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Re: st: A +ve log-likelihood!!


From   rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: A +ve log-likelihood!!
Date   Thu, 27 May 2004 09:57:42 -0500

Amani <siyama@who.int> asks:

> I am running an "streg" model with a Lognormal distribution and the
> log-likelihood of the fitted model turns positive.  I am not sure what
> happened here.

Two reasons this could happen (neither is an error):

1.  In general -ml- with continuous responses, recall that the likelihood is a
probability DENSITY, not a probability.  As such, when the scale is small 
enough, the probability density can be greater than one, hence the likelihood
is greater than one, hence the log-likelihood is positive.

2.  In -streg, lognormal()- a constant term (does not depend on the estimated
parameters) is added to the log-likelihood so as to make it directly
comparable to other log-likelihoods such as the Weibull and Exponential, which
themselves make the above constant-term adjustment so as to be invariant to
scale.

That is, we adjust Weibull and Exponential so that if you multiply the
survival times by a constant, you get the same log-likelihood.  However, in
doing so, you have to make the same adjustment to log-normal and the other
-streg- models so that you can compare log-likelihoods across all, calculate
things like AIC and BIC, etc.

In any case, if Amani wants the actual based-on-probability-density
log-likelihood, he can do the following after running -streg, dist(lnormal)-

   . gen temp = sum(_st*_d*ln(_t))

   . scalar real_ll = e(ll) - temp[_N]

   . di scalar(real_ll)

--Bobby
rgutierrez@stata.com
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