We have said that psychometric methods involving algorithms are completely objective—at least they are if the algorithm is in the form of a program for a digital computer. These objective procedures need Monte Carlo and other computer runs to determine their properties, but so do many equation-oriented techniques. The objective algorithms are flexible but not flaccid. They offer a way of dealing with complexities that formerly seemed beyond our grasp. As the computer revolution continues in psychometrics, we can expect objective algorithmic methods to become the rule rather than the exception.

The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.

Admissible probability measurement procedures utilize scoring systems with a very special property that guarantees that any student, at whatever level of knowledge or skill, can maximize his expected score if and only if he honestly reflects his degree-of-belief probabilities. Section 1 introduces the notion of a scoring system with the reproducing property and derives the necessary and sufficient condition for the case of a test item with just two possible answers. A method is given for generating a virtually inexhaustible number of scoring systems, both symmetric and asymmetric, with the reproducing property. A negative result concerning the existence of a certain subclass of reproducing scoring systems for the case of more than two possible answers is obtained. Whereas Section 1 is concerned with those instances in which the possible answers to a query are stated in the test itself, Section 2 is concerned with those instances in which the student himself must provide the possible answer(s). In this case, it is shown that a certain minor modification of a scoring system with the reproducing property yields the desired admissible probability measurement procedure.

A solutionT of the least-squares problem AT=B + E, given A and B so that trace (E′E)= minimum and T′T= I is presented. It is compared with a less general solution of the same problem which was given by Green [5]. The present solution, in contrast to Green's, is applicable to matrices A and B which are of less than full column rank. Some technical suggestions for the numerical computation of T and an illustrative example are given.

Existing analytic oblique rotation schemes proceed by optimizing a simplicity function applied to the reference structure. This article suggests optimizing a simplicity function applied to primary loadings directly. The feasibility of the suggestion is demonstrated using the quartimin criterion. An algorithm to implement the optimization is derived and the existence of an admissible solution proved. Practical comparisons with the biquartimin method are made using Thurstone's Box Problem and Holzinger and Swineford's Twenty-Four Psychological Tests Problem.

Let x be a p-component random variable having a multivariate normal distribution with covariance matrix Σ. In this paper, we consider the problem of testing hypotheses of the form H0:Σ = b1Σ1 + … + bmΣm, where bi's are unknown scalars, and Σi's are a set of known and simultaneously diagonalizable matrices. This problem has both psychometric and statistical interest, and its basic theory is developed here. Besides, the problem of obtaining likelihood-ratio statistic for testing H0 is studied, and the statistic obtained in a special case.

A method of computing communalities is presented which attempts to cluster eigenvalues about zero. Results are exhibited which show reduction of rank using computed communalities.

A previous article was concerned with simultaneous linear prediction. There one was given a set of predictor tests or items and one predicted a set of predictands (also tests or items, or perhaps criteria.) We proposed a simultaneous prediction which was a certain weighted sum of the predictors. In the present article the constraint that the prediction be a weighted sum is relaxed. We seek a general function of the predictors which will maximize the quantity chosen for measuring prediction efficiency. This quantity is the same as the one used in linear prediction and we justify this approach by showing it is the appropriate one when there is only one predictand. In order to solve the problem we restrict consideration to a vector of predictors having only a finite number of possible values, i.e., it possesses discrete probability distribution weights. This can be applied in the case of dichotomous items for instance. It may also be used in continuous distributions as an approximation, by first dividing the original range of values into a finite number of intervals. Then one attributes to the interval the weight corresponding to the probability mass it underlies in the original distribution.

A logistic model developed by Birnbaum was tested in two ways. First, plots of proportions of subjects in different score categories were examined for consistency with the assumption of a logistic trace line, and especially for departures from the logistic which seemed due to guessing in multiple choice items. The results showed that guessing seemed to have little effect. Second, an attempt was made to predict the obtained score distributions of samples of subjects on six tests from item parameters estimated on independent samples. The fits were good in all cases, despite considerable differences between the tests, and some extremely odd distributions.

This paper examines the concept of centrality with respect to small-group communication experiments. An index of centrality is presented which is based on the incidence matrix of actual communications rather than on the deviation matrix of possible communications, as in the Bavelas Index of Centrality. The index takes the value of zero for the homogeneous all-channel graph and the value of unity for the homogeneous wheel graph. The index can be computed for individuals as well as groups. Three examples are computed.

A model for the measurement of the discrepancy between two scores is presented and discussed as a paradigm for the study of growth or experimentally produced change. The model assumes two tests or measures differing in complexity, and it analyzes the true difference between the test scores into a component that is entirely dependent on the first or base-line test and a second component that is entirely independent of it. Equations for estimating both components are given and these are compared with other measurement efforts with similar goals.

It is assumed that the investigator has set up a simple structure hypothesis in the sense that he has specified the zero loadings of the factor matrix. The maximum-likelihood method is used to estimate the factor matrix and the factor correlation matrix directly without the use of rotation methods, and the likelihood-ratio technique is used to test the simple structure hypothesis. Numerical examples are presented.

An algorithm is presented for constructing from the adjacency matrix of a digraph the matrix of its simple n-sequences. In this matrix, the i, j entry, i ≠j, gives the number of paths of length n from a point vi to a point vj; the diagonal entry i, i gives the number of cycles of length n containing vi. The method is then generalized to networks—that is, digraphs in which some value is assigned to each line. With this generalized algorithm it is possible, for a variety of value systems, to calculate the values of the paths and cycles of length n in a network and to construct its value matrix of simple n-sequences. The procedures for obtaining the two algorithms make use of properties of a line digraph—that is, a derived digraph whose points and lines represent the lines and adjacency of lines of the given digraph.

A model is presented for evaluating potential effectiveness of a Bayesian classification system using the expected value of the posterior probability for true classifications as an evaluation metric. For a given set of input parameters, the value of this complex metric is predictable from a simply computed row variance metric. Prediction equations are given for several representative sets of input parameters.

It is shown that to combine readings from “points of view” configurations as assumed in Tucker and Messick's model is not to combine configurations in any simple way. Both empirical and logical consequences of the disparity are discussed.

In many decision problems the decision-maker must ask a series of questions or select cues relevant to but prior to the decision. There are many strategies which would prescribe an order for asking such questions or selecting such cues. It is the purpose of the paper to describe a manner in which the strategy or strategies of the decision-maker can be analyzed. From this model can be derived limits corresponding to the possible use of a particular strategy, and the effective or observed use of the strategy or strategies by the decision-maker.

This paper is addressed to the classical problem of estimating factor loadings under the condition that the sum of squares of off-diagonal residuals be minimized. Communalities consistent with this criterion are produced as a by-product. The experimental work included several alternative algorithms before a highly efficient method was developed. The final procedure is illustrated with a numerical example. Some relationships of minres to principal-factor analysis and maximum-likelihood factor estimates are discussed, and several unresolved problems are pointed out.

Certain aspects of point estimation are treated for two kinds of exponential latency processes. The first unitary process is represented by a simple exponential density in which the rate parameter may be viewed as an unknown constant or as a random variable. If a second, slower exponential process is grafted onto the first, there results a postulated two-component latency between stimulus and response. Moments estimators are derived for the two parameters of this latter density, and the relevance of the second parameter to decision time is emphasized.

To take account of the observed lack of fit of the power law near thresh-old intensities, two different modifications of the power law have been proposed by various investigators. In this paper, both of these two laws are derived as a special case of a generalized power function for ratio scaling. A method is presented for discriminating between the special laws which provides (i) a prescription for the manipulation of independent variables, and (ii) specification of theoretical curves to which empirical curves are to be compared. Maximum-likelihood estimators are derived for the exponents of the special laws under the assumption that the observed subjective ratios are log normal.

Two problems are considered. The first is that of rotating two factor solutions orthogonally to a position where corresponding factors are as similar as possible. A least-squares solution for transformations of the two factor matrices is developed. The second problem is that of rotating a factor matrix orthogonally to a specified target matrix. The solution to the second problem is related to the first. Applications are discussed.