Correlation in Polynomial Regression

Abstract

Polynomial regression of degree p in one independent variable χ is considered. Numerically large sample correlations between χ^α and χ^β, α < β, a, β = 1, ···, p, may cause ill-conditioning in the matrix to be inverted in application of the method of least squares. These sample correlations are investigated. It is confirmed that centering of the independent variable to have zero sample mean removes nonessential ill-conditioning. If the sample values of χ are placed symmetrically about their mean, the sample correlation between χ^α and χ^β is reduced to zero by centering when α + β is odd, but may remain large when α + β is even. Some examples and recommendations are given.

Key Words:

• Polynomial regression
• Ill-conditioning in regression
• Centering in regression
• Sample correlations
• Regression design
• Multicollinearity