[Rasch] Measures in Rasch Model

Thomas Salzberger Thomas.Salzberger at wu-wien.ac.at
Sat Jun 21 22:47:26 EST 2008

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


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 point (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  

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.

Dr. Thomas Salzberger
Österreichische Gesellschaft für Absatzwirtschaft
c/o WU Wien - University of Economics and Business, Vienna
Department of Marketing, Institute of Marketing Management
(Wirtschaftsuniversität Wien, Institut für 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

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