# [Rasch] Response to question - Permissable transformations

Fraser Rew Fraser.Rew at nzqa.govt.nz
Thu Mar 19 09:46:25 EST 2009

```Hi Anthony

What is meant by a multiplicative transform is that each element is
multiplied by itself, or a value from another data set, and would be of
the form X'=aX^2+bX+c or Z=aXY +bX+cY, for example. The important factor
here is that the new data set will have a different distribution from
the original - if X is normally distributed, X^2 will not be. Thus
whatever assumptions you needed to make about the data will very often
not hold any more.

In contrast, a linear transformation of items of a data set will lead
to an easily calculable transformation of the set's properties - if X' =
a+bX, X' will have properties mean m'=a+b*m and standard deviation
s'=b*s - this is all easy enough to calculate, and works because X' is
normally distributed. Similarly if Z=aX+bY+c the mean and sd can be
easily calculated from the mean and sd of X and Y.

I hope that this is all clear enough for you. If not you may want to
have a look a graph of a normal distribution and consider the difference
that each of the above transformations would make to it. The linear
transformations keep the same shape (moved and stretched, but still the
same shape) but the non-linear (multiplicative) ones do not.

Cheers

Fraser

Fraser Rew
Researcher/Data Analyst
Qualifications Division
DDI: 04 463 4368
Extn: 4368

www.nzqa.govt.nz

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than "Re: Contents of Rasch digest..."

Today's Topics:

1. Permissible Transformations (Anthony James)
2. Permissible Transformations (Anthony James)
3. Re: Permissible Transformations (Swank, Paul R)
4. RE: Permissible Transformations (Donald Bacon)

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Message: 1
Date: Wed, 18 Mar 2009 09:14:22 -0700 (PDT)
From: Anthony James <luckyantonio2003 at yahoo.com>
Subject: [Rasch] Permissible Transformations
To: rasch at acer.edu.au
Message-ID: <810132.83332.qm at web111414.mail.gq1.yahoo.com>
Content-Type: text/plain; charset="utf-8"

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 isnΓÇÖt multiplication permissible? IsnΓÇÖt multiplication a linear
transformation where the additive component is zero? So it must be a
linear transformation? (or probably zero isnΓÇÖt allowed to be the
multiplicative component). When a scale in linearly transformed the
distances between the objects are increased ΓÇÿaΓÇÖ times the distances
in the first scales.
I donΓÇÖt understand how interval and ratio scales are different in
relation to permissible transformations. IΓÇÖd be thankful for
Cheers
Anthony
┬á
┬á
┬á

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Message: 2
Date: Wed, 18 Mar 2009 09:14:48 -0700 (PDT)
From: Anthony James <luckyantonio2003 at yahoo.com>
Subject: [Rasch] Permissible Transformations
To: rasch at acer.edu.au
Message-ID: <412591.57752.qm at web111416.mail.gq1.yahoo.com>
Content-Type: text/plain; charset="utf-8"

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 isnΓÇÖt multiplication permissible? IsnΓÇÖt multiplication a linear
transformation where the additive component is zero? So it must be a
linear transformation? (or probably zero isnΓÇÖt allowed to be the
multiplicative component). When a scale in linearly transformed the
distances between the objects are increased ΓÇÿaΓÇÖ times the distances
in the first scales.
I donΓÇÖt understand how interval and ratio scales are different in
relation to permissible transformations. IΓÇÖd be thankful for
Cheers
Anthony
┬á
┬á
┬á

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Message: 3
Date: Wed, 18 Mar 2009 16:33:55 -0500
From: "Swank, Paul R" <Paul.R.Swank at uth.tmc.edu>
Subject: Re: [Rasch] Permissible Transformations
To: "'luckyantonio2003 at yahoo.com'" <luckyantonio2003 at yahoo.com>,
"'rasch at acer.edu.au'" <rasch at acer.edu.au>
Message-ID:
<017FB41275AE7A46988755E60E32F40101E05A782A at UTHCMS3.uthouston.edu>
Content-Type: text/plain; charset="utf-8"

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.

Paul
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 isnΓÇÖt multiplication permissible? IsnΓÇÖt multiplication a linear
transformation where the additive component is zero? So it must be a
linear transformation? (or probably zero isnΓÇÖt allowed to be the
multiplicative component). When a scale in linearly transformed the
distances between the objects are increased ΓÇÿaΓÇÖ times the distances
in the first scales.
I donΓÇÖt understand how interval and ratio scales are different in
relation to permissible transformations. IΓÇÖd be thankful for
Cheers
Anthony

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Message: 4
Date: Wed, 18 Mar 2009 16:01:07 -0600
From: Donald Bacon <dbacon at du.edu>
Subject: RE: [Rasch] Permissible Transformations
To: "Swank, Paul R" <Paul.R.Swank at uth.tmc.edu>,
"'luckyantonio2003 at yahoo.com'"	<luckyantonio2003 at yahoo.com>,
"'rasch at acer.edu.au'" <rasch at acer.edu.au>
Message-ID: <0125E40625D4414ABB79C0DCA0A8F36D0C721B at EXCH.du.edu>
Content-Type: text/plain; charset="Windows-1252"

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
ar
e 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 û

Don Bacon
Professor of Marketing
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.

Paul
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 isnÆt multiplication permissible? IsnÆt multiplication a linear
transformation where the additive component is zero? So it must be a
linear transformation? (or probably zero isnÆt allowed to be the
multiplicative component). When a scale in linearly transformed the
distances between the objects are increased æaÆ times the distances in
the first scales.
I donÆt understand how interval and ratio scales are different in
relation to permissible transformations. IÆd be thankful for comments.
Cheers
Anthony

------------------------------

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End of Rasch Digest, Vol 44, Issue 5
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