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 the [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz{document} statistic (Drasgow et al. in Br J Math Stat Psychol 38(1):67-86, 1985). Snijders (Psychometrika 66(3):331-342, 2001) derived [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz*{document}, which is the asymptotically correct version of [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz{document} 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 [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lzt{document} and [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lzt*{document}, which are extensions of [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz{document} and [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz*{document}, respectively, for the Rasch testlet model. The computation of [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lzt*{document} relies on several extensions of the Lord-Wingersky algorithm (1984) that are additional contributions of this paper. Simulation results show that [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lzt*{document} has close-to-nominal Type I error rates and satisfactory power for detecting aberrant responses. For unidimensional models, [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lzt{document} and [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lzt*{document} reduce to [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz{document} and [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}lz*{document}, 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.