# Graphical Interpretation of Logistic Regression

I am not very good in wrapping my head around statistical models, so apologies if this question is very obvious (I couldn't find an answer by googling, though). So here's the question:

I am modelling the absence/ presence of plant species along an elevational gradient. Presence is decreasing with elevation, i.e. there's an abundance decline with increasing elevation. Here's the plot of one specie as an example:

I now have hunderts of species like this & I know how to interpret the coefficients of the model output (intercept: log odds of a specie being present at an elevation of 0m, elevation variable: expected change in log odds with a one-unit-increase in elevation). But the only thing I'm actually interested in is the slope of the abundance decline to use for further analysis (i.e. the slope of the blue line in the graph). So: What is the graphical interpretation of logistic regression parameters? Can I simply use the estimate of the variable elevation as measure of abundance decline? Or do I have to exponentiate the variable?

Thanks for any help!

Assume the following simple logistic regression model

$$\mathrm{logit}(y)=\alpha + \beta\cdot x$$

The predicted probabilities are given by $$\hat{P}(Y=1)=\frac{1}{1 + \exp(-(\hat{\alpha} + \hat{\beta}\cdot x))}$$

The turning point - and the steepest slope - of the logistic curve (your red curve) is attained at $$x=-\frac{\hat{\alpha}}{\hat{\beta}}$$ where the slope is $$\hat{\beta}/4$$. Thus, the slope of the blue line in your graph is $$\hat{\beta}/4$$. Have a look at this post discussing this "divide by 4 rule" and the sources therein.

• Wow, this is amazing! Even possible to understand with my very limited math-/ stats - knowledge. But I would never have worked that out on my own - thank you very much! Commented Mar 19, 2020 at 14:02

Let's go into the derivation of slope $$= \hat \beta /4$$. In the curve there are following things to note

• Max or min (depending on the shape of the curve) of slope happens at inflection point (where slope changes in shape)
• We know second order derivate at inflection point is 0. We will use this to find x at inflection point

Derivative of the logit function gives us $$\frac{\beta\mathrm{e}^{\alpha + \beta x}}{\left(1 + \mathrm{e}^{\alpha + \beta x}\right)^{2}}$$.

Second Derivative of the logit function gives us $$\frac{\beta\mathrm{e}^{\alpha + \beta x} * (1-\mathrm{e}^{2\alpha + 2\beta x})}{\left(1 + \mathrm{e}^{\alpha + \beta x}\right)^{4}}$$. Solving this we get x = $$-\frac \alpha \beta$$

If you substitute x = $$-\frac \alpha \beta$$ in first derivative equation then you get $$\beta / 4$$.