Identification problems in parametric and non-parametric IRT models

Abstract

IRT models are widely used to analyze educational data. It is assumed that the response of an examinee p to an item i is a Bernoulli distribution conditionally on both the difficulty parameter and the ability of person p; the parameter of the Bernoulli distribution is a cumulative density function (cdf) F evaluated at a linear relationship between difficulty and ability. It is assumed that the responses of an examinee p are mutually independent conditionally his/her ability. Typically it is assumed that the distribution of the ability is known up to a scale parameter. The inference is based on the statistical model which is obtained after integrating out the abilities. In this talk we analyze the identification problem in the following cases: when F corresponds to a logistic distribution and is a mixture of two distributions, one being the logistic and the other one depending on the item; this last case is typically used to take into account the possibility that a person could answer the item buy guessing. Thereafter, it is considered a semi-parametric model, where the abilities are iid given a cdf G. In this case, the parameters of interest are the difficulty parameters and G. It is shown that under an unrealistic Data Generating Process, it is possible to identify G.