Person fit statistics are frequently used to detect deviating behavior when assuming an item response model generated the data. A common statistic, $l_z$, has been shown in previous studies to perform well under a myriad of conditions. However, it is well known that $l_z$ does not follow a standard normal distribution when using an estimated latent trait. As a result, corrections of $l_z$, called $l_z^*$, have been proposed in the literature for specific item response models. We propose a more general correction that is applicable to many types of data, namely survey or tests with multiple item types and underlying latent constructs, which subsumes previous work done by others. In addition, we provide corrections for multiple estimators of $\theta$, the latent trait, including MLE, MAP and WLE. We provide analytical derivations that justifies our proposed correction, as well as simulation studies to examine the performance of the proposed correction with finite test lengths. An applied example is also provided to demonstrate proof of concept. We conclude with recommendations for practitioners when the asymptotic correction works well under different conditions and also future directions.
A Practical Guide to Check the Consistency of Item Response Patterns in Clinical Research Through Person-Fit Statistics
