Performance of the S − χ 2 Statistic for the Multidimensional Graded Response Model

Abstract

$S-{\chi }^2$ is a popular item fit index that is available in commercial software packages such as flexMIRT. However, no research has systematically examined the performance of $S-{\chi }^2$ for detecting item misfit within the context of the multidimensional graded response model (MGRM). The primary goal of this study was to evaluate the performance of $S-{\chi }^2$ under two practical misfit scenarios: first, all items are misfitting due to model misspecification, and second, a small subset of items violate the underlying assumptions of the MGRM. Simulation studies showed that caution should be exercised when reporting item fit results of polytomous items using $S-{\chi }^2$ within the context of the MGRM, because of its inflated false positive rates (FPRs), especially with a small sample size and a long test. $S-{\chi }^2$ performed well when detecting overall model misfit as well as item misfit for a small subset of items when the ordinality assumption was violated. However, under a number of conditions of model misspecification or items violating the homogeneous discrimination assumption, even though true positive rates (TPRs) of $S-{\chi }^2$ were high when a small sample size was coupled with a long test, the inflated FPRs were generally directly related to increasing TPRs. There was also a suggestion that performance of $S-{\chi }^2$ was affected by the magnitude of misfit within an item. There was no evidence that FPRs for fitting items were exacerbated by the presence of a small percentage of misfitting items among them.