The most common type of regression analysis is simple linear regression, which is used when a predictor variable and a response variable have a linear relationship.

However, sometimes the relationship between a predictor variable and a response variable is nonlinear.

In these cases it makes sense to use **polynomial regression**, which can account for the nonlinear relationship between the variables.

The following example shows how to perform polynomial regression in SAS.

**Example: Polynomial Regression in SAS**

Suppose we have the following dataset in SAS:

/*create dataset*/ data my_data; input x y; datalines; 2 18 4 14 4 16 5 17 6 18 7 23 7 25 8 28 9 32 12 29 ; run; /*view dataset*/ proc print data=my_data;

Now suppose we create a scatter plot to visualize the relationship between the variables x and y in the dataset:

/*create scatter plot of x vs. y*/ proc sgplot data=my_data; scatter x=x y=y; run;

From the plot we can see that the relationship between x and y appears to be cubic.

Thus, we can define two new predictor variables in our dataset (x^{2} and x^{3}) and then use **proc reg** to fit a polynomial regression model using these predictor variables:

/*create dataset with new predictor variables*/ data my_data; input x y; x2 = x**2; x3 = x**3; datalines; 2 18 4 14 4 16 5 17 6 18 7 23 7 25 8 28 9 32 12 29 ; run; /*fit polynomial regression model*/ proc reg data=my_data; model y = x x2 x3; run;

From the **Parameter Estimates** table we can find the coefficient estimates and write our fitted polynomial regression equation as:

y = 37.213 – 14.238x + 2.648x^{2} – 0.126x^{3}

This equation can be used to find the expected value for the response variable based on a given value for the predictor variable.

For example if x has a value of 4 then y is expected to have a value of 14.565:

y = 37.213 – 14.238(4) + 2.648(4)^{2} – 0.126(4)^{3} = **14.565**

We can also see the polynomial regression model has an adjusted R-squared value of **0.9636**, which is extremely close to one and tells us that the model does an excellent job of fitting the dataset.

**Related:** How to Interpret Adjusted R-Squared (With Examples)

**Additional Resources**

The following tutorials explain how to perform other common tasks in SAS:

How to Perform Simple Linear Regression in SAS

How to Perform Multiple Linear Regression in SAS

How to Perform Quantile Regression in SAS