The negative hypergeometric distribution may be generalized to provide distributions involving 3, 4, or more parameters. It is shown that, in the case of the binomial error model, such distributions correspond to quadratic, cubic, etc., regressions of true scores on raw scores. Explicit formulas are given for the bivariate moments required to fit these regressions and so estimate the parameters of the generalized hypergeometric distribution. Equations are also developed for fitting the 3- and 4-parameter distributions directly, i.e., without reference to the model, and the method for generalizing these is indicated. The methods developed are applied to objective test data with satisfactory results.

Multidimensional scaling is the problem of representing n objects geometrically by n 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.

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.

An analysis of the selection and use of informational models in psychological research is undertaken. The axioms of different informational models are presented together with the deductions which have direct empirical application. The models are used to demonstrate many of the inadequacies and inaccuracies involved in the use of information theory in the psychological literature. It is also shown that certain related conceptual ambiguities are understandable in terms of confusions among various models.

It often happens that a theory specifies some variables or states which cannot be identified completely in an experiment. When this happens, there are important questions as to whether the experiment is relevant to certain assumptions of the theory. Some of these questions are taken up in the present article, where a method is developed for describing the implications of a theory for an experiment. The method consists of constructing a second theory with all of its states identifiable in the outcome-space of the experiment. The method can be applied (i.e., an equivalent identifiable theory exists) whenever a theory specifies a probability function on the sample-space of possible outcomes of the experiment. An interesting relationship between lumpability of states and recurrent events plays an important role in the development of the identifiable theory. An identifiable theory of an experiment can be used to investigate relationships among different theories of the experiment. As an example, an identifiable theory of all-or-none learning is developed, and it is shown that a large class of all-or-none theories are equivalent for experiments in which a task is learned to a strict criterion.

A model for an experimental design involving a complete set of Latin squares for testing the homogeneity of treatment effects was constructed and analyzed by Gourlay. In his analysis, however, if one or both of the preliminary F-tests are significant, the analysis cannot differentiate. He then suggests the use of a less desirable test which is biased and has fewer degrees of freedom, regardless of the number of replications (the d.f. cannot be increased by increasing the replications). Further, when heterogeneity of variance occurs, Gourlay's test procedures are in general invalid. The present paper reviews Gourlay's analysis and proposes a modified test procedure.

When several test forms are used interchangeably, they will not in practice, despite all efforts, be rigorously parallel. A definition of the standard error of measurement appropriate for this type of situation can be provided; however, it will be different from the definition in classical test theory. Appropriate formulas for the standard error of measurement and for other related quantities under both definitions are derived and compared; also sample statistics for estimating these quantities.

The Johnson-Neyman technique is a statistical tool used most frequently in educational and psychological applications. This paper starts by briefly reviewing the Johnson-Neyman technique and suggesting when it should and should not be used; then several different modifications and extensions of the Johnson-Neyman technique, all of them conceptually simple, are proposed. The close relation between confidence intervals and regions of significance of the Johnson-Neyman type is pointed out. The problem of what to do when more than two groups are being compared is considered. The situation of more than one criterion variable is also considered.

The technique of canonical correlation may be used to determine the extent to which two sets of measurements reflect the same underlying traits. However, if the two sets are not perfectly reliable, the unreliability may obscure the fact that they are essentially dependent on similar processes. If we attempt to determine sets of weights so as to maximize the correlation between linear composites after correction for attenuation in the composites, it turns out that the results may be obtained by determining the canonical correlations and canonical regression weights between the true score components of the measures making up the two sets. In addition, formulas are developed for calculating the correlations between the canonical variates and original measures, both corrected and uncorrected for attenuation. A numerical example relating the verbal to the performance subtests of the Wechsler Intelligence Scale for Children is presented.

We describe the numerical methods required in our approach to multi-dimensional scaling. The rationale of this approach has appeared previously.

The difference in factor structure resulting from the factorization of correlations, covariances, and cross products is discussed. Factoring cross products has the advantage of retaining information on both means and variances; this method of factoring learning data is recommended. The conditions under which factoring covariances and cross products lead to the same essential structure are investigated.

The problem of estimation of the parameters of the Bower three-state learning model is discussed for three forms of the model. In the simplest case, a minimum variance unbiased estimator is found and is presented with its asymptotic distribution theory and with a method of obtaining approximate confidence intervals. In the other cases, methods are discussed for obtaining the maximum likelihood estimators at least approximately. All estimation techniques are illustrated by application to a set of data obtained by Theios.

The problems of selecting the winner in a tournament, a leader in a society, or the most dominating or influential person in a group of individuals are not infrequent. Graph theory is successfully used in such situations in locating, by the use of associated matrices of graphs representing the individuals under the studied relation, the person with the greatest power to influence. In this paper one more important point is brought into consideration before selecting the leader or the most influencing personality—that is the consideration of weakness to be influenced by. The one with a nice blending of these two characters—possessing the highest power to influence a person and simultaneously having the least weakness to be influenced by—is to be selected. But, in practice, to locate such a man in a group is delicate. A solution is presented here by appealing to graph theoretic notions and using them.

A general framework for obtaining all possible factor analytic solutions, orthogonal and oblique, for a given common factor space is developed in detail. Interestingly, and seemingly paradoxically, any one of these solutions may be obtained by orthogonal transformations of selected matrices; thus an oblique solution may be determined by orthogonal transformations. Within the possible oblique solutions, two distinct categories of solutions emerge, a special case of the simpler of which apparently provides a definitive solution to the problem of independent, but correlated, clusters. Possible further specializations of the general approach to specific problems are discussed.

The correlation in a singly truncated binormal distribution is obtained in terms of Mills' ratio using the Mehler identity. A table of the correlation in the underlying distribution as a function of the correlation in the truncated distribution is presented, together with a diagram summarizing this relationship.

A ratio scale of subjective magnitude is developed from paired-comparison data. The model attempts to combine arguments of Restle, Ekman, and Luce relating data obtained from paired comparisons and corresponding data from direct psychophysical scaling methods at the ratio level. The basic set-theoretical model involves the use of imperfectly nested sets. A numerical example illustrates the application of the theory.

A similarity relation is reflexive and symmetric. A graph may be regarded as a geometric representation of an irreflexive symmetric relation; an r-graph then coincides with a similarity relation. For the realization of psychophysical phenomena, the points of a graph stand for the stimuli and a line for indistinguishability of a stimulus pair. Not all graphs, however, are likely to arise as similarity relations in a real situation. A special class of graphs, called interval graphs, does have this realistic property. These graphs are described and characterized.

A method is outlined whereby the equations resulting from multivariate analyses may be solved. Only the upper or lower triangular coefficient matrix need be employed. All solutions involving any combination(s) of the variables represented in the coefficient matrix are readily obtainable without permutation of the matrix. All solutions are given immediately without the need for back solving. One variable is added to or subtracted from the regression equation with each application of the method. The method may be applied as repeatedly as needed for any particular solution.