# Is it statistically valid to perform two linear regressions to approximate a test for a U-shaped distribution?

Consider a sample dataset with a putative U-shaped distribution. I want to test whether the U-shaped relationship is statistically significant. Specifically, if y signficantly decreases as a function of x for $$[0, \frac{n}{2}]$$ and then if y significantly increases as a function of x for $$[\frac{n}{2}, n]$$.

I think it would be simplest to simply perform two linear regressions on each half of the dataset. Is this method valid? Do I need to perform a correction on the resulting p-value if I am performing two tests (though they are on different halves of the dataset)?

• If you know the relationship is a straight line in both intervals (a big assumption), that is a valid way to test whether the slope is negative in one region and positive in the other. The test statistics are independent. Since both tests have to be significant, no adjustment for multiple testing is needed (intersection-union test). If you only found the negative slope in one side, you can't claim there is a U-shape relationship. If you know it is continuous at n/2, the usual two separate regression lines will not force it to be continuous, so a modification might be made to enforce that. – John L Apr 6 at 13:02
• Great thank you! – user317740 Apr 6 at 14:11