Psychometrika 

Abstract: A general formula (α) of which a special case is the Kuder-Richardson coefficient of equivalence is shown to be the mean of all split-half coefficients resulting from different splittings of a test. α is therefore an estimate of the correlation between two random samples of items from a universe of items like those in the test. α is found to be an appropriate index of equivalence and, except for very short tests, of the first-factor concentration in the test. Tests divisible into distinct subtests should be so divided before using the formula. The index $$\bar r_{ij} $$ , derived from α, is shown to be an index of inter-item homogeneity. Comparison is made to the Guttman and Loevinger approaches. Parallel split coefficients are shown to be unnecessary for tests of common types. In designing tests, maximum interpretability of scores is obtained by increasing the first-factor concentration in any separately-scored subtest and avoiding substantial group-factor clusters within a subtest. Scalability is not a requisite.

Abstract: An index of factorial simplicity, employing the quartimax transformational criteria of Carroll, Wrigley and Neuhaus, and Saunders, is developed. This index is both for each row separately and for a factor pattern matrix as a whole. The index varies between zero and one. The problem of calibrating the index is discussed.

Abstract: Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.

Abstract: An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that thenormal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure—although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.

Abstract: It is suggested that if Guttman's latent-root-one lower bound estimate for the rank of a correlation matrix is accepted as a psychometric upper bound, following the proofs and arguments of Kaiser and Dickman, then the rank for a sample matrix should be estimated by subtracting out the component in the latent roots which can be attributed to sampling error, and least-squares “capitalization” on this error, in the calculation of the correlations and the roots. A procedure based on the generation of random variables is given for estimating the component which needs to be subtracted.