CHAPTER 13 Simple Linear Regression

524 CHAPTER THIRTEEN Simple Linear Regression
13.3 MEASURES OF VARIATION
When using the least-squares method to determine the regression coefficients for a set of data,
you need to compute three important measures of variation. The first measure, the total sum of
squares (SST ), is a measure of variation of the Y i
values around their mean, Y . In a regression
analysis, the total variation or total sum of squares is subdivided into explained variation and
unexplained variation. The explained variation or regression sum of squares (SSR) is due to
the relationship between X and Y, and the unexplained variation, or error sum of squares
(SSE) is due to factors other than the relationship between X and Y. Figure 13.6 shows these
different measures of variation.
FIGURE 13.6
Measures of variation
Y
Y i
Error sum
of squares
n
∑ (Y i – Y i ) 2 = SSE
i =1
Y i = b 0 + b 1 X i
Total sum of squares
n
∑ (Y i – Y) 2 = SST
i =1
Regression sum
of squares
n
∑ (Y i – Y) 2 = SSR
i =1
Y
0
X i
X
Computing the Sum of Squares
The regression sum of squares (SSR) is based on the difference between Yˆ
i (the predicted value
of Y from the prediction line ) and Y (the mean value of Y). The error sum of squares (SSE)
represents the part of the variation in Y that is not explained by the regression. It is based on the
difference between Y i
and Yˆ
i . Equations (13.5), (13.6), (13.7), and (13.8) define these measures
of variation.
MEASURES OF VARIATION IN REGRESSION
The total sum of squares is equal to the regression sum of squares plus the error sum of
squares.
SST = SSR + SSE (13.5)
TOTAL SUM OF SQUARES (SST)
The total sum of squares (SST) is equal to the sum of the squared differences between each
observed Y value and Y , the mean value of Y.
SST
= Total sum of squares
n
∑
= ( Y −Y
) 2
i=
1
i
(13.6)