[Rasch] Permissible Transformations

Stephen Humphry stephen.humphry at uwa.edu.au
Thu Mar 19 16:13:32 EST 2009

Personally, I'd much rather stick to the way this has always been
understood, either implicitly or explicitly, in physics.
Let a measurement {Q} = Q:[Q] "where symbols in braces { } represent pure
numbers, and symbols in square brackets are units" (Emerson, 2008, p. 134;
Allisy, 1980) and Q is an actual quantity of some kind (e.g. an amount of
mass, force, electric potential difference, whatever it might be).
Now we can re-express the measurement (pure number) as {Q'} = {XQ} = Q:[Q']
where [Q'] = [Q]/X . That means the measurement {Q'} is simply expressed in
a unit that is 1/X of the size of the unit [Q].
For example, if [Q] is 1 cm and X is 10 then [Q]/X = cm/10 = mm. So
multiplying a measurement taken in cm by the number 10 means you have
expressed the measurement in mm. Say the quantity Q is just 1 cm so that the
measurement of the quanity is {1} = cm:[cm]. Multiplying by 10, we have
{1x10} = cm:[cm]/10 = cm:mm = 10.
It does not mean you actually measured the quantity in the unit 1 mm,
because you don't have that precision when you measure to the nearest cm:
when it comes to precision, 'changing' units is not arbitrary.
As for adding a constant, it means you have changed the origin, which is to
say you have chosen a scale on which there is a different quantity Q whose
measurement {Q''} = {XQ} + {A} becomes the pure number {0}. That quantity is
the "origin". Of course, the origin is usually chosen to some extent
arbitrarily, although often with a purpose or meaningful reference point in
Take for exmaple, X{Q} + {A} = Q:[Q'] + A:[Q'] where [Q'] is again the new
unit and A is another quantity. 
If [Q] is 1 Kelvin, X = 1, and A = -273.15 degrees Kelvin, then {XQ} + {A}
is simply a "transformation" of Kelvin to Celsius. After changing the scale
in this way, the temperature 273.15 Kelvin becomes 0 degrees Celsius; i.e.
the measure of the temperature 273.15 K in is the pure number {0} in
Celsius. A unit and origin are both implied in this pure number, though: it
is a short-hand.
X doesn't have to be 1 of course, it could be any number -- it's not 1 if
you wish to change Celsius to Fahrenheit, or vice versa.
To your question in terms of the definition of measurement that has always
been used in physics: measurements on so-called "ratio scales" are more
fundamental than measurements on so-called "interval scales". If {R} = A:B
then {R} - 1 = A:B - B:B which is the ratio of the difference A - B to the
quantity B. That is, {R} - 1 is a measurement of the difference between the
quantities in the unit [B]. Multiplying this measurement by a constant, we
can express the difference in another unit also.
On the other hand, our knowing {R} = (A - B):[Q] doesn't mean we know the
ratio A:B. We therefore don't know A:B multiplied by any number X, either.
We can go from a "ratio scale" to an "interval scale" but not the other way.

Stephen Humphry
Associate Professor, Graduate School of Education
The 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 Anthony James
Sent: Thursday, 19 March 2009 1:15 AM
To: rasch at acer.edu.au
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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