Rating Scale and Partial Credit models and the twain shall meet

For many testing situations, simple zero-one scoring is not enough and Poisson-type counts are too much. Polytomous Rasch models (*PRM*) cover the middle ground between one and infinity and allow scored responses from zero to a maximum of some small integer *m*. The integer scores must be ordered in the obvious way so that responding in category *k* implies more of the trait than responding in category *k-1*. While the scores must be consecutive integers, there is no requirement that the categories be equally spaced; that is something we can estimate just like ordinary item difficulties.

Once we admit the possibility of unequal spacing of categories, we almost immediately run into the issue, Can the thresholds (i.e., boundaries between categories) be disordered? To harken back to the baseball discussion, a four-base hit counts for more than a three-base hit, but four-bases are three or four times more frequent than three-bases. This begs an important question about whether we are observing the same aspect with three- and four-base hits, or with underused categories in general; we’ll come back to it.

To continue the archery metaphor, we now have a number, call it *m*, of concentric circles rather than just a single bull’s-eye with more points given for hitting within smaller circles. The case of *m=1* is the dichotomous model and *m*→infinity is the Poisson, both of which can be derived as limiting cases of almost any of the models that follow. The Poisson might apply in archery if scoring were based on the distance from the center rather than which one of a few circles was hit; distance from the center (in, say, *millimeters*) is the same as an infinite number of rings, if you can read your ruler that precisely.

Read on . . .Polytomous Rasch Models

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