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, Djk, 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 {Djk}. The mean item difficulty is

So that, within the item, the adjacent category dichotomies, Fjk, 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)

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 Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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