Four least-squares methods for constructing factor scores have been described in the literature. The formal properties of these scores are developed, and they are compared in terms of four generally desirable properties of constructed factor scores. In particular, it is shown that two of the methods yield scores that are conditionally unbiassed, and univocal in the sense of Guilford and Michael, though not orthogonal, while one of the other methods yields orthogonal scores.

It is shown that constructed factor scores cannot be simultaneously univocal and orthogonal, unless we choose the special basis in factor space given by Canonical Factor Analysis.

The general problem of choosing between the methods is discussed, on the basis of the theoretical relations obtained.

A maximum likelihood solution for the method of successive intervals is presented together with an iterative procedure for obtaining estimates for stimulus scale values, boundary scale values, and stimulus dispersions. The method applies without modification to incomplete data matrices. The method was programmed for ILLIAC and a brief description of the program and a numerical illustration are included.

A rank test Rn has been proposed by Cronholm and Revusky which in effect increases the number of observations per subject by the device of randomly assigning a subject to a treated group in each of a series of sub-experiments. In this note further results on the asymptotic normality of the Rn test are discussed both under the null hypothesis H0 of randomness and under a class of alternatives to H0. The limiting power of the test is also discussed.

The Gramian factorization G of a Gramian R is square and symmetric and has no negative characteristic roots. It is shown to be that square factorization that is, in the least-squares sense, most isomorphic to R, most like a scalar K, and most highly traced, and to be the necessary and sufficient relation between the oblique vectors of an oblique transformation and the orthogonal vectors of the least-squares orthogonal counterpart. A slightly modified Gramian factorization is shown to be the factorization that is most isomorphic to a specified diagonal D, and to be the main part of an iterative procedure for obtaining “simplimax,” a square factor matrix with simple structure maximized in the sense of having the largest sum of squared diagonal loadings. Several published applications of Gramian factoring are cited.

A solution is presented to the problem of determining a proper correction for spuriousness in correlation coefficients. The general case developed is the estimate of correlation between u and v, both being linear functions of the same set of variables. Special formulae relate to overlapping scales correlations, part-whole correlations, and item-total test correlations.

A new computational method for the maximum likelihood solution in factor analysis is presented. This method takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive. Instead, the maximum may be attained on the boundary of the parameter space where one or more of the unique variances are zero. It is demonstrated that such improper (Heywood) solutions occur more often than is usually expected. A general procedure to deal with such improper solutions is proposed. The proposed methods are illustrated using two small sets of empirical data, and results obtained from the analyses of many other sets of data are reported. These analyses verify that the new computational method converges rapidly and that the maximum likelihood solution can be determined very accurately. A by-product obtained by the method is a large sample estimate of the variance-covariance matrix of the estimated unique variances. This can be used to set up approximate confidence intervals for communalities and unique variances.