Polynomial Regression

Polynomial regression models the outcome as a polynomial function (quadratic, cubic and so on) of a predictor, fitted by ordinary least squares so that curved relationships can be captured.

다항회귀는 설명변수의 제곱·세제곱 항을 넣어 결과와의 곡선 관계를 최소제곱으로 적합하는 회귀모형입니다.

Does the relationship between my predictor and outcome bend, for example rise and then level off, and how do I test that?

설명변수가 커질수록 결과가 올라가다 꺾이는 것 같은데, 이 곡선 관계를 어떻게 검정하나요?

Polynomial regression adds squared (and possibly cubic) terms of a predictor to a linear model, so the fitted line can curve while estimation remains ordinary least squares. Whether curvature is needed is judged by an F test comparing the polynomial model with the straight-line model, not by the individual power terms.

When to use Polynomial Regression

• The scatterplot or residuals of a linear model show a smooth curve such as a U or inverted U shape
• You need a simple closed-form curve with a turning point that can be reported
• The predictor has many distinct values spread over the range of interest

When to use something else

• The curve has local bends or flattens at the ends; use natural cubic spline regression instead
• The relationship changes abruptly at a threshold; use segmented (piecewise) regression instead

Assumptions and requirements

• The mean of the outcome is a polynomial function of the predictor
• Errors are independent with constant variance
• Residuals are approximately normal for small-sample inference
• No extrapolation beyond the observed predictor range
• Only the first predictor receives polynomial terms; other predictors enter linearly

What the analysis reports

• Coefficients for each power term with standard errors
• R² and adjusted R² for the polynomial and linear models
• F test for the higher-order terms
• Turning point of a quadratic curve

How to report it

A quadratic regression explained [value]% of the variance in [outcome], F([df1], [df2]) = [value], p = [value]; adding the squared term improved fit over the linear model, F(1, [df2]) = [value], p = [value].

Related methods

• Natural Cubic Spline Regression
• Segmented (Piecewise) Regression
• Multiple Linear Regression

Run Polynomial Regression on Quorant