# Factor Analysis and Related Methods

By Roderick P. McDonald | Go to book overview

7
Item Response Theory

7.1. THE PROBLEM OF BINARY (DICHOTOMOUS) DATA

The question whether common factor analysis is "applicable" to items or other attributes that yield only two alternative "states" or "values"--yes/no, true/false, right/wrong--has been the subject of much confusion.

In the 1940s, the view developed that factor analyzing such binary (two- valued) data would yield misleading artifacts, known as "difficulty factors." In the context of cognitive testing, where an item is scored "right" or "wrong," an accepted (inverse) measure of the "difficulty" of an item is the proportion of the population that passes the item. Obviously, the higher the proportion that passes an item, the easier, we would say, the item is. (It would be nicer to speak of the "easiness" of an item, measured directly by the proportion passing it, but tradition is against this.) The view was put that if the difficulty levels of a set of items varied widely, a common factor analysis would yield extra factors known as difficulty factors that are not generic properties of the items but are in some sense mathematical artifacts of the analysis. The belief grew that difficulty factors would appear because the phi coefficient--a name often given to the usual correlation coefficient in the case where it has been computed from two binary variables--was not a "correct" measure of the extent of association of binary variables. The problem was considered to be that the correlation coefficient between two binary variables can never be as much as unity when the binary variables have different difficulties and hence must be "misrepresenting" the extent of the relationship. A formal mathematical justification for this belief was never given, but the view has been widely held. It has only recently been

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Factor Analysis and Related Methods

• Title Page iii
• Contents v
• Preface ix
• 1 - Introduction 1
• Glossary 40
• 2 - Exploratory Common Factor Analysis 50
• 3 - The Analysis of Covariance Structures: Confirmatory Factor Analysis and Pattern Hypotheses 96
• 4 - Models for Linear Structural Relations 113
• 5 - The Problem of Factor Scores 156
• 6 - Problems of Relationship Between Factor Analyses 171
• 7 - Item Response Theory 198
• 8 - Summary 223
• Appendix Some Matrix Algebra 232
• References 249
• Author Index 253
• Subject Index 255
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