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# RE: st: probit vs. logit

 From "Feiveson, Alan H. (JSC-SK311)" To "statalist@hsphsun2.harvard.edu" Subject RE: st: probit vs. logit Date Tue, 25 May 2010 12:53:22 -0500

```Here's another way to look at it:

If you approximate a standard normal CDF, say  PHI(x), by a logistic one

F(x) = exp(th*x)/(1 + exp(th*x))

where th = 1.701 (best nonlinear LS fit)

the maximum difference occurs at about x = 2.04 (PHI(x) = 0.979; F(x) = 0.970).

Thus unless you had enough binary observations to distinguish a "success" rate of 97.9% from one of 97.0% near x = 2.04, either model would fit equally well. A limiting case would be if all your observations were at x = 2.04 (of course you wouldn't be doing regression in that case). In this situation if you had 2000 observations with 1958 "successes" , the nominal success rate would be 0.979 with 0.970 being just outside the lower 95% confidence limit.

So a rough guideline would be that you need at least 2000 observations  to distinguish a probit model from a logit model.

Al Feiveson

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Michael N. Mitchell
Sent: Tuesday, May 25, 2010 12:05 PM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: probit vs. logit

Dear Everyone responding on this thread...

These are all great ideas about "logit" and "probit"... I think this thread has
gravitated lots of thoughts and perspectives that I see discussed "on the street" but not
written about. It is like statistical water cooler conversation.

Thanks!

Michael N. Mitchell
Data Management Using Stata      - http://www.stata.com/bookstore/dmus.html
A Visual Guide to Stata Graphics - http://www.stata.com/bookstore/vgsg.html
Stata tidbit of the week         - http://www.MichaelNormanMitchell.com

On 2010-05-25 5.00 AM, Maarten buis wrote:
> --- On Tue, 25/5/10, Michael N. Mitchell wrote:
>> I agree with Martin, that<snip>
> http://www.stata.com/statalist/archive/2010-02/msg00840.html
>
>> If someone gets picky with you and really wants to see a
>> comparison of the model fit of the two models, I think you
>> could use -estimates store- and -estimates stats- (as shown
>> below) to compare the fit of the models using the AIC and/or
>> BIC (where a smaller value means better fit). As in the
>> example below, the two values are nearly identical, and I
>> think we all expect that this would generally be the case.
>
> Michael shows that in his example he finds a BIC difference of
> .02036. To give it a bit of perspective: Adrian Raftery (1995)
> propsed the following categorization of BIC differences:
>
> 0-2  : Weak evidence
> 2-6  : Positive evidence
> 6-10 : Strong evidence
>> 10 : Very strong evidence
>
> So the kind of difference that Michael found would to all
> intends and purposes mean that the logit and probit models are
> indistinguishable.
>
> Hope this helps,
> Maarten
>
> Adrian E. Raftery (1995) "Bayesian Model Selection in Social
> Research", Sociological Methodology, Vol. 25, pp. 111-163.
>
> --------------------------
> Maarten L. Buis
> Institut fuer Soziologie
> Universitaet Tuebingen
> Wilhelmstrasse 36
> 72074 Tuebingen
> Germany
>
> http://www.maartenbuis.nl
> --------------------------
>
>
>
>
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```

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