## Regression Analysis And Analysis Of Variance

In this section we study regression analysis from the point of view of the analysis of variance and introduce the reader to an illuminating and complementary way of looking at the statistical inference problem. In Chapter 3, Section 3.5, we developed the following identity:

that is, TSS = ESS + RSS, which decomposed the total sum of squares (TSS) into two components: explained sum of squares (ESS) and residual sum of squares (RSS). A study of these components of TSS is known as the analysis of variance (ANOVA) from the regression viewpoint.

Associated with any sum of squares is its df, the number of independent observations on which it is based. TSS has n — idf because we lose 1 df in computing the sample mean Y. RSS has n — 2df. (Why?) (Note: This is true only for the two-variable regression model with the intercept fa present.) ESS has 1 df (again true of the two-variable case only), which follows from the fact that ESS = fa? Y, x2 is a function of fa only, since Y x2 is known.

Let us arrange the various sums of squares and their associated df in Table 5.3, which is the standard form of the AOV table, sometimes called the ANOVA table. Given the entries of Table 5.3, we now consider the following variable:

MSS of ESS MSS of RSS 