Binomial test models and item difficulty

In choosing a binomial test model, it is important to know exactly what conditions are imposed on item difficulty. In this paper these conditions are examined for both a deterministic and a stochastic conception of item responses. It appears that they are more restrictive than is generally understood and differ for both conceptions. When the binomial model is applied to a fixed examinee, the deterministic conception imposes no conditions on item difficulty but requires instead that all items have characteristic functions of the Guttman type. In contrast, the stochastic conception allows non- Guttman items but requires that all characteristic functions must intersect at the same point, which implies equal classically defined difficulty. The beta-binomial model assumes identical characteristic functions for both conceptions, and this also implies equal difficulty. Finally, the compound binomial model entails no restrictions on item difficulty.