Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution.

The present paper shows that the usual factor analytic structured data dispersion matrix Λ Ψ Λ’ + Δ can readily arise from a set of scores y = Λ η + ε, where the “common” (η) and “unique” (ε) factors have nonzero covariance: Γ = Cov(ε,η) ≠ 0. Implications of this finding are discussed for the indeterminacy of factor scores, and for the issue of invariance of factor analytic covariance models. The size of the problem is explored with numerical examples.

Results obtained by Guttman [1955] on the determinacy of common factors have been thought to have disturbing consequences for the common factor model. It is argued that factors must be thought of as unobservable, and uniquely defined but numerically indeterminate. It follows that Guttman's measure of indeterminacy is inconsistent with the foundations of the factor model in probability theory, and the traditional measures of factor indeterminacy used by earlier writers should be reinstated. These yield no disturbing conclusions about the model.