Regularization methods for item response and paired comparison models

1. Introduction Commonly, item response models and paired comparison models are treated as different model classes, suited for different data situations. However, there is a great similarity between item response data and paired comparisons and, accordingly, between the respective modeling approaches. Item response data appear when test persons face a certain number of items which are designed to measure a specific latent trait of the test persons. Such latent traits can, for example, be certain skills (e.g. intelligence) of the test persons or attitudes towards a specific issue (e.g. xenophobia). In the simplest case only two outcomes are possible, for example right or wrong answers or approving or disapproving of a statement. Paired comparison data occur if two objects or items compete in a certain way. The most frequent occurrence of paired comparisons is when two objects are presented and raters have to declare a preference for one or the other object. But also in other situations paired comparisons appear, as, for example, in sport competitions between two players or teams. Again, in the simple case only two outcomes are possible, namely the win/preference of one object over the other. Both in item response data and in paired comparisons, the outcome refers to the result of a specific competition between two actors. Therefore, item response data can be seen as a special type of paired comparison data. Tutz (1989) distinguishes between homogeneous and heterogeneous paired comparisons. In this sense, item response data are heterogeneous paired comparisons as the matched pairs are pairs of one item and one respondent. In contrast, homogeneous paired comparisons treat matched pairs of two objects or items. The basic and most popular models for these data are the Rasch model (RM) for item response data and the Bradley-Terry or Bradley-Terry-Luce model (BTL) for paired comparison data. The Rasch model (Rasch, 1960) assumes that the probability that a person solves an item is determined by the difference between one latent parameter representing the person and one latent parameter representing the item. Let the random variable Y pi represent the response where Y pi = 1 if person p solves item i and Y pi = 0 otherwise. With the Rasch model the probability that person p solves item i is modeled by P (Y pi = 1) = exp(θ p − β i) 1 + exp(θ p − β i) Dieses Werk ist copyrightgeschützt und darf in …