# [Rasch] Partial Credit Model vs Dichotomous Model

Svend Kreiner S.Kreiner at biostat.ku.dk
Mon Oct 20 18:16:26 EST 2008

```Yes. The sum of k dichotomous items follows the same kind of
distribution as an item from a partial credit model with k+1 response
categories. Of course, the interpretation of the threshold parameters
differs from the interpretation of the threshold parameters of a PCM
item, but mathematically the models are the same. You could also say, of
course, that scoring 8 correct responses out of 10 dichotomous items is
to get partial credit on the test.

Best regards

Svend Kreiner

Yoke Mooi Ong skrev:
> Dear all,
>
> Can we model sums of dichotomous responses by the Partial Credit Model?
>
> I am working on a dataset, a mathematics test that consists of 100
> items and a sample size of 2892.
> 14 out of 100 items, partial credit is awarded for partial response.
> Two marks are awarded for these 14 items.
> In the dataset, these items are score as dichotomous items (2 items
> i.e. (1i) and (1ii). ) These base items seems to violate the Rasch
> model assumption of local independence.
>
> So I sum the dichotomous scoring in the dataset (i.e 0 (0,0), 1(1,0),
> 2(1,1)) to form a polytomous score item).
>
> I analysed with the Partial Credit model.
>
> 7 out of 14 items indicating misfit (i.e. Infit MNSQ > 1.30).
>
> When I analysed with the dichotomous model, all the items fitted well
> within the accepted range of fit statistics.
>
> Here are 2 examples of the analysis:
>
> Item 1
> PCM
> delta1 = 1.38
> delta2 = 1.54
> Infit MNSQ = 1.40
>
> Item 1(i) -base item for Item 1
> Dichotomous
> delta = 1.38
> Infit MNSQ = 1.06
>
> Item 1(ii)
> Dichotomous
> delta = 1.65
> Infit MNSQ = 1.01
>
> Item 2
> PCM
> delta 1 = -0.69
> delta2 = -0.08
> Infit MNSQ = 1.40
>
> Item 2(i)
> Dichotomous
> delta = -1.43
> Infit MNSQ =0.99
>
> Item 2(ii)
> Dichotomous
> delta = -0.43
> Infit MNSQ = 1.16
>
> From the results of  the dichotomous and the PCM analyses, it seems to
> indicate that the dataset fitted the dichotomous model better than PCM.
>
> My questions:
>
> (1) How can I explain this? Why?
> (2) Is PCM an adequate model to describe the distribution of sums of
> dichotomous item scores? If yes, to what extent? if no, why?
>
> Thank you.
>
> Regards,
>
> Yoke Mooi
>
>
>
>
>
>
>
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--
Svend Kreiner
Professor
Department of Biostatistics
University of Copenhagen

P.O. Box 2099
DK-1014 Copenhagen K, Denmark

Email: S.Kreiner at biostat.ku.dk
Phone: (+45) 35 32 75 97

Fax: (+45) 35 32 79 07

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