When Trivial Constraints Are Not Trivial: The Choice of Uniqueness Constraints in Confirmatory Factor Analysis

Abstract

In confirmatory factor analysis, constraints on the elements of the factor pattern or factor covariance matrix or both must be imposed to achieve a factor solution that is rotationally unique. The problem of rotational uniqueness is an aspect of the general identification problem in factor analysis; the relation between these 2 problems is described. Many sets of constraints exist that are sufficient to achieve uniqueness. Ideally, all such sets should yield equivalent model fits for a fixed number of common factors. As illustrated here, however, different sets of uniqueness constraints may lead to different fit results when applied to the same data. Several examples of this phenomenon in simulated data are given, and the reasons for the variation in fit results are described. In real applications, this variation in fit results over different choices for the uniqueness constraints may mislead researchers. Some remedies for this problem are discussed.