Abstract
The four-parameter logistic model is an Item Response Theory model for dichotomous items that limit the probability of giving a positive response to an item into a restricted range, so that even people at the extremes of a latent trait do not have a probability close to zero or one. Despite the literature acknowledging the usefulness of this model in certain contexts, the difficulty of estimating the item parameters has limited its use in practice. In this paper we propose a regularized estimation approach for the estimation of the item parameters based on the inclusion of a penalty term in the log-likelihood function. Simulation studies show the good performance of the proposal, which is further illustrated through an application to a real-data set.
Highlights
Item Response Theory (IRT) provides a framework for statistical modeling of the responses to a test or questionnaire [1,2]
In the case of binary responses, the four-parameter logistic (4PL) model [3] constitutes the more flexible option, since it is able to capture the relation of the responses with the latent variable allowing for some randomness, so that people at a very low level of the latent trait have a nonzero probability of giving a positive response and people at a very high level of the latent trait have a probability of giving a positive response lower than 1
In this paper we have proposed a regularization method for the 4PL model based on penalized maximum likelihood estimation
Summary
Item Response Theory (IRT) provides a framework for statistical modeling of the responses to a test or questionnaire [1,2]. In IRT models, the probability of giving a certain response to an item depends on one or more latent variables and on some parameters related to the items. Different kinds of models have been proposed in the literature depending on the type of responses that can be given to an item. Restricting our attention to IRT, a ridge-type penalty was used for the two-parameter logistic (2PL) model [18], while a lasso penalty for the detection of differential item functioning was employed for the Rasch model [19] and for generalized partial credit models [20]. To deal with the complexity of the estimation of the 4PL model, in this paper we propose a regularization approach based on the inclusion of a penalty term in the log-likelihood function.
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