“…Including a random effect suggests that not all features that affect item difficulty are included in the model, but their net effect is a normal distribution of item difficulties with some known mean and variance. Random item models have been extended to EIRM in several different contexts including, but not limited to, explaining a construct Janssen, 2010;Janssen, Schepers, & Peres, 2004), understanding the components of item sets created using automatic item generation (Holling, Bertling, & Zeuch, 2009), predicting item difficulty (Hartig, Frey, Nold, & Klieme, 2012), understanding the impact of cognitive supports on alternative assessments (Ferster, 2013), investigating differential facet functioning (Cawthon, Kaye, Lockhart, & Beretvas, 2012), and modeling item position effects (Albano, 2013). Extending the EPCM in Equation 4, which parameterizes the model for the example considering the role of images in item difficulty for primary and secondary English-speaking students on a mathematics test, the cross-classified EPCM can be written as: 4, the only difference in Equation 6is the additional parameter of ϵ , where ϵ~(0, ), representing the random effect for residual item difficulty.…”