A Multidimensional Ideal Point Item Response Theory Model for Binary Data (original) (raw)
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A multidimensional ideal point IRT model for binary data
We introduce a multidimensional item response theory (IRT) model for binary data based on a proximity response mechanism. Under the model, a respondent at the mode of the item response function (IRF) endorses the item with probability one. The mode of the IRF is the ideal point, or in the multidimensional case, an ideal hyperplane. The model yields closed form expressions for the cell probabilities. We estimate and test the goodness of fit of the model using only information contained in the univariate and bivariate moments of the data. Also, we pit the new model against the multidimensional normal ogive model estimated using NOHARM in four applications involving (a) attitudes toward censorship, (b) satisfaction with life, (c) attitudes of morality and equality, and (d) political efficacy. The normal PDF model is not invariant to simple operations such as reverse scoring. Thus, when there is no natural category to be modeled, as in many personality applications, it should be fit separately with and without reverse scoring for comparisons.
NOHARM is a program that performs factor analysis for dichotomous variables assuming that these arise from an underlying multinormal distribution. Parameter estimates are obtained by minimizing an unweighted least squares function of the first- and second-order marginal proportions. Here, large sample standard errors for restricted as well as rotated unrestricted factor solutions are given. Also a test of the goodness of fit of the model to the first- and second-order marginals of the contingency table is proposed. In a simulation study, it was found that for small models, accurate parameter estimates, standard errors, and goodness-of-fit tests can be obtained with as few as 100 observations. Furthermore, NOHARM estimates, standard errors, and goodness-of-fit tests are comparable to those obtained using a related LISREL procedure.
Multidimensional Item Response Theory
Springer eBooks, 2009
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Educational and Psychological Measurement
Item response theory “dual” models (DMs) in which both items and individuals are viewed as sources of differential measurement error so far have been proposed only for unidimensional measures. This article proposes two multidimensional extensions of existing DMs: the M-DTCRM (dual Thurstonian continuous response model), intended for (approximately) continuous responses, and the M-DTGRM (dual Thurstonian graded response model), intended for ordered-categorical responses (including binary). A rationale for the extension to the multiple-content-dimensions case, which is based on the concept of the multidimensional location index, is first proposed and discussed. Then, the models are described using both the factor-analytic and the item response theory parameterizations. Procedures for (a) calibrating the items, (b) scoring individuals, (c) assessing model appropriateness, and (d) assessing measurement precision are finally discussed. The simulation results suggest that the proposal is ...
A Class of Multidimensional Latent Class IRT Models for Ordinal Polytomous Item Responses
Communications in Statistics - Theory and Methods, 2014
We propose a class of Item Response Theory models for items with ordinal polytomous responses, which extends an existing class of multidimensional models for dichotomously-scored items measuring more than one latent trait. In the proposed approach, the random vector used to represent the latent traits is assumed to have a discrete distribution with support points corresponding to different latent classes in the population. We also allow for different parameterizations for the conditional distribution of the response variables given the latent traits-such as those adopted in the Graded Response model, in the Partial Credit model, and in the Rating Scale model-depending on both the type of link function and the constraints imposed on the item parameters. For the proposed models we outline how to perform maximum likelihood estimation via the Expectation-Maximization algorithm. Moreover, we suggest a strategy for model selection which is based on a series of steps consisting of selecting specific features, such as the number of latent dimensions, the number of latent classes, and the specific parametrization. In order to illustrate the proposed approach, we analyze data deriving from a study on anxiety and depression as perceived by oncological patients.
The estimation of polytomous item response models with many dimensions
2002
Id entification cond itions and an im proved estim ation m ethod for a D-d im ensional m ixed coefficients m ultinom ial logit m od el are d iscussed. This m od el is a generalisation of the Ad am s and Wilson (1997) rand om coefficients m ultinom ial logit and it can be used to fit m ultd im ensional form s of a w id e range of Rasch m easurem ent m od els. The com putational d em and s of the num erical integration required in fitting such m od els have lim ited previous im plem entations to three and perhaps four-d im ensional problem s (Glas, 1992; Ad am s, Wilson and Wang, 1997). This paper illustrates a Monte Carlo integration m ethod that perm its the estim ation of m od els w ith m uch higher d im ensionality. The exam ple in this paper fits m od els of six d im ensions.
A simple descriptive method for multidimensional item response theory based on stochastic dominance
2017
In this paper we develop a descriptive concept of a (partially) ordinal joint scaling of items and persons in the context of (dichotomous) item response analysis. The developed method has to be understood as a purely descriptive method describing relations among the data observed in a given item response data set, it is not intended to directly measure some presumed underlying latent traits. We establish a hierarchy of pairs of item difficulty and person ability orderings that empirically support each other. The ordering principles we use for the construction are essentially related to the concept of first order stochastic dominance. Our method is able to avoid a paradoxical result of multidimensional item response theory models described in \cite{hooker2009paradoxical}. We introduce our concepts in the language of formal concept analysis. This is due to the fact that our method has some similarities with formal concept analysis and knowledge space theory: Both our methods as well a...
MultiLCIRT: An R package for multidimensional latent class item response models
Computational Statistics and Data Analysis, 2014
We illustrate a class of Item Response Theory (IRT) models for binary and ordinal polythomous items and we describe an R package for dealing with these models, which is named MultiLCIRT. The models at issue extend traditional IRT models allowing for (i) multidimensionality and (ii) discreteness of latent traits. This class of models also allows for different parameterizations for the conditional distribution of the response variables given the latent traits, depending on both the type of link function and the constraints imposed on the discriminating and the difficulty item parameters. We illustrate how the proposed class of models may be estimated by the maximum likelihood approach via an Expectation-Maximization algorithm, which is implemented in the MultiLCIRT package, and we discuss in detail issues related to model selection. In order to illustrate this package, we analyze two datasets: one concerning binary items and referred to the measurement of ability in mathematics and the other one coming from the administration of ordinal polythomous items for the assessment of anxiety and depression. In the first application, we illustrate how aggregating items in homogeneous groups through a model-based hierarchical clustering procedure which is implemented in the proposed package. In the second application, we describe the steps to select a specific model having the best fit in our class of IRT models.
The Fisher Information Function for Ideal Point Item Response Models for Pick Any/n Data
2006
In the last two decades, researchers have developed a number of item response models for the analysis of preference data in which the regression between latent trait θ and item responses, P (θ), is single-peaked. As opposed to the monotonic functions such as the logistic function common to IRT for dominance data, these models are probabilistic analogues of Coombs' deterministic unfolding models. One potential barrier to the wider acceptance of such models is the curious fact that most ideal point item response models have bimodal item information functions. Unfortunately, mathematically rigorous explanations for this unusual behavior have not been provided by authors. More broadly, properties of the information function of ideal point IRT models are unknown. This article proves several theorems about the IIFs of ideal point models, in particular, showing that the IIF can be bimodal, unimodal, or singular depending on qualitative characteristics of P (θ), in particular the maximum value of P (θ) and P (θ). The importance of these results for test construction is also discussed and illustrated through a simple empirical example. 1.1 Ideal Point vs. Monotonic Response Patterns To understand why these data are often better characterized by a single-peaked response, consider the following statements, which are variants of Thurstone's classic capital punishment items built according to the principles outlined in Michell (1994):