# Factor Analysis and Related Methods

By Roderick P. McDonald | Go to book overview

GLOSSARY

(Common) factor loading: Regression weight of a variable on a common factor (sometimes, common factor coefficient). The word loading is an engineering metaphor, due to Thurstone, replacing the metaphor "weight" from nineteenth-century physics.

(Common) factor pattern: The entire (n × m) matrix of regression weights of the n variables on the m common factors.

(Common) factor correlation: The correlation of a variable with a common factor.

(Common) factor structure: The entire (n × m) matrix of correlations of the n variables with the m factors.

(In these four terms the word common is commonly omitted.)

Communality: The squared multiple correlation of a variable with all the common factors--thought of as the proportion of the variance of the variable ' in common" because explained by the common factors. Often used, ambiguously, to refer to crude approximations to communality. All references to communality should be regarded with suspicion unless the context makes the use of the term absolutely unambiguous.

Uniqueness: The residual variance of a variable about its regression on the common factors. (Also unique variance.)

Unique factor loading (rare): The standard deviation of the residual of a variable. Sometimes, perhaps regrettably, the residual part is thought of as rescaled to be in standard measure; then what was its standard deviation becomes a coefficient resembling a regression weight.

Orthogonal factors: Uncorrelated factors. (The term derives from a widely used geometrical picture of correlation.)

Oblique factors: Coffelated factors. (Same remark as previous.)

Rotation: Performing arithmetic to obtain a new set of factor loadings (v-f regression weights) from a given set and perhaps to obtain also the correlation matrix of the obtained factors. (The term derives, correctly, from a geometrical picture of transformation from one set of uncorrelated factors to another but is also used, incorrectly, to describe any derivation of a set of correlated factors from another set of factors. The word transformation is to be preferred.)

Simple structure: This term has two radically different meanings that are often confused. Its primary meaning is exhibited when we say that a factor pattern (a matrix of v-f regression weights) has simple structure if (1) each row has at least one zero, (2) each column has at least as many zeros as there are common factors, (3) for every pair of columns there should be at least as many pairs of variables with a zero regression weight on one factor and a nonzero regression weight on the other, as there are common factors. The intention is to try to express in a recipe for action the notion that each

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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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