[Rasch] RSM & PCM

Rodney Staples rodstaples at ozemail.com.au
Thu Mar 11 00:52:36 EST 2010


Hi Anthony and Thomas,
There is a very full discussion of the distinction between Likert scales and
Rasch Partial Credit models in Bond And Fox, Applying the Rasch Model,
Chapter 6.

A different example drawn from a satisfaction survey is on my site at:
http://members.ozemail.com.au/~rodstaples/Measurement3.htm

Hope this helps,
Rod



___________________________________________________________________________
Dr. Rodney Staples.
e-mail: rodstaples at ozemail.com.au
Telephone: +61 3 9770 2484
Mobile: +61 4 1935 9082
Web: http://members.ozemail.com.au/~rodstaples/


-----Original Message-----
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au]On Behalf
Of Thomas Salzberger
Sent: Thursday, 11 March 2010 12:02 AM
To: rasch at acer.edu.au
Subject: Re: [Rasch] RSM & PCM

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.

Thomas




Anthony

--- 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
Anthony,
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.
Thomas



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?
Cheers
Anthony
--- 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?
Cheers
Anthony




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