Studying The Iterative Principal Axis Transformation algorithm and its correctness according to X^2-test proposed by Rippe D.D. using R program

Inhaltsverzeichnis (Table of Contents)

• What is Factor Analysis?
• Why we use Factor analysis?
• History of Factor Analysis
• Uses in psychology
• Factor analyzing in marketing
• Factor analyzing and Physical science
• Mathematical definition
• Representation of the Random Vector
• Covariance of the Random Vector
• The task of Factor analysis
• Simulation
• Algorithm
• Testing
• Results
• Conclusion
• Code

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis aims to study the Iterative Principal Axis Transformation algorithm, a fundamental tool in factor analysis, and analyze its correctness using the x²-Test proposed by Rippe. The study utilizes the R programming language to implement and test the algorithm.

• The definition and application of Factor Analysis
• The history and development of Factor Analysis, particularly in psychology, marketing, and physical sciences
• The mathematical representation and properties of the Random Vector in the context of Factor Analysis
• The Iterative Principal Axis Transformation algorithm and its implementation in R
• The correctness of the algorithm as assessed by the x²-Test

Zusammenfassung der Kapitel (Chapter Summaries)

The first chapter introduces Factor Analysis as a statistical method used to analyze relationships between a large number of variables and find underlying factors explaining those relationships. The second chapter explores the origins and applications of Factor Analysis, focusing on its initial use in psychometrics and its subsequent adoption in various fields, including social sciences, product management, and marketing. The third chapter delves into the historical development of Factor Analysis, highlighting the contributions of prominent researchers like Charles Spearman and Raymond Cattell.

The fourth chapter examines the application of Factor Analysis in psychology, specifically in intelligence research. It discusses how Factor Analysis reveals underlying similarities and correlations between different patterns, providing insights into complex psychological concepts. The fifth chapter focuses on the role of Factor Analysis in marketing, emphasizing its use in understanding the factors influencing consumer purchasing decisions.

The sixth chapter explores the application of Factor Analysis in physical sciences, such as ecology and geochemistry. The chapter illustrates how Factor Analysis helps analyze the distribution of chemical variables in water quality management and the identification of mineral associations in geochemistry.

The seventh chapter introduces the mathematical definition of Factor Analysis, defining a Random Vector X as a combination of a f-dimensional Vector Y (Factors) and a Vector Z (Errors). The chapter also defines the concepts of Loading Matrix and Individual Residual Variances.

The eighth chapter discusses the representation of the Random Vector in matrix form, emphasizing the constant nature of the Loading Matrix. The ninth chapter delves into the calculation of the covariance of the Random Vector, presenting key rules and the derivation of the formula for Cov(X).

Schlüsselwörter (Keywords)

The primary keywords and focus topics of the text include Factor Analysis, Iterative Principal Axis Transformation, x²-Test, Random Vector, Loading Matrix, Individual Residual Variances, and R programming language. These concepts underpin the study's analysis of the algorithm's correctness and its application in statistical research.