Manual Estimation of "Partial Credit" Item Difficulties and Rasch-Andrich thresholds

Computer programs are routinely used to estimate Rasch measures. But is your program functioning correctly? Are the estimates based on the intended data? Are they based on the model you think you specified? A quick manual check of the computer's results can reassure you, or point you to a discrepancy.

Estimates for partial credit items, i.e., items specified to have unique rating scales, can be awkward to verify. Here is one approach. Imagine a sample of students who respond to both a dichotomous item and a polytomous "partial credit" items. We take the reported difficulty of the dichotomy, item i, as the benchmark difficulty. From this we estimate the difficulty of the dichotomy, D_jk, between each pair of two adjacent categories, k-1 and k, of the partial credit item, j.

From their respective Rasch models,

So that

We can use this equivalence to estimate the difficulty of all the {D_jk}. The mean item difficulty is

So that, within the item, the adjacent category dichotomies, F_jk, are located at

Here is an example from Items 12 and 14 of the "Liking for Science" data (Wright and Masters, 1982), specified as "partial credit".

Paired Frequencies for "Liking for Science" items 12 and 14. (Item 12 recoded 0,1)
N00	N01	N02	N10	N11	N12
5	6	2	9	28	25

According to an analysis, the difficulty of the dichotomous item i=12, is -1.14. So that for item j=14,

In this example, the reported estimates are -.71 and 1.30, somewhat more central than the values, - 1.55 and 1.50, given by our approximation.

Andrew Stephanou, Australian Council for Educational Research

Manual estimation of partil credit item difficulties. Stephanou A. … 16:1 p.867

Manual estimation of partil credit item difficulties. Stephanou A. … Rasch Measurement Transactions, 2002, 16:1 p.867

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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