The Four Types of Estimable Functions 
Examples
A OneWay Classification Model
For the model
the general form of estimable functions
Lb is (from the previous example)
Thus,

L = (L1, L2, L3, L1L2L3)
Tests involving only the parameters A_{1}, A_{2},
and A_{3} must have an L of the form

L = (0, L2, L3, L2L3)
Since the preceding L involves only two symbols, hypotheses with
at most two degreesoffreedom can be constructed.
For example, let L2=1 and L3=0; then let L2=0 and L3=1:
The preceding L can be used to
test the hypothesis that A_{1}=A_{2}=A_{3}.
For this example, any L with two linearly independent
rows with column 1 equal to zero produces the same Sum of Squares.
For example, a pooled linear quadratic
gives the same SS.
In fact, for any L of full row rank and any
nonsingular matrix K of conformable dimensions,
Consider a threefactor main effects model
involving the CLASS variables A, B, and
C, as shown in Table 12.1.
Table 12.1: ThreeFactor Main Effects Model
Obs


A


B


C

1   1   2   1 
2   1   1   2 
3   2   1   3 
4   2   2   2 
5   2   2   2 
The general form of an estimable function is shown in
Table 12.2.
Table 12.2: General Form of an Estimable Function
for ThreeFactor Main Effects Model
Parameter


Coefficient

(Intercept)   L1 
A1   L2 
A2   L1L2 
B1   L4 
B2   L1L4 
C1   L6 
C2   L1+L2L42 ×L6 
C3   L2+L4+L6 
Since only four symbols (L1, L2, L4, and L6) are involved,
any testable hypothesis will have at most four degrees of freedom.
If you form an L matrix with four linearly independent rows
according to the preceding rules, then
In a main effects model, the usual hypothesis of interest
for a main effect is the equality of all the parameters.
In this example, it is not possible to test such a hypothesis
because of confounding. One way to proceed is to construct a
maximum rank hypothesis (MRH) involving only
the parameters of the main effect in question.
This can be done using the general form of estimable functions.
Note the following:
 To get an MRH involving only the parameters of A, the
coefficients of L associated with , B1,
B2, C1, C2, and C3 must be equated to zero.
Starting at the top of the general form,
let L1=0, then L4=0, then L6=0.
If C2 and C3 are not to be involved,
then L2 must also be zero.
Thus, A1A2 is not estimable; that is, the MRH involving
only the A parameters has zero rank and .
 To obtain the MRH involving only the
B parameters, let L1=L2=L6=0.
But then to remove C2 and C3 from the
comparison, L4 must also be set to 0.
Thus, B1B2 is not estimable and .
 To obtain the MRH involving only the
C parameters, let L1=L2=L4=0.
Thus, the MRH involving only C parameters is

C1  2 ×C2 + C3 = K (for any K)
or any multiple of the lefthand side equal to K.
Furthermore,
A Multiple Regression Model
Suppose
If the X'X matrix is of full
rank, the general form of estimable functions
is as shown in Table 12.3.
Table 12.3: General Form of Estimable Functions for a Multiple
Regression Model When X'X Matrix Is of Full Rank
Parameter


Coefficient

  L1 
  L2 
  L3 
  L4 
To test, for example, the hypothesis that
, let L1=L2=L4=0 and let L3=1.
Then SS.
In the fullrank case, all parameters, as well as
any linear combination of parameters, are estimable.
Suppose, however, that X3 = 2 ×X1+3 ×X2.
The general form of estimable functions is
shown in Table 12.4.
Table 12.4: General Form of Estimable Functions
for a Multiple Regression Model When
X'X Matrix Is Not of Full Rank
Parameter


Coefficient

  L1 
  L2 
  L3 
  2 ×L2+3 ×L3 
For this example, it is possible to test .
However, , , and
are not jointly estimable; that is,
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.