Expected Score ICC, IRF


Select by clicking on "Expected Score ICC" or from the Graphs menu. Expected Score ICC plots the model-expected item characteristic curve, also called the Item Response Function, IRF. This shows the Rasch-model prediction for each measure relative to item difficulty Its shape is always ascending monotonic. The dashed lines indicate the Rasch-half-point thresholds correspond to expected values of .5 score points. The intervals on the x-axis demarcated by dashed lines are the zones within which the expected score rounds to each observed category. To remove the dashed lines, double-click on them.


The plot is of the "model item characteristic curve (ICC)" , also called the "item response function (IRF)". This is used for inference. If we know the score on the item, we can infer the ability of the person responding. If we know the ability of the person responding, we can infer the expected score on the item.


(difficulty of item + person's score on the item on y-axis) -> person's ability on x-axis


(person's ability on x-axis - difficulty of item) -> expected score on the item on y-axis


Item discrimination: the item is more discriminating where the ICC is steeper, and less discriminating where the ICC is flatter.


Buttons are described in Graph window.



Computation of the IRF


For an item with categories k=0,1,2,...,m and Andrich Thresholds Fk, with F0 chosen to be 0, then


At location x on the latent variable (relative to the item difficulty), the probability of observing category k is


Pk(x) = exponential( k*x - sum(Fh) for h=0 to k) / sump(x)


sumpx = sum (exponential( k*x - sum(Fh) for h=0 to k)) for k=0 to m


The expected score y at location x is given by


y = sum (k*Pk(x)) for k=0 to m



Computation of the Half-Point Threshold


The formula for the half-point threshold is solved iteratively, for category j, the half-point threshold is at  point x (relative to the item difficulty) given by


j - 0.5 = sum (k*Pk(x)) for k=0 to m



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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 with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind 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
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