This book offers a unique new look at the familiar quantification theory from the point of view of mathematical symmetry and spatial symmetry. Symmetry exists in many aspects of our life―for instance, in the arts and biology as an ingredient of beauty and equilibrium, and more importantly, for data analysis as an indispensable representation of functional optimality. This unique focus on symmetry clarifies the objectives of quantification theory and the demarcation of quantification space, something that has never caught the attention of researchers.Mathematical symmetry is well known, as can be inferred from Hirschfeld’s simultaneous linear regressions, but spatial symmetry has not been discussed before, except for what one may infer from Nishisato’s dual scaling. The focus on symmetry here clarifies the demarcation of quantification analysis and makes it easier to understand such a perennial problem as that of joint graphical display in quantification theory. The new framework will help advance the frontier of further developments of quantification theory.
Many numerical examples are included to clarify the details of quantification theory, with a focus on symmetry as its operational principle. In this way, the book is useful not only for graduate students but also for researchers in diverse areas of data analysis.
Author(s): Shizuhiko Nishisato
Series: Behaviormetrics: Quantitative Approaches to Human Behavior, 12
Publisher: Springer
Year: 2022
Language: English
Pages: 198
City: Singapore
Preface
Acknowledgments
Contents
Part I Theory with Examples
1 Optimality and Symmetry
1.1 Symmetry and Graph
1.2 Optimal Quantification
References
2 Examples of Quantification
2.1 Kretschmer's Typology
2.1.1 Ordinary Analysis and Results
2.1.2 Quantification Analysis and Results
2.2 Joint Graphical Display
2.3 Singapore Data
2.3.1 Some Consideration for Ordinary Analysis
2.3.2 Quantification Analysis and Results
2.4 Graphical Display
2.5 Examples of Other Types of Graphs
2.5.1 Sorting Countries: Sorting Data
2.5.2 Seriousness of Criminal Acts: Successive Categories Data
2.5.3 Mothers' and Children's Wishes for Professions
2.5.4 Christmas Party Plans
2.5.5 Attractions of Hot Springs: Rank-Order Data
References
3 Constraints on Quantification
3.1 What Data Should We Quantify?
3.2 Some More Observations
3.3 Data in Terms of Unknown Numbers
3.4 Quantification Under Constraints
References
4 Quantification Procedures
4.1 Historical Background
4.2 Strategies
4.2.1 Quantification Through Correlation
4.2.2 Quantification Through Correlation Ratio
4.2.3 Quantification Through Cronbach's Alpha
4.2.4 Method of Reciprocal Averages: MRA
4.3 Optimal Symmetric Properties
4.4 Bilinear Expansion and Graphical Display
References
5 Mathematical Symmetry
5.1 Bi-modal Symmetry
5.1.1 Correlation
5.1.2 Correlation Ratio
5.2 Multi-modal Symmetry
5.2.1 Piecewise Method of Reciprocal Averages
5.2.2 Generalization to n Variables
References
6 Data Format and Information
6.1 Two Formats of Same Data
6.2 Further Comparisons of Data Formats
6.3 Numerical Illustration
6.3.1 Kretschmer's Typology Revisited
References
7 Space Theory and Symmetry
7.1 Spatial Symmetry
7.2 Theory of Quantification Space
7.2.1 Contingency Space
7.2.2 Dual Space: Symmetric Space
7.2.3 Pairwise Dual Subspaces
7.2.4 Total Space
7.2.5 Residual Space
7.3 Example of Space Decomposition
7.4 Recommendations
References
8 Graphical Display
8.1 Graphical Display of Rows or Columns
8.1.1 Blood Pressures and Migraines
8.2 Joint Graph: Correspondence Plot
8.3 Logically Correct Graph and Discrepancy Diagram
8.3.1 Graphs of Response-Pattern Format
8.4 Re-evaluating Correspondence Plot
8.4.1 Alternatives to Joint Graphical Display
References
Part II Gleaning in the Field
9 Forced Classification
9.1 Procedure of Forced Classification
9.1.1 Criterion-Total Correlation
9.1.2 Criterion Items Correlation
9.1.3 Partitioning of Total Space
9.1.4 Contributions of Individual Components
9.1.5 Legitimacy of Set by Set Analysis
9.1.6 An Example of Application
9.1.7 Graph in Criterion-Item Space
9.2 Generalized Forced Classification
References
10 Data with Designed Structure
10.1 Analysis of Variance of Nominal Data
10.1.1 Maximizing the Effects of α, β and γ
10.2 Quantification of Multi-way Analysis of Data
References
11 Quantifying Dominance Data
11.1 Dominance Data
11.1.1 Quantification Approaches
11.1.2 Quantification
11.1.3 Total Information
11.2 Example: Ranking of Municipal Services
11.3 Paired Comparison Data
11.3.1 Example: Wiggins' Christmas Party Plans
11.3.2 Example: Seriousness of Criminal Acts
11.3.3 Goodness of Fit
11.4 Forced Classification of Ordinal Data
11.4.1 Rank-Order and Paired Comparison Data
References
Part III Cautions for Quantification
12 Over-Quantification
12.1 Adverse Conditions of Data
12.1.1 Future of English in Hong Kong: Tung's Data
12.2 Standardized Quantification
12.2.1 Option Standardization
12.2.2 Results of Standardization
12.3 Handling Outlier Responses
12.3.1 The Method of Reciprocal Medians: MRM
12.3.2 Alernating Reciprocal Averaging and Reciprocal Medians
12.3.3 Method of Trimmed Reciprocal Averages
References
13 When Not to Analyze Data
13.1 Missing Responses and Quantification
13.2 Some Procedures
13.2.1 List-Wise Deletions
13.2.2 Extra Categories
13.2.3 Imputation
13.3 Imputation Principles
13.3.1 Principle of Maximal Internal Consistency
13.3.2 Hot-Deck Principle
13.3.3 Principle of Complete Ignorance
13.4 Decision Rules: When Not to Analyze
13.5 Towards a State-of-the-Art Framework
References
14 Epilogue
14.1 Reminiscence
14.1.1 John C. Gower
14.1.2 Jean-Paul Benzécri
14.2 Going Forward
References
Part IV Appendices
15 Stevens' Measurement Theory
15.1 Four Kinds of Measurement
15.1.1 Nominal Measurement
15.1.2 Ordinal Measurement
15.1.3 Interval Measurement
15.1.4 Ratio Measurement
15.2 Domains of Quantification
15.2.1 Full-Fledged Domain
15.2.2 Quasi-Domain
15.2.3 Outside Domain
References
16 A Numerical Example of MRA
16.1 Computing Optimal Component
16.2 Extracting More Components
References
