Quadratic regression is a type of polynomial regression that models the relationship between a continuous outcome and a single predictor using a second-degree polynomial. It captures curved relationships that cannot be modeled by a simple straight line.
Quadratic regression is still considered a linear model because it is linear in the parameters. That means the model is linear in how the coefficients (β terms) enter the equation, even if the predictors themselves are squared or transformed. The key point is that the model solves for the β values using linear algebra techniques, no matter what mathematical operations are applied to the X terms.
In a linear model, the effect of the predictor is constant across its range. In a quadratic model, the effect changes as the predictor increases, creating a curve. This property allows the model to reflect patterns such as diminishing returns or U-shaped trends.
Note: Quadratic regression typically refers to a model with a single predictor that includes both linear and squared terms. However, in broader statistical practice, analysts sometimes use the term to describe any model that includes squared terms, even when there are multiple predictors. In those cases, it’s more precise to refer to the model as a second-degree polynomial regression or to describe its structure explicitly.
Learn more about the Difference Between Linear and Nonlinear Regression Models.
Quadratic Regression Model Equation
A typical quadratic regression model is written as:
Where:
- Y is the outcome (dependent variable)
- X is the predictor (independent variable)
- β₀ is the intercept
- β₁ is the coefficient for the linear term
- β₂ is the coefficient for the squared term
- ε is the error term
Shape and Turning Point
Quadratic regression produces a curve with one bend (or turning point). The curve is U-shaped when the coefficient for the squared term (β₂) is positive, and inverted-U-shaped when it is negative. The turning point occurs at:
This point indicates where the curve changes direction and corresponds to the maximum or minimum predicted value.
Quadratic Regression Example
Suppose you’re studying crop yield as a function of fertilizer use. A straight-line model will not reflect the reality that fertilizer helps up to a point then becomes less effective or even harmful. A quadratic regression model fits this pattern well, producing a curved relationship that captures both the increase and decline.
Use my free online Regression Calculator to fit a Quadratic Model!
In the graph, the fitted model is:
This curve rises, reaches a peak, and then turns downward. You can calculate the fertilizer level that maximizes yield by using the turning point formula to find the peak of the curve:
At 3.19 kg/plot of fertilizer, the model predicts the maximum crop yield of approximately 8.20 kg/plot. This value represents the optimal fertilizer level. Applying more or less than this amount is associated with lower yields, illustrating the real-world usefulness of quadratic regression in identifying such thresholds.
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