[Rasch] Interpretation of Item Plot
Stephen Humphry
stephen.humphry at uwa.edu.au
Mon Mar 26 19:36:32 EST 2007
Mike, I provided two references -- they are included below again.
The wording is too strong? Perhaps it just requires some clarification.
Invariance is a requirement, and gives rise to sufficient statistics in
Rasch models. However, invariance may not hold empirically. Invariance is
implied by the algebraic structure of the model, but the requirement may not
be met in empirical data given the structure of the data.
Ordered thresholds and a corresponding latent deterministic Guttman response
subspace are implied analytically by the algebraic structure of the
polytomous Rasch model (Andrich, 1978, 2005).
However, ordering may not hold empirically. Perhaps you could point out
where you disagree either with my interpretation of the references or with
what is shown within them.
Ordered thresholds are a requirement in this sense. Of course, thresholds
can be estimated whether or not data conform with the model (in either the
dichotomous or polytomous case).
Hope this clarifies.
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.
_____
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Mike Linacre (RMT)
Sent: Monday, 26 March 2007 4:55 PM
To: rasch at acer.edu.au
Subject: RE: [Rasch] Interpretation of Item Plot
Stephen, Greg, et al.
As you say, Stephen, there is certainly an inferential problem if the
categories in Greg's rating scale are all intended to be modal. And the
Rasch model does require that Greg's categories are ordered qualitatively.
But your statement "The structure of the Rasch model requires they [the
Rasch-Andrich thresholds] are ordered." appears to be too strong. The
integer-scoring of ordered categories is independent of the Rasch-Andrich
thresholds - e.g., www.rasch.org/rmt/rmt131a.htm . A mathematical feature of
derivations of the Rasch Rating Scale model is that the pair-wise log-odds
of adjacent categories is unconstrained by the value of the log-odds of any
other adjacent pair of categories. Your statement introduces a constraint
on the overall pattern of adjacent-category log-odds similar to that in the
(non-Rasch) Graded Response model. If you can provide a reference to a
mathematical proof of this requirement, I would be delighted to publish it
in RMT. I did not notice it in Wright & Master's "Rating Scale Analysis",
nor in Gerhard Fischer's "The Derivation of the Polytomous Rasch Model"
(Chap. 16, "Rasch Models: ...", Fischer & Molenaar), nor in Erling
Andersen's "Rating Scale Model" (Handbook of Modern Item Response Theory).
May I suggest this wording? "The structure of the Rasch model is
inferentially more secure when they [the Rasch-Andrich thresholds] are
ordered."
Cordially,
Mike Linacre,
Editor, Rasch Measurement Transactions
At 3/26/2007, Stephen Humphry wrote:
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,
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