[Rasch] RSM & PCM

Thomas Salzberger thomas.salzberger at gmail.com
Thu Mar 11 00:02:06 EST 2010

At 13:42 10.03.2010, you wrote:
>Thanks Thomas,
>It seems that these are just a set 
>of  assumptions that we have about our data. I 
>was under the impression that when we talk about 
>unequal distances either within or across the 
>items we model the distances and weight them 
>accordingly. That is, each category gets a 
>different score depending on its difficulty. 
>Something along these lines. I think there are 
>some models which requie this, aren't there?
>So we do not need to have such complicated modelling.
>We just choose the type of the analysis 
>depending on what we think of our data. Right?

That is exactly right. Sometimes a common rating 
scale makes sense. One could at least try it.
Obviously it does not make sense when the 
categories are worded differently and it is 
impossible to run the RSM when the number of categories varies.
(That said, you can actually have several RSMs 
within your instrument with some items sharing a 
common rating scale structure and others not.)

The important thing is that weighting category 
scores (or, in general, item scores) is never 
related to the difficulty of an item (we do not 
weight difficult dichotomous items higher than 
easy ones). This is always the case, even in general IRT.

Weighting refers to discrimination. In the 2pl, 
items are weighted differently because of 
different discrimination, not because of different difficulty.

In the RSM as well as in the PCM, the 
discrimination is assumed to be equal as this is 
a key property of the Rasch model.
However, in the PCM this fact is somewhat 
obscured by the fact that different threshold 
distances between items lead to ICCs which do intersect.
But at the level of each threshold, the latent 
response curves are in fact parallel.

If it helps to illustrate the last point, I might 
send you a graph from RUMM which illustrates this nicely.


>--- On Wed, 3/10/10, Thomas Salzberger <thomas.salzberger at gmail.com> wrote:
>From: Thomas Salzberger <thomas.salzberger at gmail.com>
>Subject: Re: [Rasch] RSM & PCM
>To: rasch at acer.edu.au
>Date: Wednesday, March 10, 2010, 6:13 AM
>let us assume we have a four category item, so 
>there are three thresholds (0/1, 1/2 and 2/3, 
>referred to as tau1, tau2 and tau3, respectively)
>In the Rating scale model, the distance between 
>the thresholds tau1 and tau2 does NOT need to be 
>equal to the distance between tau2 and tau3.
>But the difference between tau1 and tau2 has to 
>be equal across all items. Likewise the 
>difference between tau2 and tau3 has to be the same for all items.
>So, no restrctions within the item but restrictions across items.
>In other words, in the PCM, each item has its 
>own rating scale structure, while in the rating 
>scale model we have a common rating scale structure across all items.
>The RSM is therefore more restrictive. Whether 
>the PCM fits statistically significantly better 
>than the RSM can be tested by a likelihood ratio test.
>What you have in mind, a model where all 
>distances between pairs of adjacent thresholds 
>are equal, would be even more restrictive than the RSM.
>At 12:39 10.03.2010, Anthony James wrote:
>>I was just wondering how PCM accomodates 
>>unequal distances when we do not model them.
>I am sorry, I don't get this statement. When we 
>do not model unequal distances (across items), 
>i.e. we model equal distances, we do not apply the PCM.
>>We just sum up correct responses on each polytomy and analyse it.
>We always do that. If it's a Rasch model, then raw score sufficiency holds.
>>A sum score is in fact given to the analysis 
>>and not modelled distances among items. Doesn't here a PCM reduce to an RSM?
>>--- On Wed, 3/3/10, Anthony James <luckyantonio2003 at yahoo.com> wrote:
>>From: Anthony James <luckyantonio2003 at yahoo.com>
>>Subject: [Rasch] RSM & PCM
>>To: rasch at acer.edu.au
>>Date: Wednesday, March 3, 2010, 2:17 AM
>>Dear All,
>>I know that this is a very old and probably a 
>>boring question for many of you. But I need to know this
>>What is the difference between rating  scale model and partial credit model?
>>What I have gathered is that in RSM the 
>>distances between the points on the scale is 
>>equal and this distance is the same for all the 
>>items in the instrument. That is, the ability 
>>difference needed to endorse 3 rather than 2 is 
>>the same as the ability difference needed to endorse 5 rather than 4. Right?
>>In PCM, however, the distances between points 
>>on the scale is unequal  both within the items 
>>and between the items in the instrument. That 
>>is, the ability increment to score 3 on an item 
>>rather than 2 is not the same as the ability 
>>increment needed to score 6 rather than 5. And 
>>these distances are unequal among  the items in the test. Right?
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>Dr. Thomas Salzberger
>Email: Thomas.Salzberger at wu.ac.at, Thomas.Salzberger at gmail.com
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