How should we assess the fit of Rasch-type models? Approximating the power of goodness-of-fit statistics in categorical data analysis.

We investigate the performance of three statistics, R1, R2 (Glas in Psychometrika 53:525-546, 1988), and M2 (Maydeu-Olivares & Joe in J. Am. Stat. Assoc. 100:1009-1020, 2005, Psychometrika 71:713-732, 2006) to assess the overall fit of a one-parameter logistic model (1PL) estimated by (marginal) maximum likelihood (ML). R1 and R2 were specifically designed to target specific assumptions of Rasch models, whereas M2 is a general purpose test statistic. We report asymptotic power rates under some interesting violations of model assumptions (different item discrimination, presence of guessing, and multidimensionality) as well as empirical rejection rates for correctly specified models and some misspecified models. All three statistics were found to be more powerful than Pearson's X(2) against two- and three-parameter logistic alternatives (2PL and 3PL), and against multidimensional 1PL models. The results suggest that there is no clear advantage in using goodness-of-fit statistics specifically designed for Rasch-type models to test these models when marginal ML estimation is used.