Applied Statistics Lesson of the Day – Polynomial Regression is Actually Just Linear Regression

June 20, 2014

(This article was originally published at The Chemical Statistician » Statistics, and syndicated at StatsBlogs.)

Continuing from my previous Statistics Lesson of the Day on what “linear” really means in “linear regression”, I want to highlight a common example involving this nomenclature that can mislead non-statisticians. Polynomial regression is a commonly used multiple regression technique; it models the systematic component of the regression model as a $p\text{th}$ -order polynomial relationship between the response variable $Y$ and the explanatory variable $x$ .

$Y = \beta_0 + \beta_1 x + \beta_2 x^2 + ... + \beta_p x^p + \varepsilon$

However, this model is still a linear regression model, because the response variable is still a linear combination of the regression coefficients. The regression coefficients would still be estimated using linear algebra through the method of least squares.

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).