Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.

Four approximate tests are considered for repeated measurement designs in which observations are multivariate normal with arbitrary covariance matrices. In these tests traditional within-subject mean square ratios are compared with critical values derived from F distributions with adjusted degrees of freedom. Two of them—the ∈ approximate and the improved general approximate (IGA) tests-behave adequately in terms of Type I error. Generally, the IGA test functions better than the ∈ approximate test, however the latter involves less computations. In regards to power, the IGA test may compete with one multivariate procedure when the assumptions of the latter are tenable.

“Determinate” solutions for the indeterminate common factor of p variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the original p variables to the original single common factor model does not determine a unique solution for the indeterminate factor of the p variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the original p variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.

This paper is concerned with the additive constant problem in metric multidimensional scaling. First the influence of the additive constant on eigenvalues of a scalar product matrix is discussed. The second part of this paper is devoted to the introduction of a new formulation of the additive constant problem. A solution is given for fixed dimensionality, by maximizing a normalized index of fit with a gradient method. An experimental computation has shown that the author's solution is accurate and easy to follow.

Tukey's scheme for finding separations in univariate data strings is described and tested. It is found that one can use the size of a data gap coupled with its ordinal position in the distribution to determine the likelihood of its having arisen by chance. It was also shown that this scheme is relatively robust for fatter-tailed-than-Gaussian distributions and has some interesting implications in multidimensional situations.

Two designs for comparing a judge's ratings with a known standard are presented and compared. Design A pertains to the situation where the judge is asked to categorize each of N subjects into one of r (known) classes with no knowledge of the actual number in each class. Design B is employed when the judge is given the actual number in each class and is asked to categorize the individuals subject to these constraints. The probability distribution of the total number of correct choices is developed in each case. A power comparison of the two procedures is undertaken.

In the applications of maximum likelihood factor analysis the occurrence of boundary minima instead of proper minima is no exception at all. In the past the causes of such improper solutions could not be detected. This was impossible because the matrices containing the parameters of the factor analysis model were kept positive definite. By dropping these constraints, it becomes possible to distinguish between the different causes of improper solutions. In this paper some of the most important causes are discussed and illustrated by means of artificial and empirical data.

Several procedures have been proposed in the statistical literature for estimating simultaneously the mean of each of k binomial populations. In terms of mental test theory, however, it is not clear that these procedures should be used when an item sampling model applies since the binomial error model is usually viewed as an oversimplification of the “true” situation. In this study we compare empirically several of these estimation techniques. Particular attention is given to situations where observations are generated according to a two-term approximation to the compound binomial distribution.

When the raters participating in a reliability study are a random sample from a larger population of raters, inferences about the intraclass correlation coefficient must be based on the three mean squares from the analysis of variance table summarizing the results: between subjects, between raters, and error. An approximate confidence interval for the parameter is presented as a function of these three mean squares.

A formulation, which is different from Guttman's is presented. The two formulations are both called the optimal scaling approach, and are proven to provide identical scale values. The proposed formulation has at least two advantages over Guttman's. Namely, (i) the former serves to clarify close relations of the optimal scaling approach to those of Slater and the vector model of preferential choice, and (ii) in addition to the stimulus scale values, it provides scores for the subjects, which indicate the degrees of response consistency (transitivity), relative to the optimum solution. The method is assumption-free and capable of multidimensional analysis.

P. M. Bentler has shown that Rao's canonical factor analysis is in effect a psychometric analysis, leading to factors that are maximally assessible from the data. He contrasts this with Kaiser and Caffrey's alpha factor analysis that leads to factors that maximally represent the true factors in the content domain. Noting the problems associated with factors that may be highly assessible, but not very representative, or vice versa, Bentler suggests the need for a technique that would, insofar as possible, be optimal with respect to both criteria. Such a technique is presented here, and is shown to resolve into a traditional scaling method, which in turn acquires a richer psychometric interpretation.

Procrustean analysis is extended here to the case where matrices have different row order. Using a definition system in the sense of facet theory, fitting transformations are determined for the set of centroids that represent common substantive equivalence classes of all or some matrices. The resulting operators are used to transform the original matrices. An index, Δ, which is consistent with this approach expresses the goodness of fit.

A method is discussed which extends principal components analysis to the situation where the variables may be measured at a variety of scale levels (nominal, ordinal or interval), and where they may be either continuous or discrete. There are no restrictions on the mix of measurement characteristics and there may be any pattern of missing observations. The method scales the observations on each variable within the restrictions imposed by the variable's measurement characteristics, so that the deviation from the principal components model for a specified number of components is minimized in the least squares sense. An alternating least squares algorithm is discussed. An illustrative example is given.

A simple proof that the squared multiple correlation of a variable with the remaining variables in the set of variables is a lower bound to the communality of that variable is presented.