Two-Lines: A Valid Alternative to the Invalid Testing of U-Shaped
Relationships with Quadratic Regressions
Abstract
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,
imposing an arbitrary functional form assumption that can lead to a
100% false-positive rate, e.g., concluding with certainty that 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 a functional form
assumption. To set the breakpoint between the lines, I introduce the
Robin Hood algorithm. It delivers higher power to detect U-shapes than
all other breakpoint setting alternatives considered. The paper includes
simulations and re-analyses of published results. The two-line test can
be performed at http://webstimate.org/twolines