I was recently talking to a researcher who had conducted a cognitive experiment that involved experimentally manipulating a variable
x, a continuous property of a stimulus, looking at the effect on a variable
p, the probability of giving a response.
The function p(x) was assumed to be a logistic function.
The researcher wanted to know how to calculate the point on
x at which the
fitted logistic regression function equalled 0.5.
Overview
Benjamin Bolker discusses function fitting in
Chapter 3 of his book
Ecological Models and Data in R.
The logistic function is
p(x) = exp(a + b * x) / (1 + exp(a + b * x)).
Inverse of the logistic function
The logistic function is monotonically increasing.
Thus, it has an inverse function.
The following algebra shows how to find the inverse of the logistic function.
p = exp(a + b * x) / (1 + exp(a + b * x)) [the initial function]
p + p * exp(a + b * x) = exp(a + b * x) [multiply both sides by
(1 + exp(a + b * x))]
p + (p - 1) * exp(a + b * x) = 0 [subtract from both sides
(exp(a + b * x))]
(p - 1) * exp(a + b * x) = - p [subtract from both sides
(p)]
exp(a + b * x) = (- p ) / (p - 1) [divide both sides by
(p - 1)]
a + b * x = ln(-p / (p - 1)) [take the natural
log of both sides]
b * x = ln(-p / (p - 1)) - a [subtract from both sides
(a)]
x = (ln(-p / (p - 1)) - a) / b [divide both sides by (b)]
When
p = .5, the function can be simplified to
x = (ln(-.5/(.5 - 1)) - a) / b
x = (ln(1) - a) / b
x = (0 - a) / b
x = -a / b
The Procedure
Thus, to estimate the the value of
x where
p = .5, the steps are:
- Estimate parameters
a and b based on the data.
- Calculate the value of
(-a / b).
For estimating parameters based on a nonlinear regression separately for a set of
participants, see
my earlier post on the topic.
