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

Abstract

A well-known person fit statistic in the item response theory (IRT) literature is the $l_z$ statistic (Drasgow et al. in Br J Math Stat Psychol 38(1):67-86, 1985). Snijders (Psychometrika 66(3):331-342, 2001) derived $l_z^{\ast }$ , which is the asymptotically correct version of $l_z$ when the ability parameter is estimated. However, both statistics and other extensions later developed concern either only the unidimensional IRT models or multidimensional models that require a joint estimate of latent traits across all the dimensions. Considering a marginalized maximum likelihood ability estimator, this paper proposes $l_{\mathrm{zt}}$ and $l_{\mathrm{zt}}^{\ast }$ , which are extensions of $l_z$ and $l_z^{\ast }$ , respectively, for the Rasch testlet model. The computation of $l_{\mathrm{zt}}^{\ast }$ relies on several extensions of the Lord-Wingersky algorithm (1984) that are additional contributions of this paper. Simulation results show that $l_{\mathrm{zt}}^{\ast }$ has close-to-nominal Type I error rates and satisfactory power for detecting aberrant responses. For unidimensional models, $l_{\mathrm{zt}}$ and $l_{\mathrm{zt}}^{\ast }$ reduce to $l_z$ and $l_z^{\ast }$ , respectively, and therefore allows for the evaluation of person fit with a wider range of IRT models. A real data application is presented to show the utility of the proposed statistics for a test with an underlying structure that consists of both the traditional unidimensional component and the Rasch testlet component.