation 44
2.2.2 Measuring the Departure from Independence 45
2.2.3 Pearson's Chi-Squared Statistic 47
2.2.4 Other 2 Measures of Association 48
2.2.5 The Power Divergence Statistic 49
2.2.6 Dealing with the Sample Size 50
2.3 The Goodman--Kruskal Tau Index 51
2.3.1 Other Measures and Issues 52
2.4 The 2 × 2 Contingency Table 52
2.4.1 Yates' Continuity Correction 53
2.5 Early Contingency Tables 54
2.5.1 The Impact of Adolph Quetelet 55
2.5.2 Gavarret's (1840) Legitimate Children Data 58
2.5.3 Finley's (1884) Tornado Data 58
2.5.4 Galton's (1892) Fingerprint Data 59
2.5.5 Final Comments 61
2.6 R Code 61
2.6.1 Expectation and Variance of the Pearson Chi-Squared Statistic 61
2.6.2 Pearson's Chi-Squared Test of Independence 62
2.6.3 The Cressie--Read Statistic 64
References 67
Part Two Correspondence Analysis of Two-Way Contingency Tables 71
3 Methods of Decomposition 73
3.1 Introduction 73
3.2 Reducing Multidimensional Space 73
3.3 Profiles and Cloud of Points 74
3.4 Property of Distributional Equivalence 79
3.5 The Triplet and Classical Reciprocal Averaging 79
3.5.1 One-Dimensional Reciprocal Averaging 80
3.5.2 Matrix Form of One-Dimensional Reciprocal Averaging 81
3.5.3 -Dimensional Reciprocal Averaging 83
3.5.4 Some Historical Comments 83
3.6 Solving the Triplet Using Eigen-Decomposition 84
3.6.1 The Decomposition 84
3.6.2 Example 85
3.7 Solving the Triplet Using Singular Value Decomposition 86
3.7.1 The Standard Decomposition 86
3.7.2 The Generalised Decomposition 88
3.8 The Generalised Triplet and Reciprocal Averaging 89
3.9 Solving the Generalised Triplet Using Gram--Schmidt Process 91
3.9.1 Ordered Categorical Variables and a priori Scores 91
3.9.2 On Finding Orthogonalised Vectors 92
3.9.3 A Recurrence Formulae Approach 94
3.9.4 Changing the Basis Vector 96
3.9.5 Generalised Correlations 97
3.10 Bivariate Moment Decomposition 100
3.11 Hybrid Decomposition 100
3.11.1 An Alternative Singly Ordered Approach 102
3.12 R Code 103
3.12.1 Eigen-Decomposition in R 103
3.12.2 Singular Value Decomposition in R 103
3.12.3 Singular Value Decomposition for Matrix Approximation 104
3.12.4 Generating Emerson's Polynomials 106
3.13 A Preliminary Graphical Summary 109
3.14 Analysis of Analgesic Drugs 112
References 115
4 Simple Correspondence Analysis 120
4.1 Introduction 120
4.2 Notation 121
4.3 Measuring Departures from Complete Independence 122
4.3.1 The 'Duplication Constant' 123
4.3.2 Pearson Ratios 123
4.4 Decomposing the Pearson Ratio 124
4.5 Coordinate Systems 126
4.5.1 Standard Coordinates 126
4.5.2 Principal Coordinates 127
4.5.3 Biplot Coordinates 132
4.6 Distances 136
4.6.1 Distance from the Origin 136
4.6.2 Intra-Variable Distances and the Metric 137
4.6.3 Inter-Variable Distances 138
4.7 Transition Formulae 140
4.8 Moments of the Principal Coordinates 141
4.8.1 The Mean of 142
4.8.2 The Variance of 142
4.8.3 The Skewness of 143
4.8.4 The Kurtosis of 143
4.8.5 Moments of the Asbestos Data 144
4.9 How Many Dimensions to Use? 145
4.10 R Code 147
4.11 Other Theoretical Issues 154
4.12 Some Applications of Correspondence Analysis 156
4.13 Analysis of a Mother's Attachment to Her Child 158
References 165
5 Non-Symmetrical Correspondence Analysis 177
5.1 Introduction 177
5.2 The Goodman--Kruskal Tau Index 180
5.2.1 The Tau Index as a Measure of the Increase in Predictability 180
5.2.2 The Tau Index in the Context of ANOVA 182
5.2.3 The Sensitivity of 182
5.2.4 A Demonstration: Revisiting Selikoff's Asbestos Data 185
5.3 Non-Symmetrical Correspondence Analysis 186
5.3.1 The Centred Column Profile Matrix 186
5.3.2 Decomposition of 187
5.4 The Coordinate Systems 188
5.4.1 Standard Coordinates 188
5.4.2 Principal Coordinates 189
5.4.3 Biplot Coordinates 193
5.5 Transition Formulae 197
5.5.1 Supplementary Points 198
5.5.2 Reconstruction Formulae 198
5.6 Moments of the Principal Coordinates 199
5.6.1 The Mean of 199
5.6.2 The Variance of 200
5.6.3 The Skewness of 201
5.6.4 The K…