Following a general approach due to Guttman, coefficientα is rederived as a lower bound on the reliability of a test. A necessary and sufficient condition under which equality is attained in this inequality and hence that α is equal to the reliability of the test is derived and shown to be closely related to the recent redefinition of the concept of parallel measurements due to Novick. This condition is then also shown to be closely related to the unit rank assumption originally adopted by Kuder and Richardson in the derivation of their formula 20. The assumption later adopted by Jackson and Ferguson and the one adopted by Gulliksen are shown to be related to the necessary and sufficient condition derived here. It is then pointed out that the statement that “coefficient α is equal to the mean of the split-half reliabilities” is true only under the restricted condition assumed by Cronbach in the body of his derivation of this result. Finally some limitations on the uses of any function of α as a measure of internal consistency are noted.

Techniques for partitioning objects into optimally homogeneous groups on the basis of empirical measures of similarity among those objects have received increasing attention in several different fields. This paper develops a useful correspondence between any hierarchical system of such clusters, and a particular type of distance measure. The correspondence gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data. In an explicitly defined sense, one method forms clusters that are optimally “connected,” while the other forms clusters that are optimally “compact.”

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.

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.

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.

Repeated measures designs have been widely employed in psychological experimentation, however, such designs have rarely been analyzed by means of permutation procedures. In the present paper certain aspects of hypothesis tests in a particular repeated measures design (one non-repeated factor (A) and one repeated factor (B) with K subjects per level of A) were investigated by means of permutation rather than sampling processes. The empirical size and power of certain normal theory F-tests obtained under permutation were compared to their nominal normal theory values. Data sets were established in which various combinations of kurtosis of subject means and intra-subject variance heterogeneity existed in order that their effect upon the agreement of these two models could be ascertained. The results indicated that except in cases of high intra-subject variance heterogeneity, the usual F-tests on B and AB exhibited approximately the same size and power characteristics whether based upon a permutation or normal theory sampling basis.

The Thurstone and the Bradley-Terry models, both initially advanced on intuitive grounds, have proved useful in the analysis of paired comparisons. The psychological meaning of these models and their relation to one another is unclear, but they fit data. Stevens has observed that there may be two basic mechanisms of discrimination 1) additive and 2) substitutive. We advance two corresponding mathematical models: that experienced sensation is 1) the sum of a large number of independent signals and 2) the maximum of a large number of independent signals. These assumptions yield 1) Thurstone's model and 2) the model of Bradley-Terry. Psychological interpretations of the various parameters, in terms of sensation, present themselves in a natural manner. Thus this paper presents a theory which unifies and interprets two paired comparison models that have proved useful in fitting experimental data.

Formulas are derived for the asymptotic variances and covariances of the maximum likelihood estimators for oblique simple structure models which are identified by prior specification of zero elements in the factor loading matrix. The formulas are expressed in terms of the various submatrices of the inverse of the required variance-covariance matrix. A numerical example using artificial data is given and problems in the application of the formulas discussed.

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.

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.

A large-sample test for the significance of the difference between two detection data points is developed based upon the assumptions of a one-parameter signal detectability model. In essence, the null hypothesis tested is that two observed data points belong to the same d′ function.

In an earlier paper [Psychometrika, 31, 1966, p. 147], Srivastava obtained a test for the Hypothesis H0 : Σ =α0Σ0 + ... +αlΣl, where Σi are known matrices,αi are unknown constants and Σ is the unknown (p × p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p × p) covariance matrix of a random variablex (with p components) the condition that Σ0, Σ1, ..., Σl form a commutative linear associative algebra and a certain vector θ, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where Σ0, Σ1, ..., Σl are the association matrices of a partially balanced association scheme.

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.

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.

A perfect Guttman scale generates a perfect Guttman simplex. This is capitalized on to get a transformation useful in a kind of non-linear factor analysis applicable to empirical correlations resembling but not necessarily conforming to a perfect simplex.

A Gramian factorization—square, symmetric, possibly of low rank but with no negative characteristic roots—of empirical correlations is transformed into latent information about nonlinear regressions of tests on a single underlying dimension. Class intervals on that continuum are defined in terms of their standard score profiles on the tests. Each class interval exhibits “local independence,” so that individual members may vary but may not covary in their profiles.

The method is flexible as to choice of diagonals, fitted rank, and exact shape of resulting non-linear regressions, so long as those regressions are essentially monotonic and have a proper progression of curvatures. Construction of the needed transformation for data of any size is outlined, and the problem of metric for the latent continuum is given several solutions.

An empirical example is provided with full and low rank solutions involving different diagonal choices.

A signed graph,S, is colorable if its point set can be partitioned into subsets such that all positive lines join points of the same subset and all negative lines join points of different subsets.S is uniquely colorable if there is only one such partition. Developed in this note is a new matrix, called the type matrix of S, which provides a classification of the way pairs of points are joined in S. Such a classification yields a criterion for colorability and unique colorability.

Variance analyses are presented for two data layouts—each corresponding to the class of all ordered pairs from a single finite set. The analysis of the dominance layout is in terms of a fixed effects linear model which includes parameters representing the scale values of the elements of the set, response bias, and pairwise interactions. A parallel parametrization is carried out for the composition layout for which corresponding point estimates and hypothesis tests are given. A joint treatment of concurrently observed dominance and composition layouts is suggested and illustrative data are presented.

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.

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.