Miller suggested ordinary least squares estimation of a constant transition matrix; Madansky proposed a relatively more efficient weighted least squares estimator which corrects for heteroscedasticity. In this paper an efficient generalized least squares estimator is derived which utilizes the entire covariance matrix of the distrubances. This estimator satisfies the condition that each row of the transition matrix must sum to unity. Madansky noted that estimates of the variances could be negative; a method for obtaining consistent non-negative estimates of the variances is suggested in this paper. The technique is applied to the hypothetical sample data used by Miller and Madansky.

Pearson's formula for univariate selection was derived with the assumption of normality of variates before and after selection. This study examined the influence of skew upon estimates from Pearson's formula under certain conditions. It was found that even with essentially symmetric distributions, a large proportion of the data is necessary to obtain reasonably precise estimates of low correlations. With increasing skew, estimates become increasingly erroneous, the direction of the error depending upon which tail of the distribution is the basis of the estimates. Difficulties in applying correction for univariate selection in several studies of the predictability of college-grades for Negroes from scores on standard aptitude tests are discussed.

The image factor analytic model (IFA), as related to Guttman’s image theory, is considered as an alternative to the traditional factor analytic model (TFA). One advantage with IFA, as compared with TFA, is that more factors can be extracted without yielding a perfect fit to the observed data. Several theorems concerning the structural properties of IFA are proved and an iterative procedure for finding the maximum likelihood estimates of the parameters of the IFA-model is given. Substantial experience with this method verifies that Heywood cases never occur. Results of an artificial experiment suggest that IFA may be more factorially invariant than TFA under selection of tests from a large battery.

When magnitude production is used to obtain a psychophysical power function for a group of Ss, the group exponent is shown to be the harmonic mean of individual-S exponents under the conditions usually employed in empirical research. The conditions are (a) the same numbers are presented to each S, (b) responses to each number are pooled over Ss by taking geometric means, and (c) the function is fit by the method of least squares in which regression is of the logarithm of Ss' responses upon the logarithm of numbers presented by E.

This paper presents a procedure for rotating an arbitrary factor matrix to maximum similarity with a specified factor pattern. The sum of squared distances between specified vectors and rotated vectors in oblique Euclidian space is minimized. An example of the application of the procedure is given.

An approximate statistical test is derived for the hypothesis that the reliability coefficients (Cronbach's α) associated with two measurement procedures are equal. Control of Type I error is investigated by comparing empirical sampling distributions of the test statistic with the theoretical model derived for it. The effect of platykurtosis in the test-score distribution on the test statistic is also considered.

A general model for describing the interrelations of common scores is derived. Guttman’s image analysis is a special case. In addition, a new factor model based upon estimation of the person product-moment matrix is described for the case in which the number of variables exceeds the number of persons.

When the covariance matrix Σ(p×P) does not satisfy the formal factor analysis model for m factors, there will be no factor matrix Λ(p×m) such that γ=(Σ-ΛΛ′) is diagonal. The factor analysis model may then be replaced by a tautology where γ is regarded as the covariance matrix of a set of “residual variates.” These residual variates are linear combinations of “discarded” common factors and unique factors and are correlated. Maximum likelihood, alpha and iterated principal factor analysis are compared in terms of the manner in which γ is defined, a “maximum determinant” derivation for alpha factor analysis being given. Weighted least squares solutions using residual variances and common variances as weights are derived for comparison with the maximum likelihood and alpha solutions. It is shown that the covariance matrix γ defined by maximum likelihood factor analysis is Gramian, provided that all diagonal elements are nonnegative. Other methods can define a γ which is nonGramian even when all diagonal elements are nonnegative.

This paper is a study of certain aspects of restricted ranking, a method intended for use by a panel of m judges evaluating the relative merits of N subjects, candidates for scholarships, awards, etc. Each judge divides the N subjects into R classes so that ni individuals receive a grade i (i = 1, 2, …, R; Σni = N) where the R numbers ni are close to N/R (ni = N/R when N is divisible by R) and are preassigned and the same for all judges. This method is superior in several respects to other likely alternatives. Under the null hypothesis that all nR = N subjects are of equal merit, four tests of significance are developed. The effectiveness of the method is investigated both theoretically by means of the asymptotic relative efficiency and more generally by simulation studies. When the numbers ni are not restricted to values close to or equal to N/R but instead are given values conforming to a normally distributed pattern, the resulting method is known as the Q-sort, so designated by certain investigators in psychotherapy. The simulation studies reveal that restricted ranking is only slightly inferior to complete ranking and generally superior in the cases considered to the Q-sort, although there are likely to be other situations when the latter is superior.

This paper demonstrates the feasibility of using a Newton-Raphson algorithm to solve the likelihood equations which arise in maximum likelihood factor analysis. The algorithm leads to clean easily identifiable convergence and provides a means of verifying that the solution obtained is at least a local maximum of the likelihood function. It is shown that a popular iteration algorithm is numerically unstable under conditions which are encountered in practice and that, as a result, inaccurate solutions have been presented in the literature. The key result is a computationally feasible formula for the second differential of a partially maximized form of the likelihood function. In addition to implementing the Newton-Raphson algorithm, this formula provides a means for estimating the asymptotic variances and covariances of the maximum likelihood estimators.

It is shown that the rank test R proposed by Cronholm and Revusky for comparing the effect of two treatments on a single group of n subjects is a linear function of Kendall’s τ-coefficient. The asymptotic results of Bennett for the distribution of R are also applicable to that of τ under a class of nonnull hypotheses.