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Assumptions. In normal logistic regressions probability has a fixed relationship with the independent variable. It either increases or stays the same as the independent variable increases OR it decreases or stays the same as the independent variable increases. It can't do both. It can't increase as the independent variable increases, and then later decrease (and vice versa).

Suppose I have a data set with a single independent variable and a binary dependent variable where an increase in the independent variable did not always come with an in increase in probability.

Real world example. Let's say youre an expert bowler and we were testing your strike probability after a certain number of warm up shots. You come in on day 1, throw 1 warmup shot and then try to roll a strike. The next day you come in and do 2 warmup shots, adding one warm up shot each day. You'd expect that as the number of warm up shots increases the probability of hitting a strike on your official attempt increases. Clearly though, if the warm up shots continue to increase, at some point your arm get worn out and the strike probability plummets. On day 1000 for example you'd be lucky to granny roll the ball down the lane for your strike attempt

Question. Assuming the independent variable was number of warm up shots and the binary dependent variable is whether or not our bowler rolled a strike after that number of warm ups, could we model strike probability by performing a polynomial regression on the dataset? Is there another more common regression approach for this situation?

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  • $\begingroup$ You can use polynomial terms in your logistic regression. $\endgroup$
    – Dave
    Feb 5, 2021 at 15:26

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Polynomial regression is one possible approach. It is not recommended in OLS because of instabilities in the tails and possible numerical convergence problems - if you simply take powers of the predictors, these will be quite collinear. I would assume the same issues hold in logistical regression.

A better approach would be to transform your predictor using , particularly restricted cubic splines or natural splines (which are linear in the tails, so you don't get the erratic behavior of polynomials). Frank Harrell's Regression Modeling Strategies has a very readable introduction to splines in the OLS situation, but it should also be helpful for logistical regression.

Needless to say, splines can indeed model non-monotonicity. Just be careful not to overfit your data. If the fit starts wobbling up and down, you have probably used too many splines/knots.

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