A combinatorial data analysis strategy is reviewed that is designed to compare two arbitrary measures of proximity defined between the objects from some set. Based on a particular cross-product definition of correspondence between these two numerically specified notions of proximity (typically represented in the form of matrices), extensions are then pursued to indices of partial association that relate the observed pattern of correspondence between the first two proximity measures to a third. The attendant issues of index normalization and significance testing are discussed; the latter is approached through a simple randomization model implemented either through a Monte Carlo procedure or distributional approximations based on the first three moments. Applications of the original comparison strategy and its extensions to partial association may be developed for a variety of methodological and substantive tasks. Besides rank correlation, we emphasize the topics of spatial autocorrelation for one variable and spatial association between two and mention the connection to the usual randomization approach for one-way analysis-of-variance.

If a loss function is available specifying the social cost of an error of measurement in the score on a unidimensional test, an asymptotic method, based on item response theory, is developed for optimal test design for a specified target population of examinees. Since in the real world such loss functions are not available, it is more useful to reverse this process; thus a method is developed for finding the loss function for which a given test is an optimally designed test for the target population. An illustrative application is presented for one operational test.

To establish the existence of his abilities, a judge is given the task of classifying each of N= rs subjects into one of r known categories, each containing s of the subjects. An incomplete design is proposed whereby the judge is presented with b groups, each one containing n= rs/b<r subjects. The n different categories corresponding to members of the group are known. Using the total number of correct classifications, this method of grouping is compared to that in which the group size is equal to the number of categories. The incomplete grouping is shown to yield a more powerful test for discriminating between the null hypothesis that the judge is guessing the classifications and the alternative hypothesis that he has some definite abilities. The incomplete design is found to be most effective (powerful) when the number of subjects in a group is limited to two or three.

In a recent article Bentler and Woodward (1983) discussed computational and statistical issues related to the greatest lower bound ρ+ to reliability. Although my work (Shapiro, 1982) was cited frequently some results presented were misunderstood. A sample estimate + of ρ + was considered and it was claimed (Bentler & Woodward) that: “Since + is not a closed form expression ... an exact analytic expression for h has not been found” (p. 247). (h is a vector of partial derivatives of ρ+ as a function of the covariance matrix.) Therefore Bentler and Woodward proposed to use numerical derivatives in order to evaluate the asymptotic variance avar ( +) of +.

Certain ambiguities in a recent paper on the computation and statistics of the greatest lower bound are clarified.

This paper brings together and compares two developments in the analysis of Likert attitude scales. The first is the generalization of latent class models to ordered response categories. The second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales. Key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of “life satisfaction” data. The way in which the latent trait model defines a unit of measurement, takes into account the order of the response categories, and scales the latent classes, is discussed. While the latent class model provides better fit to these data, this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes.

A common criticism of iterative least squares estimates of communality is that method of initial estimation may influence stabilized values. As little systematic research on this topic has been performed, the criticism appears to be based on cumulated experience with empirical data sets. In the present paper, two studies are reported in which four types of initial estimate (unities, squared multiple correlations, highest r, and zeroes) and four levels of convergence criterion were employed using four widely available computer packages (BMDP, SAS, SPSS, and SOUPAC). The results suggest that initial estimates have no effect on stabilized communality estimates when a stringent criterion for convergence is used, whereas initial estimates appear to affect stabilized values employing rather gross convergence criteria. There were no differences among the four computer packages for matrices without Heywood cases.

A Bayesian procedure is developed for the estimation of parameters in the two-parameter logistic item response model. Joint modal estimates of the parameters are obtained and procedures for the specification of prior information are described. Through simulation studies it is shown that Bayesian estimates of the parameters are superior to maximum likelihood estimates in the sense that they are (a) more meaningful since they do not drift out of range, and (b) more accurate in that they result in smaller mean squared differences between estimates and true values.

This article demonstrates that Freeman's theta, a measure of association between a nominal variable and an ordinal variable, has a sampling distribution identical with the sampling distribution of Mann-Whitney's U.

Through external analysis of two-mode data one attempts to map the elements of one mode (e.g., attributes) as vectors in a fixed space of the elements of the other mode (e.g., stimuli). This type of analysis is extended to three-mode data, for instance, when the ratings are made by more individuals. It is described how alternating least squares algorithms for three-mode principal component analysis (PCA) are adapted to enable external analysis, and it is demonstrated that these techniques are useful for exploring differences in the individuals' mappings of the attribute vectors in the fixed stimulus space. Conditions are described under which individual differences may be ignored. External three-mode PCA is illustrated with data from a person perception experiment, designed after two studies by Rosenberg and his associates whose results were used as external information.

Lord and Wingersky have developed a method for computing the asymptotic variance-covariance matrix of maximum likelihood estimates for item and person parameters under some restrictions on the estimates which are needed in order to fix the latent scale. The method is tedious, but can be simplified for the Rasch model when one is only interested in the item parameters. This is demonstrated here under a suitable restriction on the item parameter estimates.

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.

Principal components analysis can be redefined in terms of the regression of observed variables upon component variables. Two criteria for the adequacy of a component representation in this context are developed and are shown to lead to different component solutions. Both criteria are generalized to allow weighting, the choice of weights determining the scale invariance properties of the resulting solution. A theorem is presented giving necessary and sufficient conditions for equivalent component solutions under different choices of weighting. Applications of the theorem are discussed that involve the components analysis of linearly derived variables and of external variables.

We discuss a variety of methods for quantifying categorical multivariate data. These methods have been proposed in many different countries, by many different authors, under many different names. In the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data. In the second major section of the paper we introduce the notion of a duality diagram, and use this diagram to synthesize the many superficially different methods into a single method.

The purpose of this note is to show that Van den Wollenberg's method of redundancy analysis is a special case of a simultaneous linear prediction method offered by Fortier.

The properties of nonmetric multidimensional scaling (NMDS) are explored by specifying statistical models, proving statistical consistency, and developing hypothesis testing procedures. Statistical models with errors in the dependent and independent variables are described for quantitative and qualitative data. For these models, statistical consistency often depends crucially upon how error enters the model and how data are collected and summarized (e.g., by means, medians, or rank statistics). A maximum likelihood estimator for NMDS is developed, and its relationship to the standard Shepard-Kruskal estimation method is described. This maximum likelihood framework is used to develop a method for testing the overall fit of the model.

An algorithm for generating artificial data sets which contain distinct nonoverlapping clusters is presented. The algorithm is useful for generating test data sets for Monte Carlo validation research conducted on clustering methods or statistics. The algorithm generates data sets which contain either 1, 2, 3, 4, or 5 clusters. By default, the data are embedded in either a 4, 6, or 8 dimensional space. Three different patterns for assigning the points to the clusters are provided. One pattern assigns the points equally to the clusters while the remaining two schemes produce clusters of unequal sizes. Finally, a number of methods for introducing error in the data have been incorporated in the algorithm.