A Property of Well-Formulated Polynomial Regression Models

Abstract

A polynomial regression model in any number of variables that excludes hierarchically inferior terms is not well formulated. This article shows that the estimation space of a polynomial regression model is invariant under coding transformations iff the model is well formulated. Consequently, measures of goodness of fit of a not-well-formulated model may be affected by coding transformations. A data set of winter temperatures and latitude and longitude measurements for 56 locations in the United States is used for illustration. The article provides additional motivation for the use of variable selection algorithms that restrict their search to well-formulated models.

Key Words:

• Coding
• Hierarchical ordering
• Linear transformation
• Marginality
• Variable selection
• Well-formulated model