How to adjust logit function's input domain?

Question

• on the right is the typical S-curve called expit function (or sigmoidal logistic function)
• on the left is the logit function based on log-odds, which is the inverse of the above

Now consider for the left graph, that the x-axis admits for input a statistical measure $P\in [-1, 1]$ instead of the $P\in [0,1]$ shown in the picture, so that a new origin of $x=0$ cuts through the middle of the curve in the left graph, instead of the current $x=0.5$.

How must the logit function be altered to be able to admit such an input domain for the x-axis, but still retain the same curve and same $\mathbb{R}$-ranged output for the y-axis currently shown?

Answer 1

You want to

• horizontally scale by a factor of $2$, then
• shift left by $1$.

(See this random Google result for more on graph transformations.) The first step takes the domain to $[0,2]$, the second, to $[-1,1]$.

So we replace $P$ with $\frac{1}{2}(P- -1) = \frac{P+1}{2}$ everywhere it appears. $$ \ln \left( \frac{\frac{P+1}{2}}{1-\frac{P+1}{2}} \right) $$ Of course, this simplifies to $$ \ln \left( \frac{1+P}{1-P} \right) \text{.} $$

Answer 2

Unless I'm missing something, $log(\frac{1+P}{1-P})$ is what you're looking for your new variable $P$