Contribuciones a la Teoría de Respuesta al Ítem no Paramétrica

Resumen

This thesis focuses on two important aspects of non-parametric item response theory, the first is the estimation of multidimensional models and the second, the estimation of non-parametric models with monotony constraints. It begins proposing an estimator for multidimensional traits based on the empirical distribution of data functions. Then, to estimate the items response functions it presents the generalization of the Ramsay model to the multidimensional case and a model based on local polynomials. Next, it presents a unidimensional isotone model, and then a multidimensional isotone model in each component. These last models incorporate in their formulation the two variants presented when there are no constraints of monotony. In all cases, different types of response are considered: dichotomous, polytomic and continuous. Trait density estimators are defined and the complete consistency and asymptotic distribution of the estimator is demonstrated in case the trait is measured without error.The central part of the thesis is devoted to finding the hypotheses necessary for the joint consistency (weak and strong) of the trait estimators and the item characteristic curves (ICC) in the general case. When the trait is measured without error, the complete consistency and asymptotic distribution of the ICC estimator is proved. Also, methods are presented for the approximate choice of the optimum bandwidth for the density estimator and the ICC. Finally, for the isotone model the asymptotic distribution is proved when the trait is measured without error. In addition, 17 simulation studies are performed to determine the behavior of the estimators, calculating several measures of goodness of fit. The influence of the different kernels, the trait distributions, the effect of the dimension and the choice of the bandwidth in the estimators were studied. Also, the presented models are applied to two specific cases: the diagnostic evaluation of mathematics at university entrance and the CDI (Children Depression Inventory) test. It is concluded that the presented models constitute a contribution to the nonparametric item response theory, since they proved to be consistent, flexible, they worked very well in simulations studies and when they were applied to real data. They are robust to different kernels and distributions choices. In addition, the sample size necessary to obtain a good fit is moderate in low dimensions.