# [Rasch] Use of e in the Rasch model

John johnbarnard at bigpond.com
Tue Sep 6 10:43:15 EST 2011

```Quite right and may I add that I have tried to promote that we use ln
instead of log to specify that we use the natural base.

John

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Sent: Tuesday, 6 September 2011 8:48 AM
To: Thomas Salzberger; Kenji Yamazaki
Cc: Rasch at acer.edu.au
Subject: Re: [Rasch] Use of e in the Rasch model

The SLM can be written in many alternative but equivalent forms

Eg  Pr(success)=A/(A+B)

Where A is ability and B difficulty

Or

Pr(success)=ab/(1+ab)

Where a is ability and b easiness

Or

Pr(success)=exp(theta-delta)/(1+exp(theta-delta))

Or in the above replace the "exp" by another base.

The use of exp  (ie e) prevails because it is more convenient to use with a
range of statistical machinery in estimation and inference.

Ray

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Thomas Salzberger
Sent: Tuesday, 6 September 2011 8:02 AM
To: Kenji Yamazaki
Cc: rasch
Subject: Re: [Rasch] Use of e in the Rasch model

Kenji,

An interesting question, I have never really thought about.

Basically what you need is an s-shaped curve to describe/model the
relationship between the latent variable and the expected manifest response
(probability). Originally IRT used the cumulative normal distribution. For
convenience, the standard normal was used (it has to have some standard
deviation, so why not make it one). Later the logistic function replaced the
normal (logit rather than probit) as it is much easier to work with while
being virtually identical with the standard normal provided you include the
scaling constant D (roughly 1.7), as it is still done in many (all?) IRT
applications.

In Rasch modeling we do not use D, although we could. The point is that a
multiplicative constant is just a scaling factor.

But now to your question. It seems to me that a base other than e
simply means that the measures are expressed in a different unit.

If you choose a different base, let's call it f (with f > 1), then you get
the same probabilities that you get with e as the base but with a scaling
constant of a=ln(f).

Best wishes,
Thomas

2011/9/5 Kenji Yamazaki <yk0271 at yahoo.co.jp>

Hi all:

I have a question about the formula of the Rasch or IRT model.  Why is e
(=2.718) used in the formula?  Why is not it 10?  Because e is close to 3,
why isn't 3 used for the formula instead of e?  I have been having this
question for a long time, but all the Rasch and IRT books I have read treat
as given the use of e (=2.718) in the formula.  If anyone gives me an
answer, it will be very appreciated.

Kenji

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