Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items

The \(l_z\) statistic (Drasgow et al. in Br J Math Stat Psychol 38:67–86, 1985) is one of the most popular person-fit statistics (Armstrong et al. in Pract Assess Res Eval 12(16):1–10, 2007). Snijders (Psychometrika 66:331–342, 2001) derived the asymptotic null distribution of \(l_z\) when the examinee ability parameter is estimated. He also suggested the \(l^*_z\) statistic, which is the asymptotically correct standardized version of \(l_z\). However, Snijders (Psychometrika 66:331–342, 2001) only considered tests with dichotomous items. In this paper, the asymptotic null distribution of \(l_z\) is derived for mixed-format tests (those that include both dichotomous and polytomous items). The asymptotically correct standardized version of \(l_z\), which can be considered as the extension of \(l^*_z\) to such tests, is suggested. The Type I error rate and power of the suggested statistic are examined from several simulated datasets. The suggested statistic is computed using a real dataset. The suggested statistic appears to be a satisfactory tool for assessing person fit for mixed-format tests.