A common problem I encounter in the literature is authors over-interpreting the slopes of a model with quadratic terms (e.g. Y = age + age^2) at the lowest and highest ages. In variably the plot (not the confidence intervals) will seem to indicate declines (for example) at the oldest ages (example: random example off internet [1]), when really the apparent negative slope is due to the limitations of quadratic models not being able to model an asymptote.

The approach I've used, when I do not have a theoretically driven choice to work with) is using fractionated polynomials [2], e.g. x^s where s = {−2, −1, −0.5, 0, 0.5, 1, 2, 3}, and then picking a strategy to pick the best fitting polynomial while avoiding overfitting.

Its not a bad technique; I've tried others like piecewise polynomial regression, knots, etc [3],but I could not figure out how to test (for example) for a group interaction between two knotted splines). Also additive models.

[1] https://www.researchgate.net/figure/Scatter-plot-of-the-quad...) [2] https://journal.r-project.org/articles/RN-2005-017/RN-2005-0... [3] https://bookdown.org/ssjackson300/Machine-Learning-Lecture-N...