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Re: st: streg, frailty


From   rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp.)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: streg, frailty
Date   Thu, 27 Jun 2002 17:23:27 -0500

Richard Boylan <rboylan@cba.ua.edu> asked:

>> I first computed

>> streg y x, dist(weibull) robust cluster(id)
>> mfx compute, predict(mean)

>> Then to check for the imporstance of unobserved effects, I estimated

>> streg y x, dist(weibull) frailty(gamma) shared(id) 
>> mfx compute, predict(mean)

>> To my surprise all the results changed drastically.

>> What I find harder to understand is that the overal predicted times change
>> drastically:

>> with the first model:

>> mfx compute, predict(mean)

>> Marginal effects after weibull
>>       y  = predicted mean _t (predict, mean)
>>          =  3.7482296
>> with the second model:

>>  mfx compute, predict(mean)

>> Marginal effects after weibullhet
>>       y  = predicted mean _t (predict, mean)
>>          =  .50887109
>> Shouldn't I be expecting these numbers to have been similar?

Stephen Jenkins <stephenj@essex.ac.uk> responded:

> If you told us what the estimate of the Weibull shape parameter ("p") was in
> both specifications, we'd be better placed to comment.  If the frailty model
> is the 'true' one, and the hazard is rising at a fast rate, then you could
> get very different predicted means, I suspect.

I agree with Stephen.  Basically, there are two levels on which you would 
expect different answers (and in some examples, like yours, the difference
can be dramatic).

  (1) If the frailty effect is significant (i.e. theta significantly greater
  than zero), then you have two different models yielding different 
  predictions.

  (2) Note that with your second model, you are fitting a Weibull/gamma 
  shared frailty model.  When you use the -shared()- option, the prediction 
  produced by the -mfx- option -predict(mean)- is the estimated mean
  time-to-failure for a subject with predictors at their mean levels, AND 
  conditional on a frailty value of one (the mean frailty).  
  
  Thus, when comparing this to a predicted mean time that is unconditional on
  the frailty you are really comparing apples and oranges here (or, more 
  accurately, tangerines and oranges), and when you fit a standard
  (non-frailty) model you are in effect producing predictions unconditional on
  the frailty.

  On the one hand, you have an estimate of the average prediction with 
  respect to the frailty distribution (the gamma).  On the other, you have
  a prediction conditional on the mean of the frailty distribution.  In 
  general, these are not one in the same, and since the gamma distribution 
  can be quite skewed (for large theta) the difference can be substantial.

Note that the use of -robust cluster()- in the first model only effects the 
estimated standard errors of your estimates, and thus is not crucial to the
comparison of predicted values between models.

--Bobby
rgutierrez@stata.com
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