A polynomial term–a quadratic (squared) or cubic (cubed) term turns a linear regression model into a curve. But because it is X that is squared or cubed, not the Beta coefficient, it still qualifies as a linear model. This makes it a nice, straightforward way to model curves without having to model complicated non-linear models.
But how do you know if you need one–when a linear model isn’t the best model?
Well, first, a quadratic term creates a curve with one “hump”– a U or inverted U shape. The curve does not need to contain both sides of the U. In can contain just part of it.
A cubic has two humps–one facing upward and the other down. The curve goes down, back up, then back down again (or vice-versa).
There are three main situations that indicate a linear relationship may not be a good model.
1. Most important is the theoretical one. There are some relationships that a researcher will hypothesize is curvilinear. Clearly, if this is the case, include a polynomial term.
2. The second chance is during visual inspection of your variables. This is one of those reasons for always doing univariate and bivariate inspections of your data before you begin your regression analyses. (You always do this, right?) A simple scatter plot can reveal a curvilinear relationship.
3. Inspection of residuals. If you try to fit a linear model to curved data, a scatter plot of residuals (Y axis) on the predictor (X axis) will have patches of many positive residuals in the middle, but patches of negative residuals at either end (or vice versa). This is a good sign that a linear model is not appropriate, and a polynomial may do better.