[Rasch] Interpretation of Item Plot

Stephen Humphry stephen.humphry at uwa.edu.au
Mon Mar 26 15:41:32 EST 2007

```Hi Greg. You are quite correct interpreting this as signalling a problem.
Yes, it is possible it would work properly with only three categories,
although this is not guaranteed. Try recoding and look at the fit. If you
are able, look at conditional threshold probability curves (e.g. 2 given a 1
or 2). The case in which it is clearly justified to collapse two categories
is when there is 0 discrimination at a threshold.

The thresholds in your case are disordered. The parameter for threshold x
(t_x) in the model is the location at which it is equally likely a person
will response in adjacent ordered categories x-1 and x. For example, t_1 is
the location at which it is equally likely a person is equally likely to
respond in categories 0 and 1, t_2 is the location at which a person is
equally likely to respond in categories 1 and 2, etc.

In your example, taking the threshold estimates literally, the location at
which a person is equally likely to respond in categories 0 and 1 (t_1) is
*higher* than the location at which a person is equally likely to respond in
categories 1 and 2 (t_2). If the categories are ordered as you intended, t_1
(0/1 equally likely) shoud be lower than t_2 (1/2 equally likely) and then
there would be a region on the latent continuum in which x=1 is the most
likely response for any given person in the region (i.e. a region between
t_1 and t_2). There is no such region in which x=1 is the most likely in
your case, which reflects the disordering.

The disordering in your example is not pronounced, so it is possible that it
is due to instability of the threshold estimates if you don't have a large
data set. The category frequencies are not the key consideration. Rather,
the key consideration is how many people overall have estimates in the
region of the disordering.

Whether the thresholds are ordered as intended is an empirical question. The
structure of the Rasch model requires they are ordered. Ordered thresholds
are the basis for the definition of integer scoring of ordered categories,
given the formal structure of the model (Andrich, 1978, 2005). Disordered
thresholds imply the hypothesis of ordering has broken down somehow (if the
disordering is genuine).

Yes, your interpretation is correct. Yes it is possible that three
categories will work, though this is not guaranteed.

Regards,

Steve.

Andrich, D. (1978). A rating formulation for ordered response categories.
Psychometrika, 43, 357-74.

Andrich, D. (2005).  The Rasch model explained.  In Sivakumar Alagumalai,
David D Durtis, and Njora Hungi (Eds.)  Applied Rasch Measurement: A book of
exemplars.  Springer-Kluwer. Chapter 3, 308-328.

-----Original Message-----
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Petroski, Greg
Sent: Saturday, 24 March 2007 1:10 PM
To: Eric Wong; rasch at acer.edu.au
Subject: [Rasch] Interpretation of Item Plot

To keep the size of this e-mail small I put an item plot on PhotoBucket.

http://i24.photobucket.com/albums/c1/Alpha_Mutt/ItemCategoryPlot.jpg

The item is ordinal with responses 0 to 4 with larger values indicating
greater ability.  Note that two of the five catagories do not ever have the
highest probability of being endorsed.  Am I correct in interpreting this as
suggesting a defective item -- one that might work as well with only three
categories?

Greg Petrsoki
U. of Missouri - Columbia

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