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 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

The nonzero rows of the resulting matrix associated with X2 provide an L such that
The last set of nonzero rows (associated with X3) provide an L such that
Another more formalized representation of Type I generating sets for B1, B2, and B3, respectively, is
where
and

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.

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

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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