**Polynomial regression** is a technique we can use to fit a regression model when the relationship between the predictor variable(s) and the response variable is nonlinear.

A polynomial regression model takes the following form:

Y = β_{0} + β_{1}X + β_{2}X^{2} + … + β_{h}X^{h} + ε

In practice, there are three easy ways to determine if you should use polynomial regression compared to a simpler model like linear regression.

**1. Create a Scatterplot of the Predictor Variable and Response Variable**

The easiest way to determine if you should use polynomial regression is to create a simple scatterplot of the predictor variable and the response variable.

For example, suppose we’d like to use the predictor variable “hours studied” to predict the score that a student will receive on a final exam.

Before fitting a regression model, we can first create a scatterplot of hours studied vs. exam score. Suppose our scatterplot looks like the following:

The relationship between hours studied and exam score looks **linear**, so it would make sense to fit a simple linear regression model to this dataset.

However, suppose the scatterplot actually looked like the following:

This relationship looks a bit more **nonlinear**, so this tells us that it may be a good idea to fit a polynomial regression model instead.

**2. Create a Fitted Values vs. Residual Plot**

Another way to determine if you should use polynomial regression is to fit a linear regression model to the dataset and then created a **fitted values vs. residuals plot** for the model.

If there is a clear nonlinear pattern in the residuals, then this is an indication that polynomial regression could offer a better fit to the data.

For example, suppose we fit a linear regression model using hours studied as a predictor variable and exam score as a response variable, then create the following fitted values vs. residuals plot:

The residuals are randomly scattered around zero with no clear pattern, which indicates that a linear model provides an appropriate fit to the data.

However, suppose our fitted values vs. residuals plot actually looked like the following:

From the plot we can see that there is a clear nonlinear pattern in the residuals – the residuals exhibit a “U” shape.

This tells us that a linear model is not appropriate for this particular data and it could be a good idea to instead fit a polynomial regression model.

**3. Calculate the Adjusted R-Squared Value of the Model**

Another way to determine if you should use polynomial regression is to fit both a linear regression model and a polynomial regression model and calculate the adjusted R-squared values for both models.

The adjusted R-squared represents the proportion of the variance in the response variable that can be explained by the predictor variables in the model, *adjusted* for the number of predictor variables in the model.

The model with the higher adjusted R-squared represents the model that is better able to use the predictor variable(s) to explain the variation in the response variable.

**Additional Resources**

The following tutorials explain how to perform polynomial regression using different statistical software:

An Introduction to Polynomial Regression

How to Perform Polynomial Regression in R

How to Perform Polynomial Regression in Python

How to Perform Polynomial Regression in Excel