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


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: RE: fitting a gompertz curve, not in the context of survival analysis
Date   Sun, 31 May 2009 12:48:40 +0100

Thomas Love Peacock made some of his characters discuss this matter in
his novel "Crotchet Castle" (Hookham, London, 1831). 

The entire text is easily accessible at e.g.
http://www.thomaslovepeacock.net/Crotchet.html

Peacock unerringly identified several key issues in your research,
including the question of time origin. If you want to object that he
supplied neither data nor precise equations that doesn't distinguish him
from many of the economists who have discussed this matter since. On the
other hand, the allusion to modelling is naturally prescient. 

In this novel of ideas, Mr Mac Quedy is, according to various scholars,
based mostly on the Scottish economist J.R. McCulloch. Quedy is also
identified as a play on QED, quod erat demonstrandum, i.e. on the a
priori or deductive style of economising. 

What follows is an exact quotation from that URL, except that I have
corrected "desert" to "dessert". (The Project Gutenberg copy is correct
on that point.) Dr Folliott's reference to "your nation" underlines that
he is English and Mr Mac Quedy is Scottish. As an Englishman with some
Welsh antecedents married to a Scot, I want to underline that that
difference is important. 

CHAPTER VI
Theories

                      But when they came to shape the model, 
                      Not one could fit the other's noddle.

BUTLER                        

MEANWHILE the last course, and the dessert, passed by. When the ladies
had withdrawn, young Crotchet addressed the company. 
  
MR CROTCHET, JUN.
There is one point in which philosophers of all classes seem to be
agreed; that they only want money to regenerate the world. 


MR MAC QUEDY
No doubt of it. Nothing is so easy as to lay down the outlines of
perfect society. There wants nothing but money to set it going. I will
explain myself clearly and fully by reading a paper. (Producing a large
scroll.) 'In the infancy of society---' 
    


THE REV DR FOLLIOTT
Pray, Mr Mac Quedy, how is it that all gentlemen of your nation begin
every thing they write with the 'infancy of society'? 
    


MR MAC QUEDY
Eh, sir, it is the simplest way to begin at the beginning. 'In the
infancy of society, when government was invented to save a percentage;
say two and a half per cent.---' 
    


THE REV DR FOLLIOTT
I will not say any such thing. 
    


MR MAC QUEDY
Well, say any percentage you please. 
    


THE REV DR FOLLIOTT
I will not say any percentage at all. 
    


MR MAC QUEDY
'On the principle of the division of labour---' 
    


THE REV DR FOLLIOTT
Government was invented to spend a percentage. 
    


MR MAC QUEDY
To save a percentage. 
    


THE REV DR FOLLIOTT
No, sir, to spend a percentage; and a good deal more than two and a half
per cent. Two hundred and fifty per cent.; that is intelligible. 
    
<quotation ends> 

Nick 
n.j.cox@durham.ac.uk 


-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Dan Waldo
Sent: 29 May 2009 15:52
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: RE: fitting a gompertz curve, not in the context of
survival analysis

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.


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