Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 29

There is an alternate form of the simple linear regression model that is occasionally useful. Suppose that we redefine the regressor variable xi as the deviation from its own average, say . The regression model then becomes

(2.20)

Note that redefining the regressor variable in Eq. (2.20) has shifted the origin of the x’s from zero to . In order to keep the fitted values the same in both the original and transformed models, it is necessary to modify the original intercept. The relationship between the original and transformed intercept is

(2.21)

It is easy to show that the least-squares estimator of the transformed intercept is . The estimator of the slope is unaffected by the transformation. This alternate form of the model has some advantages. First, the least-squares estimators and are uncorrelated, that is, . This will make some applications of the model easier, such as finding confidence intervals on the mean of y (see Section 2.4.2). Finally, the fitted model is

(2.22)

Although Eqs. (2.22) and (2.8) are equivalent (they both produce the same value of for the same value of x), Eq. (2.22) directly reminds the analyst that the regression model is only valid over the range of x in the original data. This region is centered at .