[R] nls() fit to Kahnemann/ Tversky function

[Gabor Grothendieck]

Note that a simple logistic with a saturation level of 1 seems
to do quite well.  Below we have removed the last point in order
to avoid the singularity:

x <- p.kum[-10]
y <- felt.prob.kum[-10]
plot(log(y/(1-y)) ~ x)
abline(lm(log(y/(1-y)) ~ x), col = "red")

On 10/31/05, Mark Hempelmann <neo27 at t-online.de> wrote:
> Dear WizaRds,
>
>     I would like to fit a curve to ten points with nls() for one
> unknown parameter gamma in the Kahnemann/ Tversky function, but somehow
> it won't work and I am unable to locate my mistake.
>
> p.kum <- seq(0.1,1, by=0.1)
> felt.prob.kum <- c(0.16, 0.23, 0.36, 0.49, 0.61, 0.71, 0.85, 0.89, 0.95,
> 1) ## how to find a function that fits these points nicely?
> plot(p.kum, felt.prob.kum) ## looks a little like an "S"
>
> gamma <- rep(0.5, 10)
> nls.dataframe <- data.frame(p.kum,felt.prob.kum, gamma)
>
> nls.kurve <- nls( formula = felt.prob.kum ~
> p.kum^gamma/(p.kum^gamma+(1-p.kum)^gamma)^(1/gamma), data=nls.dataframe,
> start=c(gamma=gamma), algorithm="plinear" )
>
> summary(nls.kurve)
>
> gives: Error in La.chol2inv(x, size) : 'size' cannot exceed nrow(x) = 10
>
>     If I go with the Gauss-Newton algorithm I get an singular gradient
> matrix error, so I tried the Golub-Pereyra algorithm for partially
> linear least-squares.
>
>     It also seems the nls model tries to find ten different gammas, but
> I want only one single gamma parameter for the function. I appreciate
> your help and support. Thank you.
>
> sol lucet omnibus
> Mark Hempelmann
>
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