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if it were adjusted for, say, only variable 1. In other word, the appropriateness of

regressor variable often depend on what regressor variables are in the model with it.
The sum of squares, which is discussed on a partial test, not forming

additives contribute to the SS (Reg),

SS(1 , 2 , … , p  0 )  SS(1  0 , 2 , … ,p ) + SS(2  0 , 1 , 3 , … ,p ) + …

+ SS(j  0 , 1 , … , j-1, j+1, … ,p ) + …+ SS(p  0 , 1 , 2 , … ,p-1 ).

But they form additive partitioning of the SS (Reg), scilicet:

SS(1 , 2 , … , p  0 ) = SS(1  0 ) + SS(2  0 , 1 ) + SS(3  0 , 1 , 2 )

+ … + SS(p  0 , 1 , 2 , … ,p-1 ). (G.24)

The notation SS( .  . ) states "regression explained by ...", with the line

vertikal denoting "in the presence of ...". For example, SS(2  0 , 1 ) is an

increase in the regression sum of squares, when the regressor X2 is added to a

model that involving only X1 and the constant term.
The sequential and partial sum of squares for worked example 5.4 are

shown in Table 5.13 follows.

Table 5.13 The sequential and partial sum of squares

Sequential Partial

SS(1  0 ) = 99.145 SS(1  0 , 2 , 3 ,4 ,5) =0.299

SS(2  0 , 1 ) = 0.127 SS(2  0 , 1 , 3 ,4 ,5) =0.869

SS(3  0 , 1 , 2 ) = 4.120 SS(3  0 , 1 , 2 ,4 ,5) =0.078

SS(4  0 , 1 , 2 ,3 ) = 0.263 SS(4  0 , 1 , 2 ,3 ,5) =0.983

SS(5   ,, 1, 2 ,3 ,4) = 4.352 SS(5  0 , 1 , 2 ,3 ,4) =4.352

Partitions of sequential sum of squares is very useful when we need

information about the worth for a subset of regressors. For example, suppose,
the regression model with p = 4 regressors, we can write

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