[Rasch] Odds ratio as a scale

Svend Kreiner S.Kreiner at biostat.ku.dk
Wed Mar 9 21:04:48 EST 2011

```Anthony,

The problem is that Ability = p/(1-p) is /not/ where we start. The
starting point is p = (a/d)/(1+(a/d)) where a = ability and d =
difficulty. To compare two persons under this model we calculate the
ratio a1/a2.

Now, change the unit of a and d, e.g. u = 2a and v = 2d so that a = u/2
and d = v/2 and rewrite the probability, p =
[(u/2)/(v/2)]/[1+(u/2)/(v/2)] = (u/v)/(1+(u/v)). This is also a Rasch
model for which reason we calculate u1/u2 to compare the ability of the
persons.

But (u1/u2) = (2*a1)/2*a2) = a1/a2. After the change of the unit of the
scale on which we measure ability, the comparison of the two persons
turn out to be exactly the same as before we changed the unit. It is for
this reason that we could argue that ability is measured on a ratio
scale under Rasch's original parameterization of his model.

Svend

Den 09-03-2011 10:11, Anthony James skrev:
> Thank you Svend and others,
>
> The odds of success is defined as:
> Odds= P(X=1)/P(X=0); (Ability ) B=p/1-p
> Odds=proportion correct/proportion incorrect, such as
> Person A:  Odds=.8/.2=4 ( person A has got 8 items right and 2 wrong, out of 10 items)
> Person B: Odds=.9/.1=9
> Person C: Odds=.6/.4=1.5
> etc.
>
> Apparently these odds are ratio measures. This means that person B is more than twice as better as Person A and six times better than person C.
>
> My understanding is that everything start from Ability=p/1-p.
> Then we take the log and then exponentiate etc.
>
> Therefore the idea of p/1-p or (proportion correct/proportion incorrect) is an axiom.
>   Mike's paper in RMT also implies this: http://www.rasch.org/rmt/rmt53d.htm
>
> My question is about this axiom. Why do we accept it as a ratio scale measure?
>
> Cheers
> Anthony
>
>
> --- On Tue, 3/8/11, Mike Linacre (RMT)<rmt at rasch.org>  wrote:
>
>> From: Mike Linacre (RMT)<rmt at rasch.org>
>> Subject: Re: [Rasch] Odds ratio as a scale
>> To: rasch at acer.edu.au
>> Date: Tuesday, March 8, 2011, 4:09 PM
>>
>> Anthony:
>>
>> You may find this helpful:
>> http://www.rasch.org/rmt/rmt94b.htm
>> - it describes some of the differences between Stevens'
>> "ratio scale" and Georg Rasch's "scale of
>> ratios".
>>
>>
>> Mike L.
>>
>>
>> Mike Linacre
>>
>> Editor, Rasch Measurement Transactions
>>
>> rmt at rasch.org www.rasch.org/rmt/ Latest
>> RMT:  24:3 Winter 2010
>>
>
>
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--
Svend Kreiner
Professor
Department of Biostatistics
University of Copenhagen

P.O. Box 2099
DK-1014 Copenhagen K, Denmark

Email: S.Kreiner at biostat.ku.dk
Phone: (+45) 35 32 75 97

Fax: (+45) 35 32 79 07

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