A probabilistic unfolding model is presented which makes it possible to scale stimuli and subjects simultaneously, without assumptions concerning the distribution of subjects or of stimuli. The applicability of the model is demonstrated in areas of attitude measurement where the assumption of normality is inadequate.

When teaching elementary concepts on information theory a problem keeps reoccurring which, if only for pedagogical reasons, deserves discussion.

A model containing linear and nonlinear parameters (e. g., a spatial multidimensional scaling model) is viewed as a linear model with free and constrained parameters. Since the rank deficiency of the design matrix for the linear model determines the number of side conditions needed to identify its parameters, the design matrix acts as a guide in identifying the parameters of the nonlinear model. Moreover, if the design matrix and the uniqueness conditions constitute an orthogonal linear model, then the associated error sum of squares may be expressed in a form which separates the free and constrained parameters. This immediately provides least squares estimates of the free parameters, while simplifying the least squares problem for those which are constrained. When the least squares estimates for a nonlinear model are obtained in this way, i.e. by conceptualizing it as a submodel, the final error sum of squares for the nonlinear model will be a restricted minimum whenever the side conditions of the model become real restrictions upon its submodel. In this case the design matrix for the embracing orthogonal model serves as a guide in introducing parameters into the nonlinear model as well as in identifying these parameters. The method of overwriting a nonlinear model with an orthogonal linear model is illustrated with two different spatial analyses of a three-way preference table.

In this paper, recent developments in empirical Bayes procedures are tied in with current work in mental test theory. Point estimators of “true scores” are derived for the binomial and Rasch test models. These estimators are shown to be asymptotically optimal. Smoothing and an empirical study of the behavior of empirical Bayes estimates are taken up in the final section.

A test statistic is introduced which allows one to test the hypothesis of agreement of several judges on the ranking of items within each of two groups and between the two groups. The groups of judges may be unequal in size. A normal approximation for the test statistic is developed. The relationship to existing techniques given by Kendall, Friedman, Page, Spearman, and Lyerly is discussed. A generalization of the coefficient of concordance is presented and the extension of the method to multi-group problems is suggested.

The object of this paper is to clarify Schönemann's unfolding algorithm and, in particular, to make it clear that the equations numbered (3.2) in Schönemann's [1970] article, which define Schönemann's solutions, are not a complete set of restraints for the purpose of defining metric unfoldings. Namely, Schönemann has transformed the original equations which define an unfolding to a set of linear and non-linear equations of which he uses only the linear equations to define his solutions. Given infallible data (solution(s) exist) Schönemann's solutions will include the correct solutions. If enough data are available so that there are enough linear equations to uniquely determine a single solution, then Schönemann's solution will coincide with the correct solution. Let P and Q denote the number of elements in the two sets of points, the interset distances of which are specified by the data in the unfolding problem. Let m denote the dimensionality of the Euclidean space into which these points are to be imbedded. If only the linear equations, numbered (18) herein, are to be used, then Schönemann gives the following data requirement for the solution to be uniquely determined: If the full set of linear and nonlinear equations (18–20) are used, then the amount of data required for a solution to be locally unique is relaxed to Both of these results assume that the equations are independent, which has not been proved.

A test which allows for errors of measurement is derived for the hypothesis that all the members of a population who possess a certain skill are a sub-set of the members who possess another skill. Formulae are given for one particular case when two questions are used for each skill, and for when three questions are used for each skill. An illustrative example is given for the two-question case.

A procedure is proposed whereby questionnaire data, which is usually ordinal in nature and often error-ridden, may be transformed to reduce the error variance in the data and to improve the metric properties of the individual variables. The technique is suggested by a result of Eckart and Young. The properties of the method are investigated by means of a Monte Carlo study. Various matrices were generated representing the usual concept of “true scores”. These matrices were distorted using two levels of random errors and two kinds of categorization. The distorted matrices were in turn transformed by the proposed methods and compared to the “true scores”. In all cases an overall measure of similarity reveals the transformed matrices are better approximations to the “true scores” than the untransformed data. Some properties of the transformation are discussed and some possible applications of the general technique are suggested.

In this paper a rapid and reliable method is found for estimating the value of the Bivariate Normal Correlation Coefficient, ρ, given values of the joint probability and the normal deviates, h and k, or the related areas. This technique finds useful application in the computational approximation of the tetrachoric correlation coefficient, r, when the underlying distributions may be assumed to be normal.

It is the purpose of this paper to discuss the ‘Lens Mode’ approach to the analysis of subject performance in multiple-cue judgment tasks embedded in probabilistic environments. Various characteristics of the linear lens model are presented in matrix notation and are formulated without reference to standardized variables in order to facilitate analysis and subsequent generalization. Some new results on ‘matching’ behavior are presented. A brief outline of current problems and related work is presented.

Evidence is given to indicate that Lawley's formulas for the standard errors of maximum likelihood loading estimates do not produce exact asymptotic results. A small modification is derived which appears to eliminate this difficulty.

A general index of reliability, termed “canonical reliability,” is developed for use with profiles, or more generally, for use with vectors of random variables. Canonical reliability is defined as the ratio of the average squared distance among true scores to the average squared distance among observed scores. Based on Mahalonobis distances, canonical reliability is shown to be consistent with multivariate analogues of parallel form correlations, squared correlation between true and observed scores, and an analysis of variance formulation. The index of reliability based on Cronbach and Gleser's D2 is also derived from the general formulation. A comparison of the Mahalonobis and D2 approaches indicates that score vectors using D2 distances are more reliable; however, both methods of comparing profiles are useful depending on the nature of the information that is desired. Transforming the observed variables to independent canonical variates provides a basis for comparing profiles on maximally reliable profile dimensions. For illustrative purposes, profile reliability is calculated and interpreted for the WISC subscales for a 7 1/2 year age group.

We concern ourselves with the hypothesis that two variables have a perfect disattenuated correlation, hence measure the same trait except for errors of measurement. This hypothesis is equivalent to saying, within the adopted model, that true scores of two psychological tests satisfy a perfect linear relation. Statistical tests of this hypothesis are derived when the relation is specified with the exception of the additive constant. Two approaches are presented and various assumptions concerning the error parameters are used. Then the results are reinterpreted in terms of the possible existence of an unspecified perfect linear relation between true scores of two psychological tests. A numerical example is appended by way of illustration.

In Coombs' unidimensional unfolding theory each individual ranks all stimuli, often in terms of preference judgments. These ranks, called I scales, are used to infer the latent continuum called a J scale which is presumed to have generated the I scale rankings. A major problem concerns the inference of the J scale given a set of I scales because an I scale is not unique to any one J scale. This paper presents a procedure for estimating J scale probabilities given a set of I scales. Models for three and four or more stimuli are presented. Results of computer tests are reported.

Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.

In a manner similar to that used in the orthogonal case, formulas for the aymptotic standard errors of analytically rotated oblique factor loading estimates are obtained. This is done by finding expressions for the partial derivatives of an oblique rotation algorithm and using previously derived results for unrotated loadings. These include the results of Lawley for maximum likelihood factor analysis and those of Girshick for principal components analysis. Details are given in cases including direct oblimin and direct Crawford-Ferguson rotation. Numerical results for an example involving maximum likelihood estimation with direct quartimin rotation are presented. They include simultaneous tests for significant loading estimates.

The interdistances between thirteen places situated in different parts of the world were estimated by 60 subjects. The estimates were analysed by Kruskal's multidimensional technique and, after a cosine transformation, by factor analysis. It was found that both methods yielded the same three-dimensional solution. Also a two-dimensional configuration could describe the data, and it was shown that this configuration contained representations of unidimensional ratings obtained in other studies.