Multidimensional nonlinear descriptive analysis / Shizuhiko Nishisato.

Format:

Book

Author/Creator:

Nishisato, Shizuhiko, 1935-

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Multivariate analysis.

Correlation (Statistics).

Categories (Mathematics).

Multidimensional scaling.

Correspondence analysis (Statistics).

Physical Description:

xiii, 312 pages : illustrations ; 25 cm

Place of Publication:

Boca Raton, FL : Chapman & Hall/CRC, [2007]

Summary:

Quantification of categorical, or non-numerical, data is a problem that scientists face across a wide range of disciplines. Exploring data analysis in diverse 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.

Contents:

1 Motivation 5

1.1 Why Multidimensional Analysis? 6

1.1.1 Traditional Unidimensional Analysis 6

1.1.2 Multidimensional Analysis 11

1.2 Why Nonlinear Analysis? 14

1.2.1 Traditional Linear Analysis 16

1.2.2 Nonlinear Analysis 19

1.3 Why Descriptive Analysis? 22

2 Quantification with Different Perspectives 25

2.1 Is Likert-Type Scoring Appropriate? 25

2.2 Method of Reciprocal Averages (MRA) 29

2.3 One-Way Analysis of Variance Approach 33

2.4 Bivariate Correlation Approach 39

2.5 Geometric Approach 41

2.6 Other Approaches 44

2.6.1 The Least-Squares Approach 45

2.6.2 Approach by Cramer's and Tchuproff's Coefficients 46

2.7 Multidimensional Decomposition 47

3 Historical Overview 51

3.1 Mathematical Foundations in Early Days 52

3.2 Pioneers of MUNDA in the 20th Century 53

3.3 Rediscovery and Further Developments 55

3.3.1 Distinct Groups 56

3.3.2 Books and Papers 60

3.3.3 A Plethora of Aliases 63

3.3.4 Notes on Dual Scaling 66

3.4 Additional Notes 67

4 Conceptual Preliminaries 69

4.1 Stevens' Four Levels of Measurement 69

4.2 Classification of Categorical Data 71

4.2.1 Incidence Data 71

4.2.2 Dominance Data 74

4.3 Euclidean Space 77

4.3.1 Pythagorean Theorem 77

4.3.2 The Cosine Law 77

4.3.3 Young-Householder Theorem 78

4.3.4 Chi-Square Distance 79

4.4 Multidimensional Space 81

4.4.1 Pierce's Concept 81

4.4.2 Distance in Reduced Space 81

4.4.3 Correlation in Reduced Space 82

5 Technical Preliminaries 85

5.1 Linear Combination and Principal Space 85

5.2 Eigenvalue and Singular Value Decompositions 89

5.3 Finding the Largest Eigenvalue 92

5.3.1 Some Basics 92

5.3.2 MRA Revisited 93

5.4 Dual Relations and Rectangular Coordinates 94

5.5 Discrepancy Between Row Space and Column Space 95

5.5.1 Geometrically Correct Joint Plots (Traditional) 96

5.5.2 Symmetric Scaling 97

5.5.3 CGS Scaling 97

5.5.4 Geometrically Correct Joint Plots (New) 98

5.6 Information of Different Data Types 99

II Analysis of Incidence Data 101

6 Contingency Tables 105

6.2 Early Work 106

6.3 Some Basics 108

6.3.1 Number of Components 108

6.3.2 Total Information 108

6.3.3 Information Accounted For By One Component 109

6.4 Is My Pet a Flagrant Biter? 110

6.5 Supplementary Notes 116

7 Multiple-Choice Data 119

7.2 Early Work 120

7.3 Some Basics 121

7.4 Future Use of English by Students in Hong Kong 127

7.5 Blood Pressures, Migraines and Age Revisited 136

7.6 Further Discussion 141

7.6.1 Evaluation of alpha 141

7.6.2 Standardized Quantification 142

8 Sorting Data 145

8.2 Early Work 145

8.3 Sorting Familiar Animals into Clusters 146

8.4 Some Notes 152

9 Forced Classification of Incidence Data 155

9.1 Early Work 155

9.2 Some Basics 156

9.2.1 Principles PEP and PIC 156

9.2.2 Conditional Analysis 159

9.2.3 Alternative Formulations 161

9.2.4 Adjusted Correlation Ratio 162

9.2.5 Value of Forcing Agent 163

9.3 Age Effects on Blood Pressures and Migraines 164

9.4 Ideal Sorter of Animals 169

9.5 Generalized Forced Classification 173

III Analysis of Dominance Data 177

10 Paired Comparison Data 181

10.2 Early Work 182

10.2.1 Guttman's Formulation 182

10.2.2 Nishisato's Formulation 183

10.3 Some Basics 184

10.4 Travel Destinations 187

10.5 Criminal Acts 193

11 Rank-Order Data 199

11.2 Early Work 200

11.3 Some Basics 202

11.3.1 Slater's Formulation 204

11.3.2 Tucker-Carroll's Formulation 204

11.4 Total Information and Number of Components 205

11.5 Distribution of Information 205

11.5.1 The Case of One Judge 206

11.5.2 One-Dimensional Rank Order Data 206

11.5.3 Coomb's Unfolding and MUNDA 206

11.5.4 Goodness of Fit 208

11.6 Sales Points of Hot Springs 209

12 Successive Categories Data 217

12.2 Some Basics 217

12.3 Seriousness of Criminal Acts 219

12.4 Multidimensionality 222

12.4.1 Multidimensional Decomposition 222

12.4.2 Rank Conversion without Category Boundaries 224

12.4.3 Successive Categories Data as Multiple-Choice Data 225

IV Beyond the Basics 229

13 Further Topics of Interest 233

13.1 Forced Classification of Dominance Data 233

13.1.1 Forced Classification of Paired Comparisons: Travel Destinations 233

13.1.2 Forced Classification of Rank-Order Data: Hot Springs 235

13.2 Order Constraints on Ordered Categories 236

13.3 Stability, Robustness and Missing Responses 239

13.4 Multiway Data 240

13.5 Contingency Tables and Multiple-Choice Data 241

13.5.1 General Case of Two Variables 243

13.5.2 Statistic [delta] 244

13.5.3 Extensions from Two to Many Variables 245

13.6 Permutations of Categories and Scaling 246

14 Further Perspectives 247

14.1 Geometry of Multiple-Choice Items 247

14.2 A Concept of Correlation 248

14.3 A Statistic Related to Singular Values 250

14.4 Correlation for Categorical Variables 254

14.4.1 A New Measure [nu] 254

14.4.2 Cramer's Coefficient V 258

14.4.3 Tchuproff's Coefficient T 259

14.5 Properties of Squared Item-Total Correlation 260

14.6 Decomposition of Nonlinear Correlation 261

14.7 Interpreting Data in Reduced Dimension 266

14.8 Towards an Absolute Measure of Information 269

14.8.1 Why an Absolute Measure? 269

14.8.2 Union of Sets, Joint Entropy and Covariation 270

14.9 Final Word 273.

Notes:

Includes bibliographical references (pages 277-302) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1584886129

OCLC:

67405700

Publisher Number:

9781584886129

Online:

Publisher description