Multidimensional Nonlinear Descriptive

Quantification of categorical, or non-numerical, data is a problem that scientists face across a wide range of disciplines. Exploring data analysis in various areas of research, such as the social sciences and biology, Multidimensional Nonlinear Descriptive Analysis presents methods for analyzing categorical data that are not necessarily sampled randomly from a normal population and often involve nonlinear relations.This reference not only provides an overview of multidimensional nonlinear descriptive analysis (MUNDA) of discrete data, it also offers new results in a variety of fields. The first part of the book covers conceptual and technical preliminaries needed to understand the data analysis in subsequent chapters. The next two parts contain applications of MUNDA to diverse data types, with each chapter devoted to one type of categorical data, a brief historical comment, and basic skills peculiar to the data types. The final part examines several problems and then concludes with suggestions for future progress.Covering both the early and later years of MUNDA research in the social sciences, psychology, ecology, biology, and statistics, this book provides a framework for potential developments in even more areas of study.

Dr. Shizuhiko Nishisato is Professor Emeritus at the Ontario Institute for Studies in Education of the University of Toronto, Ontario, Canada

I Background

1

(100)

1 Motivation

5

(20)

1.1 Why Multidimensional Analysis?

6

(8)

1.1.1 Traditional Unidimensional Analysis

6

(5)

1.1.2 Multidimensional Analysis

11

(3)

1.2 Why Nonlinear Analysis?

14

(8)

1.2.1 Traditional Linear Analysis

16

(3)

1.2.2 Nonlinear Analysis

19

(3)

1.3 Why Descriptive Analysis?

22

(3)

2 Quantification with Different Perspectives

25

(26)

2.1 Is Likert-Type Scoring Appropriate?

25

(4)

2.2 Method of Reciprocal Averages (MRA)

29

(4)

2.3 One-Way Analysis of Variance Approach

33

(6)

2.4 Bivariate Correlation Approach

39

(2)

2.5 Geometric Approach

41

(3)

2.6 Other Approaches

44

(3)

2.6.1 The Least-Squares Approach

45

(1)

2.6.2 Approach by Cramér's and Tchuproff's Coefficients

46

(1)

2.7 Multidimensional Decomposition

47

(4)

3 Historical Overview

51

(18)

3.1 Mathematical Foundations in Early Days

52

(1)

3.2 Pioneers of MUNDA in the 20th Century

53

(2)

3.3 Rediscovery and Further Developments

55

(12)

3.3.1 Distinct Groups

56

(4)

3.3.2 Books and Papers

60

(3)

3.3.3 A Plethora of Aliases

63

(3)

3.3.4 Notes on Dual Scaling

66

(1)

3.4 Additional Notes

67

(2)

3.4.1 Dedications

67

(2)

4 Conceptual Preliminaries

69

(16)

4.1 Stevens' Four Levels of Measurement

69

(2)

4.2 Classification of Categorical Data

71

(6)

4.2.1 Incidence Data

71

(3)

4.2.2 Dominance Data

74

(3)

4.3 Euclidean Space

77

(4)

4.3.1 Pythagorean Theorem

77

(1)

4.3.2 The Cosine Law

77

(1)

4.3.3 Young-Householder Theorem

78

(1)

4.3.4 Chi-Square Distance

79

(2)

4.4 Multidimensional Space

81

(4)

4.4.1 Pierce's Concept

81

(1)

4.4.2 Distance in Reduced Space

81

(1)

4.4.3 Correlation in Reduced Space

82

(3)

5 Technical Preliminaries

85

(16)

5.1 Linear Combination and Principal Space

85

(4)

5.2 Eigenvalue and Singular Value Decompositions

89

(3)

5.3 Finding the Largest Eigenvalue

92

(2)

5.3.1 Some Basics

92

(1)

5.3.2 MRA Revisited

93

(1)

5.4 Dual Relations and Rectangular Coordinates

94

(1)

5.5 Discrepancy Between Row Space and Column Space

95

(4)

5.5.1 Geometrically Correct Joint Plots (Traditional)

96

(1)

5.5.2 Symmetric Scaling

97

(1)

5.5.3 CGS Scaling

97

(1)

5.5.4 Geometrically Correct Joint Plots (New)

98

(1)

5.6 Information of Different Data Types

99

(2)

II Analysis of Incidence Data

101

(76)

6 Contingency Tables

105

(14)

6.1 Example

105

(1)

6.2 Early Work

106

(2)

6.3 Some Basics

108

(2)

6.3.1 Number of Components

108

(1)

6.3.2 Total Information

108

(1)

6.3.3 Information Accounted For By One Component

109

(1)

6.4 Is My Pet a Flagrant Biter?

110

(6)

6.5 Supplementary Notes

116

(3)

7 Multiple-Choice Data

119

(26)

7.1 Example

119

(1)

7.2 Early Work

120

(1)

7.3 Some Basics

121

(6)

7.4 Future Use of English by Students in Hong Kong

127

(9)

7.5 Blood Pressures, Migraines and Age Revisited

136

(5)

7.6 Further Discussion

141

(4)

7.6.1 Evaluation of alpha

141

(1)

7.6.2 Standardized Quantification

142

(3)

8 Sorting Data

145

(10)

8.1 Example

145

(1)

8.2 Early Work

145

(1)

8.3 Sorting Familiar Animals into Clusters

146

(6)

8.4 Some Notes

152

(3)

9 Forced Classification of Incidence Data

155

(22)

9.1 Early Work

155

(1)

9.2 Some Basics

156

(8)

9.2.1 Principles PEP and PIC

156

(3)

9.2.2 Conditional Analysis

159

(2)

9.2.3 Alternative Formulations

161

(1)

9.2.4 Adjusted Correlation Ratio

162

(1)

9.2.5 Value of Forcing Agent

163

(1)

9.3 Age Effects on Blood Pressures and Migraines

164

(5)

9.4 Ideal Sorter of Animals

169

(4)

9.5 Generalized Forced Classification

173

(4)

III Analysis of Dominance Data

177

(52)

10 Paired Comparison Data

181

(18)

10.1 Example

181

(1)

10.2 Early Work

182

(2)

10.2.1 Guttman's Formulation

182

(1)

10.2.2 Nishisato's Formulation

183

(1)

10.3 Some Basics

184

(3)

10.4 Travel Destinations

187

(6)

10.5 Criminal Acts

193

(6)

11 Rank-Order Data

199

(18)

11.1 Example

199

(1)

11.2 Early Work

200

(2)

11.3 Some Basics

202

(3)

11.3.1 Slater's Formulation

204

(1)

11.3.2 Tucker-Carroll's Formulation

204

(1)

11.4 Total Information and Number of Components

205

(1)

11.5 Distribution of Information

205

(4)

11.5.1 The Case of One Judge

206

(1)

11.5.2 One-Dimensional Rank Order Data

206

(1)

11.5.3 Coomb's Unfolding and MUNDA

206

(2)

11.5.4 Goodness of Fit

208

(1)

11.6 Sales Points of Hot Springs

209

(8)

12 Successive Categories Data

217

(12)

12.1 Example

217

(1)

12.2 Some Basics

217

(2)

12.3 Seriousness of Criminal Acts

219

(3)

12.4 Multidimensionality

222

(11)

12.4.1 Multidimensional Decomposition

222

(2)

12.4.2 Rank Conversion without Category Boundaries

224

(1)

12.4.3 Successive Categories Data as Multiple-Choice Data

225

(4)

IV Beyond the Basics

229

(48)

13 Further Topics of Interest

233

(14)

13.1 Forced Classification of Dominance Data

233

(3)

13.1.1 Forced Classification of Paired Comparisons: Travel Destinations

233

(2)

13.1.2 Forced Classification of Rank-Order Data: Hot Springs

235

(1)

13.2 Order Constraints on Ordered Categories

236

(3)

13.3 Stability, Robustness and Missing Responses

239

(1)

13.4 Multiway Data

240

(1)

13.5 Contingency Tables and Multiple-Choice Data

241

(5)

13.5.1 General Case of Two Variables

243

(1)

13.5.2 Statistic δ

244

(1)

13.5.3 Extensions from Two to Many Variables

245

(1)

13.6 Permutations of Categories and Scaling

246

(1)

14 Further Perspectives

247

(30)

14.1 Geometry of Multiple-Choice Items

247

(1)

14.2 A Concept of Correlation

248

(2)

14.3 A Statistic Related to Singular Values

250

(4)

14.4 Correlation for Categorical Variables

254

(6)

14.4.1 A New Measure upsilon

254

(4)

14.4.2 Cramér's Coefficient V

258

(1)

14.4.3 Tchuproff's Coefficient T

259

(1)

14.5 Properties of Squared Item-Total Correlation

260

(1)

14.6 Decomposition of Nonlinear Correlation

261

(5)

14.7 Interpreting Data in Reduced Dimension

266

(3)

14.8 Towards an Absolute Measure of Information

269

(4)

14.8.1 Why an Absolute Measure?

269

(1)

14.8.2 Union of Sets, Joint Entropy and Covariation

270

(3)

14.9 Final Word

273

(4)

References

277

(26)

Author index

303

(6)

Subject index

309

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Multidimensional Nonlinear Descriptive Analysis

Summary

Author Biography

Table of Contents

Supplemental Materials