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, R <Subscript>1</Subscript>, R <Subscript>2</Subscript> (Glas in Psychometrika 53:525–546, <CitationRef CitationID="CR11">1988</CitationRef>), and M <Subscript>2</Subscript> (Maydeu-Olivares & Joe in J. Am. Stat. Assoc. 100:1009–1020, <CitationRef CitationID="CR23">2005</CitationRef>, Psychometrika 71:713–732, <CitationRef CitationID="CR24">2006</CitationRef>) to assess the overall fit of a one-parameter logistic model (1PL) estimated by (marginal) maximum likelihood (ML). R <Subscript>1</Subscript> and R <Subscript>2</Subscript> were specifically designed to target specific assumptions of Rasch models, whereas M <Subscript>2</Subscript> 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 <Superscript>2</Superscript> 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. Copyright The Psychometric Society 2013