The Four Types of Estimable Functions: General Form of an Estimable Function

General Form of an Estimable Function

This section demonstrates a shorthand technique for displaying the generating set for any estimable $\text{[math]}$ . Suppose

$\text{[math]}$

$\text{[math]}$ is a generating set for $\text{[math]}$ , but so is the smaller set

$\text{[math]}$

$\text{[math]}$ is formed from $\text{[math]}$ by deleting duplicate rows.

Since all estimable $\text{[math]}$ s must be linear functions of the rows of $\text{[math]}$ for $\text{[math]}$ to be estimable, an $\text{[math]}$ for a single-degree-of-freedom estimate can be represented symbolically as

$\text{[math]}$

For this example, $\text{[math]}$ is estimable if and only if the first element of $\text{[math]}$ is equal to the sum of the other elements of $\text{[math]}$ or if

$\text{[math]}$

is estimable for any values of $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ .

If other generating sets for $\text{[math]}$ 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 $\text{[math]}$ to produce an identity matrix in the first $\text{[math]}$ submatrix of the resulting matrix

$\text{[math]}$

then $\text{[math]}$ is also a generating set for $\text{[math]}$ . An estimable $\text{[math]}$ generated from $\text{[math]}$ can be represented symbolically as

$\text{[math]}$

Note that, again, the first element of $\text{[math]}$ 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 $\text{[math]}$ symbolically. Clearly, a generating set containing a minimum of rows (of full row rank) and a maximum of zero elements is desirable.

The generalized $\text{[math]}$ -inverse $\text{[math]}$ of $\text{[math]}$ computed by the modified sweep operation (Goodnight; 1979) has the property that $\text{[math]}$ usually contains numerous zeros. For this reason, in PROC GLM the nonzero rows of $\text{[math]}$ are used to represent $\text{[math]}$ symbolically.

If the generating set represented symbolically is of full row rank, the number of symbols $\text{[math]}$ represents the maximum rank of any testable hypothesis (in other words, the maximum number of linearly independent rows for any $\text{[math]}$ 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.