The manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the Goodman association model (RC) and of correspondence analysis (CA) is considered.

Estimation of the row and column scores is presented in this context by developing a unified framework that includes both models. Incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables, depending on the topology of the graph. An example of doubly additive scores in the analysis of a Burt subtable is given.

This paper discusses the application of a class of Rasch models to situations where test items are grouped into subsets and the common attributes of items within these subsets brings into question the usual assumption of conditional independence. The models are all expressed as particular cases of the random coefficients multinomial logit model developed by Adams and Wilson. This formulation allows a very flexible approach to the specification of alternative models, and makes model testing particularly straightforward. The use of the models is illustrated using item bundles constructed in the framework of the SOLO taxonomy of Biggs and Collis.

We study a proportional reduction in loss (PRL) measure for the reliability of categorical data and consider the general case in which each of N judges assigns a subject to one of K categories. This measure has been shown to be equivalent to a measure proposed by Perreault and Leigh for a special case when there are two equally competent judges, and the correct category has a uniform prior distribution. We consider a general framework where the correct category is assumed to have an arbitrary prior distribution, and where classification probabilities vary by correct category, judge, and category of classification. In this setting, we consider PRL reliability measures based on two estimators of the correct category—the empirical Bayes estimator and an estimator based on the judges' consensus choice. We also discuss four important special cases of the general model and study several types of lower bounds for PRL reliability.

Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matrices X with columns x1, ..., xr. These functions are h1(x)=Σk (x′Akx)(x′Ckx)−1, H1(X)=Σk tr (X′AkX)(X′CkX)−1, h1(X)=Σk Σl (x′lAkxl) (x′lCkxl)−1 with X constrained to be columnwise orthonormal, h2(x)=Σk (x′Akx)2(x′Ckx)−1 subject to x′x=1, H2(X)=Σk tr (X′AkX)(X′AkX)′(X′CkX)−1 subject to X′X=I, and h2(X)=Σk Σl (x′lAkxl)2 (x′lCkXl)−1 subject to X′X=I. In these functions the matrices Ck are assumed to be positive definite. The matrices Ak can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of the x2 test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.

Many of the “classical” multivariate data analysis and multidimensional scaling techniques call for approximations by lower dimensional configurations. A model is proposed, in which different sets of linear constraints are imposed on different dimensions in component analysis and “classical” multidimensional scaling frameworks. A simple, efficient, and monotonically convergent algorithm is presented for fitting the model to the data by least squares. The basic algorithm is extended to cover across-dimension constraints imposed in addition to the dimensionwise constraints, and to the case of a symmetric data matrix. Examples are given to demonstrate the use of the method.

The concept of an ordinal instrumental probabilistic comparison is introduced. It relies on an ordinal scale given a priori and on the concept of stochastic dominance. It is used to define a weakly independently ordered system, or isotonic ordinal probabilistic (ISOP) model, which allows the construction of separate “sample-free” ordinal scales on a set of “subjects” and a set of “items”. The ISOP-model is a common nonparametric theoretical structure for unidimensional models for quantitative, ordinal and dichotomous variables.

Fundamental theorems on dichotomous and polytomous weakly independently ordered systems are derived. It is shown that the raw score system has the same formal properties as the latent system, and therefore the latter can be tested at the observed empirical level.

It is demonstrated in this paper that two major tests for 2 × 2 talbes are highly related from a Bayesian perspective. Although it is well-known that Fisher's exact and Pearson's chi-square tests are asymptotically equivalent, the present analysis shows that a formal similarity also exists in small samples. The key assumption that leads to the resemblance is the presence of a continuous parameter measuring association. In particular, it is shown that Pearson's probability can be obtained by integrating a two-moment approximation to the posterior distribution of the log-odds ratio. Furthermore, Pearson's chi-square test gave an excellent approximation to the actual Bayes probability in all 2×2 tables examined, except for those with extremely disproportionate marginal frequencies.