Yito itsexpected value:
my¼b 0 þ
Xk
j¼ 1
bjxji
In particular, the method of least squares requires that we consider thesum of
squared deviations:
S¼
Xn
i¼ 1
Yib 0
Xk
j¼ 1
bjxji
! 2
According to the method of least squares, the good estimates ofb 0 andbi’s
are valuesb 0 andbi’s, respectively, whichminimizethe sumS. The method is
the same, but the results are much more di‰cult to obtain; fortunately, these
results are provided by most standard computer programs, such as Excel and
SAS. In addition, computer output also provides standard errors for all esti-
mates of regression coe‰cients.
8.2.6 Analysis-of-Variance Approach
The total sum of squares,
SST¼
X
ðYiYÞ^2
and its associated degree of freedom (n1) are defined and partitioned the
same as in the case of simple linear regression. The results aredisplayedin the
form of ananalysis-of-variance(ANOVA)table(Table 8.6) of the same form,
wherekis the number of independent variables. In addition:
- The coe‰cient of multiple determination is defined as
R^2 ¼
SSR
SST
It measures the proportionate reduction of total variation inYassociated
with the use of the set of independent varables. As forr^2 of the simple
TABLE 8.6
Source of
Variation SS df MS FStatistic pValue
Regression SSR k MSR¼SSR=kF¼MSR=MSE p
Error SSE nk1 MSE¼SSE=ðnk 1 Þ
Total SST n 1
MULTIPLE REGRESSION ANALYSIS 297