On a multiple-choice test in which each item has k alternative responses, the test taker is permitted to choose any subset which he believes contains the one correct answer. A scoring system is devised that depends on the size of the subset and on whether or not the correct answer is eliminated. The mean and variance of the score per item are obtained. Methods are derived for determining the total number of items that should be included on the test so that the average score on all items can be regarded as a good measure of the subject's knowledge. Efficiency comparisons between conventional and the subset selection scoring procedures are made. The analogous problem of r > 1 correct answers for each item (with r fixed and known) is also considered.

This paper traces the course of the consequences of viewing test responses as simply providing dichotomous data concerning ordinal relations. It begins by proposing that the score matrix is best considered to be items-plus-persons by items-plus-persons, and recording the wrongs as well as the rights. This shows how an underlying order is defined, and was used to provide the basis for a tailored testing procedure. It also was used to define a number of measures of test consistency. Test items provide person dominance relations, and the relations provided by one item can be in one of three relations with a second one: redundant, contradictory, or unique. Summary statistics concerning the number of relations of each kind are easy to get and provide useful information about the test, information which is related to but different from the usual statistics. These concepts can be extended to form the basis of a test theory which is based on ordinal statistics and frequency counts and which invokes the concept of true scores only in a limited sense.

To date, virtually all techniques appropriate for ordinal data are based on the uniform probability distribution over the permutations. In this paper we introduce and examine an alternative probability model for the distribution of ordinal data. Preliminary to deriving the expectations of Spearman's rho and Kendall's tau under this model, we show how to compute certain conditional expectations of rho and tau under the uniform distribution. The alternative probability model is then applied to ordinal test theory, and the calculation of true scores and test reliability are discussed.

Inference models motivated by the combinatorial chance literature and the concept of object matching may be used in the analysis of a contingency table if the conditional assumption of fixed row and column totals is imposed. More specifically, by developing a matching reinterpretation for several problems of interest in the prediction analysis of cross-classifications—as defined by Hildebrand, Laing and Rosenthal, appropriate significance tests can be given that may differ from those justified by the more common multinomial models. In the course of the paper the distinction between a degree-1 statistic (based on the relationship between single objects) and a degree-2 statistic (based on the relationship between object pairs) is reviewed in some detail. Also, several specializations are presented to topics of current methodological importance in psychology; for instance, a number of references are made to the measurement of nominal scale response agreement between two raters.

Based on a simple nonparametric procedure for comparing two proximity matrices, a measure of concordance is introduced that is appropriate when K independent proximity matrices are available. In addition to the development of a general concept of concordance and specific techniques for its evaluation within and between the subsets of a partition of the K matrices, several methods are also suggested for comparing and/or for fitting a particular structure to the given data. Finally, brief indications are provided as to how the well-known notion of concordance for K rank orders can be included within the more general framework.

This paper details a Bayesian alternative to the use of least squares and equal weighting coefficients in regression. An equal weight prior distribution for the linear regression parameters is described with regard to the conditional normal regression model, and resulting posterior distributions for these parameters are detailed. Some interesting connections between this Bayesian procedure and several other methods for estimating optimal weighting coefficients are discussed. In addition, results are presented of a Monte Carlo investigation which compared the effectiveness of the Bayesian procedure relative to least squares, equal weight, ridge, and Bayesian exchangeability estimations.

This paper is concerned with the development of a measure of the precision of a multidimensional euclidean structure. The measure is a precision index for each point in the structure, assuming that all the other points are precisely located. The measure is defined and two numerical methods are presented for its calculation. A small Monte Carlo study of the measure's behavior is performed and findings discussed.

A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.

A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.

A method is presented to provide estimates of parameters of specified nonlinear equations from ordinal data generated from a crossed design. The analytic method, NOPE, is an iterative method in which monotone regression and the Gauss-Newton method of least squares are applied alternatively until a measure of stress is minimized. Examples of solutions from artificial data are presented together with examples of applications of the method to experimental results.

Many authors have demonstrated for idealized item configurations that equal item weights are often virtually as good for a particular predictive purpose as the item weights that are theoretically optimal. What has not been heretofore clear, however, is what happens to the similarity between weighted and unweighted composites of the same items when the item configuration's variance structure is complex.

The method of scoring is used to obtain maximum likelihood estimates of the parameters in the White and Clark learning hierarchy validation model. From the estimate of the proportion of the population possessing only the superordinate skill in a pair of hierarchical skills, and its variance, the hypothesis of inclusion is tested. An illustrative example of the procedure is given.

The issue of factor indeterminacy, and its meaning and significance for factor analysis, has been the subject of considerable debate in recent years. Interestingly, the identical issue was discussed widely in the literature of the late 1920's and early 1930's, but this early discussion was somehow lost or forgotten during the development and popularization of multiple factor analysis. There are strong parallels between the arguments in the early literature, and those which have appeared in recent papers. Here I review the history of this early literature, briefly survey the more recent work, and discuss these parallels where they are especially illuminating.

When comparing examinees to a control, the examiner usually does not know the probability of correctly classifying the examinees based on the number of items used and the number of examinees tested. Using ranking and selection techniques, a general framework is described for deriving a lower bound on this probability. We illustrate how these techniques can be applied to the binomial error model. New exact results are given for normal populations having unknown and unequal variances.

The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in which t-statistics based upon these jackknife estimates of the standard errors are computed. The validity of the t-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackknife may be greatly enhanced in the factor analysis model.

A modified beta binomial model is presented for use in analyzing ramdom guessing multiple choice tests and certain forms of taste tests. Detection probabilities for each item are distributed beta across the population subjects. Properties for the observable distribution of correct responses are derived. Two concepts of true score estimates are presented. One, analogous to Duncan’s empirical Bayes posterior mean score, is appropriate for assessing the subject’s performance on that particular test. The second is more suitable for predicting outcomes on similar tests.

A new subroutine has been developed for calculating the terachoric correlation coefficient. Recent advances in computing inverse normal and bivariate normal distributions have been utilized. The iterative procedure is started with an approximation with an error less than ±.0135.

Guttman's assumption underlying his definition of “total images” is rejected: Partial images are not generally convergent everywhere. Even divergence everywhere is shown to be possible. The convergence type always found on partial images is convergence in quadratic mean; hence, total images are redefined as quadratic mean-limits. In determining the convergence type in special situations, the asymptotic properties of certain correlations are important, implying, in some cases, convergence almost everywhere, which is also effected by a countable population or multivariate normality or independent variables. The interpretations of a total image as a predictor, and a “common-factor score”, respectively, are made precise.

A Monte Carlo study was carried out in order to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures when portions of the data are missing. All sets of simulated proximity measures were based on 30 stimuli and three dimensions, and selection of missing elements was done randomly. Properties of the simulated data varied according to (a) the number of individuals, (b) the level of random error, (c) the proportion of missing data, and (d) whether the same entries or different entries were deleted for each individual. Results showed that very accurate recovery of true distances, stimulus coordinates, and weight vectors could be achieved with as much as 60% missing data as long as sample size was sufficiently large and the level of random error was low.