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The Four Types of Estimable Functions |
This section demonstrates a shorthand technique for displaying the generating set for any estimable L. Suppose
Since all estimable Ls must be linear functions of the rows of X^{*} for to be estimable, an L for a single-degree-of-freedom estimate can be represented symbolically as
If other generating sets for L are represented symbolically, the symbolic notation looks different. However, the inherent nature of the rules is the same. For example, if row operations are performed on X^{*} to produce an identity matrix in the first 3 ×3 submatrix of the resulting matrix
With the thousands of generating sets available, the question arises as to which one is the best to represent L symbolically. Clearly, a generating set containing a minimum of rows (of full row rank) and a maximum of zero elements is desirable. The generalized inverse of X'X computed by the GLM procedure has the property that (X'X)^{-}X'X usually contains numerous zeros. For this reason, PROC GLM uses the nonzero rows of (X'X)^{-}X'X to represent L symbolically.
If the generating set represented symbolically is of full row rank, the number of symbols (L1, L2, ... ) represents the maximum rank of any testable hypothesis (in other words, the maximum number of linearly independent rows for any L matrix that can be constructed). By letting each symbol in turn take on the value of 1 while the others are set to 0, the original generating set can be reconstructed.
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