# [Rasch] FW: c-parameter

Mon Jun 4 16:16:01 EST 2012

```I've always thought it unhelpful to speak of "the" discrimination of an item.   The discriminating power of an item (even under the Rasch model) is a function of ability.  ie  item has discrimination x and ability level y.   It is best to think about the item discrimination in terms of the information function which is the derivative of the expected score curve with respect to ability.

SLM items have information functions that are equivalent in shape and they peak at a value of 0.25, when ability equals the item difficulty.  No items are highly discriminating when they are not well targeted.

For PCM items the information functions are not of equivalent shape, so as Margaret implies the idea of "equal" discrimination doesn't have a lot of use with PCM items. The property that PCM items satisfy is that the area under the item information function is equal to one less than the number of categories in the item. So the area under the area under the information functions for dichotomous items equals 1.  Under the Rasch model all items with k categories have an equal amount of total information (=k-1) the distribution of which over the ability dimension is a function of the  item parameters.  For PCM items, the highest peaks in the function typically occur at disordered Andrich thresholds.  So if you do like to speak of discrimination of PCM items (I don't), the items with disordered parameter estimates are the most discriminating.

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf Of David Andrich
Sent: Monday, 4 June 2012 1:31 PM
To: rasch
Subject: Re: [Rasch] FW: c-parameter

Margaret. It might help if you define discrimination for a partial credit item? We know the discrimination in dichotomous item. I am not sure the two are compatible. Also, what is your formulation of a "generalized 2-parameter analysis"?  Maybe you can give us the equation in an attachment.

Thanks

David

David Andrich, BSc MEd W.Aust., PhD Chic, FASSA
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From: rasch-bounces at acer.edu.au<mailto:rasch-bounces at acer.edu.au> [mailto:rasch-bounces at acer.edu.au] On Behalf Of Margaret Wu
Sent: Sunday, 3 June 2012 6:27 PM
To: rasch at acer.edu.au<mailto:rasch at acer.edu.au>
Subject: [Rasch] FW: c-parameter

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_software/oplm.<http://www.cito.com/research_and_development/pyschometrics/psychometric_software/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> [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<mailto: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_software/oplm.<http://www.cito.com/research_and_development/pyschometrics/psychometric_software/oplm.aspx>aspx

<http://www.cito.com/research_and_development/pyschometrics/psychometric_software/oplm.aspx>Mike L.

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