# [Rasch] Units of measurement in the social sciences

Andrew Kyngdon akyngdon at lexile.com
Tue Oct 23 07:06:29 EST 2007

```My two cents...

If an attribute possesses additive structure, then it is theoretically
possible it can be measured with an infinite number of types of units.
The prime example of this is length which can be measured using cubits,
millimetres, Planck lengths, feet, inches, kilometres or light years.
This is because measurement involves not just a unit and a magnitude,
but a conception of additivity (Michell, 1993) or symmetry (Luce, 2001)
of the attribute. An infinite number of compositional relations exist
for quantitative attributes, all of which satisfy the formal
requirements for additive structure (Michell, 1993).

Not all units are useful for all measurement purposes (e.g. Planck
lengths or lightyears when you are measuring your waistline). The unit
of measurement in the Lexile reading scale is defined as 1/1000th of the
difference in comprehensibility between basal primer texts and Grolier's
(1986) Encyclopaedia (Stenner, 1986). As an interval scale, it has to be
unique up to linear transformations (i.e. y = m(x) + b). We add a
constant to theoretical logit measures of text difficulty, multiply this
sum (x) by another constant (m), then add another constant (b) to this
product to obtain the Lexile scale (for person reading ability in
Lexiles the same equation is used except empirical Rasch person logit
measures are substituted for the theoretical text logits). The unit is
arbitrary as it is in thermometry, where the unit of the Celsius scale,
for example, is defined as 1/100th of the difference in temperature
between the boiling and freezing points of water at sea level. In both
situations other equally valid scales are possible with different units.
The selection of one unit of measurement does not bestow ontological
privilege on it over the other possible units (Michell, 1993). There is
really nothing special about units in and of themselves.

Cheers,

Andrew

Refs:

Luce, R.D. (2001) Conditions equivalent to unit representations of
ordered relational structures. Journal of Mathematical Psychology, 45,
81-98.

Michell, J. (1993). Numbers, ratios and structural relations.
Australasian Journal of Philosophy, 71, 325-332.

Stenner, A.J. (1996) Measuring reading comprehension with the Lexile
framework. Paper presented to the California Comparability Symposium
(available at www.lexile.com <http://www.lexile.com/> ).

________________________________

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On
Behalf Of Anthony James
Sent: Sunday, October 21, 2007 2:25 PM
To: rasch at acer.edu.au
Subject: Re: [Rasch] Units of measurement in the social sciences

regarding my question I found something which seems relevent:

http://rasch.org/rmt/rmt32b.htm

Anthony James <luckyantonio2003 at yahoo.com> wrote:

Dear folks,

Another dumb question...

I'll be grateful for any comments.

When we are talking about "units of measurement" in the physical
sciences we are talking about some tangible things. (I avoide the word
"concrete" because I know you don't like it in this context and argue
that all measures are abstractions). However, I mean kilo or meter , for
example, are understandable attributes that have a concerete existence.
A "sample meter" ,i.e., a rod of 1 meter can be aligned with any object
and count the number of alignments. Or we can put some potatos on the
pan of a scale and put enough weaights on the other pan until the beam
is balanced.

1. How does the Rasch model make this "sample meter" or the
one-kilo weight to compare the performance of the students against?

2. Whose performance  is considered as the unit and how is it
constructed?

3. What's the defenition of a logit?

Cheers

Anthony

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