# [Rasch] FW: c-parameter

Margaret Wu wu at edmeasurement.com.au
Sun Jun 3 20:27:29 EST 2012

```Rense writes: One down (guessing), one to go (discrimination)

Mike writes: Answer: OPLM with its fixed, but different, discrimination
coefficients.
http://www.cito.com/research_and_development/pyschometrics/psychometric_soft
ware/oplm.
<http://www.cito.com/research_and_development/pyschometrics/psychometric_sof
tware/oplm.aspx>

There isn't such a divide between the Rasch model and the two parameter
model. Let's consider an example. Suppose we have three Rasch partial credit
items. Item 1 has scores 0,1,2,3,4. Item 2 has scores 0,1,2, and Item 3 has
scores 0,1. So Item 1 has the highest weight in the test. That is, Item 1
has twice the weight of Item 2 (it counts 4 points in the test, while Item 2
only counts 2 points), and four times the weight of Item 1 in the test. How
does one decide on the weight of a partial credit item? Or, the question
could be, how does one decide on the maximum score of an item? Contrary to
common perception that the maximum score of a partial credit relates to its
difficulty, actually the maximum score of an item (or the weight of an item)
depends on its discrimination. This makes sense. If an item does not
discriminate, we want to weigh it down in the test. If an item discriminates
highly, we want to increase the weight of the item in the test.

Suppose we run a generalized 2-parameter analysis, the scores of Item 1 may
come out to be 0,  0.8,  1.1, 1.7, 2.2. These scores suggest that, instead
of scoring Item 1 with 0, 1, 2, 3, 4, we should score these five categories
as 0,  0.8,  1.1, 1.7, 2.2. Because the scores are not integer, we no longer
have the Rasch model. However, if we round the scores to integers, and score
the five categories, 0, 1, 1, 2, 2, (that is, we collapse original
categories 1 and 2 as 1, and collapse categories 3 and 4 as 2), our new
partial credit scoring is now 0, 1, 2. This is still a Rasch partial credit
item. The consequence of this re-scoring will make the item fit the Rasch
model better, and potentially increase the test reliability, because now we
put more weight on "good" items (more discriminating ones), and less weight
on "poor" items, and we still stay in the Rasch family.

In practice, many of us using the Rasch model have already been doing this.
We examine the item analysis and decide on how to re-score or collapse
categories to improve fit and reliability, without actually running a
2-parameter model . What we are doing is already trying to find the best
weight for each item. So whenever you are using Rasch partial credit model,
you are already giving different items different weights, so, in fact, you
are already moving into the 2-parameter model concept.

The technical difference between a 2-parameter model and a Rasch partial
credit model is that for a partial credit model, the category scores must be
integer and there must not be jumps (e.g., we can't have 0,1,3,4). The
2-parameter model allows for non-integer scores. But by rounding the scores,
you can have the best of the two worlds, and that's how OPLM can bring the
best out of the one-parameter model.

Certainly, whenever you are using Rasch partial credit models, you are
already using a special case of the 2-parameter model. We should realize
that when items have different maximum scores, you are already providing
different weights to the items, and that's the idea of 2-parameter models.
But we can still stay within the Rasch family when we incorporate the item
discrimination information.

Margaret

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Mike Linacre
Sent: Sunday, 3 June 2012 5:29 AM
To: rasch at acer.edu.au
Subject: Re: [Rasch] c-parameter

Rense writes: One down (guessing), one to go (discrimination)

Answer: OPLM with its fixed, but different, discrimination coefficients.
http://www.cito.com/research_and_development/pyschometrics/psychometric_soft
ware/oplm.
<http://www.cito.com/research_and_development/pyschometrics/psychometric_sof
tware/oplm.aspx> aspx
<http://www.cito.com/research_and_development/pyschometrics/psychometric_sof
tware/oplm.aspx>

Mike L.

Mike Linacre
rmt at rasch.org www.rasch.org/rmt/ Latest RMT: 25:4 Spring 2012

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