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44 SIMPLE LINEAR REGRESSION
correlation coefficient between Y and Y, which is given by
(2.42)
where y is the mean of the response variable Y and 11 is the mean of the
fitted valu:s. In fact, the scatter plot of Y versus X and the scatter plot of
Y versus Y are redundant because the patterns of points in the two graphs
are identical. The two corresponding values of the correlation coefficient are
related by the following equation:
Cor(Y, Y) = ICor(Y, X) I. (2.43)
Note that Cor(Y, Y) cannot be negative (why?), but Cor(Y, X) can be positive
or negative [-1 ~ Cor(Y, X) ~ 1]. Therefore, in simple linear regression,
the scatter plot of Y versus Y is redundant. However, in multiple regression,
the scatter plot of Y versus Y is not redundant. The graph is very useful
because, as we shall see in Chapter 3, it is used to assess the strength of the
relationship between Y and the set of predictor variables Xl, X2,'" ,Xp,
4. Although scatter plots of Y versus Y and Cor(Y, Y) are redundant in simple
linear regression, they give us an indication of the quality of the fit in both
simple and multiple regression. Furthermore, in both simple and multiple
regressions, Cor(Y, Y) is related to another useful measure of the quality of
fit of the linear model to the observed data. This measure is developed as
follows. After we compute the least squares estimates of the parameters of a
linear model, let us compute the following quantities:
SST ~)Yi - y)2,
SSR ~)Yi - y)2, (2.44)
SSE = L)Yi - Yi)2,
where SST stands for the total sum of squared deviations in Y from its mean
y, SSR denotes the sum of squares due to regression, and SSE represents
the sum of squared residuals (errors). The quantities (Yi - V), (Yi - V), and
(Yi - Yi) are depicted in Figure 2.7 for a typical point (Xi, Yi). The line
Yi = /30 + /31 Xi is the fitted regression line based on all data points (not
shown on the graph) and the horizontal line is drawn at Y = y. Note that
for every point (Xi, Yi), there are two points, (Xi, Yi), which lies on the fitted
line, and (Xi, y) which lies on the line Y = y.
A fundamental equality, in both simple and multiple regressions, is given by
SST = SSR + SSE. (2.45)
This equation arises from the description of an observation as
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