[Rasch] Units of measurement in the social sciences

Mark Moulton MarkM at eddata.com
Wed Oct 17 05:07:00 EST 2007

```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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