Abilities are usually assumed to exist in a “positive manifold.” Experimental manipulations of physiological variables, however, suggest that negative relationships exist between certain of the neural processes contributing to simple perceptual-motor vs. perceptual-restructuring tasks. First-order correlative evidence of this phenomenon cannot be obtained because the between-individual differences in general ability level tend to exceed the behavioral effects of the intra-individual opposition between neural processes. Also, since statistical removal of the “g” variance induces bipolarity in the remaining variance, the second-order negative correlations are necessarily regarded as artifactual. A combined correlational-experimental approach is suggested to overcome this difficulty.

How can an investigator choose a good team of raters to use for measuring a continuous variable when each available rater produces only dichotomous responses? We formulate an underlying model, define an index of goodness for rater teams in terms of average mean square error of the estimate, develop a new estimator and derive the optimal rater terms. The optimal raters have characteristic curves which are linear in form and satisfy the requirements for a Guttman scale.

In a previous paper [Elashoff 1969], we derived optimal rater teams for a particular formulation of the dichotomous rater problem. Here, we describe a computer-based procedure for selecting good rater teams in practice; we apply the procedure to the selection of items for a psychological inventory.

Let x denote one of the variables in a factor analysis model. In 1935 Roff proved that the communality of x is greater than or equal to the squared multiple correlation coefficient of x with the other variables in the model. In this paper it is shown that the inequality is the first of an infinite sequence of inequalities, each sharper than the one before, and that the condition for equality is the same for all members of the sequence. Also, the limiting inequality is considered and the condition for equality derived. As this condition is always satisfied in a factor analysis model, the inequality is really an identity.

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.

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.

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.