Page 88 - Basic Statistics

83

SS(Total)uncorr = SS(Model) + SS(Res) ( R6 )

Total sum of squares is partitioned over the explained sum of squares (SS

(Model)) and the unexplained sum of squares (SS (Res)). The sum of squares on

both sides of this partition is the sum of the squares that have not been
corrected. This partition can then be made into a sum of squares corrected

partition. Corrections were made on both sides of the equation by a correction

2
factor nY .

ˆ
 Y j – n Y = (  Y - n Y ) +  ( Yj-Y )
2
2
2
ˆ 2
2
j
j
SS(Total) = SS(Reg) + SS(Res) ( R7 )

ˆ
If the estimated value β is used, the sum of squares regression on this partition
1
can be written as follows.

SS(Total) = SS(Reg) + SS(Res)

ˆ
2
2
ˆ 2
2
2
 Y j – n Y = (  Y - n Y ) +  ( Yj-Y )
j
j
ˆ
2
2
ˆ 2
 Y j – n Y = ( β  (Xj - X ) ) +  ( Yj- Y ) 2 ( R8 )
2
j
1

Degrees of freedom associated with the sum of the squares is determined
by the sample size and the number of parameters in the model (p). Each degree
of freedom of the corrected sum of squares is always reduced by 1 as a result of
correction factors. Degrees of freedom associated with SS(total) is n - 1. Degrees
of freedom associated with SS(Reg) is the degrees of freedom of the SS(model)
minus one. Degrees of freedom associated with SS(Model) is equal to the

number of parameters in the regression model, that is p = 2. Thus the degrees of

freedom associated with SS(Reg) is p-1 = 2-1. Degrees of freedom associated
with SS(Res) is the n-p = n-2.

~~* CHAPTER 5 LINEAR REGRESSION MODEL *~~