[Rasch] thresholds

Stephen Humphry stephen.humphry at uwa.edu.au
Tue Jul 10 11:24:46 EST 2007


Hi Muy and Andres. There is a lot of misunderstanding on this issue. The
term Rasch-Thurstone threshold is really an oxymoron. The Thurstone model is
incompatible with the Rasch model.

The Thurstone thresholds are obtained by summing the probabilities of
responses of adjacent categories. Summing probabilities of adjacent
categories means the model is no longer a Rasch model. Rasch pointed this in
1966, Anderson showed algebraically the conditions when such probilities
could be summed, and in 1978 Andrich showed that the relevant algebraic
conditions implied the discrimination at the thresholds was 0. (In the
dichotomous response case, the item difficulty is the threshold. In
polytomous cases these thresholds in the Rasch model arise naturally from
the model as the point where the probability of a response in each of a pair
of adjacent categories is 0.50.) Thus when a response in one of an adjacent
pair of categories is random, then the probabilties of two adjacent
categories can be summed to form a new category in the Rasch model. However,
this is the only instance in which the probabilities can simply be summed.
Doing so is otherwise incompatible with the Rasch model.

It is not an explanation to say that "Disordered Rash-Andrich thresholds
means that the categories define a very narrow interval on the latent
variable" because in this statement the 'interval' is defined in Thurstone
terms, not in terms of the Rasch model and its thresholds.

When the Andrich-Rasch thresholds are reversed there is something wrong in
the ordering of the thresholds given the operationalisation of the
categories, something that needs a qualitative explanation. The intended
interval on the latent continuum is undefined!

Regards,

Steve.

Dr Stephen Humphry
Graduate School of Education
University of Western Australia
35 Stirling Highway
CRAWLEY  WA  6009
Mailbox M428
P: (08) 6488 7008
F: (08) 6488 1052
 
-----Original Message-----
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Andrés Burga León
Sent: Saturday, 7 July 2007 10:59 PM
To: rasch at acer.edu.au; rmt at rasch.org
Subject: RE: [Rasch] thresholds


Hello Muy:

Disordered Rash-Andrich thresholds means that the categories define a very
narrow interval on the latent variable. You could check
http://www.rasch.org/rn2.htm for some guidelines about rating scale.

Sometimes when I've collapsed Partial Credit items I get better fit indexes,
and the separation reliability didn’t get worse. If by collapsing I get
worse separation reliability and the fit indexes didn't improve, I prefer
not to do it so.

-----


I know that is harder to use rating / or partial credit items for equating
and that you usually anchor only one o some of he thresholds. Here are a
couple of questions: 
- How do you choose which threshold to use?
- What happens if in Winsteps instead of using the SAFILE to anchor some
thresholds, you use an IAFILE specifying the measure for a partial credit
item?
- If you have disordered thresholds in a sample, is it wise to use them, for
example in anchoring in order to equate different tests?


Andres

-----Mensaje original-----
De: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] En nombre
de Mike Linacre (RMT) Enviado el: Sábado, 07 de Julio de 2007 03:12 a.m.
Para: rasch at acer.edu.au
Asunto: Re: [Rasch] thresholds

Dear Muy Ignoto:

Welcome to the Rasch list, and thank you for your questions.

To clarify, Rasch-Thurstone thresholds are always ordered, and are the
intersections of the cumulative probabilities, for example: 0 vs 1+2+3
points,  0+1 vs 2+3 points, 0+1+2 vs 3 points. Rasch-Andrich thresholds can
be disordered, and are the intersections of the category probabilities, for
example: 0 vs 1 point,  1 vs 2 points, 2 vs 3 points.

>Your question: Do we need to collapse the categories if the threshold 
>are disordered?
Reply: This depends on your purpose. For example, are you designing a new
instrument or trying to make sense of an old dataset familiar to your
audience? Also is the threshold disordering an accident of the current
dataset or structural to the instrument? For instance, in survey
instruments, Likert scales are sometimes printed as the response mechanism
to true-false items. This confuses the respondents and such items are
obvious candidates for collapsing categories.

Your objective is to produce Rasch measures that are useful and easy to
communicate. The reason for linear measurement (which Rasch constructs) is
to assist with clear thinking. So if collapsing categories makes the
measures easier to understand, then do it. If collapsing categories makes
the measures more obscure, then consider carefully the advantages and
disadvantages.

Cordially,
Mike Linacre
rmt at rasch.org

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