Peter H. Schonemann
The minimum average correlation between equivalent sets of uncorrelated factors
Psychometrika, 1971, 36, 221-30.
A simplied proof of a lemma by Lederman (1938), which lies at the core of the factor indeterminacy issue, is presented. It leads to a representation of an orthogonal matrix T, relating equivalent factor solutions, which is different from Lederman's (1938) and Guttman's (1955). T is used to evaluate bounds on the average correlation between equivalent sets of uncorrelated factors.
It is found that the minmum average correlation is independent of the data.
This was the opening salvo of a series of papers devoted to the factor indeterminacy problem. As I was soon to find out, my fascination with this problem was was not shared, let alone appreciated, by the psychometric establishment. Psychometricians may have had a vested interest in covering up this "Achilles heel" (Guttman) of the factor model because it casts doubt on the very foundations of the whole intelligence testing enterprise.
As I only discovered long after I had obtained my Ph. D. at the UofI (which at that time was widely regarded as the Mecca of psychometrics), the problem was by no means new. E. B. Wilson, a polymath held in awe even by Thurstone, raised it for the first time in a book Review (Science, 1928, LXVII, 244-248) of Spearman's (1927) Abilities of Man . During all my years in graduate school, I was never told of E.B Wilson, or factor indeterminacy, nor were they mentioned in any of the texts then in use.
In hindsight the basic problem is simple: Since the factor model postulates more factors than observed variables, factor scores are not uniquely defined. Wilson showed how one could find two equivalent sets of minimally correlated factor scores which explain all the observed data equally well. One reason this issue was treated like a dirty diaper was probably that it vitiates the very purpose the factor model was designed to serve: To "objectively determine and measure" general intelligence. Hence, as Wechsler had noticed: "Psychometrics, without it [the g factor] loses its basic prop" (Wechsler, 1939).
The basic result of the paper states that the average minimum correlation between equivalent factor solutions (where the average is taken over all factors, m common plus p unique factors) is given by the ratio (p-m)/(p+m). It thus does not depend on the observed correlations at all. Usually, on finds that m is approximately p/3 so that, in this case, the average is 1/2. This means at least one factor predicts no more than 25% of the variance of its perfectly equivalent twin. Empirical values for common factors obtained for a number of published studies are given in Schonemann and Wang (1972).