A Didactic Presentation of Snijders's 'l[subscript z]*' Index of Person Fit with Emphasis on Response Model Selection and Ability Estimation

This paper focuses on two likelihood-based indices of person fit, the index "l[subscript z]" and the Snijders's modified index "l[subscript z]*". The first one is commonly used in practical assessment of person fit, although its asymptotic standard normal distribution is not valid when true abilities are replaced by sample ability estimates. The "l[subscript z]*" index is a generalization of "l[subscript z]", which corrects for this sampling variability. Surprisingly, it is not yet popular in the psychometric and educational assessment community. Moreover, there is some ambiguity about which type of item response model and ability estimation method can be used to compute the "l[subscript z]*" index. The purpose of this article is to present the index "l[subscript z]*" in a simple and didactic approach. Starting from the relationship between "l[subscript z]" and "l[subscript z]*", we develop the framework according to the type of logistic item response theory (IRT) model and the likelihood-based estimators of ability. The practical calculation of "l[subscript z]*" is illustrated by analyzing a real data set about language skill assessment. (Contains 2 tables and 2 figures.)