Two-Lines: The First Valid Test of U-Shaped Relationships
Many psychological theories predict u-shaped relationships: x is good in low quantities, but bad in high quantities, or vice-versa. These predictions are almost exclusively tested via quadratic regressions; here I show such approach is essentially never valid. I introduce a new test: estimating a regression with two separate lines, one for ‘low’ and one for ‘high’ values of x. A u-shape is present if the two slopes have opposite sign and are individually significant. A procedure to set the breakpoint for ‘low’ vs ‘high’ is proposed and shown to be superior to all alternatives considered, including maximizing fit. The problems with the quadratic and advantages of the two-lines are demonstrated via simulations and examples from published psychological research.
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