Getting ready

Polynomial models should be applied where the relationship between response and explanatory variables is curvilinear. Sometimes, polynomial models can also be used to model a non-linear relationship in a small range of explanatory variable. A polynomial quadratic (squared) or cubic (cubed) term converts a linear regression model into a polynomial curve. However, since it is the explanatory variable that is squared or cubed and not the beta coefficient, it is still considered as a linear model. This makes it a simple and easy way to model curves, without needing to create big non-linear models. Let's consider the following diagram:

We can see that there is a natural curve to the pattern of data points. This linear model is unable to capture this. Let's see what a polynomial model would look like:

The dotted line represents the linear regression model, and the solid line represents the polynomial regression model. The curviness of this model is controlled by the degree of the polynomial. As the curviness of the model increases, it gets more accurate. However, curviness adds complexity to the model as well, making it slower. This is a trade-off: you have to decide how accurate you want your model to be given the computational constraints.