[Rasch] Use of e in the Rasch model
Anthony James
luckyantonio2003 at yahoo.com
Wed Sep 7 19:35:32 EST 2011
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>
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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