The Four Types of Estimable Functions

Type III hypotheses are constructed by working directly with the general form of estimable functions. The following steps are used to construct a hypothesis for an effect F1:

By once again observing the Type II hypotheses being tested in the balanced factorial, it is possible to verify that the A and A * B hypotheses are orthogonal and also that the B and A * B hypotheses are orthogonal. This principle of orthogonality between an effect and any effect that contains it holds for all balanced designs. Thus, construction of Type III hypotheses for any design is a logical extension of a process that is used for balanced designs.

The Type III hypotheses are precisely the hypotheses being tested by programs that reparameterize using the usual assumptions (for example, constraining all parameters for an effect to sum to zero). When no missing cells exist in a factorial model, Type III SS coincide with Yates’ weighted squares-of-means technique. When cells are missing in factorial models, the Type III SS coincide with those discussed in Harvey (1960) and Henderson (1953).

The following discussion illustrates the construction of Type III estimable functions for a factorial with no missing cells.

To obtain the A * B interaction hypothesis, start with the general form and equate the coefficients for effects , A, and B to zero, as shown in Table 15.8.

The last column in Table 15.8 represents the form of the MRH for A * B.

To obtain the Type III hypothesis for A, first start with the general form and equate the coefficients for effects and B to zero (let L1 = L4 = 0). Next let L6 = K L2, and find the value of K that makes the A hypothesis orthogonal to the A * B hypothesis. In this case, K = 0.5. Each of these steps is shown in Table 15.9.

In Table 15.9, the fourth column (under L6 = K L2) represents the form of all estimable functions not involving , B1, or B2. The prime difference between the Type II and Type III hypotheses for A is the way K is determined. Type II chooses K as a function of the cell frequencies, whereas Type III chooses K such that the estimable functions for A are orthogonal to the estimable functions for A * B.

An example of Type III estimable functions in a factorial with unequal cell frequencies and missing diagonals is given in Table 15.10 ( through represent the nonzero cell frequencies).

For any nonzero values of through , the Type III estimable functions for each effect are shown in Table 15.11.

By once again looking at the Type II hypotheses being tested in the balanced factorial (see Table 15.6), you can see another characteristic of the hypotheses employed for balanced designs: the coefficients of lower-order effects are averaged across each higher-level effect involving the same subscripts. For example, in the A hypothesis, the coefficients of AB11 and AB12 are equal to one-half the coefficient of A1, and the coefficients of AB21 and AB22 are equal to one-half the coefficient of A2. With this in mind, the basic concept used to construct Type IV hypotheses is that the coefficients of any effect, say F1, are distributed equitably across higher-level effects that contain F1. When missing cells occur, this same general philosophy is adhered to, but care must be taken in the way the distributive concept is applied.

Construction of Type IV hypotheses begins as does the construction of the Type III hypotheses. That is, for an effect F1, equate to zero all coefficients in the general form that do not belong to F1 or to any other effect containing F1. If F1 is not contained in any other effect, then the Type IV hypothesis (and Type II and III) has been found. If F1 is contained in other effects, then simplify, if necessary, the coefficients associated with F1 so that they are all free coefficients or functions of other free coefficients in the F1 block.

To illustrate the method of resolving the free coefficients outside the F1 block, suppose that you are interested in the estimable functions for an effect A and that A is contained in AB, AC, and ABC. (In other words, the main effects in the model are A, B, and C.)

With missing cells, the coefficients of intermediate effects (here they are AB and AC) do not always have an equal distribution of the lower-order coefficients, so the coefficients of the highest-order effects are determined first (here it is ABC). Once the highest-order coefficients are determined, the coefficients of intermediate effects are automatically determined.

The following process is performed for each free coefficient of A in turn. The resulting symbolic vectors are then added together to give the Type IV estimable functions for A.

An example of a factorial with four missing cells ( through represent positive cell frequencies) is shown in Table 15.12.

The Type IV estimable functions are shown in Table 15.13.

For the vast majority of designs, Type III and Type IV hypotheses for a given effect are the same. Specifically, they are the same for any effect F1 that is not contained in other effects for any design (with or without missing cells). For factorial designs with no missing cells, the Type III and Type IV hypotheses coincide for all effects. When there are missing cells, the hypotheses can differ. By using the GLM procedure, you can study the differences in the hypotheses and then decide on the appropriateness of the hypotheses for a particular model.

The Type III hypotheses for three-factor and higher completely nested designs with unequal Ns in the lowest level differ from the Type II hypotheses; however, the Type IV hypotheses do correspond to the Type II hypotheses in this case.

When missing cells occur in a design, the Type IV hypotheses might not be unique. If this occurs in PROC GLM, you are notified, and you might need to consider defining your own specific comparisons.