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Tabel 5.7 The Splus output

*** Linear Model ***

Coefficients:
Value Std. Error t value Pr(>|t|)
(Intercept) -0.2949 0.9986 -0.2953 0.7701
radiation 0.0013 0.0011 1.2080 0.2379
temperature 0.0456 0.0137 3.3429 0.0025
wind -0.0280 0.0302 -0.9268 0.3626

Residual standard error: 0.5909 on 26 degrees of freedom
Multiple R-Squared: 0.4583
F-statistic: 7.332 on 3 and 26 degrees of freedom, the p-value is 0.00102

5.2.3 FORMS OF LINEAR FUNCTION OF Y

Here it can be shown that the statistics (estimator) in multiple regression
ˆ
ˆ
is a linear function of Y. These statistics include β (coefficient estimators), Y
(predicted value), and e (residual).

ˆ
β = ( X’X ) ( X’Y )
-1

= [( X’X ) X’] Y (G.5)
-1

ˆ
Vector β is a linear function of Y, with coefficients [( X’X ) X’ ] .
-1

The vector of estimated means of dependent variable Y for values the

ˆ
ˆ
independent variables in the data set is computed as Y = X β . This is the
ˆ
simplest way to compute Y and useful to express as a linear function of Y.
ˆ
ˆ
Y = X β
= X ( X’X ) ( X’Y )
-1
= [ X ( X’X ) X’] Y
-1
= H Y (G.6)

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