Peter H. Schönemann
Professor Emeritus • Department of Psychological Sciences • Purdue University

Peter H. Schonemann and Ming-Mei Wang

Some new results on factor indetermincay

Psychometrika, 1972, 37, 61-91.


Some relations between maximum liklihood factor analysis and factor indeterminacy are discussed.

Bounds are derived for the minimum average correlation between equivalent sets of correlated factors which depend on the latent roots of the factor correlations matrix Psi.

Empirical examples are presented to illustrate the theory and indicate the extent to which it can be expected to be relevant in practice.


A follow-up paper to Schonemann (1971), this paper focuses on the degree of indeterminacy of the common factors. We found that the average minimum correlation of the common factors, though no longer independent of the data, can be easily computed from the latent roots of the observed  correlation matrix after it has been rescaled by the inverse of U (the square roots of the unique variances). The largest m roots relate to the average indeterminacy of the m orthogonal common factors, and the remaining p-m roots to the likelihood ratio test for evaluating the fit for m factors (p. 64).

In the empirical part, we computed the actual minimum correlations for all m common factors for some 14 previously published studies. They were often negative once the number of common factors m was raised to achieve an acceptable fit. This means that a person who was assigned a far above average  "intelligence" score for one solution of the "intelligence factor"  (g) might be assigend a far below average intelligence score for another, empirically equivalent solution.

Postpublication History

Shortly after the paper had been published, a psychometrician, who previously had made a name for himself showing how to "estimate" the indeterminate factor scores optimally, submitted a spirited critique of our findings to Psychometrika.  His main point was that Guttman's and our views  were "fundamentally mistaken", and our results  "trivial".

Not lacking a sense of humor, he prefaced his work with a  limerick:

    "The work of a numer of "man(n)s"
    Isn't quite in accord with their plans.
    The results that seemed strong
    Are vacuous or wrong.
    Factor scoring is under no bans.

[The other intended "man(n)s", besides Guttman and Schonemann, were Kestelman and Heerman]. In his paper, this critic  introduced a novel type of factors which he called "non-trivial Kestelman factors". We pointed out in our review that such factors do not exist because their definition contained a contradiction. (For example, odd numbers divisible by 4 do not exist). Another reviewer was Louis Guttman. He recommended that the limerick be published without the rest of the paper.

The Psychometrika editor, on the other hand, was enthusiastic about the paper. Dismissing our criticisms as "philosophical", he advised us of his decision to publish the paper, but without inviting Guttman or us to comment on it in a rebuttal . However,  by that time it had dawned on the author that his misbegotten non-trivial Kestelman factors might turn into an embarrassment. He, therefore, embarked on a seemingly interminable sequence of revisions, progressively reducing the frequency of the word "trivial"in the process. Eventually he even changed the title. In the end, he arrived at a conclusion diametrically opposed to what  the editor had so warmly endorsed endorsed initially:

"Thus, the question of factor score indeterminacy cannot be thought of as a 'metaphysical' problem, divorced from the practice of factor analysis. It actually concerns the quite real question whether we can draw ... variables which yield consistent factor structures ... If we cannot, the factor we seek does not exist."

Guttman, who had  become increasingly frustrated with the conduct of the  editor, wrote on July 18, 1976 to the chairman of the Editorial Council of Psychometrika requesting

"that [it] suspend [the current editor] from being editor until completion of a thoroughgoing investigation of the current refereeing process of Psychometrika."

The Council responded by appointing an "Ad Hoc Committee to Consider Certain Charges" of psychometric elders who issued their Final Report on June 1, 1978. Unsurprisingly, they found

    "much in  [the editor's] behavior that is praiseworthy" (p. 58).

When Guttman in 1975 submitted a paper, "The Green-McDonald proof of the nonexistence of factor analysis",  the same Psychometrika editor rejected it without review. It eventually appeared post-humously in S. Levy (ed.): Louis Guttman on Theory and Methodology: Selected Writings. Aldershot: Dartmouth, 1994, pp. 361-369. It stands as a befitting  tribute to this "editor of  Psychometrika [who] was an active collaborator in the development of the published G-M arguments" (p. 368). All too often, behind the scenes contributions of editors of main stream journals go unacknowledged.

As a result of this experience I dropped my Psychometric Society membership.