The Four Types of Estimable Functions

This section demonstrates a shorthand technique for displaying the generating set for any estimable . Suppose

is a generating set for , but so is the smaller set

is formed from by deleting duplicate rows.

Since all estimable s must be linear functions of the rows of for to be estimable, an for a single-degree-of-freedom estimate can be represented symbolically as

or

For this example, is estimable if and only if the first element of is equal to the sum of the other elements of or if

is estimable for any values of L1, L2, and L3.

If other generating sets for 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 to produce an identity matrix in the first submatrix of the resulting matrix

then is also a generating set for . An estimable generated from can be represented symbolically as

Note that, again, the first element of is equal to the sum of the other elements.

With multiple generating sets available, the question arises as to which one is the best to represent 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 computed by the modified sweep operation (Goodnight 1979) has the property that usually contains numerous zeros. For this reason, in PROC GLM the nonzero rows of are used to represent symbolically.

If the generating set represented symbolically is of full row rank, the number of symbols represents the maximum rank of any testable hypothesis (in other words, the maximum number of linearly independent rows for any 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.