Applied Statistics and Probability for Engineers

526 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

combination for a particular contrast. The column headings for Table 14-12 are the main ef-

fects Aand B, the ABinteraction, and I, which represents the total. The row headings are the

treatment combinations. Note that the signs in the ABcolumn are the product of signs from

columns Aand 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 a 4  3 b 4  3 ab 4 ab 112 ab.

Contrasts are used in calculating both the effect estimates and the sums of squares for A,

B, and the ABinteraction. The sums of squares formulas are

(14-14)

SSAB

3 ab 112 ab 42

4 n

SSB

3 baba 11242

4 n

SSA

3 aabb 11242

4 n

The analysis of variance is completed by computing the total sum of squares SST(with 4n 1

degrees of freedom) as usual, and obtaining the error sum of squares SSE[with 4(n1)

degrees of freedom] by subtraction.

EXAMPLE 14-3 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 pro-

cessing 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 intro-

duced 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 2^2 factorial design with n4 replicates using the

factors Adeposition time and Barsenic flow rate. The two levels of deposition time

areshort and  long, and the two levels of arsenic flow rate are  55% and 

59%. The response variable is epitaxial layer thickness (m). We may find the estimates of

the effects using Equations 14-11, 14-12, and 14-13 as follows:

AB

1

2142

3 59.15656.08159.29955.686 4 0.032

AB

1

2 n

3 ab 112 ab 4



1

2142

3 55.68659.15659.29956.081 4 0.067

B

1

2 n

3 baba 1124



1

2142

3 59.29959.15655.68656.081 4 0.836

A

1

2 n

3 aabb 1124

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