Confidence Interval Estimation for RMSD and MD Item Fit Statistics

Item response theory (IRT) models are widely used in the social sciences to analyze multivariate discrete data that include cognitive test items. In many applications, the performance of two groups is compared using IRT modeling. The assessment of differential item functioning (DIF) plays a central role in this context, as it evaluates whether specific items function differently across groups; that is, whether their item parameters differ between groups. DIF detection is commonly based on statistical inference using item fit statistics. The mean deviation (MD) and root mean square deviation (RMSD) statistics are two widely used item fit measures. However, in the literature and in empirical research, these statistics are typically treated only as effect size measures (i.e., point estimates), and formal statistical inference for them is largely lacking. To address this gap, this article proposes confidence interval (CI) estimation for the MD and RMSD statistics based on asymptotic theory and a computationally efficient parametric bootstrap method. A simulation study was conducted to evaluate the proposed CI estimation approaches and demonstrated their validity. Across both item fit statistics, for DIF and non-DIF items, and across all simulation conditions, the results indicate that CI estimation based on the parametric bootstrap using empirical percentiles performed best and outperformed both the parametric bootstrap with normal distribution-based CIs and the asymptotic theory-based approach. It is therefore recommended that CI estimation for MD and RMSD statistics be routinely reported in addition to point estimates in empirical research.