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# Latent and manifest monotonicity in item response models

**Brian W. Junker and Klaas Sijtsma**

### Abstract:

*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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