Summary Sheet
How do we test for non-linear association?
Background
Sometimes data is strongly correlated, but there is a non-linear pattern. Take the following example:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 16 | 81 | 256 | 625 | 1296 |
There is a perfect relationship between the data: \(y=x^4\). Yet the correlation, calculated from a linear fit, doesn’t quite reflect that: \(\rho=\) 0.896.
What we’re in need of is an improvement of Pearson’s correlation. We’d still like it to describe what happens to \(Y\) when \(X\) increases, but relax the assumption that the relationship needs to be linear.