[Rasch] Permissible Transformations

Donald Bacon dbacon at du.edu
Thu Mar 19 09:01:07 EST 2009

   As a practical matter, you might use interval measures as independent variables in a regression model, where the effects are additive.  It should be clear that by adding a constant to one variable, or by multiplying one variable by a constant, the correlations among variables would not change.
   If you model anything like an interaction term (the product of two variables, not just a variable multiplied by a constant), ratio measures may be necessary (especially if the variables are not just 0-1 variables).  For example, the PV=nRT equation from physics (where P = pressure, V = volume, and T = temperature) only works if the measures are ratio.  This problem has been studied in multi-attribute attitude models.  When two interval variables are multiplied together, some sort of adjustment may be necessary for the model to make sense.  A good basic paper on this topic is

Holbrook, Morris B. (1977).  Comparing Multiattribute Attitude Models by Optimal Scaling. The Journal of Consumer Research, 4(3), 165-171.

Hope this helps –

Don Bacon
Professor of Marketing
Daniels College of Business
University of Denver
From: rasch-bounces at acer.edu.au [rasch-bounces at acer.edu.au] On Behalf Of Swank, Paul R [Paul.R.Swank at uth.tmc.edu]
Sent: Wednesday, March 18, 2009 3:33 PM
To: 'luckyantonio2003 at yahoo.com'; 'rasch at acer.edu.au'
Subject: Re: [Rasch] Permissible Transformations

You can add a costant to a interval scale but not to a ratio scale because it changes the absolute zero. You can multiply either by a constant.

Dr. Paul R. Swank
Sent using BlackBerry

From: rasch-bounces at acer.edu.au
To: rasch at acer.edu.au
Sent: Wed Mar 18 11:14:22 2009
Subject: [Rasch] Permissible Transformations
Dear folks,
It is said that linear transformations in the from of X' = a + bX are permissible for interval scales. What does this mean? If we multiply a set of numbers by a constant and add them with another constant we will get a new different set. What properties does this new set have and how is it related to the first set that makes it a linear transformation?
Why isn’t multiplication permissible? Isn’t multiplication a linear transformation where the additive component is zero? So it must be a linear transformation? (or probably zero isn’t allowed to be the multiplicative component). When a scale in linearly transformed the distances between the objects are increased ‘a’ times the distances in the first scales.
I don’t understand how interval and ratio scales are different in relation to permissible transformations. I’d be thankful for comments.

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