Optimal Quantification and Symmetry

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. 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