# Interpreting a Quadratic Term in Binary Logistic Regression

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?

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.

• Thank you so much for your explanation. That makes sense :) Jul 28, 2018 at 20:17
• Thanks for this explanation, I was wondering what would happen if the coefficient for the quadratic is also negative? In that case there is no turning point right? There is just an accelerating negative effect on the odds no? Also is it allowed to interpret the other way around? Decrease in x leads to an increase in odds if the coefficients are negative? I can't wrap my head around the shape of the effects. Aug 13, 2022 at 22:16
• Suppose the coefficient of the quadratic term was negative. The log-odds function is still a parabola. Since log is an increasing function, the argmax of the log-odds is the same as the argmax of the odds function. Thus there is still a turning point. Aug 15, 2022 at 2:48

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.