Introduction to Linear Regression Analysis, 5th Edition

CHAPTER 7

POLYNOMIAL REGRESSION MODELS

7.1 INTRODUCTION

The linear regression model y = Xβ + ε is a general model for fitting any relationship that is linear in the unknown parameters β. This includes the important class of polynomial regression models. For example, the second-order polynomial in one variable

and the second-order polynomial in two variables

are linear regression models.

Polynomials are widely used in situations where the response is curvilinear, as even complex nonlinear relationships can be adequately modeled by polynomials over reasonably small ranges of the x's. This chapter will survey several problems and issues associated with fitting polynomials.

7.2 POLYNOMIAL MODELS IN ONE VARIABLE

7.2.1 Basic Principles

As an example of a polynomial regression model in one variable, consider

Figure 7.1 An example of a quadratic polynomial.

This model is called a second-order model in one variable. It is also sometimes called a quadratic model, since the expected value of y is

which describes a quadratic function. A typical example is shown in Figure 7.1. We often call