The Four Types of Estimable Functions

Given two effects F1 and F2, F1 is said to be contained in F2 provided that the following two conditions are met:

Note that the intercept effect is contained in all pure CLASS effects, but it is not contained in any effect involving a continuous variable. No effect is contained by .

Type II, Type III, and Type IV estimable functions rely on this definition, and they all have one thing in common: the estimable functions involving an effect F1 also involve the parameters of all effects that contain F1, and they do not involve the parameters of effects that do not contain F1 (other than F1).

The Type II estimable functions for an effect F1 have an (before reduction to full row rank) of the following form:

In these submatrices,

For the model

class A B;
model Y = A B A*B;

the Type II SS correspond to

for effects A, B, and A * B, respectively. For the model

class A B C;
model Y =  A B(A) C(A B);

the Type II SS correspond to

for effects A, and , respectively. For the model

model Y = x x*x;

the Type II SS correspond to

for x and , respectively.

Note that, as in the situation for Type I tests, PROC MIXED and PROC GLIMMIX compute Type I hypotheses by sweeping , but their test statistics are not necessarily equivalent to the results of sequentially fitting with those procedures models that contain successively more effects; while PROC TRANSREG computes tests labeled as being Type II by leaving out each effect in turn, but the specific linear hypotheses associated with these tests might not be precisely the same as the ones derived from successively sweeping .

For a factorial with w observations per cell, the general form of estimable functions is shown in Table 5. Any nonzero values for L2, L4, and L6 can be used to construct vectors for computing the Type II SS for A, B, and A * B, respectively.

Table 5: General Form of Estimable Functions for Factorial

Effect		Coefficient
		L1
A1		L2
A2		L1 – L2
B1		L4
B2		L1 – L4
AB11		L6
AB12		L2 – L6
AB21		L4 – L6
AB22		L1 – L2 – L4 + L6

For a balanced factorial with the same number of observations in every cell, the Type II estimable functions are shown in Table 6.

Table 6: Type II Estimable Functions for Balanced Factorial

		Coefficients for Effect
Effect		A		B		A * B
		0		0		0
A1		L2		0		0
A2		–L2		0		0
B1		0		L4		0
B2		0		–L4		0
AB11		0.5 L2		0.5 L4		L6
AB12		0.5 L2		–0.5 L4		–L6
AB21		–0.5 L2		0.5 L4		–L6
AB22		–0.5 L2		–0.5 L4		L6

Now consider an unbalanced factorial with two observations in every cell except the AB22 cell, which contains only one observation. The general form of estimable functions is the same as if it were balanced, since the same effects are still estimable. However, the Type II estimable functions for A and B are not the same as they were for the balanced design. The Type II estimable functions for this unbalanced factorial are shown in Table 7.

Table 7: Type II Estimable Functions for Unbalanced Factorial

		Coefficients for Effect
Effect		A		B		A * B
		0		0		0
A1		L2		0		0
A2		–L2		0		0
B1		0		L4		0
B2		0		–L4		0
AB11		0.6 L2		0.6 L4		L6
AB12		0.4 L2		–0.6 L4		–L6
AB21		–0.6 L2		0.4 L4		–L6
AB22		–0.4 L2		–0.4 L4		L6

By comparing the hypothesis being tested in the balanced case to the hypothesis being tested in the unbalanced case for effects A and B, you can note that the Type II hypotheses for A and B are dependent on the cell frequencies in the design. For unbalanced designs in which the cell frequencies are not proportional to the background population, the Type II hypotheses for effects that are contained in other effects are of questionable value.

However, if an effect is not contained in any other effect, the Type II hypothesis for that effect is an MRH that does not involve any parameters except those associated with the effect in question.

Thus, Type II SS are appropriate for the following models:

In addition to the preceding models, Type II SS are generally accepted by most statisticians for purely nested models.