What is factor score indeterminacy?

       A latent factor is a novel variate constructed from the manifest variates (measured variables) included in the factor analysis. In the principal component model each factor, referred to as a component, is uniquely constructed and considered determinate in nature. In most varieties of the common factor model, however, the factors are not uniquely constructed and are hence indeterminate. The scores on the common factors are equivalently indeterminate, meaning that an infinite number of sets of factor scores can be computed for any given analysis that satisfy the stipulations of the common factor model. For example, consider a researcher who conducts an iterated principal axis factor analysis on a particular data set. The researcher extracts two common factors from the correlation matrix, computes the pattern coefficients and factor correlations, and then labels the first factor as “agreeableness / disagreeableness”.  The indeterminate nature of the factor scores makes it possible to compute an infinite number of sets of such scores that would all be consistent with the pattern coefficients. Moreover, under certain conditions an individual in the analysis with a high ranking on agreeableness according to one set of factor scores could receive a low ranking on agreeableness according to another set of factor scores and the researcher would have no way of deciding which ranking is “true” based upon the obtained results. In other words, there would be no way of telling whether the individual is “truly” agreeable or disagreeable. This mathematical problem, referred to as factor indeterminacy or factor score indeterminacy, was discovered in the 1920s and has been a source of considerable controversy ever since. Unfortunately, most psychologists appear to be unaware of the indeterminacy problem and fail to realize the importance of examining the degree of indeterminacy in any given common factor. They also appear to be unaware of the importance of evaluating the properties of their estimated factor scores; for instance, assessing the degree to which the estimated factor scores correlate with their respective factors or with one another.