[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
mind.
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.
Regards,
Steve
Stephen Humphry
Associate Professor, Graduate School of Education
The University of Western Australia
35 Stirling Highway
CRAWLEY WA 6009
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.
Cheers
Anthony
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