[Rasch] FW: c-parameter

Stephen Humphry stephen.humphry at uwa.edu.au
Mon Jun 4 19:22:59 EST 2012

Margaret, you are right that there isn't such a divide but having said that, the 2PL is a degenerate case in which you cannot possibly, in principle, estimate person and item locations in a common unit. Also, the 2PL only allows items to affect discrimination whereas empirically, person factors can affect discrimination (Humphry, 2010) and so can other factors, such as rater characteristics.

If the rationale for using a Rasch model (other than the Poisson) is tenable, the magnitude of discrimination is inversely proportion to the magnitude of the unit. So unless there can somehow be measurement without a measurement unit, discrimination is pretty fundamental. See the first two references below for a detailed explanation of this, including an analysis of the overlap and distinctions between the 2PL and Rasch model with a discrimination parameter for item sets (but not individual items).

I disagree that when you are using the polytomous Rasch model / PCM that you are using a special case of the 2-parameter model. The discrimination at each threshold (1 conditional on 0 or 1, 2 conditional on 1 or 2, etc) is equal in the polytomous Rasch model. David Andrich showed this (implicitly at least) is necessary for sufficiency and for the justification for using the unweighted raw score in his original paper on the Polytomous Rasch model. Nothing has changed. I don't think this is "discrimination" as you're referring to it. It is readily shown that discrimination at each threshold is inversely proportional to the unit of measurement in the Polytmous RM as it is in the dichotomous model. However, I probably don't disagree much with the implications -- there is not a great divide between them in many respects.


Humphry, S. M. (2011). The role of the unit in physics and psychometrics. Measurement: Interdisciplinary Research and Perspectives, 9, 1-24.

Humphry, S. M. (2010). Modeling the Effects of Person Group Factors on Discrimination. Educational and Psychological Measurement, 70, 215-231

Humphry, S.M. & Andrich, D. (2008). Understanding the Unit in the Rasch Model. Journal of Applied Measurement, 9(3), 249-264.

From: rasch-bounces at acer.edu.au [rasch-bounces at acer.edu.au] On Behalf Of Margaret Wu [wu at edmeasurement.com.au]
Sent: Sunday, 3 June 2012 6:27 PM
To: 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.


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_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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