Here is a data set:
df <- data.frame('y' = c(81,67,54,49,41,25), 'x' =c(-50,-30,-10,10,30,50))
So far, I know how to fit a sigmoidal curve and display it on screen:
plot(df$y ~ df$x)
fit <- nls(y ~ SSlogis(x, Asym, xmid, scal), data = df)
summary(fit)
lines(seq(-100, 100, length.out = 100),predict(fit, newdata = data.frame(x = seq(-100,100, length.out = 100))))
I now want to find a point on the sigmoidal curve when y = 50. How do I do this?
The function that SSlogis is fitting is given in the help for the function as:
Asym/(1+exp((xmid-input)/scal))
For simplicity of notation, let's change input
to x
and we'll set this function equal to y
(which is fit
in your code):
y = Asym/(1+exp((xmid - x)/scal))
We need to invert this function to get x
alone on the LHS so that we can calculate x
from y
. The algebra to do that is at the end of this answer.
First, let's plot your original fit:
plot(df$y ~ df$x, xlim=c(-100,100), ylim=c(0,120))
fit <- nls(y ~ SSlogis(x, Asym, xmid, scal), data = df)
lines(seq(-100, 100, length.out = 100),predict(fit, newdata = data.frame(x = seq(-100,100, length.out = 100))))
Now, we'll create a function to calculate the x value from the y value. Once again, see below for the algebra to generate this function.
# y is vector of y-values for which we want the x-values
# p is the vector of 3 parameters (coefficients) from the model fit
x.from.y = function(y, p) {
-(log(p[1]/y - 1) * p[3] - p[2])
}
# Run the function
y.vec = c(25,50,75)
setNames(x.from.y(y.vec, coef(fit)), y.vec)
25 50 75 61.115060 2.903734 -41.628799
# Add points to the plot to show we've calculated them correctly
points(x.from.y(y.vec, coef(fit)), y.vec, col="red", pch=16, cex=2)
Work through algebra to get x
alone on the left side. Note that in the code below p[1]=Asym, p[2]=xmid, and p[3]=scal (the three parameters calculated by SSlogis
).
# Function fit by SSlogis
y = p[1] / (1 + exp((p[2] - x)/p[3]))
1 + exp((p[2] - x)/p[3]) = p[1]/y
exp((p[2] - x)/p[3]) = p[1]/y - 1
log(exp((p[2] - x)/p[3])) = log(p[1]/y - 1)
(p[2] - x)/p[3] = log(p[1]/y - 1)
x = -(log(p[1]/y - 1) * p[3] - p[2])
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