Quantifying Item Dependency

Some items are more closely related than others. Comprehension items that address the same written text and arithmetic items requiring the same operation are examples. In Rasch analysis, this is no major source for concern provided there is no extreme systematic dependency between two or a few items. When such dependency does exist, combining the separate items into one partial-credit item may be useful. Low values of the Rasch INFIT statistic signal dependency. When a diagnostically specific investigation of inter-item dependency is required, Fisher's Z is useful.

1)Obtain the Rasch-based standardized residuals for each observation of Person n on item i:

zni = (Observedni - Expectedni)/Standard Errorni

2)Correlate the standardized residuals for all pairs of items i, j across all N persons:

rij = (sum(zni znj) - sum(zni)sum(znj)/N) / N SD(zni) SD(znj)

3)Use Fisher's Z to linearize Pearson correlations:

Zij = 1/2 Log_e( (1+rij)/(1-rij) )

4)Draw a histogram of the Fisher Z values. Noticeable positive skew (see Figures) indicates the presence of sets of dependent items. A cluster analysis of the Z matrix will identify the sets. Outlying positive Z values flag pairs or sets of highly dependent items for further investigation.

Quantifying item dependency. Shen L. … Rasch Measurement Transactions, 1996, 10:1 p.485

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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