Latent and manifest monotonicity in item response models

Brian W. Junker and Klaas Sijtsma


Nonparametric item response theory (IRT) provides tools for analyzing sets of test items under the assumptions of IRT, but without assuming any particular parametric form for the item response functions (IRF's) or latent ability distribution. A central feature of most (parametric as well as nonparametric) item response models is monotonicity of the IRF's, since this (1) facilitates interpretation of the items as ``measuring'' an underlying trait or feature; and (2) allows a general theory of nonparametric inference for the underlying trait to be built up, based on MLR properties of the likelihood of the item responses given ability. Thus an important part of data analysis with nonparametric item response models is checking the monotonicity assumption. Various authors have advocated checking monotonicity in the model by exploring the monotonicity of the nonparametric regression of individual items on the total test score. In this paper we review some positive and negative results in checking monotonicity of a particular item by regressing individual item scores on the total test score and on the ``rest score'' obtained by omitting the item from the total test score. We show that, for some familiar dichotomous item response models satisfying monotonicity of IRF's, the item-total regressions can exhibit nonmonotonicities that become worse as the test length increases; on the other hand item-rest regressions never exhibit nonmonotonicities under the nonparametric monotone unidimensional item response model. We discuss the implications of these results for exploratory analysis of dichotomous item response data, and we also briefly consider the application of these results to polytomous item response data.

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