Probabilities from measures
Request: I'm trying to obtain the probabilities by working backwards from the parameters Winsteps estimated, to see if probabilities I calculate correspond to those used to estimate the expected value for each person-item encounter.
Yes, if you have the reported person ability, item difficulty, and the Rasch-Andrich threshold values (rating scale structure), then you can calculate exactly the same probabilities as Winsteps.
For dichotomous responses,
Probability of 0 = 1 / (1 + exp(ability-difficulty))
Probability of 1 = 1 / (1 + exp(difficulty-ability))
These probabilities always sum to 1.
An easy way to do this for polytomous (rating-scale, partial-credit) responses is:
Value of bottom category v0 = 1
Value of next category v1 = v0 * exp ( ability - difficulty - threshold1)
Value of next category v2 = v1 * exp ( ability - difficulty - threshold2)
Value of top category vm = v(m-1) * exp ( ability - difficulty - threshold m)
Probability of category 0 = p0 = v0/ sum(v0+v1+...+vm)
Probability of category 1 = p1 =v1 / sum(v0+v1+...+vm)
Probability of category m = pm = vm/sum(v0+v1+...+vm).
These probabilities always sum to 1.
Please compare your probabilities with the graphed values.
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|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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