[Rasch] Estimation with ICC

Mike Linacre (RMT) rmt at rasch.org
Tue Dec 16 10:31:50 EST 2008

Anthony, thank you for asking about the logistic ogives.

It appears that you are doing the same investigation that Georg Rasch does 
on page 69ff. of his book "Probabilistic Models".

It is much easier to think in terms of fitting straight-lines rather than 
fitting ogives, so let's transform the x- and y-axes so we are looking at 
straight lines.

Also, let's assume we have a complete rectangular dataset:

For each ability group j, compute: xj = log(average raw score of group / 
(maximum raw score - average raw score of group)).

For each ability group j and item i, compute yji = log (proportion 
incorrect / proportion correct)

Plot all the (xj, yji) points for each i, they will form a series of 
approximate straight lines, one for each i.

The set of parallel diagonal straight-lines, one for each item, will 
approximate the exact Rasch logistic ogives. Your plot will look like p. 87 
of Rasch's book.

Once you have identified the straight lines, you can convert to ogives by 
transforming the y-axis: y = exp(y)/(1+exp(y))

 From the straight lines (or the ogives) you can predict the performance of 
any person (with a known raw score) on any item, or, if you know the 
success-rate of a new sample on an item, you can predict their overall 
performance on the test.

How does this sound?

Mike L.

At 12/15/2008, you wrote:
>In fitting the Rasch model we need to know the ability estimates a priori 
>and make j ability groups. On the y axis we represent the proportion of 
>correct responses in each ability group. Then arbitrary values as item 
>measures are suggested and the probabilities of correct replies for every 
>ability level are estimated using the logistic function. These 
>probabilities are then compared with the proportions correct for each 
>ability level. If they don't agree the value of the item measure is 
>adjusted until they agree.
>I was always under the impression that both ability and item measures can 
>be read off from the curves. But it seems that the ICC has only a cosmetic 
>or a didactic role in Rasch and IRT and everything is done with the 
>algebra. Without ability estimates drawing ICCs is not possible. Beside an 
>ICC is drawn for an item and estimating person abilities which are based 
>on taking several items is not possible.
>Id be grateful for some explanations on the importance of the curves and 
>the curve fitting endeavor in Rasch analysis, since I cannot appreciate 
>the centrality of the curve and curve fitting.

Mike Linacre
Editor, Rasch Measurement Transactions
rmt at rasch.org www.rasch.org/rmt/ Latest RMT:  22:1 Summer 2008
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