It is proved for the common factor model with r common factors that under certain condition s which maintain the distinctiveness of each common factor a given common factor will be determinate if there exists an unlimited number of variables in the model each having an absolute correlation with the factor greater than some arbitrarily small positive quantity.

The factor analysis (FA) model does not permit unique estimation of the common and unique factor scores. This weakness is notorious as the factor indeterminacy in FA. Luckily, some part of the factor scores can be uniquely determined. Thus, as a whole, they can be viewed as a sum of determined and undetermined parts. The paper proposes to select the undetermined part, such that the resulting common factor scores have the following feature: the rows (i.e., individuals) of the common factor score matrix are as well classified as possible into few clusters. The clear benefit is that we can easily interpret the factor scores simply by focusing on the clusters. The procedure is called clustered common factor exploration (CCFE). An alternating least squares algorithm is developed for CCFE. It is illustrated with real data examples. The proposed approach can be viewed as a parallel to the rotation techniques in FA. They exploit another FA indeterminacy, the rotation indeterminacy, which is resolved by choosing the rotation that transforms the loading matrix into the ‘most’ interpretable one according to a pre-specified criterion. In contrast to the rotational indeterminacy, the factor indeterminacy is utilized to achieve well-clustered factor scores by CCFE. To the best of our knowledge, such an approach to the FA interpretation has not been studied yet.

The assumptions of the model for factor analysis do not exclude a class of indeterminate covariances between factors and error variables (Grayson, 2003). The construction of all factors of the model for factor analysis is generalized to incorporate indeterminate factor-error covariances. A necessary and sufficient condition is given for indeterminate factor-error covariances to be arbitrarily small, for mean square convergence of the regression predictor of factor scores, and for the existence of a unique determinate factor and error variable. The determinate factor and error variable are uncorrelated and satisfy the defining assumptions of factor analysis. Several examples are given to illustrate the results.

The issue of factor indeterminacy, and its meaning and significance for factor analysis, has been the subject of considerable debate in recent years. Interestingly, the identical issue was discussed widely in the literature of the late 1920's and early 1930's, but this early discussion was somehow lost or forgotten during the development and popularization of multiple factor analysis. There are strong parallels between the arguments in the early literature, and those which have appeared in recent papers. Here I review the history of this early literature, briefly survey the more recent work, and discuss these parallels where they are especially illuminating.

A theorem is presented which gives the range of possible correlations between a common factor and an external variable (i.e., a variable not included in the test battery factor analyzed). Analogous expressions for component (and regression component) theory are also derived. Some situations involving external correlations are then discussed which dramatize the theoretical differences between components and common factors.

Several sufficient conditions are available for mean square convergence of factor predictors. A necessary and sufficient condition is given in the Heywood case with respect to (confirmatory) factor analysis. This condition generalizes that of Krijnen (2006) and performs better than a signal-to-noise type of condition (Schneeweiss & Mathes, 1995).

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γp = Φ1/2Λ′pΨp−1ΛpΦ1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γp or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors.

A new factor analysis (FA) procedure has recently been proposed which can be called matrix decomposition FA (MDFA). All FA model parameters (common and unique factors, loadings, and unique variances) are treated as fixed unknown matrices. Then, the MDFA model simply becomes a specific data matrix decomposition. The MDFA parameters are found by minimizing the discrepancy between the data and the MDFA model. Several algorithms have been developed and some properties have been discussed in the literature (notably by Stegeman in Comput Stat Data Anal 99:189–203, 2016), but, as a whole, MDFA has not been studied fully yet. A number of new properties are discovered in this paper, and some existing ones are derived more explicitly. The properties provided concern the uniqueness of results, covariances among common factors, unique factors, and residuals, and assessment of the degree of indeterminacy of common and unique factor scores. The properties are illustrated using a real data example.

Parceling—using composites of observed variables as indicators for a common factor—strengthens loadings, but reduces the number of indicators. Factor indeterminacy is reduced when there are many observed variables per factor, and when loadings and factor correlations are strong. It is proven that parceling cannot reduce factor indeterminacy. In special cases where the ratio of loading to residual variance is the same for all items included in each parcel, factor indeterminacy is unaffected by parceling. Otherwise, parceling worsens factor indeterminacy. While factor indeterminacy does not affect the parameter estimates, standard errors, or fit indices associated with a factor model, it does create uncertainty, which endangers valid inference.