Four parameters logistic regression derivative

Question

Good Evening,

I fitted a four parameter logistic curve using R nls function with the following equation:

y = alpha + lambda/(1+exp(-beta(x-mu))

I would like to determine the maximum slope of this curve and for this I would like to compute the derivative of the function. Do you know how I can find the derivative of this function and use it to determine the maximum slope or the maximum derivative value?

Thank you in advance,

Rohan

I find the regular sigmoid equation y = 1/1+e-x and its derivative but not with the parameters. I am expecting some help with the derivative of my equation and a piece of script that can help me to find the maximum value.

Answer 1

Let us say that the parameters you have calculated are alpha = 1, lambda = 2, beta = 3 and mu = 4. Then create the derivative function fder and use optimize to find its maximum. Evidently the maximum slope occurs at mu and equals 1.5 or substituting x = mu into the derivative fder we have that the derivative at the maximum equals lambda * beta / 4.

fder <- function(x, alpha = 1, lambda = 2, beta = 3, mu = 4) {}
body(fder) <- D(expression(alpha + lambda/(1+exp(-beta*(x-mu)))), "x")
optimize(fder, c(-10, 10), maximum = TRUE)
## $maximum
## [1] 3.99999
##
## $objective
## [1] 1.5

Answer 2

Starting from @G.Grothendieck's answer, here's a logical explanation of why the maximum derivative is lambda*beta/4.

• The maximum derivative of the unscaled logistic function is 1/4, at x=0
• The maximum derivative of 1/(1+exp(-beta*x)) is beta/4 at x=0 (you can look this up on Wikipedia
• adjusting the midpoint (e.g. 1/(1+exp(-beta*(x-mu)))) shifts the location of the maximum derivative to x=mu but doesn't change its value
• shifting the curve up by adding alpha (alpha + 1/(1+exp(-beta*(x-mu)))) doesn't change the max slope or its location
• scaling the curve by lambda (alpha + lambda/(1+exp(-beta*(x-mu)))) scales the max derivative by lambda (beta/4 → lambda*beta/4)