Interpreting a Quadratic Term in Binary Logistic Regression

Question

Apologies in advance for my limited stats knowledge. I hope someone can help. I am trying to understand how to interpret the coefficients of both the linear and quadratic term in a binary logistic regression model.

When I fit the model, I get the following coefficients as:

x: 0.0265

x^2: -0.000462

Both coefficients are significant. I have other terms in my model, but I won't include them here. Taking the exponential of each coefficient, I get:

x: 1.0269

x^2: 0.9995

Now I understand if I had only the coefficient for x in my model, I would interpret this as the odds of a positive result in response variable y increasing by 2.69% for every 1 unit increase in x. But I'm not sure how to interpret the coefficient for the squared term. Is this saying that the increase in odds decreases by 0.05% for every 1 unit increase in x? i.e. the increase in odds is 2.69%, then 2.64%, then 2.59%, and so on, each time x increases by 1.

That is, the odds of a positive result in y are increasing as x increases but the rate of this growth is slowing down and eventually the odds will start to decrease? Or have I got this totally wrong?

Thanks in advance.

Answer 1

So your model is something like

$$ \log(\frac{p}{1-p}) = -0.000462 x^2 + 0.0265 x + c $$

implying that

$$ \frac{p}{1-p} = e^{-0.000462 x^2}e^{0.0265 x} e^c $$

You can interpret $ e^c $ as the "baseline odds".

Then, note that the positive and negative effects are zero when $ x = \frac{0.0265}{0.000462} \approx 57.36 $. This is the point where odds begin to decrease; thus, you can say for values of $ x \in (-\infty, 57.36) $ the odds are increasing as $ x $ increases, but afterwards odds decrease as $ x $ decreases.

You could go further and see where the maximum odds are attained, and maybe if you take the derivative of this function you can see the rate of odds increase/decrease at each point, but it gets pretty contrived and case-specific.

Answer 2

Just a small edit to Kevin's answer: I think there's a small typo as the derivative of the expression $\frac{p}{1-p}$ written above reaches a stationary point at $x=\frac{0.0265}{2 * 0.000462}$. So $57.36$ should be divided by 2.