Page 114 - Basic Statistics

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the defining matrices. Because of its two defining matrices are idempotent

matrices, will rank equally with its trace, i.e

-1
Rank (H) = tr(H) = tr(X ( X’X ) X’)
= tr( X’X ) X’ X ) (theory in matrix algebra tr(ABC) = tr(BCA) )
-1
= tr ( Ip+1) = p +1 ( p + 1 = number of columns of the matrix X )

Rank(I - H) = tr(I - H) = tr(I) – tr(H)

= n-(p+1)

Degrees of freedom for SS(Model) is p +1, and the degrees of freedom for

SS(Res) is n-(p +1).

How to count the sum of squares through the matrix notation are

SS(Total) = Y’ Y

ˆ
ˆ
ˆ
ˆ
SS(Model) =Y ’ Y = (X β )’(X ( X’X ) X’Y)= β ’ X’Y
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ˆ
SS(Sisa) = Y’ Y - β ’ X’Y. (G.16)
In regression analysis, we need the knowledge of the contribution of a
group of independent variables to the variation of Y around the mean value.
The size information can be seen from the difference between the SS(Model),

which contains independent variables with SS(Model) without independent

variables. SS(Model) without independent variables is called a correction factor,

and written SS(). The difference between the SS(model) with SS() is called the

sum of squares regression, written SS (Reg).

SS(Reg) = SS(Model) – SS()

To get SS(), written in the linear additive model Y = X*  + , where
the matrix X* = 1. Matrix 1 is only a column vector with all elements 1 or the
first column of the matrix X.

ˆ
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β = (( X*)’ X* ) (X*)’Y = (1’ 1 ) 1’Y = (1/n) 1’Y = Y , so that
-1

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