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9.4 EVALUATING THE REGRESSION EQUATION 427
260
240
220
Unexplained
deviation
(y i – y^i )
200
Deep abdominal AT area (cm 2 ), Y
180
160
140
120
100
80
y – = 101.89
y^ = _ 216 + 3.46x
Total deviation
(y i – y – )
Explained
deviation
(y^i – y_ )
60
40
20
0
0 60 65 70 75 80 85 90 95 100 105 110 115 120 125
Waist circumference (cm), X
FIGURE 9.4.4 Scatter diagram showing the total, explained, and unexplained
deviations for a selected value of Y, Example 9.3.1.
Unexplained Sum of Squares The unexplained sum of squares is a measure
of the dispersion of the observed Y values about the regression line and is sometimes
called the error sum of squares, or the residual sum of squares (SSE). It is this quantity
that is minimized when the least-squares line is obtained.
We may express the relationship among the three sums of squares values as
SST = SSR + SSE
The numerical values of these sums of squares for our illustrative example appear in
the analysis of variance table in Figure 9.3.2. Thus, we see that SST = 354531,
SSR = 237549, SSE = 116982, and
354531 = 237549 + 116982
354531 = 354531
Calculating r 2 It is intuitively appealing to speculate that if a regression equation
does a good job of describing the relationship between two variables, the explained
or regression sum of squares should constitute a large proportion of the total sum of