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Figure 14-15 Plot of residuals versus deposition time.
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Figure 14-14 Normal probability plot of residuals for the epitaxial process experiment.
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Figure 14-16 rate.
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The methods presented in the previous section for factorial designs with k 2 factors each at two levels can be easily extended to more than two factors. For example, consider k 3 factors, each at two levels. This design is a 23 factorial design, and it has eight runs or treatment combinations. Geometrically, the design is a cube as shown in Fig. 14-18(a), with the eight runs forming the corners of the cube. Figure 14-18(b) lists the eight runs in a table, with each row representing one of the runs are the and settings indicating the low and high levels
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Figure 14-17 The standard deviation of epitaxial layer thickness at the four runs in the 22 design.
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abc Run 1 2 A + + + + B + + + + C + + + +
C ab b (1) + a B
3 4 5 6 7 8
Figure 14-18 design.
The 23
(a) Geometric view
(b) The 23 design matrix
for each of the three factors. This table is sometimes called the design matrix. This design allows three main effects to be estimated (A, B, and C ) along with three two-factor interactions (AB, AC, and BC ) and a three-factor interaction (ABC ). The main effects can easily be estimated. Remember that the lowercase letters (1), a, b, ab, c, ac, bc, and abc represent the total of all n replicates at each of the eight runs in the design. As seen in Fig. 14-19(a), the main effect of A can be estimated by averaging the four treatment combinations on the right-hand side of the cube, where A is at the high level, and by
+ +
A B Main effects (a) C
+ + + + +
AB AC Two-factor interactions (b) BC
Figure 14-19 Geometric presentation of contrasts corresponding to the main effects and interaction in the 23 design. (a) Main effects. (b) Twofactor interactions. (c) Three-factor interaction.
C B A ABC Three-factor interaction (c)
= + runs = runs
subtracting from this quantity the average of the four treatment combinations on the left-hand side of the cube where A is at the low level. This gives A yA a yA ab ac 4n 112
b 4n
This equation can be rearranged as
1 3a 4n
In a similar manner, the effect of B is the difference in averages between the four treatment combinations in the back face of the cube (Fig. 14-19a), and the four in the front. This yields
yB yB 1 3b ab 4n
The effect of C is the difference in average response between the four treatment combinations in the top face of the cube in Figure 14-19(a) and the four in the bottom, that is,
yC yC 1 3c ac 4n
The two-factor interaction effects may be computed easily. A measure of the AB interaction is the difference between the average A effects at the two levels of B. By convention, one-half of this difference is called the AB interaction. Symbolically, B High ( ) Low ( ) Difference Average A Effect 3 1abc 5 1ac 3abc bc2 1ab b2 4 2n c2 3a 112 4 6 2n bc ab b ac 2n
112 4
Since the AB interaction is one-half of this difference,
1 3abc 4n
112 4
We could write Equation 14-18 as follows: AB abc ab c 4n 112 bc b 4n ac a
In this form, the AB interaction is easily seen to be the difference in averages between runs on two diagonal planes in the cube in Fig. 14-19(b). Using similar logic and referring to Fig. 14-19(b), we nd that the AC and BC interactions are
1 3 112 4n 1 3 112 4n
ab ab
ac ac
bc bc
abc4 abc4
(14-19) (14-20)
The ABC interaction is de ned as the average difference between the AB interaction for the two different levels of C. Thus, ABC or 1 5 3abc 4n bc4 3ac c4 3ab b4 3a 112 4 6
1 3abc 4n
112 4
As before, we can think of the ABC interaction as the difference in two averages. If the runs in the two averages are isolated, they de ne the vertices of the two tetrahedra that comprise the cube in Fig. 14-19(c). In Equations 14-15 through 14-21, the quantities in brackets are contrasts in the treatment combinations. A table of plus and minus signs can be developed from the contrasts and is shown in Table 14-15. Signs for the main effects are determined directly from the test matrix in Figure 14-18(b). Once the signs for the main effect columns have been established, the signs for the remaining columns can be obtained by multiplying the appropriate