[Rasch] Re: Rasch Digest, Vol 35, Issue 12

Stephen Humphry shumphry at cyllene.uwa.edu.au
Tue Jun 24 01:00:26 EST 2008


Fraser, to clarify, the "item difficulty for each of the responses"  
you refer to in the first paragraph below is the Thurstone threshold.  
This is Not the threshold to which I was referring.

I am not entirely clear what you are referring to as the "crossing  
point". If you mean the point at which a response in two adjacent  
categories is equally probable, this is the parameter I was referring  
to as the threshold that becomes the item difficulty in the  
dichotomous case.

The Thurstone thresholds are not parameters of the polytomous Rasch model.

Steve


Quoting Fraser Rew <Fraser.Rew at nzqa.govt.nz>:

> Picking up on Steve Humphry's comments, my understanding (and I'm a
> relative newcomer to all this so someone may need to correct me) is that
> in the item difficulty for each of the responses is the point at which
> that response or a higher one becomes more likely than any lower
> response, rather than being simply the point at which they cross. In the
> dichotomous case these two are obviously the same; in the polytomous
> case they are not.
>
> Looking at examination data we have found many cases, particularly
> where the difficulty of two adjacent items is similar or (using the
> two-parameter model) that the discrimination is low that the crossing
> point is actually well removed from the difficulty parameter. In some
> cases a given response is not the most likely for ANY ability score and
> in these cases the curve's crossing point for the higher grade will
> clearly be at a lower ability score than its crossing point in the lower
> grade. As a digression we weren't able to directly calculate crossing
> points based on item parameters, and used Newton-Raphson instead.
>
> In practice the difference between the parameter and the crossing point
> is often negligible, but in many cases it can be surprisingly large.
>
> Regards
>
> Fraser
>
> Fraser Rew
> Researcher/Data Analyst
> Qualificiations Division
> DDI: 04 463 4368
> Extn: 4368
>
> www.nzqa.govt.nz
>
>>>> <rasch-request at acer.edu.au> 22/06/2008 2:00 p.m. >>>
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> Today's Topics:
>
>    1. Measures in Rasch Model (Juanito Talili)
>    2. Re: Measures in Rasch Model (Thomas Salzberger)
>    3. Re: Measures in Rasch Model (Stephen Humphry)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Fri, 20 Jun 2008 20:26:01 -0700 (PDT)
> From: Juanito Talili <talilij at yahoo.com>
> Subject: [Rasch] Measures in Rasch Model
> To: rasch at acer.edu.au
> Message-ID: <963004.46958.qm at web46205.mail.sp1.yahoo.com>
> Content-Type: text/plain; charset="iso-8859-1"
>
> Dear Rasch Modelers,
> á
> For a dichotomous case, the item measure is called "item difficulty"
> while the person measure is called "person ability".á I have a
> difficulty of understanding the measuresáfor theápolytomous(rating
> scale)ácase.á Can somebody have a kind heart to explain these things?
> á
> Thank you.
> J. Talili
>
>
>
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> Message: 2
> Date: Sat, 21 Jun 2008 14:47:26 +0200
> From: Thomas Salzberger <Thomas.Salzberger at wu-wien.ac.at>
> Subject: Re: [Rasch] Measures in Rasch Model
> To: rasch at acer.edu.au
> Message-ID: <20080621144726.jn0yubnd6kgcg0cw at imp.wu-wien.ac.at>
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> Quoting Juanito Talili <talilij at yahoo.com>:
>
>> Dear Rasch Modelers,
>>  For a dichotomous case, the item measure is called "item
>> difficulty"  while the person measure is called "person ability".  I
>
>> have a  difficulty of understanding the measures for the
>> polytomous(rating  scale) case.  Can somebody have a kind heart to
>> explain these things?
>>  Thank you.
>> J. Talili
>
> Juanito,
>
> what is the item difficulty (delta) in a dichotomous item? It is the
> location on the latent continuum where a positive response
> (correct/affirmative answer, coded 1) and a negative response (coded
> 0) have the same probability. Below that point (beta < delta) a
> negative response is more likely and above that poi
> nt (beta > delta) a
>
> positive answer is more likely. So, it is a transition point where
> (going from minus infinity to plus infinity) the most likely response
>
> changes from 0 to 1. For two responses to distinguish between one item
>
> parameter is needed. This parameter is the threshold between response
>
> 0 and response 1.
>
> Generalising this idea to the polytomous case is relatively
> straighforward. If you have three response options, two such
> transition points (thresholds) have to be modelled. With m categories,
>
> you need m-1 thresholds to be modelled and that, of course, is done by
>
> the threshold parameters. Consequently, the interpretation of the item
>
> location in the dichotomous case and the threshold parameters in the
> polytomous model is, in principle, the same.
>
> Since a set of, say, six thresholds (needed for a seven category
> response scale) is a little bit hard to handle if you compare items,
> it is quite common to calculate the mean of the item's threshold and
> interpret it as a sort of overall-difficulty. If your items have all
> have a similar spread of threshold locations (in the rating scale
> model they have the same by definition) this should be okay. If
> thresholds vary considerably between items the overall location can be
>
> misleading.
>
> In the context of item difficulty, I also want to point out that the
> term difficulty makes many, uninitiated, people think the Rasch model
>
> (or IRT models in general) can only be applied to achievement tests. I
>
> don't know what field of science you are in, but I can tell you that
> in marketing research, where we are mostly concerned with attitudinal
>
> or attitude-like constructs, it can be hopeless to explain people that
>
> item difficulty is a technical term and that it should not always be
> interpreted literally. Using a neutral term like 'item location' or
> something like endorsability may help convey the idea.
>
> Thomas
> _______________________________________________________
> Dr. Thomas Salzberger
> ?sterreichische Gesellschaft f?r Absatzwirtschaft
> c/o WU Wien - University of Economics and Business, Vienna
> Department of Marketing, Institute of Marketing Management
> (Wirtschaftsuniversit?t Wien, Institut f?r Marketing-Management)
> Augasse 2-6, 1090 Wien
> Email: Thomas.Salzberger at gmail.com, Thomas.Salzberger at wu-wien.ac.at
> WWW: http://www2.wu-wien.ac.at/marketing/user/salzberger/
> MBC: http://www.matildabayclub.net
>
>
>
> ------------------------------
>
> Message: 3
> Date: Sat, 21 Jun 2008 20:55:21 +0800
> From: Stephen Humphry <shumphry at cyllene.uwa.edu.au>
> Subject: Re: [Rasch] Measures in Rasch Model
> To: rasch at acer.edu.au
> Message-ID: <20080621205521.vxps3diask0wcc84 at webmail-2.ucs.uwa.edu.au>
> Content-Type: text/plain;	charset=ISO-8859-1;	DelSp="Yes";
> 	format="flowed"
>
>
> Hi. For dichotomous items, there is a single threshold at which it is
>
> equally probably a person will respond incorrectly or correctly. This
>
> threshold is the item difficulty.
>
> For polytomous item, the item parameters are also thresholds. Each
> threshold is the location at which it is equally likely a person's
> response will be in one of two adjacent categories, or be judged in
> one of two adjacent categories. At the first threshold, the scores 0
> and 1 are equally likely, at the second threshold, the scores 1 and 2
>
> are equally likely, and so on.
>
> The polytomous model is sometimes written with an item difficulty and
>
> parameters that represent the distance of each threshold from the item
>
> difficutly. In this case, item difficulty is by definition the average
>
> threshold location.
>
> However, to understand how the parameters of the polytomous model are
>
> an extension of the single parameter of the dichotomous model, it is
> probably simpler for you to think about one threshold (item
> difficulty) for dichotomous items, and more than one threshold for
> polytomous items with 3 or more categories.
>
> Item difficulty is simply the single threshold in the special case in
>
> which there is only
>  one threshold and two categories, such as
> incorrect and correct responeses. Different terms may be used to refer
>
> to the thresholds but the concept remains the same whatever these
> parameters are called.
>
> Hope that helps,
>
> Steve
>
>
> Quoting Juanito Talili <talilij at yahoo.com>:
>
>> Dear Rasch Modelers,
>> á
>> For a dichotomous case, the item measure is called "item difficulty"
>
>> while the person measure is called "person ability".á I have a
>> difficulty of understanding the measuresáfor theápolytomous(rating
>> scale)ácase.á Can somebody have a kind heart to explain these
> things?
>> á
>> Thank you.
>> J. Talili
>>
>>
>>
>
>
>
>
>
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> End of Rasch Digest, Vol 35, Issue 12
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