A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.

The validity conditions for univariate repeated measures designs are described. Attention is focused on the sphericity requirement. For a v degree of freedom family of comparisons among the repeated measures, sphericity exists when all contrasts contained in the v dimensional space have equal variances. Under nonsphericity, upper and lower bounds on test size and power of a priori, repeated measures, F tests are derived. The effects of nonsphericity are illustrated by means of a set of charts. The charts reveal that small departures from sphericity (.97 ≤ ε < 1.00) can seriously affect test size and power. It is recommended that separate rather than pooled error term procedures be routinely used to test a priori hypotheses.

This paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach, Alternating Least Squares and Optimal Scaling. Using these two principles, my colleagues and I have extended a variety of analysis procedures originally proposed for quantitative (interval or ratio) data to qualitative (nominal or ordinal) data, including additivity analysis and analysis of variance; multiple and canonical regression; principal components; common factor and three mode factor analysis; and multidimensional scaling. The approach has two advantages: (a) If a least squares procedure is known for analyzing quantitative data, it can be extended to qualitative data; and (b) the resulting algorithm will be convergent. Three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented.

Bayesian least squares techniques are adapted to estimation of stimulus-response curves, rather broadly conceived. Illustrative examples deal with estimation of person characteristic curves and item characteristic curves in the context of mental testing, and estimation of a stimulus-response curve using data from a psychophysical experiment.

Most psychophysicists neglect to consider how error should be characterized in applications of the power law. Failures of the power law to agree with certain theoretical predictions may be due to bias of parameter estimates. A power law with lognormal product structure is proposed and approximately unbiased parameter estimates are given for several common estimation situations.

For assigning subjects to treatments the point of intersection of within-group regression lines is ordinarily used as the critical point. This decision rule is critized and, for several utility functions and any number of treatments, replaced by optimal monotone, nonrandomized (Bayes) rules. Both treatments with and without mastery scores are considered. Moreover, the effect of unreliable criterion scores on the optimal decision rule is examined, and it is illustrated how qualitative information can be combined with aptitude measurements to improve treatment assignment decisions. Although the models in this paper are presented with special reference to the aptitude-treatment interaction problem in education, it is indicated that they apply to a variety of situations in which subjects are assigned to treatments on the basis of some predictor score, as long as there are no allocation quota considerations.

A nonparametric test of dispersion with paired replicates data is described which involves jackknifing logarithmic transformations of the ratio of variance estimates for the pre- and post-treatment populations. Results from a Monte Carlo simulation show that the test performs well under Ho and has good power properties. Examples are given of applying the procedure on psychiatric data.

A maximum likelihood estimation procedure is developed for multidimensional scaling when (dis)similarity measures are taken by ranking procedures such as the method of conditional rank orders or the method of triadic combinations. The central feature of these procedures may be termed directionality of ranking processes. That is, rank orderings are performed in a prescribed order by successive first choices. Those data have conventionally been analyzed by Shepard-Kruskal type of nonmetric multidimensional scaling procedures. We propose, as a more appropriate alternative, a maximum likelihood method specifically designed for this type of data. A broader perspective on the present approach is given, which encompasses a wide variety of experimental methods for collecting dissimilarity data including pair comparison methods (such as the method of tetrads) and the pick-M method of similarities. An example is given to illustrate various advantages of nonmetric maximum likelihood multidimensional scaling as a statistical method. At the moment the approach is limited to the case of one-mode two-way proximity data, but could be extended in a relatively straightforward way to two-mode two-way, two-mode three-way or even three-mode three-way data, under the assumption of such models as INDSCAL or the two or three-way unfolding models.

Asymptotic distribution theory of Brogden's form of biserial correlation coefficient is derived and large sample estimates of its standard error obtained. Its efficiency relative to the biserial correlation coefficient is examined. Other modifications of the statistic are evaluated, and on the basis of these results, recommendations for choice of estimator of biserial correlation are presented.

On distractor-identification tests students mark as many distractors as possible on each test item. A grading scale is developed for this type testing. The scale is optimal in that it is the unique scale giving an unbiased estimate of the student's “true score”, i.e., the score that would result if no guessing occurred. If the test is administered as a usual multiple choice test and graded using the usual correction for guessing scale, the expected item score is the same as for the distractor-identification testing using the optimal grading scale. However, the variance of the item score is shown to be less for distractor-identification testing than for usual multiple choice testing under certain conditions.

A linear utility model is introduced for optimal selection when several subpopulations of applicants are to be distinguished. Using this model, procedures are described for obtaining optimal cutting scores in subpopulations in quota-free as well as quota-restricted selection situations. The cutting scores are optimal in the sense that they maximize the overall expected utility of the selection process. The procedures are demonstrated with empirical data.

A new method is proposed for a simultaneous factor analysis of dichotomous responses from several groups of individuals. The method makes it possible to compare factor loading pattern, factor variances and covariances, and factor means over groups. The method uses information from first and second order proportions and estimates the model by generalized least-squares. Hypotheses regarding different degrees of invariance over groups may be evaluated by a large-sample chi-square test.

Limitations and extensions of Feldt's approach to testing the equality of Cronbach's alpha coefficients in independent and matched samples are discussed. In particular, this approach is used to test equality of intraclass correlation coefficients.

This paper attempts to clarify the nature of redundancy analysis and its relationships to canonical correlation and multivariate multiple linear regression. Stewart and Love introduced redundancy analysis to provide non-symmetric measures of the dependence of one set of variables on the other, as channeled through the canonical variates. Van den Wollenberg derived sets of variates which directly maximize the between set redundancy. Multivariate multiple linear regression on component scores (such as principal components) is considered. The problem is extended to include an orthogonal rotation of the components. The solution is shown to be identical to van den Wollenberg's maximum redundancy solution.

Procedures are described for the analysis of profiles of means in repeated measures designs under order restriction for patterns of mean change. The exact likelihood ratio is derived for the case of two groups of subjects, and a computationally simple alternative to the exact likelihood ratio test is provided for designs involving more than two groups. Tables of critical values are provided for the case of simple-order alternatives.

A model for four-mode component analysis is developed and presented. The developed model, which is an extension of Tucker's three-mode factor analytic model, allows for the simultaneous analysis of all modes of a four-mode data matrix and the consideration of relationships among the modes. An empirical example based upon viewer perceptions of repetitive advertising shows the four-mode model applicable to real data.

Various recent works have developed “feature” or “aspect” models of similarity and preference. These models are more concerned with the fine detail of the judgment process than were prior models, but nevertheless they have not in general developed an underlying stochastic process compatible with the assumed structure. In this paper, we show that a particular class of multivariate stochastic processes, namely those associated with the Marshall-Olkin multivariate exponential distribution, generates several of these models. In particular, such stochastic processes (appropriately interpreted) yield Tversky's elimination by aspects model, Edgell and Geisler's (normal) additive random aspects model, and Shepard and Arabie's additive cluster model.

A simple stochastic model is formulated in order to determine the optimal time between the first test and the second test when the test-retest method of assessing reliability is used. A forgetting process and a change in true score process are postulated. The optimal time between tests is derived by maximizing the probability that the respondent has not remembered the response on the first test and has not had a change in true score. The resulting test-retest correlation is then found to be a linear function of the true reliability of the test, where the slope of this function is the key probability of not remembering and having no change in true score. Some numerical examples and suggestions for using the results in empirical studies are given. Specific recommendations are presented for improved design and analysis of intentions data.