Algorithm

Statistics

Applied mathematics

Set (abstract data type)

Computer science

Sign (mathematics)

Regression

Mathematics

Quadratic equation

Mathematical analysis

False positive paradox

Geometry

Combinatorics

Point (geometry)

Programming language

Advances in Methods and Practices in Psychological Science

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 .

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