Factor Analysis

An Applied Approach

Edward E. Cureton, Ralph B. D'Agostino

• Publisher: Psychology Press
• Year: 2013
• Print ISBN: 9781138179240
• eBook ISBN: 9781317759645
• Topic: Psychology
• Subtopic: History & Theory in Psychology
• Index: Psychology

This book is written primarily as a text for a course in factor analysis at the advanced undergraduate or graduate level. It is most appropriate for students of the behavioral and social sciences, though colleagues and students in other disciplines also have used preliminary copies.

1

Introduction and Spearman Approach

1.1 Introduction

Factor analysis consists of a collection of procedures for analyzing the relations among a set of random variables observed or counted or measured for each individual of a group.

An individual may be a person, some other organism, an object, or in general any entity. The group consists of one class of such entities, for example, sixth-grade pupils, college students, members of a legislative body, a particular species, white rats of a given strain, plots of ground in a forest or field, counties in a state or nation, or boxes. It may be a population (all entities of the defined class) or more commonly a sample from such a defined population.

The random variables of the set to be analyzed may consist of any attributes on which the members of the group differ. For example, with groups of humans the random variables may be sex, age, vote (yes or no) on an issue, time to solve a particular problem, height, weight, or score on an aptitude, personality, or educational achievement test. With agricultural problems the group may consist of counties, and the random variables may be average yields per acre of different crops. With botanical problems the groups may be plots in different kinds of terrain, and the random variables may be numbers of plants of different species per plot. In experimental psychology, the group may consist of white rats, and the random variables may be the number of seconds required to solve each of a number of problems using different types of apparatus. For a group consisting of a collection of boxes, the random variables might be perimeters (of top, side, and end), diagonals (of top, side, and end), squares (of length, width, and height), and the like. For a factor analysis, several different random variables must be observed or counted or measured for each member of the group.

Because factor analysis deals with the relations among the random variables, we must first obtain a score on each variable, and then compute the correlation between the scores on each pair of variables. Two-point variables are scored arbitrarily by assigning the number 0 or 1 to each individual of the group, for example, 0 for no or 1 for yes on a vote; and 0 for failure or 1 for success with a problem. The number of plants of a particular species found in a plot is recorded directly as a score. Average yields are often scored as bushels per acre. Age may be scored in years (for adults), in months (for children), or in weeks or even days (for infants or small animals). Aptitude and achievement test scores are usually the number of right answers, or perhaps the number right minus a fraction of the number wrong, to minimize the effects of differences in guessing tendency on scores that are intended to represent aptitude or achievement. Height may be scored in centimeters or inches; weight in pounds or grams. Measures of boxes (perimeter, diagonal of top, and the like) will also be scored in centimeters or inches.

In the case of ordered categories (e.g., strongly agree, agree, neutral, disagree, or strongly disagree with a proposition), the assignment of the scores 5,4, 3, 2, 1 may be insufficient. In this case, substitution of normalized standard scores is recommended. And in general, wherever the distribution of a set of raw scores is substantially skewed, substitution of either normalized standard scores or ranks is recommended (Section 4.6.1).

The scores on any variable will be treated as interval-scale measurements, that is, as measurements whose successive scale-intervals are equal, as are inches or pounds, even though they may in fact be scores on variables whose units are larger at the extremes than near the middle. With such scores, as with ranks, Baker, Hardyck, and Petrinovich (1966) have shown that normalization is unnecessary; and ranking, where this can be done without many ties and particularly without any large multiple ties, may be acceptable as an approximation, to substitute for normalization in order to get rid of skewness.

The original variables, representing actual observations or counts or measurements, are termed the manifest variables. Historically, the first manifest variables to be studied by the methods of factor analysis were psychological and educational test scores. We use the terms “variable” and “test” (often interchangeably) throughout this book rather than the more cumbersome “manifest variable.”

The purpose of a factor analysis is to account for the intercorr...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Tables and Figures
• Preface
• 1. Introduction and Spearman Approach
• 2. Centroid Method; Rotation in Two Dimensions
• 3. Elements of Matrix Algebra
• 4. Linear Computations
• 5. The Principal-Axes Method
• 6. Rotation of Axes
• 7. Extended Vectors; Varieties of Simple Structure
• 8. Orthogonal Transformations
• 9. Oblique Transformations
• 10. Refinement Transformations
• 11. Second-Order and Hierarchical Analysis
• 12. Component Analysis
• 13. Factor Scores
• 14. Cluster Analysis of Variables
• 15. Simplex Analysis
• 16. Some Special Problems
• Appendices
• References
• Author Index
• Subject Index