Suppose a collection of standard tests is given to all subjects in a random sample, but a different new test is given to each group of subjects in nonoverlapping subsamples. A simple method is developed for displaying the information that the data set contains about the correlational structure of the new tests. This is possible to some extent, even though each subject takes only one new test. The method uses plausible values of the partial correlations among the new tests given the standard tests in order to generate plausible simple correlations among the new tests and plausible multiple correlations between composites of the new tests and the standard tests. The real data example included suggests that the method can be useful in practical problems.

A factor analysis model consists of a random sequence of variates defined on a probability space and satisfying the usual descriptive equations of the common-factor analysis in which the common-factor scores are dimensionally independent. Necessary and sufficient conditions are given for a model to exist with essentially unique and hence determinate common factor scores. Parallel results are given for the existence of models with nonunique and hence indeterminate scores. It is then próved that two models cannot exist with essentially unique but different scores for the same common factors. The meaning and application of these results are discussed.

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.

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.

Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.

The relationship between variables in applied and experimental research is often investigated by the use of extreme (i.e., upper and lower) groups. Recent analytical work has provided an extreme groups procedure that is more powerful than the standard correlational approach for all values of the correlation and extreme group size. The present article provides procedures to optimize power by determining the relative number of subjects to use in each of two stages of data collection given a fixed testing budget.

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.

Researchers in the past ten years have studied various parameters involved in nonmetric multidimensional scaling by utilizing Monte Carlo procedures. This paper develops stress distributions using Kruskal's second stress formula based upon a null hypothesis of equal likelihood in the ranking of a set of proximities. These distributions can serve to determine whether a set of data has other than random structure.

This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an “analytic” solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other “analytic” procedures. The procedure, which we call SUMSCAL (subjective metric scaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.

For ordinal measurement the concept of an individual propensity distribution is developed. For any given individual the mean of this distribution is his true score, for which estimation procedures are discussed. Two measures of individual dispersion are considered and their distributions derived in the null case. These measures are shown to be counterparts at the individual level of Kendall’s tau and Spearman’s rho. Estimation of the two dispersion measures from sample data is investigated, and the relation of these estimates to the variance of the individual propensity distribution is derived.

“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.

A family of models for the representation and assessment of individual differences for multivariate data is embodied in a hierarchically organized and sequentially applied procedure called PINDIS. The two principal models used for directly fitting individual configurations to some common or hypothesized space are the dimensional salience and perspective models. By systematically increasing the complexity of transformations one can better determine the validities of the various models and assess the patterns and commonalities of individual differences. PINDIS sheds some new light on the interpretability and general applicability of the dimension weighting approach implemented by the commonly used INDSCAL procedure.

Cohen's kappa index is reformulated for multiple classifications based on exchangeable random variables. It is found that kappa is between 0 and 1 inclusive. Two characterizations for kappa are stated in terms of the relationship between such random variables. Within the normal test score model, kappa increases with test reliability and test length. Furthermore, when based on binary classifications, kappa is an inverse U-shaped function of the cutoff score. These trends also hold for the beta-binomial test score model.

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.

The problem of comparing two sociometric matrices, as originally discussed by Katz and Powell in the early 1950’s, is reconsidered and generalized using a different inference model. In particular, the proposed indices of conformity are justified by a regression argument similar to the one used by Somers in presenting his well-known measures of asymmetric ordinal association. A permutation distribution and an associated significance test are developed for the specific hypothesis of “no conformity” reinterpreted as a random matching of the rows and (simultaneously) the columns of one sociometric matrix to the rows and columns of a second. The approximate significance tests that are presented and illustrated with a simple numerical example are based on the first two moments of the permutation distribution, or alternatively, on a random sample from the complete distribution.

Several ways of using the traditional analysis of variance to test heterogeneity of spread in factorial designs with equal or unequal n are compared using both theoretical and Monte Carlo results. Two types of spread variables, (1) the jackknife pseudovalues of s2 and (2) the absolute deviations from the cell median, are shown to be robust and relatively powerful. These variables seem to be generally superior to the Z-variance and Box-Scheffé procedures.

A method of introducing a controlled degree of skew and kurtosis for Monte Carlo studies was derived. The form of such a transformation on normal deviates [X ~ N(0, 1)] is Y = a + bX + cX2 + dX3. Analytic and empirical validation of the method is demonstrated.

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.

A multiple-answer multiple-choice test item has a certain number of alternatives, any number of which might be keyed. The examinee is also allowed to mark any number of alternatives. This increased flexibility over the one keyed alternative case is useful in practice but raises questions about appropriate scoring rules. In this article a certain class of item scoring rules called the binary class is considered. The concepts of standard scoring rules and equivalence among these scoring rules are introduced in the “misinformation” model for which the traditional “knowledge” model is a special case. The examinee’s strategy with respect to a scoring rule is examined. The critical role of a quantity called the scoring ratio is emphasized. In the case of examinee uncertainty about the number of correct alternatives on an item, a Bayes and a minimax strategy for the examinee are developed. Also an appropriate response for the examiner to the minimax strategy is outlined.

A special case of Bloxom's version of Tucker's three-mode model is developed statistically. A distinction is made between modes in terms of whether they are fixed or random. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. The identification problem is discussed, and unknown parameters of the model are estimated by a weighted least squares method based upon a Gauss-Newton algorithm. A goodness-of-fit statistic is presented. An example based upon self-report and peer-report measures of personality shows that the model is applicable to real data. The model represents a generalization of Thurstonian factor analysis; weighted least squares estimators and maximum likelihood estimators of the factor model can be obtained using the proposed theory.

Incorporating temporal decline in predictive validity into mental testing theory is outlined. Starting from multivariate regression of criterion on repeated measurements, an analytic extension results in a weighting function for repeated measurements, replacing the beta weights. Besides optimizing, the procedure permits an evaluation of any particular prognosis setting: In cases of exponentially declining predictive validity, prognostic range can be extended if concurrent validity is nonperfect by optimal weighting (“predictive filtering”) of repeated measurements. Considerable gain in prognostic range over the traditional approach can be achieved if predictive validity declines concavely downwards.