[Rasch] Why are Rasch measures linear?

Paul Barrett pbarrett at hoganassessments.com
Tue Sep 4 10:20:37 EST 2007

The issue as ever is concerned with the relation of a Rasch "unit" of
measurement to that unit of the proposed psychological or educational
In practice, I think many assume these to be synonymous with one
another. As Robert Wood (1978) demonstrated, this can be a dangerously
incorrect assumption.
However, maybe researchers simply propose that the Rasch unit is simply
THE unit for the constructed variable. As David indicated, the Rasch
latent variable is rendered linear by the properties given to it by the
Importantly, as Joel Michell argued in 2004, there is nothing in
IRT/Rasch models which equate an arbitrary Rasch unit with a proposed
underlying psychological variable "unit". 
So, the linearity issue is perhaps best settled by first considering
whether appropriate evidence exists which supports the Rasch model
actually fitting the dataset under examination, and then deducing what
outcomes might be examined which would help indicate that the variable
being measure is indeed linearly structured i.e. what predictions might
be made which would confirm/disconfirm the notion that the variable
being measured does vary linearly in magnitude?
Perhaps it is useful to contrast asserting linearity via statistical
methodology vs discovering linearity by more "empirical" means
(experimentation, manipulations, theoretical deductions and predictions
Perhaps it is a two-stage process - construct a Rasch variable, then
seek to confirm the linear properties via other kinds of empirical
observations where linear magnitudes of the variable should yield
specific, related, magnitudes of other outcomes - in contrast to simple
sumscore or other kinds of magnitude scoring (ordinal classes etc.).
I note the recent paper by Cherneyshenko et al which compared
ideal-point approaches with IRT magnitude scales, and the dreaded simple
sum scores - and showed little or no difference in their validity to
predict real-world behaviors. It's not necessarily a Rasch-focussed
application, but given Michell's arguments, exactly what one might
expect when "measuring"  intrinsically non-quantitative psychological
variables; "non-linear" sum-scores are no better/worse than linear IRT
scores under these conditions - as both methods are "good enough" at
approximating magnitudes of an essentially "messy ordinal structure"
non-linear variable. 
Regards ... Paul
Cherneyshenko, O.S., Stark, S., Drasgow, F., & Roberts, B.W. (2007)
Constructing personality scales under the assumption of an Ideal Point
response process: toward increasing the flexibility of personality
measures. Psychological Assessment, 19, 1, 88-106.
Michell, J. (2004) Item Response Models, pathological science, and the
shape of error. Theory and Psychology, 14, 1, 121-129.
Wood, R. (1978) Fitting the rasch model - a heady tale. British Journal
of Mathematical and Statistical PSychology, 31, , 27-32.
Regards .. Paul
Paul Barrett                                       918.749-0632 x 326
Chief Research Scientist                          Skype: pbar088
Hogan Assessment Systems Inc.
2622 East 21st St., Tulsa, OK 74114          
Email: pbarrett at hoganassessments.com
<mailto:pbarrett at hoganassessments.com>  
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