# [Rasch] Why are Rasch measures linear?

David Andrich david.andrich at uwa.edu.au
Mon Sep 3 14:27:42 EST 2007

```Anthony. There are different levels at which to consider the linearity. The
simplest is just that in the model, the difficulty and ability parameters
are additive in giving a probability of a response, which in the dichotomous
case is a 0 or 1. Then it is a consequence of the model that the total score
is sufficient for the person parameter. This parameter can be eliminated in
estimating the item parameters. The item parameters enter into the
estimation equation additively. If you are comparing person parameters, they
also enter additively in the equation. The relationship between the estimate
of the ability and each raw score for a particular set of items is non
linear. Hence it is said that the parameters are additve, and the raw scores
are not.
Thus while one can get into issues of additivity and fundamental measurement
and the Rasch model, it is not necessary to understand additivity in the
model with respect to the parameters, and the non-additivity of the raw
scores. That can be understood within the algebra of the model.

Hope this helps.

David

David Andrich, BSc., MEd. (UWA), PhD (Chic), FASSA
Chapple Chair, Graduate School of Education
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From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Mike Linacre (RMT)
Sent: Monday, 3 September 2007 8:00 AM
To: rasch at acer.edu.au
Subject: Re: [Rasch] Why are Rasch measures linear?

Anthony et al.:

It is always useful when considering this type of question to think about
the same situation in physical measurement. How would you prove that
"meters" are linear or "grams" or "temperature degrees"? Whatever method you
would choose to use, we can think of a parallel situation with Rasch
measures.

For instance, if you choose "concatenation" (putting sticks end-to-end for
length, or piling up bricks for weight) as physicist Norman Campbell did,
then we can do an analogous operation with Rasch measures:
http://www.rasch.org/rmt/rmt21b.htm - Unfortunately this is not "in laymen
terms" because, at some point, mathematics become central.

OK?

Cordially,
Mike L.

At 9/2/2007, you wrote:

I have a basic question on the Rasch model.
We always hear that Rasch measures are linear while raw scores aren't.
Linearity is defined as "equal increments to the difficulty or ability
measures when one more unit is added to a person's or an item's location".
What I can't grasp is why Rasch measures are linear? What's magic about the
Rasch measures. Once I heard  "Rasch measures are cnstructed to be linear".
But how?
I'd be grateful if someone explains in laymen terms that why Rasch measures
are linear.

Mike Linacre
Editor, Rasch Measurement Transactions
rmt at rasch.org  <http://www.rasch.org/rmt/> www.rasch.org/rmt/ Latest RMT:
21:1 Summer 2007

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