The Probability of Exceedance as a Nonparametric Person-Fit Statistic for Tests of Moderate Length

To classify an item score pattern as not fitting a nonparametric item response theory (NIRT) model, the probability of exceedance (PE) of an observed response vector x can be determined as the sum of the probabilities of all response vectors that are, at most, as likely as x, conditional on the test's total score. Vector x is to be considered not fitting when its PE is smaller than a prespecified level. Although this concept is not new, it is hardly if ever applied in practice. In the present paper, the authors show how the PE of a response vector x can be computed in a NIRT context and how misfitting response patterns are detected using the exact distribution of PE. Results from two empirical applications are discussed. A simulation study is conducted to investigate the robustness of the PE against violation of the invariant item ordering condition. Finally, considerations over possible asymptotic distributions of PE are discussed.