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The Four Types of 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
Using the Type I generating set G_{2} (for example), if an L is formed from linear combinations of the rows of G_{2} such that L is of full row rank and of the same row rank as G_{2}, 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 G_{1}, G_{2}, and G_{3}, 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
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