Association models constitute an attractive alternative to the usual log-linear models for modeling the dependence between classification variables. They impose special structure on the underlying association by assigning scores on the levels of each classification variable, which can be fixed or parametric. Under the general row-column (RC) association model, both row and column scores are unknown parameters without any restriction concerning their ordinality. However, when the classification variables are ordinal, order restrictions on the scores arise naturally. Under such restrictions, we adopt an alternative parameterization and draw inferences about the equality of adjacent scores using the Bayesian approach. To achieve that, we have constructed a reversible jump Markov chain Monte Carlo algorithm for moving across models of different dimension and estimate accurately the posterior model probabilities which can be used either for model comparison or for model averaging. The proposed methodology is evaluated through a simulation study and illustrated using actual datasets.

In a multiple regression analysis with three or more predictors, every set of alternate weights belongs to an infinite class of “fungible weights” (Waller, Psychometrica, in press) that yields identical SSE (sum of squared errors) and R2 values. When the R2 using the alternate weights is a fixed value, fungible weights (ai) that yield the maximum or minimum cosine with an OLS weight vector (b) are called “fungible extrema.” We describe two methods for locating fungible extrema and we report R code (R Development Core Team, 2007) for one of the methods. We then describe a new approach for populating a class of fungible weights that is derived from the geometry of alternate regression weights. Finally, we illustrate how fungible weights can be profitably used to gauge parameter sensitivity in linear models by locating the fungible extrema of a regression model of executive compensation (Horton & Guerard, Commun. Stat. Simul. Comput. 14:441–448, 1985).

Human abilities in perceptual domains have conventionally been described with reference to a threshold that may be defined as the maximum amount of stimulation which leads to baseline performance. Traditional psychometric links, such as the probit, logit, and t, are incompatible with a threshold as there are no true scores corresponding to baseline performance. We introduce a truncated probit link for modeling thresholds and develop a two-parameter IRT model based on this link. The model is Bayesian and analysis is performed with MCMC sampling. Through simulation, we show that the model provides for accurate measurement of performance with thresholds. The model is applied to a digit-classification experiment in which digits are briefly flashed and then subsequently masked. Using parameter estimates from the model, individuals’ thresholds for flashed-digit discrimination is estimated.

Computerized adaptive testing (CAT) is a mode of testing which enables more efficient and accurate recovery of one or more latent traits. Traditionally, CAT is built upon Item Response Theory (IRT) models that assume unidimensionality. However, the problem of how to build CAT upon latent class models (LCM) has not been investigated until recently, when Tatsuoka (J. R. Stat. Soc., Ser. C, Appl. Stat. 51:337–350, 2002) and Tatsuoka and Ferguson (J. R. Stat., Ser. B 65:143–157, 2003) established a general theorem on the asymptotically optimal sequential selection of experiments to classify finite, partially ordered sets. Xu, Chang, and Douglas (Paper presented at the annual meeting of National Council on Measurement in Education, Montreal, Canada, 2003) then tested two heuristics in a simulation study based on Tatsuoka’s theoretical work in the context of computerized adaptive testing. One of the heuristics was developed based on Kullback–Leibler information, and the other based on Shannon entropy. In this paper, we showcase the application of the optimal sequential selection methodology in item selection of CAT that is built upon cognitive diagnostic models. Two new heuristics are proposed, and are compared against the randomized item selection method and the two heuristics investigated in Xu et al. (Paper presented at the annual meeting of National Council on Measurement in Education, Montreal, Canada, 2003). Finally, we show the connection between the Kullback–Leibler-information-based approaches and the Shannon-entropy-based approach, as well as the connection between algorithms built upon LCM and those built upon IRT models.

Latent class models for cognitive diagnosis often begin with specification of a matrix that indicates which attributes or skills are needed for each item. Then by imposing restrictions that take this into account, along with a theory governing how subjects interact with items, parametric formulations of item response functions are derived and fitted. Cluster analysis provides an alternative approach that does not require specifying an item response model, but does require an item-by-attribute matrix. After summarizing the data with a particular vector of sum-scores, K-means cluster analysis or hierarchical agglomerative cluster analysis can be applied with the purpose of clustering subjects who possess the same skills. Asymptotic classification accuracy results are given, along with simulations comparing effects of test length and method of clustering. An application to a language examination is provided to illustrate how the methods can be implemented in practice.

In component analysis solutions, post-multiplying a component score matrix by a nonsingular matrix can be compensated by applying its inverse to the corresponding loading matrix. To eliminate this indeterminacy on nonsingular transformation, we propose Joint Procrustes Analysis (JPA) in which component score and loading matrices are simultaneously transformed so that the former matrix matches a target score matrix and the latter matches a target loading matrix. The loss function of JPA is a function of the nonsingular transformation matrix and its inverse, and is difficult to minimize straightforwardly. To deal with this difficulty, we reparameterize those matrices by their singular value decomposition, which reduces the minimization to alternately solving quartic equations and performing the existing multivariate procedures. This algorithm is assessed in a simulation study. We further extend JPA for cases where targets are linear functions of unknown parameters. We also discuss how the application of JPA can be extended in different fields.

The clique partitioning problem (CPP) requires the establishment of an equivalence relation for the vertices of a graph such that the sum of the edge costs associated with the relation is minimized. The CPP has important applications for the social sciences because it provides a framework for clustering objects measured on a collection of nominal or ordinal attributes. In such instances, the CPP incorporates edge costs obtained from an aggregation of binary equivalence relations among the attributes. We review existing theory and methods for the CPP and propose two versions of a new neighborhood search algorithm for efficient solution. The first version (NS-R) uses a relocation algorithm in the search for improved solutions, whereas the second (NS-TS) uses an embedded tabu search routine. The new algorithms are compared to simulated annealing (SA) and tabu search (TS) algorithms from the CPP literature. Although the heuristics yielded comparable results for some test problems, the neighborhood search algorithms generally yielded the best performances for large and difficult instances of the CPP.

The selection of a subset of variables from a pool of candidates is an important problem in several areas of multivariate statistics. Within the context of principal component analysis (PCA), a number of authors have argued that subset selection is crucial for identifying those variables that are required for correct interpretation of the components. In this paper, we adapt the variable neighborhood search (VNS) paradigm to develop two heuristics for variable selection in PCA. The performances of these heuristics were compared to those obtained by a branch-and-bound algorithm, as well as forward stepwise, backward stepwise, and tabu search heuristics. In the first experiment, which considered candidate pools of 18 to 30 variables, the VNS heuristics matched the optimal subset obtained by the branch-and-bound algorithm more frequently than their competitors. In the second experiment, which considered candidate pools of 54 to 90 variables, the VNS heuristics provided better solutions than their competitors for a large percentage of the test problems. An application to a real-world data set is provided to demonstrate the importance of variable selection in the context of PCA.

Considering a dyad as a dynamic system whose current state depends on its past state has allowed researchers to investigate whether and how partners influence each other. Some researchers have also focused on how differences between dyads in their interaction patterns are related to other differences between them. A promising approach in this area is the model that was proposed by Gottman and Murray, which is based on nonlinear coupled difference equations. In this paper, it is shown that their model is a special case of the threshold autoregressive (TAR) model. As a consequence, we can make use of existing knowledge about TAR models with respect to parameter estimation, model alternatives and model selection. We propose a new estimation procedure and perform a simulation study to compare it to the estimation procedure developed by Gottman and Murray. In addition, we include an empirical example based on interaction data of three dyads.