Factor Analysis

Purposes

Factor analysis has three fundamental purposes. First, it is useful for measuring constructs that cannot readily be observed in nature. For example, we cannot hear, see, smell, taste, or touch intelligence, but it can be inferred from the assessment of observable variables such as performance on specific ability tests. Factor analysis is also helpful in the development of scales to measure attitudes or other such latent constructs by assessing responses to specific questions. Second, factor analysis is useful for summarizing a large amount of observations into a smaller number of factors. For example, there exist thousands of personality descriptors in the English language. Through factor analysis, researchers have been able to reduce the number of distinct factors needed to describe the structure of personality. Third, factor analysis is useful for providing evidence of construct validity (e.g., factorial, convergent, and discriminant validity). For example, if certain observable variables are theoretically related to one another, then factor analysis should demonstrate these theoretical relationships, simultaneously demonstrating that the same variables are reasonably uncorrelated with variables from other latent factors. All three of these uses of factor analysis can be employed in the development and testing of psychological theories.

Basic Factor Model

The basic factor analysis problem takes a number of observable variables and explains their interrelationships in a manner that is analogous to a regression equation. The common factor model is a regression equation in which the common factors act as predictors of the observed X variables. The basic factor model is depicted in Equation 1.

In this equation, X is the matrix of observed variables, L is the matrix of factor loadings or regression weights, f is the matrix of common factors, and u is the matrix of residuals. The goal is to explain the interrelationships among the X variables by the common factors, f, and the residual error terms, called uniqueness. The variance in X is partitioned into common and specific components. Unlike regression, however, the predictors, f, are unknown.

To provide a fictional example of this problem, suppose a number of supervisors are asked to rate the relevance of six personality characteristics to effective job performance by subordinates. The characteristics assessed are organized, systematic, careless, creative, [Page 244]intellectual, and imaginative.Table 1 depicts the hypothetical correlation matrix for these variables. The factor analysis problem is to explain the relationships among these variables with fewer than six underlying latent factors. Organized, systematic, and careless are all correlated with one another, but they are not correlated with creative, intellectual, and imaginative. Likewise, creative, intellectual, and imaginative are all correlated with one another, but they are not correlated with organized, systematic, and careless. There are two sets of correlations reflecting two underlying factors.

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