Page 113 - Basic Statistics

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5.2.5 PARTITIONING OF THE SUM OF SQUARES

To match the linear additive model, the vector of observation for the

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dependent variable Y, partitioned into vector Y plus residual vector e. namely

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Y = Y + e

Similar partitions are used to sum of the squares of the dependent variable Y.

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Y’ Y = (Y + e )’ (Y + e )
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= Y ’ Y + Y ’ e + e’ Y + e’ e

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Substituting Y = H Y, dan e = [I - H] Y gives

Y’ Y = (H Y)’(H Y) + (H Y)’ ([I - H] Y) + ([I - H] Y)’(H Y) + ([I - H] Y)’( [I - H] Y)

= Y’ H’ H Y + Y’ (H’ [I - H] )Y + Y’ ( [I - H]’H ) Y + Y’ ( [I - H]’ [I - H] ) Y

Both H and [I - H] is symmetric and idempotent, so that H’ H = H and

[I - H]’ [I - H] = [I - H]. The two middle term are zero because the two quadratic

forms are orthogonal to each other, ie H’ [I - H] = [H - H] = 0. Furthermore

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Y’ Y = Y’ H Y + Y’ [I - H] Y = Y ’ Y + e’ e (G.14)

SS(Total) = SS(Model) + SS (Res) (G.15)

Total sum of squares was partitioned into two sums of squares, which is

the model sum of squares and residual sum of squares. Both of the sums of

squares consecutive states as explained component and unexplained
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components of the model. SS (Model) =Y ’ Y = Y’ H Y has a defining matrix H,
and SS (Res) = e’ e = Y’ [I - H] Y has a defining matrix [I - H].
If the two defining matrices are multiplied, H [I - H] = 0, so the two sums

of squares are mutually orthogonal, then forming additive partition. Degrees of

freedom for both the sum of the squares of each is determined by the rank of

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