Another way to think of item difficulty for each score point of a PCM item is to calculate the expected score as a function of ability, and then find the ability at which the expected score is 0, 1, 2, etc.
Between the ability range of -0.73 and 1.93, the expected score is between 0.5 and 1.5. We may call this ability range of the score 1 region. Below -0.73, the expected score is closer to 0 than closer to 1, so this region is the score 0 region. Above an ability of 1.93, the expected score is closer to 2 than closer to 1, so this may be the score 2 region. In this way, -0.73 and 1.93 may be regarded as item difficulty measures for a PCM item.
How do we derive the expected score of 0.5 for a person with ability -0.73?
We calculate the probability of their scores in each category, multiply by the category score, and then sum them. So, theta = -0.73, δ1 = -0.5 and δ2 = 1.7.
\[
Prob(X=1) = \frac{\exp(\theta - \delta_1)}{1+\exp(\theta - \delta_1) + \exp(2\theta - \delta_1 - \delta_2)}
\tag{26.1}\]
\[
Prob(X=1) = \frac{\exp(-0.73 - -0.5)}{1+\exp(-0.73 - -0.5) + \exp(2* -0.73 - -0.5 - 1.7)}
\tag{26.2}\]
\[
Prob(X=1) = \frac{0.7945336}{1+0.7945336 + 0.06994822}
\tag{26.3}\]
\[
Prob(X=1) = 0.4261418
\tag{26.4}\]
\[
Prob(X=2) = \frac{\exp(2\theta - \delta_1 - \delta_2)}{1+\exp(\theta - \delta_1) + \exp(2\theta - \delta_1 - \delta_2)}
\tag{26.5}\]
\[
Prob(X=2) = \frac{\exp(2*-0.73 - -0.5 - 1.7)}{1+\exp(-0.73 - -0.5) + \exp(2*-0.73 * - -0.5 - 1.7)}
\tag{26.6}\]
\[
Prob(X=2) = \frac{0.06994822}{1+0.7945336 + 0.08803683}
\tag{26.7}\]
\[
Prob(X=2) = 0.0371557
\tag{26.8}\]
We can then add the probabilities to derive the expected score:
\[
E (-0.7)) = 0.4261418 + (2 * 0.0371557) = 0.5
\tag{26.9}\]
Extension activity
Now do the same for E = 1.93
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