[Rasch] Use of e in the Rasch model

Stephen Humphry stephen.humphry at uwa.edu.au
Mon Sep 12 23:03:46 EST 2011

```I doubt you'll find much sense on this issue. Rasch argued for the "model" explicitly in terms of physical analogy. In physical analogy, we can substitute actual/real magnitudes for a and d. For example, say they are lengths and a/d = 3 m / 1 m. It's then common to "define" a = exp(a*). There can be no analogy with physical measurement here, because for example exp(3 m) is nonsense.

This doesn't mean either model is untenable, it just means they cannot be the same and also be understand in a way that is consistent with physical measurement. In P = exp(b*-d*)/(1+exp(b*-d*)), b*-d* must be a "pure number" as typically stated in metrology (e.g. in SI and VIM documents). To remain consistent with physics, this implies that b*-d* is actually the ratio of a difference to a unit. For example, the ratio (5 m - 3 m) / 1 m is said to be the "pure number" 2. The term exp(2) makes sense and, therefore, the term exp[(5 m - 3 m) / 1 m] does make sense in a way that is consistent with physical measurement. It doesn't matter whether it is length, time, mass, or some other quantity involved.

One option is to acknowledge this. Another option is to argue for an entirely different conception of measurement. (Well, a third is to ignore the question of what measurement means entirely, of course). It is fascinating to me to see that so many people fail to acknowledge the above, yet also fail to argue for a different conception of measurement. I'm yet to see someone explain to me how exp(1 metre) can mean anything, albeit one person at least tried!

Steve

________________________________
From: rasch-bounces at acer.edu.au [rasch-bounces at acer.edu.au] On Behalf Of Anthony James [luckyantonio2003 at yahoo.com]
Sent: Wednesday, 7 September 2011 5:35 PM
To: Rasch at acer.edu.au
Subject: Re: [Rasch] Use of e in the Rasch model

Some experts told me that Georg Rasch expressed his model as
p = (a/d)/(1+(a/d)) where a = ability and d = difficulty.

Why do we need 'e'' in the first place? Can't we continue with this expression and estimate the parameters without exponentiation and then taking the log to get rid of it again?

Anthony

--- On Tue, 9/6/11, Mark Moulton <markhmoulton at gmail.com> wrote:

From: Mark Moulton <markhmoulton at gmail.com>
Subject: Re: [Rasch] [BULK] Use of e in the Rasch model
To: "Kenji Yamazaki" <yk0271 at yahoo.co.jp>
Cc: Rasch at acer.edu.au
Date: Tuesday, September 6, 2011, 1:36 PM

Kenji,

I remember Mike Linacre showing me how using natural logs simplifies the Rasch standard error formula.  I asked him how Rasch standard errors were all reported in logits when the formula used to calculate them is based on probabilities:

SE[i][in logits???] = 1 / sum[across row i](p[i] * (1 - p[i]))

His answer was something along the lines that this is a property of taking the derivative of the natural log.  Any other base would require using additional constants of some sort.

Unfortunately I'm foggy on the algebra this morning -- it wasn't hard.  Maybe someone remembers how it goes.  But the upshot is that any log will work for the Rasch model -- the only visible difference is that the logit scale will tend to be more or less dispersed.  However, the natural log does simplify some of the algebra under the hood.  Plus, it is used by convention throughout the statistical and scientific world, as was noted.

Mark Moulton

2011/9/5 Kenji Yamazaki <yk0271 at yahoo.co.jp<https://webmail.staff.uwa.edu.au/owa/UrlBlockedError.aspx>>

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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