[R] Generalized Logistic and Richards Curve

Ben Bolker bbolker at gmail.com
Thu Jul 7 15:49:04 CEST 2011

Vincy Pyne <vincy_pyne <at> yahoo.ca> writes:

>  Dear R helpers, I am not a statistician and right now struggling
> with Richards curve. Wikipedia says
> (http://en.wikipedia.org/wiki/Generalised_logistic_function) The
> "generalized logistic curve or function", also known as Richard's
> curve is a widely-used and flexible sigmoid function for growth
> modelling, extending the well-known logistic curve.  Now I am
> confused and will like to know if the Generalized Logistic
> distribution as described in lmomco package is same as what
> wikipedia is describing. In other words, is Generalized Logistic
> Function same as Generalized logistic distribution?  I do understand
> there is separate R package "richards' for dealing with Richards
> curve.  Kindly guide Vincy [[alternative HTML version deleted]]

I think not quite.  In general it's unlikely that something described as a
"function" will necessarily be the same as something described as a
"distribution", since the latter (or at least its density function) has to
integrate to 1 and the former
doesn't ...

Looking at 'cdfglo' in the lmomco manual gives

F(x)=1/(1+exp(-y))
y = -k^{-1} log(1-k(x-xi)/alpha) (for k not equal 0)

whereas wikipedia gives

Y(t) = A + { K-A \over (1 + Q e^{-B(t - M)}) ^ {1 / \nu} }

In order to carefully check whether these are the same (e.g.
whether the density function (rather than the CDF which is given
in the lmomco manual) is the same as the Richards curve) you would
have to match up terms.
One tip-off that they can't be identical is that 'cdfglo' has
3 parameters (location parameter xi, scale parameter alpha, shape
parameter k) while the Richards has 5 (location M, scale B,
scale ? Q, shape nu, lower value A, upper value K).

I think they would be *nearly* equivalent for A=0, K=1 in the
Richards (then M=xi,B=k/alpha, nu=alpha) but not quite.  A little
more algebra is required.

If you tell us more about what you're trying to do you might