The purpose of this book is to thoroughly prepare diverse areas of researchers in quantification theory. As is well known, quantification theory has attracted the attention of a countless number of researchers, some mathematically oriented and others not, but all of them are experts in their own disciplines. Quantifying non-quantitative (qualitative) data requires a variety of mathematical and statistical strategies, some of which are quite complicated. Unlike many books on quantification theory, the current book places more emphasis on preliminary requisites of mathematical tools than on details of quantification theory. As such, the book is primarily intended for readers whose specialty is outside mathematical sciences. The book was designed to offer non-mathematicians a variety of mathematical tools used in quantification theory in simple terms. Once all the preliminaries are fully discussed, quantification theory is then introduced in the last section as a simple application of those mathematical procedures fully discussed so far. The book opens up further frontiers of quantification theory as simple applications of basic mathematics.
Author(s): Shizuhiko Nishisato
Series: Behaviormetrics: Quantitative Approaches to Human Behavior, 16
Publisher: Springer
Year: 2023
Language: English
Pages: 213
City: Singapore
Preface
Acknowledgments
Contents
Part I Measurement
1 Information for Analysis
1.1 An Overview
1.2 Introduction
1.2.1 Fundamental Arithmetic Operations
1.2.2 Data Types
1.3 Stevens' Theory of Measurement
1.3.1 Nominal Measurement
1.3.2 Ordinal Measurement
1.3.3 Interval Measurement
1.3.4 Ratio Measurement
1.4 Concluding Remarks on Measurement
1.5 Task of Quantification Theory
References
2 Data Analysis and Likert Scale
2.1 Two Examples of Uninformative Reports
2.1.1 Number of COVID Patients
2.1.2 Number of Those Vaccinated
2.2 Likert Scale, a Popular but Misused Tool
2.2.1 How Does Likert Scale Work?
2.2.2 Warnings on Inappropriate Use of Likert Scale
References
Part II Mathematics
3 Preliminaries
3.1 An Overview
3.2 Series and Limit
3.2.1 Examples from Quantification Theory
3.3 Differentiation
3.4 Derivative of a Function of One Variable
3.5 Derivative of a Function of a Function
3.6 Partial Derivative
3.7 Differentiation Formulas
3.8 Maximum and Minimum Value of a Function
3.9 Lagrange Multipliers
3.9.1 Example 1
3.9.2 Example 2
References
4 Matrix Calculus
4.1 Different Forms of Matrices
4.1.1 Transpose
4.1.2 Rectangular Versus Square Matrix
4.1.3 Symmetric Matrix
4.1.4 Diagonal Matrix
4.1.5 Vector
4.1.6 Scaler Matrix and Identity Matrix
4.1.7 Idempotent Matrix
4.2 Simple Operations
4.2.1 Addition and Subtraction
4.2.2 Multiplication
4.2.3 Scalar Multiplication
4.2.4 Determinant
4.2.5 Inverse
4.2.6 Hat Matrix
4.2.7 Hadamard Product
4.3 Linear Dependence and Linear Independence
4.4 Rank of a Matrix
4.5 System of Linear Equations
4.6 Homogeneous Equations and Trivial Solution
4.7 Orthogonal Transformation
4.8 Rotation of Axes
4.9 Characteristic Equation of the Quadratic Form
4.10 Eigenvalues and Eigenvectors
4.10.1 Example: Canonical Reduction
4.11 Idempotent Matrices
4.12 Projection Operator
4.12.1 Example 1: Maximal Correlation
4.12.2 Example 2: General Decomposition Formula
References
5 Statistics in Matrix Notation
5.1 Mean
5.2 Variance-Covariance Matrix
5.3 Correlation Matrix
5.4 Linear Regression
5.5 One-Way Analysis of Variance
5.6 Multiway Analysis of Variance
5.7 Discriminant Analysis
5.8 Principal Component Analysis
References
6 Multidimensional Space
6.1 Introduction
6.2 Pierce's Description
6.2.1 Pythagorean Theorem
6.2.2 The Cosine Law
6.2.3 Young–Householder Theorem
6.2.4 Eckart–Young Theorem
6.2.5 Chi-Square Distance
6.3 Distance in Multidimensional Space
6.4 Correlation in Multidimensional Space
References
Part III A New Look at Quantification Theory
7 General Introduction
7.1 An Overview
7.2 Historical Background and Reference Books
7.3 First Step
7.3.1 Assignment of Unknown Numbers
7.3.2 Constraints on the Unknowns
7.4 Formulations of Different Approaches
7.4.1 Bivariate Correlation Approach
7.4.2 One-Way Analysis of Variance Approach
7.4.3 Maximization of Reliability Coefficient Alpha
7.5 Multidimensional Decomposition
7.6 Eigenvalues and Singular Values Decompositions
7.7 Finding the Largest Eigenvalue
7.8 Method of Reciprocal Averages
7.9 Problems of Joint Graphical Display
7.10 How Important Data Formats Are
7.11 A New Framework: Two-Stage Analysis
References
8 Geometry of Space: A New Look
8.1 Background
8.2 Geometric Space Theory
8.3 Rorschach Data
8.3.1 Major Dual Space or Contingency Space
8.3.2 Residual Space of Response-Pattern Table
8.3.3 Minor Dual Space of Response-Pattern Table
8.3.4 Dual Space
8.4 Dual Subspace, A Bridge Between Data Types
8.4.1 A Shortcut for Finding Exact Coordinates
8.5 Conclusions
References
9 Two-Stage Quantification: A New Look
9.1 Barley Data
9.1.1 Stage 1 Analysis
9.1.2 Stage 2 Analysis
9.2 Rorschach Data
9.2.1 Stage 1 Analysis
9.2.2 Stage 2 Analysis
9.3 Squared Distance Matrix in Dual Space
9.4 Summary of Two-Stage Quantification
9.5 Concluding Remarks
References
10 Joint Graphical Display
10.1 Toward a New Horizon
10.2 Correspondence Plots and Exact Plots
10.2.1 Rorschach Data
10.2.2 Barley Data
10.2.3 Kretschmer's Typology Data
10.3 Multidimensional Joint Display
10.3.1 Readers' Tasks: Rorschach Data
10.3.2 Readers' Tasks: Barley Data
10.3.3 Readers' Tasks: Kretschmer's Data
10.4 Discussion on Joint Graphical Display
10.5 Cluster Analysis as an Alternative
10.6 Final Notes
References
11 Beyond the Current Book
11.1 Selected Problems
11.1.1 Geometric Space Theory for Many Variables
11.1.2 Non-symmetric Quantification Analysis
11.1.3 Forced Classification Analysis
11.1.4 Projection Operators and Quantification
11.1.5 Robust Quantification
11.1.6 Biplots
11.1.7 Multidimensional Joint Graphs
11.1.8 Cluster Analysis as an Alternative
11.1.9 Computer Programs
11.1.10 Inferential Problems
11.1.11 Nishisato's Quandary on Multidimensional Analysis
11.1.12 Gleaning in the Field of Quantification
11.2 Final Words: Personal Notes
References
