Polynomial regression is still linear regression

Correcting a common misunderstanding in applied statistics

The term "polynomial regression" is somewhat of a misnomer, because it can mislead a newcomer to think that it is mathematically different from "linear regression". In fact, polynomial regression is still linear regression.

Polynomial regression is a common technique in multiple regression; it models the systematic component of the regression model as a pth-order polynomial relationship between the response variable and the explanatory variable.

\(Y = \beta_0 + \beta_1x + \beta_2x^2 + \dots + \beta_px^p + \varepsilon\)

However, polynomial regression is still linear regression, because the response variable is still a linear combination of the regression coefficients. We still use linear algebra and the method of least squares to estimate the coefficients.

Remember: The “linear” in linear regression refers to the linearity between the response variable and the regression coefficients, NOT between the response variable and the explanatory variable(s). This is a major contrast between linear regression and linear functions.

• In linear regression, the linearity is between Y and the β's.

• In linear functions, the linearity is between y and x.