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Re: st: RE: fitting a gompertz curve, not in the context of survival analysis


From   Dan Waldo <dan_waldo@yahoo.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: RE: fitting a gompertz curve, not in the context of survival analysis
Date   Fri, 29 May 2009 07:52:07 -0700 (PDT)

Thanks to Maarten, Stephen, and Nick for very helpful comments. In particular, the -nl- suggestion was spot on ... I must have been experiencing one of those moments when you bypass the simple and go directly to the complicated.

Nick asked about the tumor-growth reference. I confess up front that I got this from Wikipedia: Laird AK. "Dynamics of tumor growth." British Journal of Cancer 18:490-502, 1964. The model is nice in that it is expressed in terms of the change in tumor size (in my case, revenue as a proportion of GDP).

Nick also expressed puzzlement over my intent -- a feeling that I often share. The hypothesis behind the work is that government revenue as a proportion of GDP is limited over time, either by political sentiment or by characteritstics of the evolving economy. Using the Gompertz curve is a simplification of a number of the things, and it could very well be that it so oversimplifies things as to become unworkable.
 
Both Stephen and Nick asked about the natural time zero in this model, a point very well taken. I guess that one could establish as time zero the year in which the government were first created, if one were estimating the Gompertz function itself, although the analogy to tumors is a bit strong for my taste. But to take the time derivative of Laird's formulation (as shown in that Wikipedia article), I think that the time scale itself becomes irrelevant (not being a mathematician, I have asked a colleague to confirm this is so):

dX(t) = a*log(K/X(t))*X(t)


Unfortunately, this latter representation produces results that are inconsistent with the data, which could suggest that the model is not appropriate, or that my data are messy enough that the model cannot distinguish the "true" pattern. Estimating the function itself requires specifying a bae year, as Stephen pointed out. This estimation turns out to be exquisitely sensitive to the choice of base year and produces funky results, too -- leading me to think that the hypothesis is not borne out by the data.

Dan



      
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