# [Rasch] Units of measurement in the social sciences

Stephen Humphry stephen.humphry at uwa.edu.au
Thu Oct 18 12:01:44 EST 2007

```Anthony, good question. Mark, you make some good oversvations about what I
think are key considerations.

In the physical sciences, units are specific amounts of specific kinds of
quantites. For example, 1 cm is a unit of length, 1 kg a unit of mass, 1
newton a unit of force and so on.

In the physical sciences, the size of the unit of an instrument depends upon
the empirical construction of the instrument and the conditions under which
it is used. Clearly, the empirical construction is also designed so that a
particular kind of quantity is measured.

Similarly, when the Rasch model is applied, the kind of quantity measured
depends upon the way the instrument is designed. The size of the unit should
also depend upon features of the empirical construction of the instrument
and the conditions under which it is used. In more general terms, using a
term emphasized by Rasch, the kind of quantity and the size of the unit
should both depend upon the frame of reference for measurement.

Returning to a physical example, if a particular distance or length is D = 5
cm, then 5 is a measurement of a specific kind of quanity (distance) in a
particular unit (cm). As a measurement, the value 5 has an implied unit of
measurement of distance; i.e. a cm. The unit depends upon the empirical
construction of the instrument. To obtain measurments in that unit also
depend upon the way the instrument is used and the conditions under which it
is used.

A logit value is a number that has a probabilistic interpretation with
respect to specific well-defined events (e.g. success on an item). I do not
think it is useful to read 2 logits as 2 units, such as 2 cm or 2 kg. Logit
values may relate to different kinds of quantities. Logit is a generic term
for values that relate to many different kinds of quantities. In addition,
for a given kind, the size of the unit depends on relevant characteristics
of the frame of reference.

For a more complete and precise explanation of this analysis of the unit in
the Rasch model, see
http://wwwlib.murdoch.edu.au/adt/browse/view/adt-MU20050830.95143  A shorter
and more accessible version should be appearing in a JAM article next year
though.

Cheers,

Steve.

Dr Stephen Humphry
Graduate School of Education
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 Mark Moulton
Sent: Wednesday, 17 October 2007 3:07 AM
To: 'Anthony James'
Cc: Rasch at acer.edu.au
Subject: RE: [Rasch] Units of measurement in the social sciences

Anthony,

The logit difference between a person and an item (Person Ability - Item
Difficulty = logn(p/(1-p)) ) is a linearized probability of success of a
person on an item.  By convention we say that any probability greater than
0.50 (logit difference > 0.00) means that the person will probably be
successful with the item.  It is important to note that there is no absolute
Rasch ability or item difficulty, no absolute reference.  There is simply
ability relative to a given item, and difficulty relative to a given person,
or relative to some arbitrary reference point such as the item mean.

If the item were, say, lifting a kilo weight off the ground, then persons
would be measured in terms of their (linearized) probability of lifting that
kilo weight.  A probability greater than 0.50 (logit difference from item >
0.00) would mean the person is considered to be, in general, strong enough
to lift the kilo weight.

If we introduce a 2 kilo item,  then the probability of success will go
down, and the 2 kilo item will be more difficult (have a higher logit value)
than the 1 kilo item.  Does this mean that the 2 kilo item will have twice
as many logits as the 1 kilo item, or some similar relationship?  No.  There
is no theoretical a priori relationship between the logit spacing and the
kilo spacing.  This is due in part to the fact that the logit spacing
between items, their relative probabilities, is to some degree driven by
forces (measurement error) unique to the experimental situation.

However, if you were actually to graph a number of kilo weight logit values
against their weights in kilos, they would form an approximately straight
line.  As long as you know the formula for that straight line relationship,
you could then convert logits to kilos.  This straight line relationship
would hold so long as you always "anchored" subsequent analyses to the logit
values of some of the persons or items from the original analysis.

Mark Moulton

-----Original Message-----
From: Anthony James [mailto:luckyantonio2003 at yahoo.com]
Sent: Tuesday, October 16, 2007 11:27 AM
To: Rasch at acer.edu.au
Subject: [Rasch] Units of measurement in the social sciences

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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