Testing the Rasch model with the conditional likelihood ratio test: sample size requirements and bootstrap algorithms

Abstract

Background: The Rasch model allows for a conditional likelihood ratio goodness of fit test. The speed of approximation of the test statistic to the limiting distribution as a function of sample size and test length has not been analyzed so far. Three bootstrap simulation methods are analyzed with respect to their performance in providing a proper distribution of the test statistic under the null- and the alternative hypothesis. Results: We found a stable approximation to the limiting χ 2 -distribution for sample sizes of at least 500 and 10 items. The three bootstrap algorithms rendered consistent results for the H 0 -cases but not for the H 1 -cases. Conclusion: A sequential probability sampling scheme proves sufficiently apt for generating samples under the alternative hypothesis. This superiority can be justified from a theoretical point of view.