﻿ Probabilities from measures

# 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)

then

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.

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