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112 2n a 112 4 (14-12)
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Finally, the AB interaction is estimated by taking the difference in the diagonal averages in Fig. 14-12, or 112
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The quantities in brackets in Equations 14-11, 14-12, and 14-13 are called contrasts. For example, the A contrast is ContrastA a ab b 112
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In these equations, the contrast coef cients are always either 1 or 1. A table of plus and minus signs, such as Table 14-12, can be used to determine the sign on each treatment
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Table 14-12 Signs for Effects in the 22 Design Treatment Combination 112 a b ab Factorial Effect I A B AB
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CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS
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combination for a particular contrast. The column headings for Table 14-12 are the main effects A and B, the AB interaction, and I, which represents the total. The row headings are the treatment combinations. Note that the signs in the AB column are the product of signs from columns A and B. To generate a contrast from this table, multiply the signs in the appropriate column of Table 14-12 by the treatment combinations listed in the rows and add. For example, contrastAB 31124 3 a4 3 b4 3ab4 ab 112 a b. Contrasts are used in calculating both the effect estimates and the sums of squares for A, B, and the AB interaction. The sums of squares formulas are 3a 3b 3ab 112 4 2 112 4 2 b4 2 (14-14)
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The analysis of variance is completed by computing the total sum of squares SST (with 4n degrees of freedom) as usual, and obtaining the error sum of squares SSE [with 4(n degrees of freedom] by subtraction. EXAMPLE 14-3
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An article in the AT&T Technical Journal (Vol. 65, March/April 1986, pp. 39 50) describes the application of two-level factorial designs to integrated circuit manufacturing. A basic processing step in this industry is to grow an epitaxial layer on polished silicon wafers. The wafers are mounted on a susceptor and positioned inside a bell jar. Chemical vapors are introduced through nozzles near the top of the jar. The susceptor is rotated, and heat is applied. These conditions are maintained until the epitaxial layer is thick enough. Table 14-13 presents the results of a 22 factorial design with n 4 replicates using the factors A deposition time and B arsenic ow rate. The two levels of deposition time are short and long, and the two levels of arsenic ow rate are 55% and 59%. The response variable is epitaxial layer thickness ( m). We may nd the estimates of the effects using Equations 14-11, 14-12, and 14-13 as follows: A 1 3a ab b 112 4 2n 1 359.299 59.156 2142 1 3b ab a 112 4 2n 1 355.686 59.156 2142 1 3ab 112 2n 1 359.156 2142 a b4 59.299 55.6864 0.032
AB AB
14-7 2k FACTORIAL DESIGNS
Table 14-13 The 22 Design for the Epitaxial Process Experiment Treatment Combination 112 a b ab Design Factors B AB 14.037 14.821 13.880 14.888 Thickness ( m) Total Average 56.081 14.020 59.299 14.825 55.686 13.922 59.156 14.789
Thickness ( m) 14.165 13.972 14.757 14.843 13.860 14.032 14.921 14.415
13.907 14.878 13.914 14.932
The numerical estimates of the effects indicate that the effect of deposition time is large and has a positive direction (increasing deposition time increases thickness), since changing deposition time from low to high changes the mean epitaxial layer thickness by 0.836 m. The effects of arsenic ow rate (B) and the AB interaction appear small. The importance of these effects may be con rmed with the analysis of variance. The sums of squares for A, B, and AB are computed as follows: SSA SSB SSAB SST 3a 3b 3ab ab ab b 16 a 16 112 4 2 112 4 2 b4 2 36.6884 2 16 2.7956 0.0181 0.0040 p 16 59.1562 2