Applied Statistics and Probability for Engineers

554 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

Computer Solution

Fractional factorial designs are usually analyzed with a software package. Table 14-26 shows

the effect estimates obtained from Minitab for Example 14-7. They are in agreement with the

hand calculation reported earlier.

Normal Probability Plot of Effects

The normal probability plot is very useful in assessing the significance of effects from a frac-

tional factorial design, particularly when many effects are to be estimated. We strongly rec-

ommend examining this plot. Figure 14-30 presents the normal probability plot of the effects

from Example 14-7. This plot was obtained from Minitab. Notice that the A, D, and ADinter-

action effects stand out clearly in this graph.

Residual Analysis

The residuals can be obtained from a fractional factorial by the regression model method

shown previously. Note that the Minitab output for Example 14-7 in Table 14-26 shows the

regression coefficients. The residuals should be graphically analyzed as we have discussed

before, both to assess the validity of the underlying model assumptions and to gain additional

insight into the experimental situation.

Projection of the 2k^1 Design

If one or more factors from a one-half fraction of a 2kcan be dropped, the design will project

into a full factorial design. For example, Fig. 14-31 presents a 2^3 ^1 design. Notice that this de-

sign will project into a full factorial in any two of the three original factors. Thus, if we think

that at most two of the three factors are important, the 2^3 ^1 design is an excellent design for

identifying the significant factors. This projection propertyis highly useful in factor screen-

ing, because it allows negligible factors to be eliminated, resulting in a stronger experiment in

the active factors that remain.

In the 2^4 ^1 design used in the plasma etch experiment in Example 14-7, we found that two

of the four factors (Band C) could be dropped. If we eliminate these two factors, the remain-

ing columns in Table 14-25 form a 2^2 design in the factors Aand D, with two replicates. This

design is shown in Fig. 14-32. The main effects of Aand Dand the strong two-factor ADin-

teraction are clearly evident from this graph.

Table 14-26 Effect Estimates from Minitab,

Example 14-7

Fractional Factorial Fit

Estimated Effects and Coefficients for Etch Rt

Term Effect Coef

Constant 756.00

Gap 127.00 63.50

Pressure 4.00 2.00

F. 11.50 5.75

Power 290.50 145.25

Gap*Pressure 10.00 5.00

Gap*F. 25.50 12.75

Gap*Power 197.50 98.75

_ 200

_1.0

_1.5

_0.5

0.0

0.5

1.0

1.5

_ 100 0 100 200 300

AD

D

A

Effect

Normal score

Figure 14-30 Normal probability plot of

the effects from Minitab, Example 14-7.

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