A model for a score based on an interview is presented which identifies the effect due to the subject, to the manner in which the interviewer tends to conduct his interviews, to the criteria he tends to use in scoring subjects' responses, to the compromises he tends to adopt between the demands of interviewing and those of scoring, and to chance errors. A suggested experimental design calls for each of K investigators to interview a different sample of N subjects, but for all investigators to score each subject. The drawing of inferences when interest is only in the K participants in the reliability study is considered, and a numerical example is given.

A method for fitting a perfect simplex [Guttman, 1954] is suggested which, in contrast to Kaiser's [1962], is independent of the order of the manifest variables. It is based on a procedure for scaling a set of points from their pairwise distances [Torgerson, 1958; Young & Householder, 1938] which is reviewed in compact notation in the Appendix. The method is extended to an iterative algorithm for fitting a quasi-simplex. Some empirical results are included.

An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations.

Prior to the statistical treatment of experimentally derived data the experimenter must obtain the data. In many studies in psychology, the content of group interactions is obtained by a data coding scheme. Coding schemes are shown to possess hierarchical structures and can be characterized by a hierarchy of fields. Three examples of coding schemes as hierarchies of fields are given. Techniques for extracting numbers from the set are presented. Problems associated with the improper assignment of elements of a data coding scheme to a theoretical structure are examined. The main result is that there will be errors in assignment of data to theory unless the structure of the data coding scheme is identical with that structure which is relevant to the theory being tested. Whenever the structures are not equal, the results are hard to interpret. In fact, any size of assignment error can be obtained. Since these errors precede the usual statistical treatment of data, it would appear that reported empirical findings are only valid when there exists no assignment errors.

Measures of test parsimony and factor parsimony are defined. Minimizing their weighted sum produces a general rotation criterion for either oblique or orthogonal rotation. The quartimax, varimax and equamax criteria are special cases of the expression. Two new criteria are developed. One of these, the parsimax criterion, apparently gives excellent results. It is argued that one of the most important factors bearing on the choice of a rotation criterion for a particular problem is the amount of information available on the number of factors that should be rotated.

Employing simulated data, several methods for estimating correlation and variance-covariance matrices are studied for observations missing at random from data matrices. The effect of sample size, number of variables, percent of missing data and average intercorrelations of variables are examined for several proposed methods.

Data are reported which show the statistical relation between the sample and population characteristic vectors of correlation matrices with squared multiple correlations as communality estimates. Sampling fluctuations were found to relate only to differences in the square roots of characteristic roots and to sample size. A principle for determining the number of factors to rotate and interpret after rotation is suggested.

Item characteristic curves are estimated without restrictive assumptions about their mathematical form. The resulting curves are compared with estimates obtained under the assumption that all curves are of logistic form. Surprising agreement is found between the curves obtained by the two unrelated methods.

An explicit solution is given to the problem of assigning relative lengths to the subtests of a test so as to maximize the correlation of the unit weight composite with a specified criterion when the total testing time is fixed. This solution is valid and unique whenever it specifies nonnegative times for all variables. A step-down procedure is suggested for cases in which some of the testing times are zero. This procedure does not necessarily provide an optimal allocation. However in examples studied it is found to provide near optimum results. Algorithms are also developed for the determination of the least total testing time required to attain specified multiple and composite correlations. A numerical example is given illustrating the use of the unit weight procedure in combination with the regression weight algorithm.

In many areas of research in which the experimenter deals with a set of dichotomous dimensions it is often desirable to specify them in terms of marginal distributions and some conditional distributions. This paper develops a set of systematic procedures for determination of joint distributions with an arbitrary number of dimensions. Most of these procedures require only addition and/or subtraction. An extension of the procedures permits easy determination of covariance and correlation matrices.

A method of estimating the parameters of the normal ogive model for dichotomously scored item-responses by maximum likelihood is demonstrated. Although the procedure requires numerical integration in order to evaluate the likelihood equations, a computer implemented Newton-Raphson solution is shown to be straightforward in other respects. Empirical tests of the procedure show that the resulting estimates are very similar to those based on a conventional analysis of item “difficulties” and first factor loadings obtained from the matrix of tetrachoric correlation coefficients. Problems of testing the fit of the model, and of obtaining invariant parameters are discussed.

The problem of locating two sets of points in a joint space, given the Euclidean distances between elements from distinct sets, is solved algebraically. For error free data the solution is exact, for fallible data it has least squares properties.

A model for direct multidimensional ratio scaling is presented, based on the concepts “common” and “difference” of the “halos” of two percepts. Measures of halos and their differences are proportional to lengths of corresponding percept vectors and their distance in subjective space. Ekman type scaling judgements are assumed to reflect the ratio measure of the common/measure of the standard's halo. The model is supposed to yield results that are in line with the results of distance models of multidimensional ratio scaling since negative scalar products of percept vectors are admitted.

The degree of metric determinancy afforded by nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the true dimensionality of the data being scaled, and the amount of error contained in the data being scaled. It was found 1) that if the ratio of the degrees of freedom of the data to that of the coordinates is sufficiently large then metric information is recovered even when random error is present; and 2) when the number of points being scaled increases the stress of the solution increases even though the degree of metric determinacy increases.

Greeno and Steiner have shown that a three state Markov Chain with a single absorbing state is equivalent to many of the current formalizations of All-or-None learning theories. Distribution statistics and other summary statistics are derived from the general model. Expressions for the maximum likelihood estimators of its parameters and the sampling variances of the estimates are presented. Likelihood ratio tests for several different null hypotheses are derived. These tests permit one to evaluate the usual null hypotheses in terms of the parameters of a process model.

Latent partition analysis enables an examination of the modal population categorizations of a set of items. The initial model proposed by Wiley is reformulated, extended and generalized. Firstly, the effects of judges are regarded as random, rather than fixed. In a second model, several latent partitions, rather than one, are allowed, resulting in a type of multidimensional approach. Finally, both the single and multiple partition models are shown to be specializations of a less restrictive model, called the category focusing model. In each of these three cases, properties implicit in the model are used to derive appropriate solution procedures.

Measures of individual differences are considered, and the application of the procedure to experimental work in concept formation is discussed. Several sets of data are used to demonstrate the characteristics of the techniques evolved.

The quartimax and varimax algorithms for orthogonal rotation attempt to maximize particular simplicity criteria by a sequence of two-factor rotations. Derivations of these algorithms have been fairly complex. A simple general theory for obtaining “two factor at a time” algorithms for any polynomial simplicity criteria satisfying a natural symmetry condition is presented. It is shown that the degree of any symmetric criterion must be a multiple of four. A basic fourth degree algorithm, which is applicable to all symmetric fourth degree criteria, is derived and applied using a variety of criteria. When used with the quartimax and varimax criteria the algorithm is mathematically identical to the standard algorithms for these criteria. A basic eighth degree algorithm is also obtained and applied using a variety of eighth degree criteria. In general the problem of writing a basic algorithm for all symmetric criteria of any specified degree reduces to the problem of maximizing a trigonometric polynomial of degree one-fourth that of the criteria.

For certain kinds of structure consisting of quantitative dimensions superimposed on a discrete class structure, spatial representations can be viewed as being composed of two subspaces, the first of which reveals the discrete classes as isolated clusters and the second of which contains variation along the quantitative attributes. A numerical method is presented for rotating a multi-dimensional configuration or factor solution so that the first few axes span the space of classes and the remaining axes span the space of quantitative variation. The use of this method is then illustrated in the analysis of some experimental data.

By proposing that the latent or true nature of subjects is identified with a limited number of response patterns (the Guttman scale patterns), the probability of an observed response pattern can be written as the sum of products of the probability of the true type multiplied by the chance of sufficient response error to cause the observed pattern to appear. This model contains the proportions of the true types as parameters plus some misclassification probabilities as parameters. Maximum likelihood methods are used to make estimates and test the fit for some examples.