Nonparametric item response theory (IRT) models consist of assumptions that restrict the joint item-score distribution. These assumptions imply stochastic ordering properties that allow ordering of respondents and items using the simple sum score and item mean score, respectively, and imply observable data properties that are useful for investigating model fit. In this paper, we investigate these properties for two-level nonparametric IRT. We introduce four two-level nonparametric IRT models. Two models pertain to respondents nested in groups: The MHM-1, useful for ordering respondents and groups, and the DMM-1, useful for ordering respondents, groups, and items. Two models pertain to groups rated by multiple respondents: The MHM-2, useful for ordering groups, and the DMM-2, useful for ordering groups and items. We define the model assumptions, derive implied stochastic ordering properties, and derive observable data properties that are useful for model fit investigation. Relations between models and properties are also presented.

A new observable consequence of the property of invariant item ordering is presented, which holds under Mokken’s double monotonicity model for dichotomous data. The observable consequence is an invariant ordering of the item-total regressions. Kendall’s measure of concordance W and a weighted version of this measure are proposed as measures for this property. Karabatsos and Sheu proposed a Bayesian procedure (Appl. Psychol. Meas. 28:110–125, 2004), which can be used to determine whether the property of an invariant ordering of the item-total regressions should be rejected for a set of items. An example is presented to illustrate the application of the procedures to empirical data.

We propose three latent scales within the framework of nonparametric item response theory for polytomously scored items. Latent scales are models that imply an invariant item ordering, meaning that the order of the items is the same for each measurement value on the latent scale. This ordering property may be important in, for example, intelligence testing and person-fit analysis. We derive observable properties of the three latent scales that can each be used to investigate in real data whether the particular model adequately describes the data. We also propose a methodology for analyzing test data in an effort to find support for a latent scale, and we use two real-data examples to illustrate the practical use of this methodology.