# Non-linear logistic regression, one predictor with non-linear effect?

I'm a medical student and for a research project, I'm trying to predict the success of a medical procedure. An independent variable of interest is the amount of prior experience the doctor has performing the procedure. This effect is probably non-linear. In other words, you learn more the first time you try something then the 100th time. I would like to quantify this effect, but regular logistic models assume a linear effect of a continuous variable (as far as my textbooks explain).

Is there a method that allows for non-linear logistic regression?

• Yes, you can enter the amount of experience (I assume this is the number of times performed the procedure) and amount of experience squared. Alternatively, given that this is a count variable, presumably with no zeros, you could enter the natural log as the predictor. This actually has a nice interpretation (going from 1-2 and 10-20 have the same effect on the natural log scale -- a doubling of experience). – dbwilson Apr 3 '18 at 21:08
• You could also model the effect of experience with a spline, see stats.stackexchange.com/questions/122212/…, stats.stackexchange.com/questions/129739/…, stats.stackexchange.com/questions/206073/… for some examples. – kjetil b halvorsen Apr 3 '18 at 21:30
• @dbwilson And if the effect is neither quadratic, nor logarithmic, but still nonlinear? Nonparametric and semi-parametric regression models (e.g., generalized additive models) are precisely for instances where some of the independent variables have an effect on the dependent variable with an unknown functional form. – Alexis Apr 3 '18 at 23:35
• I think your understanding is incorrect. In a logistic model, $$\Pr(Y=1 \vert E) = \frac{\exp \{a + b\cdot E \}}{1+\exp \{a + b\cdot E \}},$$ so the effect of experience on the probability of success is nonlinear. This will not handle all kinds nonlinearities, but may be sufficient to pick up the diminishing marginal effect of additional experience that you have in mind. The beginning of this post may help clarify things. – Dimitriy V. Masterov Apr 4 '18 at 0:12