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 The Four Types of Estimable Functions

## Examples

### A One-Way Classification Model

For the model
the general form of estimable functions Lb is (from the previous example)

Thus,

L = (L1, L2, L3, L1-L2-L3)
Tests involving only the parameters A1, A2, and A3 must have an L of the form
L = (0, L2, L3, -L2-L3)
Since the preceding L involves only two symbols, hypotheses with at most two degrees-of-freedom 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 A1=A2=A3. 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,

### A Three-Factor Main Effects Model

Consider a three-factor main effects model involving the CLASS variables A, B, and C, as shown in Table 12.1.

Table 12.1: Three-Factor 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 Three-Factor Main Effects Model
 Parameter Coefficient (Intercept) L1 A1 L2 A2 L1-L2 B1 L4 B2 L1-L4 C1 L6 C2 L1+L2-L4-2 ×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, A1-A2 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, B1-B2 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 left-hand 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 full-rank 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,

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