Year of Award

2023

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Experimental Psychology

Department or School/College

Department of Psychology

Committee Chair

Daniel J. Denis

Commitee Members

Yoonhee Jang, Rachel Severson, Allen Szalda-Petree, Mark Cracolice

Keywords

Dimensionality reduction, Eigen analysis, PCA, Teaching of statistics

Publisher

University of Montana

Abstract

Despite leaving graduate school with a functional understanding of eigen decomposition, many students of the social sciences lack a conceptual or intuitive understanding of what eigen objects are or how to adapt the utility of these objects to new situations and properly interpret them. Conceptual understanding is beneficial and desirable in several ways: It endows users with higher accuracy in how they apply a given method, as well as the means to challenge improper or misguided use of that method. The objective of this dissertation is to lay out the derivation of simple instructional tools that can be incorporated into a multivariate statistics curriculum with little interruption to the existing course and without extensive and timely diversions into the realm of linear algebra. These tools rely on a geometric presentation of eigen analysis in order to create visual aids that can be included alongside the typical and customary algebraic presentation. The standard textbook approach to teaching eigen analysis is reviewed and gaps in the sequence of operations that can lead to confusion about the nature and derivation of eigen objects are emphasized, including a discussion of how a geometric interpretation of the determinant can help rather than hinder comprehension. The role of eigenvectors as rotators of data and components as geometric results of that rotation are emphasized. We conclude with ideas for how the techniques presented in this dissertation can be evaluated in the typical multivariate curriculum.

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