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The Four Types of Estimable Functions |
When an effect is contained in another effect, the Type II hypotheses for that effect are dependent on the cell frequencies. The philosophy behind both the Type III and Type IV hypotheses is that the hypotheses tested for any given effect should be the same for all designs with the same general form of estimable functions.
To demonstrate this concept, recall the hypotheses being tested by the Type II SS in the balanced 2 ×2 factorial shown in Table 12.6. Those hypotheses are precisely the ones that the Type III and Type IV hypotheses employ for all 2 ×2 factorials that have at least one observation per cell. The Type III and Type IV hypotheses for a design without missing cells usually differ from the hypothesis employed for the same design with missing cells since the general form of estimable functions usually differs.
If F1 is not contained in any other effect, this step defines the Type III hypothesis (as well as the Type II and Type IV hypotheses). If F1 is contained in other effects, go on to step 2. (See the section "Type II SS and Estimable Functions" for a definition of when effect F1 is contained in another effect.)
The Type III hypotheses are precisely the hypotheses being tested by programs that reparameterize using the usual assumptions (for example, all parameters for an effect summing 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 steps illustrate the construction of Type III estimable functions for a 2 ×2 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 12.8.
Table 12.8: Type III Hypothesis for A*B InteractionEffect | General Form | L1=L2=L4=0 | ||
L1 | 0 | |||
A1 | L2 | 0 | ||
A2 | L1-L2 | 0 | ||
B1 | L4 | 0 | ||
B2 | L1-L4 | 0 | ||
AB11 | L6 | L6 | ||
AB12 | L2-L6 | -L6 | ||
AB21 | L4-L6 | -L6 | ||
AB22 | L1-L2-L4+L6 | L6 |
The last column in Table 12.8 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 12.9.
In Table 12.9, 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.
Table 12.9: Type III Hypothesis for AEffect | General Form | L1=L4=0 | L6=K*L2 | K=0.5 | ||||
L1 | 0 | 0 | 0 | |||||
A1 | L2 | L2 | L2 | L2 | ||||
A2 | L1-L2 | -L2 | -L2 | -L2 | ||||
B1 | L4 | 0 | 0 | 0 | ||||
B2 | L1-L4 | 0 | 0 | 0 | ||||
AB11 | L6 | L6 | K*L2 | 0.5*L2 | ||||
AB12 | L2-L6 | L2-L6 | (1-K)*L2 | 0.5*L2 | ||||
AB21 | L4-L6 | -L6 | -K*L2 | -0.5*L2 | ||||
AB22 | L1-L2-L4+L6 | -L2+L6 | (K-1)*L2 | -0.5*L2 |
An example of Type III estimable functions in a 3 ×3 factorial with unequal cell frequencies and missing diagonals is given in Table 12.10 (N_{1} through N_{6} represent the nonzero cell frequencies).
Table 12.10: A 3 ×3 Factorial Design with Unequal Cell Frequencies and Missing DiagonalsB | ||||
1 | 2 | 3 | ||
1 | N_{1} | N_{2} | ||
A | 2 | N_{3} | N_{4} | |
3 | N_{5} | N_{6} |
For any nonzero values of N_{1} through N_{6}, the Type III estimable functions for each effect are shown in Table 12.11.
Table 12.11: Type III Estimable Functions for 3 ×3 Factorial Design with Unequal Cell Frequencies and Missing DiagonalsEffect | A | B | A*B | |||
0 | 0 | 0 | ||||
A1 | L2 | 0 | 0 | |||
A2 | L3 | 0 | 0 | |||
A3 | -L2-L3 | 0 | 0 | |||
B1 | 0 | L5 | 0 | |||
B2 | 0 | L6 | 0 | |||
B3 | 0 | -L5-L6 | 0 | |||
AB12 | 0.667*L2+0.333*L3 | 0.333*L5+0.667*L6 | L8 | |||
AB13 | 0.333*L2-0.333*L3 | -0.333*L5-0.667*L6 | -L8 | |||
AB21 | 0.333*L2+0.667*L3 | 0.667*L5+0.333*L6 | -L8 | |||
AB23 | -0.333*L2+0.333*L3 | -0.667*L5-0.333*L6 | L8 | |||
AB31 | -0.333*L2-0.667*L3 | 0.333*L5-0.333*L6 | L8 | |||
AB32 | -0.667*L2-0.333*L3 | -0.333*L5+0.333*L6 | -L8 |
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 of 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.
B | ||||
1 | 2 | 3 | ||
1 | N_{1} | N_{2} | ||
A | 2 | N_{3} | N_{4} | |
3 | N_{5} |
The Type IV estimable functions are shown in Table 12.13.
Table 12.13: Type IV Estimable Functions for 3 ×3 Factorial Design with Four Missing CellsEffect | A | B | A*B | |||
0 | 0 | 0 | ||||
A1 | -L3 | 0 | 0 | |||
A2 | L3 | 0 | 0 | |||
A3 | 0 | 0 | 0 | |||
B1 | 0 | L5 | 0 | |||
B2 | 0 | -L5 | 0 | |||
B3 | 0 | 0 | 0 | |||
AB11 | -0.5*L3 | 0.5*L5 | L8 | |||
AB12 | -0.5*L3 | -0.5*L5 | -L8 | |||
AB21 | 0.5*L3 | 0.5*L5 | -L8 | |||
AB22 | 0.5*L3 | -0.5*L5 | L8 | |||
AB33 | 0 | 0 | 0 |
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 may not be unique. If this occurs in PROC GLM, you are notified, and you may need to consider defining your own specific comparisons.
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