Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items
Authors: Sinharay Sandip 
Published: May 2015

The <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz\documentclass[12pt]{minimal}
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\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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz\documentclass[12pt]{minimal}
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\begin{document}$$l_z$$\end{document} when the examinee ability parameter is estimated. He also suggested the <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz∗\documentclass[12pt]{minimal}
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\begin{document}$$l^*_z$$\end{document} statistic, which is the asymptotically correct standardized version of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz\documentclass[12pt]{minimal}
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\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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz\documentclass[12pt]{minimal}
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\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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz\documentclass[12pt]{minimal}
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\begin{document}$$l_z$$\end{document}, which can be considered as the extension of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content">lz∗\documentclass[12pt]{minimal}
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\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.