# Entering a quadratic term in a logistic regression model

I’d like to know how to plot my quadratic relationship. I ran a logistic regression model that included a quadratic term for age on support for helping at-risk juveniles (1=want to help, 0=does not). I found that age was significant (negative direction) and age*age was also significant (positive direction).

The problem I am having is figuring out how to plot this to see at what ages people change in their likelihood of support (i.e., at what age the curve occurs). The SPSS "Observed Groups and Predicted Probabilities" is not much help. Since this is logistic regression, I am not sure using the method to plot curvilinear effects in OLS would be appropriate.

The mean age is 46.7 and the range is 21 to 92 years old. The initial significant effect of age suggests a negative relationship. As age increases, there are reduced odds of support for helping at-risk youth. However, at some point (what I am trying to determine) as the significant age*age variable indicates, the odds of support for helping at-risk youth begins to increase.

If age and its square are the only predictors, then you should have a coefficient of log-odds on age, call it $$b_1$$, and one on age squared, call it $$b_2$$. Hence those are just two terms like those of any quadratic
$$b_1 x + b_2 x^2$$
so that a maximum with $$x$$ will be observed whenever its derivative $$b_1 + 2 b_2 x$$ is $$0$$, so long as $$b_2 <0$$. So the age with maximum log-odds is $$-b_1 / 2b_2$$. The position of this age is not affected by going from log-odds to predicted probabiity, which is a monotonic transformation.