When plotting predictions over linear predictors, what's a "informative" range for linear predictor values?

When plotting predictions over linear predictors, what's a "informative" range for linear predictor values?

The reason to ask is this:

This is a very common plot in logistic regression, but isn't the range of -4 to 4 a bit uninformative? Like what does e.g. lin. predictor == -4 mean?

So is this the only way to plot this?

Is the x-axis inferable in some "informative" way?

• It depends on what one intends by "informative" which is not clear. Are you asking about something someone else has said? If so, some context might reveal more about the intended sense of the word. Oct 11, 2016 at 22:16
• @Glen_b No, by "informative" I mean something that is more understandable than "lin. predictor == -4". I.e. something that makes the x-axis have informative meaning. Oct 12, 2016 at 5:12

1. When presenting logistic regression generically, as in an introductory class setting, I might show a plot ranging from $-4$ to $4$ because that's where the 'action' is. There isn't much to see beyond those limits. When the linear predictor is $<-4$, the probability is $\approx 0$, and when it's $>4$, it's $\approx 1$. In truth, it will not exactly equal $0$ or $1$, and will always be getting closer to those values, but it will be hard for students to see.
2. It also might help to think about what the 'linear predictor' is. Modeling the probability of a binary outcome is tricky, because $X$-variables are not necessarily bounded, but probabilities must lie within $[0, 1]$. The way this is dealt with in logistic regression is that we don't quite model the probability of 'success' (i.e., $Pr(Y=1)$) in a straightforward sense, but instead model the log odds of success (i.e., $\ln(^{\text{odds}(Y=1)}/_{1+\text{odds}(Y=1)})$). With that in mind, the linear predictor is the result of multiplying your coefficients by the $X$-values and summing them. It is the log odds of success.