Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items

Abstract

The lz \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_z$$\end{document} 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 lz \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_z$$\end{document} when the examinee ability parameter is estimated. He also suggested the lz∗ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^*_z$$\end{document} statistic, which is the asymptotically correct standardized version of lz \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_z$$\end{document} . However, Snijders (Psychometrika 66:331–342, 2001) only considered tests with dichotomous items. In this paper, the asymptotic null distribution of lz \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_z$$\end{document} is derived for mixed-format tests (those that include both dichotomous and polytomous items). The asymptotically correct standardized version of lz \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_z$$\end{document} , which can be considered as the extension of lz∗ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^*_z$$\end{document} 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.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 40 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.