What is Polynomial Regression?
Polynomial regression is a type of linear regression analysis that models the relationship between a continuous outcome and one or more predictors using a polynomial function. It allows the model to fit curved relationships, rather than assuming a straight-line fit.
Despite the curvature, statisticians consider this type of model to be a linear model because it is linear in the parameters. That means the model estimates linear coefficients (betas) for each term, such as x, x², or x³, even though the predictor variable is raised to a power.
In a typical linear regression, the effect of the predictor is constant across its range. For example, increasing x by 1 unit always increases the outcome by the same amount. In polynomial regression, that’s no longer true. The effect of x changes depending on its value, producing a curved relationship.
Learn more about the Difference Between Linear and Nonlinear Regression Models.
Common Forms of Polynomial Regression
The highest exponent in a polynomial regression model determines the number of bends the fitted line can have. The number of bends, or turning points, in the curve is one less than the highest power of the predictor.
Use my free online Regression Calculator to fit the following model types!
Quadratic Regression
Quadratic regression includes the squared term (x²). The curve has a single bend—either a U-shape or an inverted U.
Cubic Regression
Cubic regression includes the cubic term (x³). This allows for two bends, such as an S-shaped curve.
Higher-Degree Polynomials
Quartic terms (x4) and higher are possible, but they can lead to overfitting and unreliable estimates and predictions.
Polynomial Regression Example
Suppose you’re modeling crop yield as a function of fertilizer use. A linear model may not capture the diminishing returns that occur at higher levels of fertilizer. In this case, a polynomial regression model is used to reflect that relationship.
The quadratic fitted equation is the following:
The graph below shows the fitted line with the observed data points.
This polynomial regression model shows that yield initially increases with fertilizer but eventually levels off and begins to decline. This captures the real-world effect where excessive fertilizer reduces crop efficiency. The curve provides a better fit to the data than a straight line.
In this context, this type of model better matches the real-world curvature while maintaining the familiar framework of linear modeling.
« Back to Glossary Index