Nested logit models have been presented as an alternative to multinomial logistic models for multiple-choice test items (Suh and Bolt in "Psychometrika" 75:454-473, 2010) and possess a mathematical structure that naturally lends itself to evaluating the incremental information provided by attending to distractor selection in scoring. One potential…

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is…

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…

Most dichotomous item response models share the assumption of latent monotonicity, which states that the probability of a positive response to an item is a nondecreasing function of a latent variable intended to be measured. Latent monotonicity cannot be evaluated directly, but it implies manifest monotonicity across a variety of observed scores,…

In this paper, optimal designs will be derived for estimating the ability parameters of the Rasch model when difficulty parameters are known. It is well established that a design is locally D-optimal if the ability and difficulty coincide. But locally optimal designs require that the ability parameters to be estimated are known. To attenuate this…

In some popular test designs (including computerized adaptive testing and multistage testing), many item pairs are not administered to any test takers, which may result in some complications during dimensionality analyses. In this paper, a modified DETECT index is proposed in order to perform dimensionality analyses for response data from such…

A multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the Dutch Identity (Holland in "Psychometrika" 55:5-18, 1990). In the model, latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item…

Starting from an explicit scoring rule for time limit tasks incorporating both response time and accuracy, and a definite trade-off between speed and accuracy, a response model is derived. Since the scoring rule is interpreted as a sufficient statistic, the model belongs to the exponential family. The various marginal and conditional distributions…

Conditional independence is a fundamental principle in latent variable modeling and item response theory. Violations of this principle, commonly known as local item dependencies, are put in a test information perspective, and sharp bounds on these violations are defined. A modeling approach is proposed that makes use of a mixture representation of…

This paper presents a noncompensatory latent trait model, the multicomponent latent trait model for diagnosis (MLTM-D), for cognitive diagnosis. In MLTM-D, a hierarchical relationship between components and attributes is specified to be applicable to permit diagnosis at two levels. MLTM-D is a generalization of the multicomponent latent trait…

Item replenishing is essential for item bank maintenance in cognitive diagnostic computerized adaptive testing (CD-CAT). In regular CAT, online calibration is commonly used to calibrate the new items continuously. However, until now no reference has publicly become available about online calibration for CD-CAT. Thus, this study investigates the…

Three plausible assumptions of conditional independence in a hierarchical model for responses and response times on test items are identified. For each of the assumptions, a Lagrange multiplier test of the null hypothesis of conditional independence against a parametric alternative is derived. The tests have closed-form statistics that are easy to…

We propose three latent scales within the framework of nonparametric item response theory for polytomously scored items. Latent scales are models that imply an invariant item ordering, meaning that the order of the items is the same for each measurement value on the latent scale. This ordering property may be important in, for example,…

This paper presents a study of the impact of prior structure on paradoxical results in multidimensional item response theory. Paradoxical results refer to the possibility that an incorrect response could be beneficial to an examinee. We demonstrate that when three or more ability dimensions are being used, paradoxical results can be induced by…

A comprehensive analysis of difficulty for multiple-choice items requires information at different levels: the test, the items, and the alternatives. This paper introduces a new parameterization of the nominal categories model (NCM) for analyzing difficulty at these three levels. The new parameterization is referred to as the NE-NCM and is…