Volume IV of The American Solider presents the first complete account of the scale analysis approach of Guttman and the latent structure approach of Lazarsfeld to the problem of attitude measurement. This review has been prepared for the purpose of providing an expository account of the models proposed by Guttman and Lazarsfeld, together with an indication of the places which call for additional clarification and research.

For the point distribution model of Lazarsfeld's latent structure analysis, the general matrix equation is stated which relates the manifest data in the form of joint occurrence matrices to the latent parameters. The relationship of the item responses and these joint occurrence matrices is also indicated in matrix form. A general solution for the latent parameters is then presented, which is based on the notion of factoring two joint occurrence matrices. The solution is valid under certain conditions which will usually be fulfilled. The solution assumes that estimates are available for the elements in the joint occurrence matrices with recurring subscripts, analogous to item communality or reliability. Some alternative methods of obtaining these estimates are discussed. Finally a fictitious 3-class, 8-item example is presented in detail.

Twenty years have now elapsed since Professor Thurstone’s ingenuity pulled the factor problem out of its tetrad difference quagmire. Most of us have watched Thurstone’s brain-child grow. One might say that the fledgling was so precocious as to reach maturity during the first ten years of its life. Its development was fostered not only by Thurstone but also by the resistance it encountered. Indeed, even as an infant, it was forced to throw aside its swaddling clothes in order to kick back at the Spearmans, the Tryons, the Anastasis, and others who sometimes rightly, ofttimes wrongly, pointed a finger at supposed imperfections. The youngster weathered more or less successfully the many storms it had stirred up, and by the end of its tenth year of life had in the hands of Thurstone made significant contributions and was ready for the sober coming-of-age evaluation of its strengths and weaknesses by Dael Wolfle. We suspect that the facts of life pointed out by Wolfle had already been whispered to the growing adolescent by none other than Thurstone himself.

Thurstone's Case V of the method of paired comparisons assumes equal standard deviations of sensations corresponding to stimuli and zero correlations between pairs of stimuli sensations. It is shown that the assumption of zero correlations can be relaxed to an assumption of equal correlations between pairs with no change in method. Further the usual approach to the method of paired comparisons Case V is shown to lead to a least squares estimate of the stimulus positions on the sensation scale.

An extension of Dwyer's “square root” method has been made to the problem of selecting a minimum set of variables in a multiple regression problem. The square root method of selection differs from the Wherry-Doolittle method primarily in that (1) the computations required are more compact, (2) an F ratio criterion is used which leads to the selection of fewer variables. The method provides solutions for the problems of test selection, item analysis, analysis of variance with disproportionate frequencies, and other problems requiring the rejection of superfluous variables. In a subsequent article a worked example will be given, and the square root and Wherry-Doolittle methods compared.

Non-spurious methods are needed for estimating the coefficient of equivalence for speeded tests from single-trial data. Spuriousness in a split-half estimate depends on three conditions; the split-half method may be used if any of these is demonstrated to be absent. A lower-bounds formula, rc, is developed. An empirical trial of this coefficient and other bounds proposed by Gulliksen demonstrates that, for moderately speeded tests, the coefficient of equivalence can be determined approximately from single-trial data. It is proposed that the degree to which tests are speeded be investigated explicitly, and an index τ is advanced to define this concept.

A battery of 46 tests was given to 237 college men. A factor analysis using the Thurstone technique revealed eight clearly interpretable first-order factors, one dubious factor, and a residual factor. The factors were interpreted as induction, deduction, flexibility of closure, speed of closure, space, verbal comprehension, word fluency, and number. Four second-order factors were abstracted from the matrix of first-order correlations. The presence of induction, deduction, and flexibility of closure on the first second-order factor, interpreted as an analytic factor, confirmed previous indications of relationships between the reasoning and closure factors. A second bipolar factor is interpreted as a speed of association factor. The third factor is interpreted as facility in handling meaningful verbal materials—perhaps an ability to do abstract thinking. The fourth factor is possibly a second-order closure factor—perhaps an ability to do concrete thinking.

The correlations among the thirteen personality scores yielded by the Guilford schedule for factors STDCR, and the Guilford-Martin schedules for factors GAMIN, and O, Ag, and Co, as reported by Lovell, were factored by the centroid method. The purpose was to see how many factors were represented by the thirteen scores; therefore the test reliabilities were used in the diagonal cells. It was found that the scores represent not more than nine linearly independent factors. The orthogonal factor matrix was rotated to oblique simple structure. Seven of the oblique factors were given tentative interpretation. Two factors were regarded as residual factors because of the small variance which they represent. The seven factors have been named Active, Vigorous, Impulsive, Dominant, Stable, Sociable, and Reflective.

Having given a fixed amount of total testing time it is important to know how long each test in the battery should be so that the correlation of the battery with the criterion will be a maximum. The precise solution for the test lengths will depend on a particular set of conditions which may be specified. The writer has previously presented solutions for two sets of conditions. This article presents the solution for a third set of conditions. These are: (1) The total number of items or testing time is fixed. (2) The score is the total number of items correctly answered. (3) The test lengths are determined in such a way that the correlation of total score with the criterion is a maximum. The solutions for the two previous sets of conditions, together with the current set, are summarized. A set of experimental data is submitted to each solution and the three sets of results are compared.

The nature of psychological measurements in relation to mathematical structures and representations is examined. Some very general notions concerning algebras and systems are introduced and applied to physical and number systems, and to measurement theory. It is shown that the classical intensive and extensive dimensions of measurements with their respective ordinal and additive scales are not adequate to describe physical events without the introduction of the notions of dimensional units and of dimensional homogeneity. It is also shown that in the absence of these notions, the resulting systems of magnitudes have only a very restricted kind of isomorphism with the real number system, and hence have little or no mathematical representations. An alternative in the form of an extended theory of measurements is developed. A third dimension of measurement, the supra-extensive dimension, is introduced; and a new scale, the multiplicative scale, is associated with it. It is shown that supra-extensive magnitudes do constitute systems isomorphic with the system of real numbers and that they alone can be given mathematical representations. Physical quantities are supra-extensive magnitudes. In contrast, to date, psychological quantities are either intensive or extensive, but never of the third kind. This, it is felt, is the reason why mathematical representations have been few and without success in psychology as contrasted to the physical sciences. In particular, the Weber-Fechner relation is examined and shown to be invalid in two respects. It is concluded that the construction of multiplicative scales in psychology, or the equivalent use of dimensional analysis, alone will enable the development of fruitful mathematical theories in this area of investigation.

Most indexes of item validity and difficulty vary systematically with changes in the mean and variance of the group. Formulas are presented showing how certain item parameters will vary with these alterations in group mean and variance. Item parameters are also suggested which should remain invariant under such changes. These parameters are developed under two different assumptions: first, the assumption that the total distribution of the item ability variable is normal, and, second, that the distribution of the item ability variable for each array of the explicit selection variable is normal.

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 \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar r_{ij} $$\end{document}, 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.

If customary methods of solution are used on the method of paired comparisons for Thurstone's Case V (assuming equal standard deviations of sensations for each stimulus), when in fact one or more of the standard deviations is aberrant, all stimuli will be properly spaced except the one with the aberrant standard deviation. A formula is given to show the amount of error due to the aberrant stimulus.

The conventional scoring formula to “correct for guessing” is derived and is compared with a regression method for scoring which has been recently proposed by Hamilton. It is shown that the usual formula, S = R – W/(n−1), yields a close approximation (correct within one point) to the maximum-likelihood estimate of an individual's “true score” on the test, if we assume that the individual “knows” or “does not know” the answer to each item, that guessing at unknown items is random, and that success at guessing is governed by the binomial law. It is also shown that the usual scoring formula yields an unbiased estimate of the individual's “true score,” when the true score is defined as the mean score over an indefinitely large number of independent attempts at the test or at equivalent (parallel) tests.

A procedure for estimating the reliability of sets of ratings, test scores, or other measures is described and illustrated. This procedure, based upon analysis of variance, may be applied both in the special case where a complete set of ratings from each of k sources is available for each of n subjects, and in the general case where k1, k2, ..., kn ratings are available for each of the n subjects. It may be used to obtain either a unique estimate or a confidence interval for the reliability of either the component ratings or their averages. The relations of this procedure to others intended to serve the same purpose are considered algebraically and illustrated numerically.

Formulas are developed for estimating a point-biserial r or a tetrachoric r from an obtained phi coefficient. The estimate of a tetrachoric r, which is called rϕ, is shown to be equivalent to that obtained from first-order use of the tetrachoric r series. A tabulation is made of corrections needed to make rϕ equivalent numerically to the tetrachoric r. In spite of its greater generality than estimates of tetrachoric r by previous methods, there are limitations, which are pointed out.

The square root method of selection has been explained in a previous article. In the present article a worked example is given which illustrates the compactness of the procedure. The square root method is compared with the Wherry-Doolittle method.

A test of goodness of fit is developed for Thurstone's method of paired comparisons, Case V.

The test involves the computation of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ^2= n\Sigma(\theta '' - \theta ')^2 /821,$$\end{document}, where n is the number of observations per pair, and θ″ and θ′ are the angles obtained by applying the inverse sine transformation to the fitted and the observed proportions respectively. The number of degrees of freedom is (k−1) (k−2)/2.

The problem considered is the use of a set of measurements on an individual to decide from which of several populations he has been drawn. It is assumed that in each population there is a probability distribution of the measurements. Principles for choosing the rule of classification are based on costs of misclassification. Optimum procedures are derived in general terms. If the measurements are normally distributed, the procedures use one discriminant function in the case of two populations and several discriminant functions in the cases of more populations. The numerical example given involves three normal populations.