[Rasch] What in Winsteps tells how uni dimensional a fit is?

Mike Linacre mike at winsteps.com
Mon Sep 2 18:34:53 EST 2013

A profound question, Michael: "What in Winsteps tells how unidimensional 
a fit is?"

The Rasch measures estimated by unidimensional Rasch models are forced 
to be unidimensional. Off-dimensional aspects of the data are in the 
part of the data not explained by the Rasch measures, i.e., the Rasch 
residuals. The Rasch residuals decompose into (a) the randomness 
predicted by the Rasch model, and (b) components on dimensions other 
than the unidimensional Rasch variable, (c) off-dimensional  noise, such 
as random guessing.

In empirical data, (a) and (c) usually dominate (b), so that item-level 
or person-level fit statistics tend to be insensitive to 
multidimensionality, as R.P. Macdonald (1985) reports. Accordingly we 
must focus on techniques that quantify (b), such as PCA of residuals. If 
the eigenvalues reported by PCA approximate the size predicted by the 
Rasch model, then the data are effectively unidimensional. Otherwise, 
the bigger the eigenvalues, the less unidimensional are the data.

Does this help?

Mike L.

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