Asymptotically Correct Person Fit z-Statistics For the Rasch Testlet Model

A well-known person fit statistic in the item response theory (IRT) literature is thelzstatistic (Drasgowet al. in Br J Math Stat Psychol 38(1):67-86, 1985). Snijders (Psychometrika 66(3):331-342, 2001) derivedl & lowast;z, which is the asymptotically correct version oflzwhen the ability parameter is estimated. However,both statistics and other extensions later developed concern either only the unidimensional IRT models ormultidimensional models that require a joint estimate of latent traits across all the dimensions. Considering amarginalized maximum likelihood ability estimator, this paper proposeslztandl & lowast;zt, which are extensions oflzandl & lowast;z, respectively, for the Rasch testlet model. The computation ofl & lowast;ztrelies on several extensions of theLord-Wingersky algorithm (1984) that are additional contributions of this paper. Simulation results showthatl & lowast;zthas close-to-nominal Type I error rates and satisfactory power for detecting aberrant responses. Forunidimensional models,lztandl & lowast;ztreduce tolzandl & lowast;z, respectively, and therefore allows for the evaluationof person fit with a wider range of IRT models. A real data application is presented to show the utility of theproposed statistics for a test with an underlying structure that consists of both the traditional unidimensionalcomponent and the Rasch testlet component