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RE: st: A rose by any other name?


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: A rose by any other name?
Date   Fri, 14 Nov 2008 12:37:54 -0000

The references given in my reference give many further details.
(Seemingly, no one actually read it.) I read somewhere that the logistic
is just a special case of a Riccati equation, which would make its
implicit discovery an 18th century matter. I never followed that up; I
do note that several quite different things have been called Riccati
equations. 

But implicit discovery aside, it's my understanding that Verhulst easily
deserves as much credit as anyone. 

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

Lachenbruch, Peter

Wikipedia gives Verhulst as the originator of the logistic curve, but I
recall that Adolphe Quetelet had also done some work in this area.  Both
were contemporaries (early 1800s) - Verhulst lived from about 1804 to
1849 and Quetelet from about 1809 to 1874 - I'm not sure about these
dates even though I just looked them up on Wikipedia (laziness  to not
go back).  Interestingly, there is no mention of the logistic curve in
the Quetelet article.  

Can anyone expand on my impression?

BTW, Quetelet is credited with developing the Body Mass Index, BMI

jverkuilen

I believe you are right about the logistic curve predating the
distribution From what I recall it was first derived (using a firstorder
nonlinear differential equation) by Verhulst to model population as an
elaboration of Malthus' model, which is verbal but corresponds to the
first order linear differential equation for exponential growth. Been a
while since I read any of that stuff so my memory may be faulty.
Verhulst's equation is a popular example for a nonlinear ODE that can be
solved analytically. There are so few... 

Nick Cox

As a matter of history, I believe that logistic as a growth curve came
long before the logistic as a CDF, but as Jay implies, between friends
it's the same equation. 

There are some historical references on this within 

SJ-8-1  gr0032  . . . . . . .  Stata tip 59: Plotting on any transformed
scale
        . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  N.
J. Cox
        Q1/08   SJ 8(1):142--145                                 (no
commands)
        tip on how to graph data on a transformed scale

Verkuilen, Jay

>>To be more precise, the proposed model is a gamma density kernel, not
a
bonafide gamma density ,which integrates on 1.  Of course in this
context, the function is used to model nonlinear trend, not a
probability distribution of some random variable.>>

Right, and thus it's not dissimilar from using the logistic CDF as a
model for growth between asymptotes, which is often done using, say,
Gaussian errors around the curve itself. 

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