On the Generalized S − X 2 –Test of Item Fit: Some Variants,Residuals,and a Graphical Visualization

Abstract

The generalized $\mathrm{S}-{\mathrm{X}}^{\mathit{2}}$ –test is a test of item fit for items with polytomous responses format. The test is based on a comparison of the observed and expected number of responses in strata defined by the test score. In this article, we make four contributions. We demonstrate that the performance of the generalized $\mathrm{S}-{\mathrm{X}}^{\mathit{2}}$ –test depends on how sparse cells are pooled. We propose alternative implementations of the test within the framework of limited information testing. We derive the distribution of the $\mathrm{S}-{\mathrm{X}}^{\mathit{2}}$ –residuals that can be used for post hoc analyses. We suggest a diagnostic plot that visualizes the form of the misfit. The performance of the alternative implementations is investigated in a simulation study. The simulation study suggests that the alternative implementations are capable of controlling the Type-I error rate well and have high power. An empirical application concludes this article.