Page 271 - Statistics for Environmental Engineers

L1592_Frame_C30 Page 274 Tuesday, December 18, 2001 2:49 PM

The matrix of independent variables is:

1 – 1 – 1 – 1 +1 1 1 – 1
1 1 – 1 – 1 – 1 – 1 1 1
1 – 1 1 – 1 – 1 1 – 1 1
X = 1 1 1 – 1 +1 – 1 – 1 – 1
1 – 1 – 1 1 1 – 1 – 1 1
1 1 – 1 1 – 1 1 – 1 – 1
1 – 1 1 1 – 1 – 1 1 – 1
1 1 1 1 1 1 1 1

3
Notice that this matrix is the same as the model matrix for the 2 factorial shown in Table 27.3.
To calculate b and Var(b) we need the transpose of X, denoted as X′. The transpose is created by
making the first column of X the first row of X′; the second column of X becomes the second row of
X′, etc. This is shown below. We also need the product of X and X′, denoted as X′X, the and the inverse
−1
of this, which is (X′ X) .
The transpose of the X matrix is:

1 1 1 1 1 1 1 1
– 1 1 – 1 1 – 1 1 – 1 1
– 1 – 1 1 1 – 1 – 1 11
X′ = – 1 – 1 – 1 – 1 1 1 1 1
1 – 1 – 1 1 1 – 1 – 1 1
1 – 1 1 – 1 – 1 1 – 1 1
1 1 – 1 – 1 – 1 – 1 11
– 1 1 1 – 1 1 – 1 – 1 1

The X′X matrix is:

8 0 0 0 0 0 0 0
0 8 0 0 0 0 0 0
0 0 8 0 0 0 0 0
X′X = 0 0 0 8 0 0 0 0
0 0 0 0 8 0 0 0
0 0 0 0 0 8 0 0
0 0 0 0 0 0 8 0
0 0 0 0 0 0 0 8

The inverse of the X′X matrix is called the variance-covariance matrix. It is:

1/8 0 0 0 0 0 0 0
0 1/8 0 0 0 0 0 0
0 0 1/8 0 0 0 0 0
( X′X) – 1 = 0 0 0 1/8 0 0 0 0
0 0 0 0 1/8 0 0 0
0 0 0 0 0 1/8 0 0
0 0 0 0 0 0 1/8 0
0 0 0 0 0 0 0 1/8
© 2002 By CRC Press LLC