A survey of the current state of multidimensional scaling using the city-block metric is presented. Topics include substantive and theoretical issues, recent algorithmic developments and their implications for seemingly straightforward analyses, isometries with other metrics, links to graph-theoretic models, and future prospects.

A logistic regression model is suggested for estimating the relation between a set of manifest predictors and a latent trait assumed to be measured by a set of k dichotomous items. Usually the estimated subject parameters of latent trait models are biased, especially for short tests. Therefore, the relation between a latent trait and a set of predictors should not be estimated with a regression model in which the estimated subject parameters are used as a dependent variable. Direct estimation of the relation between the latent trait and one or more independent variables is suggested instead. Estimation methods and test statistics for the Rasch model are discussed and the model is illustrated with simulated and empirical data.

This paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights. Global optimality is obtained from inequalities rather than from partial derivatives of a Lagrangian function. Applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances, in Procrustes analysis, and in ridge regression analysis. The similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed.

The option characteristic curve, the relation between ability and probability of choosing a particular option for a test item, can be estimated by nonparametric smoothing techniques. What is smoothed is the relation between some function of estimated examinee ability rankings and the binary variable indicating whether or not the option was chosen. This paper explores the use of kernel smoothing, which is particularly well suited to this application. Examples show that, with some help from the fast Fourier transform, estimates can be computed about 500 times as rapidly as when using commonly used parametric approaches such as maximum marginal likelihood estimation using the three-parameter logistic distribution. Simulations suggest that there is no loss of efficiency even when the population curves are three-parameter logistic. The approach lends itself to several interesting extensions.

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set of n × n matrices that have a rank less than n has zero volume. Kruskal pointed out that a 2 × 2 × 2 array has rank three or less, and that the subsets of those 2 × 2 × 2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2 × n × n arrays. Incidentally, it is shown that two n × n matrices can be diagonalized simultaneously with positive probability.

The polytomous unidimensional Rasch model with equidistant scoring, also known as the rating scale model, is extended in such a way that the item parameters are linearly decomposed into certain basic parameters. The extended model is denoted as the linear rating scale model (LRSM). A conditional maximum likelihood estimation procedure and a likelihood-ratio test of hypotheses within the framework of the LRSM are presented. Since the LRSM is a generalization of both the dichotomous Rasch model and the rating scale model, the present algorithm is suited for conditional maximum likelihood estimation in these submodels as well. The practicality of the conditional method is demonstrated by means of a dichotomous Rasch example with 100 items, of a rating scale example with 30 items and 5 categories, and in the light of an empirical application to the measurement of treatment effects in a clinical study.

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified models M1 and M2 when fitting the more restricted model M0. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters of M1 and M2 (which characterizes model M12), is a necessary and sufficient condition for the local equivalence of M1 and M2. This condition is satisfied, in practice, when the analysis dealing with the fitting of M0, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting of M1 or M2, or M12 are all equal for any set of sample data, except on differences due to rounding errors.

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.

In pairwise multidimensional scaling, a spatial representation for a set of objects is determined from comparisons of the dissimilarity of any two objects drawn from the set to the dissimilarity of other pairs of objects drawn from that set. In pairwise conjoint scaling, comparisons among the joint effects produced by pairs of objects, where the objects in a pair are drawn from separate sets, are used to determine numerical representations for the objects in each set. Monte Carlo simulations of both pairwise dissimilarities and pairwise conjoint effects show that Johnson's algorithm can provide good metric recovery in the presence of high levels of error even when only a small percentage of the complete set of pairwise comparisons are tested.

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.