# Make Nonlinear Smooth Interpretable in Logistic GAM Regression

I have a nonlinear smooth fit in a logistic regression from the package mgcv in R. Visualizing the smooth, the y-axis I get using either plot(mymod) or predict.gam(mymod, type="terms") is in log-odds. I would like to change the y-axis to be something more interpretable.

If this was a linear regression and there was just one linear coefficient to interpret, I would calculate the average marginal effect for that coefficient. However, since the effect is nonlinear (it is a smooth spline), and I am trying to interpret the y-value at each given x-value, I do not think a marginal effect (change in x-value from 0 to 1) is exactly what I am looking for.

To put it more concretely, I have the following plot:

I could estimate the average marginal effect by calculating, for every observation, the average change in the probability of the outcome as the Predictor Variable changes from 0 to 1. But this tells me nothing about the effect on the probability of the outcome when Predictor Variable is equal to -2. How can I convert that change in log-odds when Predictor Variable is equal to -2 (0.1629) into a more interpretable value, like the change in the probability of the outcome?

This becomes more difficult when you have additional covariates in the model if you want to look at the effect of varying covariate $$x$$ on the response. In that situation you need to hold the other covariates at some representative value and then predict() (on the type = "link" scale if want to compute a confidence interval from the standard error, and then back transform to the response scale).
Log-odds isn't always so immediately convenient, but some pointers can make these plots easier to understand. 0 represents the overall mean, so if log-odds are positive for some values of the covariate $$x$$, the probability of the event is higher that average. If the log-odds are 0, or close to it, for a some values of $$x$$ the probability would be unchanged, and similarly negative log-odds would indicate probability is below the average for those covariate values. But if you really want a probability value, then you would need to predict() and hold any other covariates in the model at some representative value.