We investigate the performance of three statistics, R [subscript 1], R [subscript 2] (Glas in "Psychometrika" 53:525-546, 1988), and M [subscript 2] (Maydeu-Olivares & Joe in "J. Am. Stat. Assoc." 100:1009-1020, 2005, "Psychometrika" 71:713-732, 2006) to assess the overall fit of a one-parameter logistic model…

Maydeu-Olivares and Joe (J. Am. Stat. Assoc. 100:1009-1020, "2005"; Psychometrika 71:713-732, "2006") introduced classes of chi-square tests for (sparse) multidimensional multinomial data based on low-order marginal proportions. Our extension provides general conditions under which quadratic forms in linear functions of cell residuals are…

Recently, there has been increasing interest in reporting subscores. This paper examines reporting of subscores using multidimensional item response theory (MIRT) models (e.g., Reckase in "Appl. Psychol. Meas." 21:25-36, 1997; C.R. Rao and S. Sinharay (Eds), "Handbook of Statistics, vol. 26," pp. 607-642, North-Holland, Amsterdam, 2007; Beguin &…

It is common practice in IRT to consider items as fixed and persons as random. Both, continuous and categorical person parameters are most often random variables, whereas for items only continuous parameters are used and they are commonly of the fixed type, although exceptions occur. It is shown in the present article that random item parameters…

This paper studies three models for cognitive diagnosis, each illustrated with an application to fraction subtraction data. The objective of each of these models is to classify examinees according to their mastery of skills assumed to be required for fraction subtraction. We consider the DINA model, the NIDA model, and a new model that extends the…

Bounds are established for log odds ratios (log cross-product ratios) involving pairs of items for item response models. First, expressions for bounds on log odds ratios are provided for one-dimensional item response models in general. Then, explicit bounds are obtained for the Rasch model and the two-parameter logistic (2PL) model. Results are…

Goodness of fit tests for the Rasch model are typically large-sample, global measures. This paper offers suggestions for small-sample exploratory techniques for examining the fit of item data to the Rasch model. (Author/JKS)

A person fit test based on the Lagrange multiplier test is presented for three item response theory models for polytomous items: the generalized partial credit model, the sequential model, and the graded response model. The test can also be used in the framework of multidimensional ability parameters. It is shown that the Lagrange multiplier…

Latent trait and latent class analyses of Likert-type data are compared. Key similarities and differences between these methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of "life satisfaction" data. (Author/NSF)

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order "r." They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent…

The assumptions of the Rasch model are discussed and the Rasch model is reformulated as a quasi-independence model. Using ordinary contingency table methods, the Rasch model can be tested generally or against less restrictive quasi-loglinear models to investigate specific violations of its assumptions. (Author/BW)

This paper shows that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. Parameters of this latent class model are estimated using an application of the Gibbs sampler, and model fit is investigated using posterior predictive checks. (SLD)

Discusses the Linear Logistic Test Model (LLTM) and demonstrates that there are many equivalent ways to specify a model. Analyzed a real data set (300 responses to 5 analogies) using a Lagrange multiplier test for the specification of the model, and demonstrated that there may be many ways to change the specification of an LLTM and achieve the…

The practicality of using the EM algorithm for maximum likelihood estimation of item parameters in the marginal distribution is presented. The EM procedure is shown to apply to general item-response models. (Author/JKS)

A simple approximation to the conditional distribution of goodness-of-fit statistics for the Rasch model is presented that is used when item difficulties are known. The approximation, which is easily programmed, gives relatively accurate assessments of conditional p-values for tests of 10 or more items. (Author/SLD)