# [Rasch] Permissible Transformations

Andrew Kyngdon akyngdon at lexile.com
Thu Mar 19 09:57:17 EST 2009

```Anthony,

The correct changing units of measurement is the germ of truth in Stevens' (1946) theory of scale types. What Stevens meant by "permissible" is that for "interval scales", measurements of a quantity in a particular unit of measurement remain invariant under linear (affine) transformations. With interval scales, if you want to change from one unit of measurement to another then a linear transformation must be used.

The reason why concerns the structure of the attribute you are measuring. The use of an interval scale means the differences between the degrees of an attribute are known to be quantitative. Hence it is these differences which must remain invariant under the transformation of a unit. Positive similarities transformations (multiplication by a positive real number) do no maintain this invariance. For example, the difference between 40 degrees Celsius and 20 degrees Celsius is simply 20 degrees Celsius. Multiplying both measurements by 3, for example, results in a 60 degree difference, which violates measurement invariance.

However, if you have a "ratio scale", then measurements in a particular unit of measurement do remain invariant under positive similarties transformations. For example, if length is being measured in the inch unit, changing to millimetres requires multiplication of the magnitude in inches by 25.4. This transformation does not change the length magnitude, it merely changes the unit. This is not to say that there are no empirical consequences of changing units. As David Andrich has pointed out, going from a larger to a smaller unit increases precision at the expense of increasing measurement variation (i.e. instead of obtaining "one" measurement with the larger unit, you obtain a distribution of measurements with the smaller one).

Just to complicate matters, an interval scale is really just a ratio scale of differences between the degrees of a quantitative attribute. Hence if an attribute is genuinely quantitative, it can be measured using either an interval or ratio scale. If the best you can do is an interval scale, then your knowledge of the attribute is incomplete. Why we had interval scales of temperature measurement (e.g. Celsius, Fahrenheit, Reamur) before we had the Kelvin scale was that temperature was simply not as well understood.

Hope this helps,

Andrew

Dr Andrew Kyngdon
Director of Pacific Rim Operations
MetaMetrics, Inc.
P.O. Box 754 North Sydney NSW 2059 Australia
Phone: 0401768090 (Int. +61401768090)
Email: akyngdon at lexile.com
Web: www.lexile.com <http://www.lexile.com/>
Lexile Teacher's Lounge: http://lexile-teachers.blogspot.com/

________________________________

From: rasch-bounces at acer.edu.au on behalf of Anthony James
Sent: Thu 19-Mar-09 3:14 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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