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RE: st: Fitting the logistic curve to a time-series


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: Fitting the logistic curve to a time-series
Date   Fri, 21 Jul 2006 12:01:03 +0100

I should spell out an equivocation here. 
-logit()- is mathematically possible because
the response lies within [0,1]. That doesn't 
mean that applying logit models makes any 
assumptions behind those apply if they don't 
otherwise do so. The one graph I used does
suggest that the model is not crazy, but 
only Serguei can comment on the substantive
or scientific fit.  

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

Nick Cox
 
> An alternative is to pass your -q- through -logit()- and 
> feed the thus-transformed response to -regress-. 
> 
> . gen logit_q = logit(q) 
> . twoway connected logit_q year 
> . regress logit_q year
> . predict logit_q_hat 
> . gen q_hat = invlogit(logit_q_hat) 
> 
> The error assumptions are naturally different, but 
> if the model is any good, that should not matter 
> too much. 
> 
> This doesn't fix the asymptote, but the by-product of 
> being able to check for linearity on the logit scale 
> could be useful.
> 
> Yet another alternative, even simpler, is 
> 
> -glm, link(logit) family(gaussian)-
> 
> even though a glance at the help for -glm- might lead
> you to suppose that this combination was not allowed. 
> 
> I read in your data, did
> 
> . glm q year, link(logit) family(gaussian) 
> 
> and followed with 
> 
> . regplot, bands(200) 
> 
> where -regplot- comes from -modeldiag- (-search 
> modeldiag- for download information). 
> 
> The error assumptions are different again, but 
> I make the same assertion. 
> 
> The graph shows that your data all seem to fall
> on the accelerating part of the curve: the asymptote
> can not be glimpsed, let alone touched. 
> 
> Those interested in the history of ideas might know
> that Berkson played with logistic curves before he 
> came up with what we now know as logit models. He 
> was not the first to do that either. 
> 
> Nick 
> n.j.cox@durham.ac.uk 
> 
> Maarten buis
>  
> > --- Serguei Kaniovski asked:
> > > How can I fit the logistic curve to a time-series "q" so that 
> > > I can control the asymptotes, i.e. I what the fitted curve to 
> > > level out at the value of q=0.04?
> > 
> > Serguei:
> > You can do that with the -nl- command which estimates a nonlinear
> > least squares model. See the example below. You don't have to fix
> > the minimum and maximum, you can also estimate them. Even if you
> > want to fix them this may be useful as a model to compare with
> > for e.g. a likelihood ratio test, BIC and AIC, or just graphically,
> > as is also shown in the example below.
> > 
> > The parameter in my example called {min} is the lower asymptote, 
> > {max} is the higher asymptote, {b1} is the slope, and {b2} is the
> > year where the curve is halfway between {min} and {max}.
> > 
> > HTH,
> > Maarten
> > 
> > 
> > *---------------begin example----------------
> > set more off
> > capture drop _all
> > input year	q
> > 1981	.01102246
> > 1982	.01127048
> > 1983	.01152055
> > 1984	.0118841
> > 1985	.01211535
> > 1986	.01247724
> > 1987	.0129496
> > 1988	.01321121
> > 1989	.01319597
> > 1990	.01362598
> > 1991	.014358
> > 1992	.01417304
> > 1993	.01437101
> > 1994	.01509818
> > 1995	.01539195
> > 1996	.01586584
> > 1997	.01686939
> > 1998	.01767208
> > 1999	.01880662
> > 2000	.01914837
> > 2001	.02034989
> > 2002	.02122596
> > 2003	.02201964
> > 2004	.02243478
> > 2005	.02346839
> > 2006	.02428688
> > end
> > 
> > twoway line q year 
> > 
> > nl ( q = .04/(1+exp(-{b1}*(year-{b2=2000}))) )
> > est store const
> > predict qconst
> > label var qconst "min and max constrained"
> > 
> > nl ( q = {min=0} + {max=.04}/(1+exp(-{b1}*(year-{b2=2000}))) )
> > nl log4: q year
> > est store minmax
> > predict qminmax
> > label var qminmax "min and max not constrained"
> > 
> > twoway line q qconst qminmax year
> > /* The graph shows that the unconstrained
> >    model fits better */
> > 
> > lrtest const minmax
> > /* hypothesis that min = 0 and
> >    max = .4 is rejected */
> > 
> > est stats const minmax
> > /* AIC and BIC also show that the
> >    unconstrained model fits better */
> > *----------------end example ----------------

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