Type I SS and Estimable Functions

The Four Types of Estimable Functions

Type I SS and Estimable Functions

The Type I SS and the associated hypotheses they test are by-products of the modified sweep operator used to compute a generalized inverse of X'X and a solution to the normal equations. For the model E(Y) = X1 ×B1+X2 ×B2+X3 ×B3, the Type I SS for each effect correspond to

Effect		Type I SS
B1		R(B1)
B2		R(B2\|B1)
B3		R(B3\|B1, B2)

The Type I SS are model-order dependent; each effect is adjusted only for the preceding effects in the model.

There are numerous ways to obtain a Type I hypothesis matrix L for each effect. One way is to form the X'X matrix and then reduce X'X to an upper triangular matrix by row operations, skipping over any rows with a zero diagonal. The nonzero rows of the resulting matrix associated with X1 provide an L such that

${SS}(H_0\colon L {\beta}= 0) = R(B1)$

The nonzero rows of the resulting matrix associated with X2 provide an L such that

${SS}(H_0\colon L {\beta}= 0) = R(B1| B2)$

The last set of nonzero rows (associated with X3) provide an L such that

${SS}(H_0\colon L {\beta}= 0) = R(B3| B1, B2)$

Another more formalized representation of Type I generating sets for B1, B2, and B3, respectively, is

$G_1 & = & ( & X_1'X_1 & | & X_1'X_2 & | & X_1'X_3 & ) \ G_2 & = & ( & 0 & | & X... ...X}_2 & | & X_2'M_2{X}_3 & ) \ G_3 & = & ( & 0 & | & 0 & | & X_3'M_3{X}_3 & ) \$

where

and

$M_2 & = & M_1 - M_1{X}_2(X_2'M_1{X}_2)^-X_2'M_1$

Using the Type I generating set G₂ (for example), if an L is formed from linear combinations of the rows of G₂ such that L is of full row rank and of the same row rank as G₂, then SS $(H_0: L {\beta}=0)=R(B2| B1)$ .

In the GLM procedure, the Type I estimable functions displayed symbolically when the E1 option is requested are

$G_1^* & = & (X_1' X_1)^-G_1 \ G_2^* & = & (X_2'M_1{X}_2)^-G_2 \ G_3^* & = & (X_3'M_2{X}_3)^-G_3$

As can be seen from the nature of the generating sets G₁, G₂, and G₃, only the Type I estimable functions for B3 are guaranteed not to involve the B1 and B2 parameters. The Type I hypothesis for B2 can (and usually does) involve B3 parameters. The Type I hypothesis for B1 usually involves B2 and B3 parameters.

There are, however, a number of models for which the Type I hypotheses are considered appropriate. These are

balanced ANOVA models specified in proper sequence (that is, interactions do not precede main effects in the MODEL statement and so forth)
purely nested models (specified in the proper sequence)
polynomial regression models (in the proper sequence).

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