# The probability of exceedance as a nonparametric person-fit statistic for tests of moderate length

Jorge N. Tendeiro, Rob R. Meijer

August 2013
### Abstract

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.

Publication

*Applied Psychological Measurement, 37*