Page 125 - Basic Statistics

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The sum of squares for liear hypotesis H0 : K’  = m is computed by

(Searle, 1971),

ˆ
ˆ
-1
-1
Q = (K’ β - m)’ (K’ ( X’X ) K) (K’ β - m) (G.30)

ˆ
This is quadratic form in K’ β - m with defining matrix A = (K’ ( X’X ) K) .
-1
-1
The defining matrix, except 1/ , is a inverse of the varince-covariance matrix
2
ˆ
of linear functions K’ β - m. Thus, tr(AV) = tr(Ik) = k . Furthermore,
2
expectation of Q

2
-1
-1
E(Q) = k + (K’ - m)’ (K’ ( X’X ) K) (K’  - m) (G.31)

With the assumption of normality, Q/ is distributed as a noncentral Chi-
2
squares random variable with k degrees of freedom. In Rawlings (1988), F-test

for hypothesis H0 : K’  = m is

Q k /
F = (G.32)
S 2

Worked Example 5.5:

For example problems using data from environmental studies in Table
5.6. By using the statistical software S-Plus, obtained matrix (X’X) as follows.
-1

Matrix ( X’X ) :
-1

[,1] [,2] [,3] [,4]

[1,] 2.8559225951 6.173356e-004 -0.03535068677 -4.090973e-002
[2,] 0.0006173356 3.341412e-006 -0.00001807261 1.409908e-008
[3,] -0.0353506868 -1.807261e-005 0.00053385411 1.439104e-004
[4,] -0.0409097295 1.409908e-008 0.00014391044 2.613921e-003

Test the contribution of 2 regressors, namely radiation (X1) and wind (X3) on the

model that includes variable temperature (X2).

~~* CHAPTER 5 THE MULTIPLE LINEAR REGRESSION MODEL *~~