Two Lines Test - Testing for Quadratic Regressions
From Simonsohn (2017): “Many psychological theories predict u-shaped relationships: the effect of x is positive for low values of x, but negative for high values, or vice-versa. Despite implying merely a change of sign, hypotheses about u-shapes are tested almost exclusively via quadratic regressions. The unnecessary and arbitrary quadratic assumption can lead to a 100% false-positive rate, e.g., concluding y=log(x) is u-shaped. Estimating two regression lines, one for low and one high values of x, allows testing for a sign change without functional form assumptions.”
Amazing app that will test the shape of the data for you (efficiently, quickly, and for those who are not R savvy): http://webstimate.org/twolines/
Paper calling for the two lines test using the Robin Hood algorithm: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3021690