Applied Statistics and Probability for Engineers

528 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

since the only active variable is deposition time, which is represented by a coded variable x 1.

The low and high levels of deposition time are assigned values x 1 1 and x 1 1,

respectively. The least squares fitted model is

where the intercept is the grand average of all 16 observations ( ) and the slope is one-

half the effect estimate for deposition time. (The regression coefficient is one-half the effect

estimate because regression coefficients measure the effect of a unit change in x 1 on the mean

of Y, and the effect estimate is based on a two-unit change from1 to1.)

This model can be used to obtain the predicted values at the four points that form the cor-

ners of the square in the design. For example, consider the point with low deposition time

(x 1 1) and low arsenic flow rate. The predicted value is

and the residuals for the four runs at that design point are

e 1 14.03713.9710.066

e 2 14.16513.9710.194

e 3 13.97213.9710.001

e 4 13.90713.9710.064

The remaining predicted values and residuals at the other three design points are calculated in

a similar manner.

A normal probability plot of these residuals is shown in Fig. 14-14. This plot indicates

that one residual e 15 0.392 is an outlier.Examining the four runs with high deposition

time and high arsenic flow rate reveals that observation y 15 14.415 is considerably smaller

than the other three observations at that treatment combination. This adds some additional

evidence to the tentative conclusion that observation 15 is an outlier. Another possibility is

that some process variables affect the variabilityin epitaxial layer thickness. If we could dis-

cover which variables produce this effect, we could perhaps adjust these variables to levels

that would minimize the variability in epitaxial layer thickness. This could have important im-

plications in subsequent manufacturing stages. Figures 14-15 and 14-16 are plots of residuals

versus deposition time and arsenic flow rate, respectively. Apart from that unusually large

residual associated with y 15 , there is no strong evidence that either deposition time or arsenic

flow rate influences the variability in epitaxial layer thickness.

Figure 14-17 shows the standard deviation of epitaxial layer thickness at all four runs in

the 2^2 design. These standard deviations were calculated using the data in Table 14-13. Notice

that the standard deviation of the four observations with Aand Bat the high level is consider-

ably larger than the standard deviations at any of the other three design points. Most of this

difference is attributable to the unusually low thickness measurement associated with y 15. The

standard deviation of the four observations with Aand Bat the low level is also somewhat

larger than the standard deviations at the remaining two runs. This could indicate that other

process variables not included in this experiment may affect the variability in epitaxial layer

thickness. Another experiment to study this possibility, involving other process variables,

could be designed and conducted. (The original paper in the AT&T Technical Journalshows

that two additional factors, not considered in this example, affect process variability.)

yˆ14.389a

0.836

2

b 1  12 13.971 m

ˆ 0 y ˆ 1

yˆ14.389a

0.836

2

b x 1

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