A model is proposed that describes subject behavior on repeat paired comparison preference tests. The model extends prior work in this area in that it explicitly allows for abstentions and permits the derivation of individual true scores for discrimination ability as well as conditional estimates of proportionate preference. With these results, the properties of a paired comparison test can be thoroughly explored. An empirical example is presented, and test design issues are considered. In particular, repeat paired comparison preference tests are shown to be inherently less efficient discrimination tests than are pick 1 of 2 tests.

A stochastic postulate is given for the multiple-item, successive-intervals scaling of populations. The logistic equivalent of this postulate provides an aggregate item response model in which a unidimensional submodel may be nested. This reduction provides a subtractive conjoint measurement of several items and stimuli on the same latent scale. Generalized-least-squares methods are used to estimate and test the multiple-item model, and its unidimensional reduction, on aggregate survey responses. The entire procedure is illustrated with an analysis of semantic-differential attitude data. This analysis exhibits an item selection procedure that is applicable to various social constructs.

This paper deals with two-group classification when a unidimensional latent trait, θ, is appropriate for explaining the data, X. It is shown that if X has monotone likelihood ratio then optimal allocation rules can be based on its magnitude when allocation must be made to one of two groups related to θ. These groups may relate to θ probabilistically via a non-decreasing function p(θ), or may be defined by all subjects above or below a selected value on θ.

In the case where the data arise from dichotomous items, then only the assumption that the items have nondecreasing item characteristic functions is enough to ensure that the unweighted sum of responses (the number-right score or raw score) possesses this fundamental monotone likelihood ratio property.

Canonical analysis of two convex polyhedral cones consists in looking for two vectors (one in each cone) whose square cosine is a maximum. This paper presents new results about the properties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of scalings of an ordinal variable is a convex polyhedral cone, which thus plays an important role in optimal scaling methods for the analysis of ordinal data. Monotone analysis of variance, and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace. Optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given. We discuss these three situations and illustrate them by examples.

Two kinds of measures of multivariate association, based on Wilks' Λ and the Bartlett-Nanda-Pillai trace criterion V, respectively, are compared in terms of properties of the univariate R2 which they generalize. A unified set of derivations of the properties is provided which are self-contained and not restricted to decompositions in canonical variates. One conclusion is that a symmetric index based on Λ allows generalization of the multiplicative decomposition of R2 in terms of squared partial correlations, but not the additive decomposition in terms of squared semipartial correlations, while the reverse is true for an asymmetric index based on V.

The present paper is concerned with testing the fit of the Rasch model. It is shown that this can be achieved by constructing functions of the data, on which model tests can be based that have power against specific model violations. It is shown that the asymptotic distribution of these tests can be derived by using the theoretical framework of testing model fit in general multinomial and product-multinomial models. The model tests are presented in two versions: one that can be used in the context of marginal maximum likelihood estimation and one that can be applied in the context of conditional maximum likelihood estimation.

Two coefficients are proposed for measuring the extent of overlap in distributions as a direct function of the variance between the arithmetic means (“disco” and “odisco”). They are designed to answer such questions as: “Given the value of a numerical variable x, to which population should an individual be assigned so that minimum error would be incurred?” This is just the reverse of the question addressed by ANOVA. These coefficients are shown to be analytic in x and they are related to Pearson's eta and Fisher's F. Extensions of these coefficients (designed for univariate, one-way discrimination) to k-way and multivariate discriminant analysis and measurement of “interaction” are suggested.

Correspondence analysis can be described as a technique which decomposes the departure from independence in a two-way contingency table. In this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi-independence model. This form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended, for example, in case of missing data or structural zeros. It is shown that Nora's reconstitution of order zero, a procedure well-known in the French literature, is formally identical to our correspondence analysis of incomplete tables. Therefore, reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi-independence model. Furthermore, correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis.

Goodman's (1979, 1981, 1985) loglinear formulation for bi-way contingency tables is extended to tables with or without missing cells and is used for exploratory purposes. A similar formulation is done for three-way tables and generalizations of correspondence analysis are deduced. A generalized version of Goodman's algorithm, based on Newton's elementary unidimensional method is used to estimate the scores in all cases.

An extension of component analysis to longitudinal or cross-sectional data is presented. In this method, components are derived under the restriction of invariant and/or stationary compositing weights. Optimal compositing weights are found numerically. The method can be generalized to allow differential weighting of the observed variables in deriving the component solution. Some choices of weightings are discussed. An illustration of the method using real data is presented.

This paper describes the authors' FORTRAN algorithm FACAIC for choosing the number of factors for an orthogonal factor model using Akaike's Information Criterion. FACAIC utilizes the IMSL subroutine OFCOMM.

A method is presented for investigating two aspects of the sensitivity of a linear composite to the values of its weights.

Most of the currently used analytic rotation criteria for simple structure in factor analysis are summarized and identified as members of a general symmetric family of quartic criteria. A unified development of algorithms for orthogonal and direct oblique rotation using arbitrary criteria from this family is given. These algorithms represent fairly straightforward extensions of present methodology, and appear to be the best methods currently available.

A macro for calculating the Hubert and Arabie (1985) adjusted Rand statistic is presented. The adjusted Rand statistic gives a measure of classification agreement between two partitions of the same set of objects. The macro is written in the SAS macro language and makes extensive use of SAS/IML software (SAS Institute, 1985a, 1985b). The macro uses two different methods of handling missing values. The default method assumes that each object that has a missing value for the classification category is in its own separate category or cluster for that classification. The optional method places all objects with a missing value for the classification category into the same category for that classification.

Consider the class of two parameter marginal logistic (Rasch) models, for a test of m True-False items, where the latent ability is assumed to be bounded. Using results of Karlin and Studen, we show that this class of nonparametric marginal logistic (NML) models is equivalent to the class of marginal logistic models where the latent ability assumes at most (m + 2)/2 values. This equivalence has two implications. First, estimation for the NML model is accomplished by estimating the parameters of a discrete marginal logistic model. Second, consistency for the maximum likelihood estimates of the NML model can be shown (when m is odd) using the results of Kiefer and Wolfowitz. An example is presented which demonstrates the estimation strategy and contrasts the NML model with a normal marginal logistic model.

A new method is proposed for the statistical analysis of dyadic social interaction data measured over time. The data to be studied are assumed to be realizations of a social network of a fixed set of actors interacting on a single relation. The method is based on loglinear models for the probabilities for various dyad (or actor pair) states and generalizes the statistical methods proposed by Holland and Leinhardt (1981), Fienberg, Meyer, & Wasserman (1985), and Wasserman (1987) for social network data. Two statistical models are described: the first is an “associative” approach that allows for the study of how the network has changed over time; the second is a “predictive” approach that permits the researcher to model one time point as a function of previous time points. These approaches are briefly contrasted with earlier methods for the sequential analysis of social networks and are illustrated with an example of longitudinal sociometric data.

A method is proposed for empirically testing the appropriateness of using tetrachoric correlations for a set of dichotomous variables. Trivariate marginal information is used to get a set of one-degree of freedom chi-square tests of the underlying normality. It is argued that such tests should preferrably preceed further modeling of tetrachorics, for example, modeling by factor analysis. The assumptions are tested in some real and simulated data.