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Testing Linear Models for Ability Parameters in Item Response Models

Methods for testing hypotheses concerning the regression parameters in linear models for the latent person parameters in item response models are presented. Three tests are outlined: A likelihood ratio test, a Lagrange multiplier test and a Wald test. The tests are derived in a marginal maximum likelihood framework. They are explicitly formulated for the 3-parameter logistic model, but it is shown that the approach applies to a broad class of item response models. Since the distributions of the test statistics are derived asymptotically, simulation studies were performed to assess the Type I error rates of the tests for small realistic sample sizes. Overall, the Type I error rates for the null hypothesis that a regression coefficient equals zero, were close to the nominal significance level. A number of power studies were conducted. It is argued that on theoretical grounds the power of the Lagrange multiplier test might be less than the power of the other two tests, but this expectation was not corroborated. The robustness of the tests to violation of the item response model was investigated with simulation studies of the power and Type I error rate. The results showed that the performance of the tests was acceptable in the cases where local independence and the constancy of the discrimination parameters over treatment groups were violated to the same extent for all treatment groups. The simulation studies also showed that the tests were biased if local independence was violated for one of the treatment groups.