Applied Statistics and Probability for Engineers

418 CHAPTER 12 MULTIPLE LINEAR REGRESSION

Note that there are pk1 normal equations in pk1 unknowns (the values of

Furthermore, the matrix XXis always nonsingular, as was assumed above,

so the methods described in textbooks on determinants and matrices for inverting these matri-

ces can be used to find. In practice, multiple regression calculations are almost al-

ways performed using a computer.

It is easy to see that the matrix form of the normal equations is identical to the scalar form.

Writing out Equation 12-12 in detail, we obtain

If the indicated matrix multiplication is performed, the scalar form of the normal equations

(that is, Equation 12-10) will result. In this form it is easy to see that is a (pp) sym-

metric matrix and is a (p1) column vector. Note the special structure of the ma-

trix. The diagonal elements of are the sums of squares of the elements in the columns of

X,and the off-diagonal elements are the sums of cross-products of the elements in the

columns of X.Furthermore, note that the elements of are the sums of cross-products of

the columns of Xand the observations

The fitted regression model is

(12-14)

In matrix notation, the fitted model is

The difference between the observation yiand the fitted value is a residual,say,

The (n1) vector of residuals is denoted by

(12-15)

EXAMPLE 12-2 In Example 12-1, we illustrated fitting the multiple regression model

where yis the observed pull strength for a wire bond, x 1 is the wire length, and x 2 is the

die height. The 25 observations are in Table 12-2. We will now use the matrix approach

y 0  1 ̨x 1  2 x 2 

eyyˆ

eiyiyˆi.

yˆi

yˆXˆ

yˆiˆ 0  a

k

j 1

̨ˆj ̨xij i1, ̨2,p, ̨ n

5 yi 6.

Xy

XX

Xy XX

XX

H

ˆ 0

ˆ 1

o

ˆk

XH

a

n

i 1

yi

a

n

i 1

xi 1 yi

o

a

n

i 1

xik ̨yi

H X

n a

n

i 1

xi (^1) a
n
i 1
xi 2 p a
n
i 1
xik
a
n
i 1
xi (^1) a
n
i 1
x^2 i (^1) a
n
i 1
xi 1 xi 2 p a
n
i 1
xi 1 xik
oooo
a
n
i 1
xik a
n
i 1
xikxi (^1) a
n
i 1
xikxi 2 p a
n
i 1
x^2 ik
X
1 X¿X 2 ^1
ˆ 0 , ˆ 1 ,p, ˆk 2.
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