Introductory Biostatistics

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

Yi
b 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

ðYi
YÞ^2

and its associated degree of freedom (n
1) 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:

1. 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 n
k
1 MSE¼SSE=ðn
k
1 Þ

Total SST n
1

MULTIPLE REGRESSION ANALYSIS 297