An analytic method is presented for optimally classifying individuals into two subgroups on the basis of a cutting score on a test or test composite. The development assumes the test and criterion scores to be normally distributed, and the correlation surface to be bivariate normal. It is further assumed that individuals belong to the first or second sub-group depending on whether their criterion score is above or below a specified value. The predictor cutting score is determined so as to maximize the expected value of the decision procedure, taking gains and losses associated with correct and incorrect assignments into account.

The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years. These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications.

In this paper an effective iterative method for obtaining the exact solution is developed. By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown. Newton's method is suggested for solving this equation. Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations.

Between-sample shrinkage of validity from sample errors is compounded when usual multiple regression techniques are employed to estimate weights for new battery components. A rationale is described for increasing prediction weight validity through a combination of a reduced-rank regression technique and a method for determining maximal factored congruence between two sets of measures. A numerical illustration is based on data drawn from a problem in academic prediction.

Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.

It has been observed that the number of different ways in which a graph with p points can be labelled is p! divided by the number of symmetries, and that this holds regardless of the species of structure at hand. In this note, a simple group-theoretic proof is provided.

Significance tests are developed for evaluating the absolute invariance of the dominance and composition scales presented in a previous paper [Bechtel, 1967]. These tests derive from the multivariate normality of the estimated scale values—this multinormality stemming from that of the observations upon which these estimated scale parameters are based. The scales compared by means of the present techniques will typically be constructed under distinct experimental conditions, which may take the concrete form of treatments, occasions, individuals, or groups. Thus the present tests possess a wide range of applicability including various experimental designs in which scale configuration is the dependent variable. Illustrative data are used in statistically comparing the scale structures of two different individuals.

Latent partition analysis has been formulated to study the relationships between two or more partitions of the same set of items. The major structural hypothesis is that a latent partition underlies the manifest partitions; that is, it is assumed that each item belongs to a latent category and that the manifest categories are derived by dividing and combining the latent categories. We have found that by examining manifest categories it is possible to reconstruct information about the latent partition and about its relation to the manifest partitions.

Usual system: take as scale value the normal deviate corresponding to the area p″ = (p − p0)/(1 − p0), where p is the proportion of correct answers and p0 is the chance proportion. If p ≦ p0, the item difficulty is unscalable. Proposed system: take as scale value the normal deviate corresponding to the area p′ = px, with x so chosen that p′ = .5 when p is half-way between p0 and 1. Table gives D (=500 + 100z) for p = .00(.01)1.00 and for a (= number of alternatives) 2,3,4,5,6,7, ∞.

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.