[Rasch] Why are Rasch measures linear?

Stephen Humphry stephen.humphry at uwa.edu.au
Tue Sep 4 11:30:28 EST 2007

A unit is implicit within the parameters of the Rasch model, when applied in
a given frame of reference. As mentioned by Svend Kreiner, the role of the
unit becomes more apparent when we measure the same trait in more than one
frame of refernece, each having a different unit. This is also what Wood
(1978) pointed out in informal terms.
When the unit is made explicit within the parameters of the Rasch model, its
role becomes clear, and so does the linearity of the scale on which
parameters are estimated. Additivity follows if estimates of parameters are
relative to the same unit and origin.
These points are explained more fully in Maintaining a Common Unit in Social
Measurement. See
Dr Stephen Humphry
Graduate School of Education
University of Western Australia
35 Stirling Highway
Mailbox M428
P: (08) 6488 7008
F: (08) 6488 1052

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Paul Barrett
Sent: Tuesday, 4 September 2007 8:21 AM
To: rasch at acer.edu.au
Subject: RE: [Rasch] Why are Rasch measures linear?

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
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 math. 
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:  <mailto:pbarrett at hoganassessments.com> pbarrett at hoganassessments.com

Web:    <http://www.hoganassessments.com/> www.hoganassessments.com 
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