14 - The Set of Feasible Solutions for Reliability and Factor Analysis

Abstract

This chapter focuses on the set of feasible solutions for reliability and factor analysis. The mathematical model underlying factor analysis has the same structure as classical test theory. Both in factor analysis and in test theory, a convex set of “feasible tautologies” has been defined. In factor analysis, the set contains all the nonnegative diagonal matrices Ψ of unique variances which entail a reduced covariance matrix Σ −Ψ with no negative eigenvalues. In reliability theory, the set contains all non-negative diagonal matrices of error variances which entail a true score covariance matrix with no negative eigenvalues. Points in the feasible set are considered which have direct psychometric interpretations. The most reliable point in the test theory has principal component analysis as a factor analysis counterpart. Similarly, the least reliable point, associated with the greatest lower bound to reliability, corresponds to minimum trace factor analysis. Other factor solutions in the set are minimum rank factor analysis and a novel quadratic variant of it.