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E-raamat: Introduction to Correspondence Analysis

(University of Newcastle, Australia), (University of Campania Luigi Vanvitelli, Italy)
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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.
Preface xiii
1 Introduction
1(28)
1.1 Data Visualisation
1(2)
1.2 Correspondence Analysis in a "Nutshell"
3(1)
1.3 Data Sets
4(4)
1.3.1 Traditional European Food Data
4(2)
1.3.2 Temperature Data
6(1)
1.3.3 Shoplifting Data
6(1)
1.3.4 Alligator Data
7(1)
1.4 Symmetrical Versus Asymmetrical Association
8(2)
1.5 Notation
10(2)
1.5.1 The Two-way Contingency Table
10(1)
1.5.2 The Three-way Contingency Table
11(1)
1.6 Formal Test of Symmetrical Association
12(5)
1.6.1 Test of Independence for Two-way Contingency Tables
12(1)
1.6.2 The Chi-squared Statistic for a Two-way Table
13(1)
1.6.3 Analysis of the Traditional European Food Data
13(2)
1.6.4 The Chi-squared Statistic for a Three-way Table
15(1)
1.6.5 Analysis of the Alligator Data
16(1)
1.7 Formal Test of Asymmetrical Association
17(5)
1.7.1 Test of Predictability for Two-way Contingency Tables
17(1)
1.7.2 The Goodman-Kruskal tau Index
17(1)
1.7.3 Analysis of the Traditional European Food Data
18(1)
1.7.4 Test of Predictability for Three-way Contingency Tables
19(1)
1.7.5 Marcotorchino's Index
19(1)
1.7.6 Analysis of the Alligator Data
20(1)
1.7.7 The Gray-Williams Index and Delta Index
21(1)
1.8 Correspondence Analysis and R
22(3)
1.9 Overview of the Book
25(4)
Part I Classical Analysis of Two Categorical Variables
29(70)
2 Simple Correspondence Analysis
31(40)
2.1 Introduction
31(1)
2.2 Reducing Multi-dimensional Space
32(7)
2.2.1 Profiles Cloud of Points
32(1)
2.2.2 Profiles for the Traditional European Food Data
33(1)
2.2.3 Weighted Centred Profiles
33(6)
2.3 Measuring Symmetric Association
39(2)
2.3.1 The Pearson Ratio
39(1)
2.3.2 Analysis of the Traditional European Food Data
40(1)
2.4 Decomposing the Pearson Residual for Nominal Variables
41(5)
2.4.1 The Generalised SVD of Uij - 1
41(3)
2.4.2 SVD of the Pearson Ratio's
44(1)
2.4.3 GSVD and the Traditional European Food Data
44(2)
2.5 Constructing a Low-Dimensional Display
46(4)
2.5.1 Standard Coordinates
46(1)
2.5.2 Principal Coordinates
47(3)
2.6 Practicalities of the Low-Dimensional Plot
50(9)
2.6.1 The Two-Dimensional Correspondence Plot
50(4)
2.6.2 What is NOT Being Shown in a Two-Dimensional Correspondence Plot?
54(3)
2.6.3 The Three-Dimensional Correspondence Plot
57(2)
2.7 The Biplot Display
59(4)
2.7.1 Definition
59(1)
2.7.2 Isometric Biplots of the Traditional European Food Data
60(3)
2.7.3 What is NOT Being Shown in a Two-Dimensional Biplot?
63(1)
2.8 The Case for No Visual Display
63(1)
2.9 Detecting Statistically Significant Points
64(5)
2.9.1 Confidence Circles and Ellipses
64(1)
2.9.2 Confidence Ellipses for the Traditional European Food Data
65(4)
2.10 Approximate p-values
69(1)
2.10.1 The Hypothesis Test and its p-value
69(1)
2.10.2 P-values and the Traditional European Food Data
70(1)
2.11 Final Comments
70(1)
3 Non-Symmetrical Correspondence Analysis
71(28)
3.1 Introduction
71(1)
3.2 Quantifying Asymmetric Association
72(4)
3.2.1 The Goodman-Kruskal tau Index
72(1)
3.2.2 The r Index and the Traditional European Food Data
72(1)
3.2.3 Weighted Centred Column Profile
73(1)
3.2.4 Profiles of the Traditional European Food Data
73(3)
3.3 Decomposing πi|j for Nominal Variables
76(3)
3.3.1 The Generalised SVD of πi|j
76(1)
3.3.2 GSVD and the Traditional Food Data
77(2)
3.4 Constructing a Low-Dimensional Display
79(3)
3.4.1 Standard Coordinates
79(1)
3.4.2 Principal Coordinates
79(3)
3.5 Practicalities of the Low-Dimensional Plot
82(7)
3.5.1 The Two-Dimensional Correspondence Plot
82(3)
3.5.2 The Three-Dimensional Correspondence Plot
85(4)
3.6 The Biplot Display
89(3)
3.6.1 Definition
89(1)
3.6.2 The Column Isometric Biplot for the Traditional Food Data
90(1)
3.6.3 The Three-Dimensional Biplot
91(1)
3.7 Detecting Statistically Significant Points
92(4)
3.7.1 Confidence Circles and Ellipses
92(1)
3.7.2 Confidence Ellipses for the Traditional Food Data
93(3)
3.8 Final Comments
96(3)
Part II Ordinal Analysis of Two Categorical Variables
99(44)
4 Simple Ordinal Correspondence Analysis
101(24)
4.1 Introduction
101(1)
4.2 A Simple Correspondence Analysis of the Temperature Data
102(2)
4.3 On the Mean and Variation of Profiles with Ordered Categories
104(7)
4.3.1 Profiles of the Temperature Data
104(1)
4.3.2 Defining Scores
105(2)
4.3.3 On the Mean of the Profiles
107(1)
4.3.4 On the Variation of the Profiles
108(1)
4.3.5 Mean and Variation of Profiles for the Temperature Data
108(3)
4.4 Decomposing the Pearson Residual for Ordinal Variables
111(4)
4.4.1 The Bivariate Moment Decomposition of Vijk - 1
111(2)
4.4.2 BMD and the Temperature Data
113(2)
4.5 Constructing a Low-Dimensional Display
115(5)
4.5.1 Standard Coordinates
115(1)
4.5.2 Principal Coordinates
116(3)
4.5.3 Practicalities of the Ordered Principal Coordinates
119(1)
4.6 The Biplot Display
120(4)
4.6.1 Definition
120(1)
4.6.2 Ordered Column Isometric Biplot
120(1)
4.6.3 Ordered Row Isometric Biplot
120(1)
4.6.4 Ordered Isometric Biplots for the Temperature Data
121(3)
4.7 Final Comments
124(1)
5 Ordered Non-symmetrical Correspondence Analysis
125(18)
5.1 Introduction
125(1)
5.2 The Goodman-Kruskal tau Index Revisited
126(2)
5.3 Decomposing for Ordinal and Nominal Variables
128(5)
5.3.1 The Hybrid Decomposition of πj|j
128(3)
5.3.2 Hybrid Decomposition and the Shoplifting Data
131(2)
5.4 Constructing a Low-Dimensional Display
133(2)
5.4.1 Standard Coordinates
133(1)
5.4.2 Principal Coordinates
134(1)
5.5 The Biplot
135(6)
5.5.1 An Overview
135(1)
5.5.2 Column Isometric Biplot
135(1)
5.5.3 Column Isometric Biplot of the Shoplifting Data
135(2)
5.5.4 Row Isometric Biplot
137(1)
5.5.5 Row Isometric Biplot of the Shoplifting Data
137(3)
5.5.6 Distance Measures and the Row Isometric Biplots
140(1)
5.6 Some Final Words
141(2)
Part III Analysis of Multiple Categorical Variables
143(58)
6 Multiple Correspondence Analysis
145(18)
6.1 Introduction
145(1)
6.2 Crisp Coding and the Indicator Matrix
146(6)
6.2.1 Crisp Coding
146(1)
6.2.2 The Indicator Matrix
146(1)
6.2.3 Crisp Coding and the Alligator Data
147(1)
6.2.4 Application of Multiple Correspondence Analysis using the Indicator Matrix
148(4)
6.3 The Burt Matrix
152(4)
6.4 Stacking
156(5)
6.4.1 A Definition
156(1)
6.4.2 Stacking and the Alligator Data - Lake(Size) × Food
156(3)
6.4.3 Stacking and the Alligator Data - Food(Size) × Lake
159(2)
6.5 Final Comments
161(2)
7 Multi-way Correspondence Analysis
163(38)
7.1 An Introduction
163(1)
1.2 Pearson's Residual Yij - 1 and the Partition of X2
164(3)
7.2.1 The Pearson Residual
164(1)
7.2.2 The Partition of X2
165(1)
7.2.3 Partition of X2 for the Alligator Data
165(2)
7.3 Symmetric Multi-way Correspondence Analysis
167(8)
7.3.1 Tucker3 Decomposition of yijk - 1
167(3)
7.3.2 T3D and the Analysis of Two Variables
170(1)
7.3.3 On the Choice of the Number of Components
171(1)
7.3.4 Tucker3 Decomposition of yijk - 1 and the Alligator Data
171(4)
7.4 Constructing a Low-Dimensional Display
175(13)
7.4.1 Principal Coordinates
175(1)
7.4.2 The Interactive Biplot
176(5)
7.4.3 Column-Tube Interactive Biplot for the Alligator Data
181(4)
7.4.4 Row Interactive Biplot for the Alligator Data
185(3)
7.5 The Marcotorchino Residual πi|j,k and the Partition of τM
188(3)
7.5.1 The Marcotrochino Residual
188(1)
7.5.2 The Partition of τM
189(1)
7.5.3 Partition of τM for the Alligator Data
190(1)
7.6 Non-symmetrical Multi-way Correspondence Analysis
191(3)
7.6.1 Tucker3 Decomposition of πi|j,k
191(2)
7.6.2 Tucker3 Decomposition of πi|j,k and the Alligator Data
193(1)
1.1 Constructing a Low-Dimensional Display
194(5)
7.7.1 On the Choice of Coordinates
194(1)
1.1.2 Column-Tube Interactive Biplot for the Alligator Data
195(4)
7.8 Final Comments
199(2)
References 201(12)
Author Index 213(4)
Subject Index 217
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.
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