[wiley series in probability and statistics] methods and applications of linear models (regression and the analysis of variance) polynomial models and qualitative predictors精品.pdf

Methods and Applications ofLinear Models: Regression and the Analysis of Variance,2nd Edition. Ronald R. Hocking Copyright 0 2003 John Wiley Sons, Inc. ISBN: 0-471-23222-X 7 Polynomial Models and Qualitative Predictors In this chapter we discuss polynomial models with more than one predictor. We also introduce the concept of indicator variables. The discussion of polynomial models includes the application of such models in the analysis of response surfaces, Qualitative (indicator) variables are a powerkl tool for fitting Iinear models. Indicator variables are used to improve the fit in regression models and to make inferences about means of populations. We give a brief discussion of fitting segmented regression models that gives an alternative to fitting complex models. The concepts are illustratedwith numerical and conceptual examples. 7.1 POLYNOMIAL MODELS 7.1.1 Quadratic Model with Two Predictors In Examples 2.1 and 2.2 we considered the introduction of powers of our predictor variable to improve the fit of the prediction equation. This same concept can be applied to models with several predictors. The motivation, as in the case of one variable, is that a polynomial may provide an approximation to the true form of the expected-value function. The dangers, cited in Chapter 2, of fitting powers much greater than two or three are still applicable. We will see that fitting second-order models, that is, models including all terms of degree two, provides a usehl approximation for an iterative examination of the actual response function. The essential concepts are revealed by a discussion of a quadratic model in two predictors. We use the following extension of Example 2.1 t