Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.

A unidimensional latent trait model for continuous ratings is developed. This model is an extension of Andrich's rating formulation which assumes that the response process at latent thresholds is governed by the dichotomous Rasch model. Item characteristic functions and information functions are used to illustrate that the model takes ceiling and floor effects into account. Both the dichotomous Rasch model and a linear model with normally distributed error can be derived as limiting cases. The separability of the structural and incidental parameters is demonstrated and a procedure for estimating the parameters is outlined.

The problem of comparing two sociometric relations or measurements (A and B) recorded in distinct sociomatrices was originally discussed by Katz and Powell in the early 1950's and Hubert and Baker in the late 1970's. The problem is considered again using a probabilistic model designed specifically for discrete-valued network measurements. The model allows for the presence of various structural tendencies, such as reciprocity and differential popularity. A parameter that isolates the tendency for actors to choose other actors on both relations simultaneously is introduced, and estimated conditional on the presence of other parameters that reflect additional important network properties. The parameter is presented as a symmetric index but is also generalized to the predictive (A on B or B on A) situation. This approach to the problem is illustrated with the same data used by the earlier solutions, and the unique nature of the two relations in the data set (A = received choices, B = perceived choices), as it affects the modeling, is discussed. Significance tests for the parameter and related parameters are described, as well as an extension to more than two relations.

Kroonenberg and de Leeuw (1980) have developed an alternating least-squares method TUCKALS-3 as a solution for Tucker's three-way principal components model. The present paper offers some additional features of their method. Starting from a reanalysis of Tucker's problem in terms of a rank-constrained regression problem, it is shown that the fitted sum of squares in TUCKALS-3 can be partitioned according to elements of each mode of the three-way data matrix. An upper bound to the total fitted sum of squares is derived. Finally, a special case of TUCKALS-3 is related to the Carroll/Harshman CANDECOMP/PARAFAC model.

Both the elements and the eigenvalues of the Pearson correlation matrix of dichotomous Guttman-scalable items can be expressed as simple functions of the number of items if the score distribution is uniform and there is an equal number of items at each difficulty level. Even when these special conditions do not hold, the correlations can often be expressed in a simple form by assuming a particular score distribution.

In the continuous response (CR) task, the subject is given the name of a familiar category and must respond with category examples within a fixed time limit. The response sequence and the interresponse times are the measured variables. A general model is proposed for performance in the CR task. The time taken to generate a response is identified with the shortest search time from among a set of simultaneous search processes. Each response has a generation probability that is a function of the parameters governing the search process durations. Data involving the category “states in the United States” are used to test the model. The model accounts for the performance of nearly all individuals in these data.

The information criterion AIC was introduced to extend the method of maximum likelihood to the multimodel situation. It was obtained by relating the successful experience of the order determination of an autoregressive model to the determination of the number of factors in the maximum likelihood factor analysis. The use of the AIC criterion in the factor analysis is particularly interesting when it is viewed as the choice of a Bayesian model. This observation shows that the area of application of AIC can be much wider than the conventional i.i.d. type models on which the original derivation of the criterion was based. The observation of the Bayesian structure of the factor analysis model leads us to the handling of the problem of improper solution by introducing a natural prior distribution of factor loadings.

This article discusses alternative procedures to the standard F-test for ANCOVA in case the covariate is measured with error. Both a functional and a structural relationship approach are described. Examples of both types of analysis are given for the simple two-group design. Several cases are discussed and special attention is given to issues of model identifiability. An approximate statistical test based on the functional relationship approach is described. On the basis of Monte Carlo simulation results it is concluded that this testing procedure is to be preferred to the conventional F-test of the ANCOVA null hypothesis. It is shown how the standard null hypothesis may be tested in a structural relationship approach. It is concluded that some knowledge of the reliability of the covariate is necessary in order to obtain meaningful results.

A model (called the rule space model) which permits measuring cognitive skill acquisition, diagnosing cognitive errors, detecting the weaknesses and strengths of knowledge possessed by individuals was introduced earlier. This study further discusses the theoretical foundation of the model by introducing “bug distribution” and hypothesis testing (Bayes' decision rules for minimum errors) for classifying subjects into their most plausible latent state of knowledge. The model is illustrated with the domain of fraction arithmetic and compared with the results obtained from a conventional artificial intelligence approach.

A review of model-selection criteria is presented, with a view toward showing their similarities. It is suggested that some problems treated by sequences of hypothesis tests may be more expeditiously treated by the application of model-selection criteria. Consideration is given to application of model-selection criteria to some problems of multivariate analysis, especially the clustering of variables, factor analysis and, more generally, describing a complex of variables.

A cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described. Statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method/degree of smoothing. These include: (a) asymptotic significance tests that compare no equating and linear equating to equipercentile equating; (b) a scheme for estimating total equating error and for dividing total estimated error into systematic and random components. The smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test.

In this paper, linear structural equation models with latent variables are considered. It is shown how many common models arise from incomplete observation of a relatively simple system. Subclasses of models with conditional independence interpretations are also discussed. Using an incomplete data point of view, the relationships between the incomplete and complete data likelihoods, assuming normality, are highlighted. For computing maximum likelihood estimates, the EM algorithm and alternatives are surveyed. For the alternative algorithms, simplified expressions for computing function values and derivatives are given. Likelihood ratio tests based on complete and incomplete data are related, and an example on using their relationship to improve the fit of a model is given.

During the last fifteen years, Akaike's entropy-based Information Criterion (AIC) has had a fundamental impact in statistical model evaluation problems. This paper studies the general theory of the AIC procedure and provides its analytical extensions in two ways without violating Akaike's main principles. These extensions make AIC asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the “true” models. These selection criteria are called CAIC and CAICF. Asymptotic properties of AIC and its extensions are investigated, and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different Monte Carlo experiments under different conditions.

A non-forced choice model is developed that describes subject behavior on repeat trial discrimination tests of the “pick 1 of k” form. The model is developed from the Dirichlet distribution, and it allows for the derivation of individual true scores and of sampling properties for various constructs of interest. These results permit the analysis and comparison of test designs. The model is applied to issues such as forced vs. non-forced choice formats, the best number of alternatives at a choice point, and the selection of expert panels.

A discussion of alternative constraint systems has been lacking in the literature on correspondence analysis and related techniques. This paper reiterates earlier results that an explicit choice of constraints has to be made which can have important effects on the resulting scores. The paper also presents new results on dealing with missing data and probabilistic category assignment.

Following the works of Guttman (1953) and Kaiser (1976), we show that the image and anti-image covariance matrices can be derived from a singular correlation matrix, by making use of the orthogonal projector and a weaker generalized inverse matrix.

Test items are often evaluated and compared by contrasting the shapes of their item characteristics curves (ICC's) or surfaces. The current paper develops and applies three general (i.e., nonparametric) comparisons of the shapes of two item characteristic surfaces: (i) proportional latent odds, (ii) uniform relative difficulty, and (iii) item sensitivity. Two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items, and no assumption about the dimensionality of the latent variable. Also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations.

A new method of multiple discriminant analysis was developed that allows a mixture of continuous and discrete predictors. The method can be justified under a wide class of distributional assumptions on the predictor variables. The method can also handle three different sampling situations, conditional, joint and separate. In this method both subjects (cases or any other sampling units) and criterion groups are represented as points in a multidimensional euclidean space. The probability of a particular subject belonging to a particular criterion group is stated as a decreasing function of the distance between the corresponding points. A maximum likelihood estimation procedure was developed and implemented in the form of a FORTRAN program. Detailed analyses of two real data sets were reported to demonstrate various advantages of the proposed method. These advantages mostly derive from model evaluation capabilities based on the Akaike Information Criterion (AIC).

Three methods for estimating reliability are studied within the context of nonparametric item response theory. Two were proposed originally by Mokken (1971) and a third is developed in this paper. Using a Monte Carlo strategy, these three estimation methods are compared with four “classical” lower bounds to reliability. Finally, recommendations are given concerning the use of these estimation methods.