Item response theory models applied to data allowing examinee choice

Examinations that permit students to choose a subset of the items are popular despite the potential thar students may take examinations of varying difficulty us rr result of their choices. We provide a set of conditions for the validity of inference for Item Response theory (IRT) models applied to data collected from choice-bused examinations. Valid likelihood and Bayesian inference using standard estimation methods require (except in estraordinary circumstances) that there is no dependence, after conditioning on the observed item responses, between the examinees choices and their (potential but unobserved) responses ro omitted items, as well as their latent abilities. These independence assumptions are typical of those required in much more general settings. Common low-dimensional IRT models estimated by standard methods, though potentially useful tools for educational data, do not resolve the difficult problems posed by choice-based data.