Two Lines: A Valid Alternative to the Invalid Testing of U-Shaped Relationships With Quadratic Regressions

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-shaped functions are tested almost exclusively via quadratic regressions, an approach that imposes an arbitrary functional-form assumption that in some scenarios can lead to a 100% rate of false positives (e.g., the incorrect conclusion that y = log(x) is U shaped). Estimating two regression lines, one for low and one for high values of x, allows testing for a sign change without a functional-form assumption. I introduce the Robin Hood algorithm as a way to set the break point between the lines. This algorithm delivers higher power to detect U shapes than all the other break-point-setting alternatives I compared with it. The article includes simulations demonstrating the performance of the two-lines test and reanalyses of published results using this test. An app for running the two-lines test is available at http://webstimate.org/twolines.