6: II 7: III
5.5 Alternative Screening Designs
In resolution III fractional factorial 2k−p designs some main effects will be completely confounded with two-factor interactions, and in resolution IV 2k−p designs some two-factor interactions will be completely confounded with other two-factor interactions. The confounding makes the analysis of these designs tricky, and follow-up experiments may be required to get independent esti- mates of the important main effects and two-factor interactions.
For experiments with 6, 7, or 8 factors, alternative screening designs in 16 runs require no more runs than resolution III or resolution IV 2k−p designs.
However, these designs do not have complete confounding among main effects and two-factor interactions. They have complex confounding which means that main effects are only partially confounded with two-factor interactions.
Models fit to designs with complex confounding can contain some main effects and some two-factor interactions. An appropriate subset of main effects and interactions can usually be found be using forward stepwise regression.
Hamada and Wu[37] and Jones and Nachtsheim[50] suggested using a for- ward stepwise regression that enforces model hierarchy. In other words, when an interaction is the next term to enter the model, the parent linear main effect(s) are automatically included, if they are not already in the model. The number of steps in the forward selection procedure is the number of steps until the last term entered is still significant. When the term entered at the current step is not significant, revert to the previous step to obtain the final model.
This procedure greatly simplifies the model fitting for alternative screening designs.
The Altscreen() function in the daewr package can be used to recall 16 run alternative screening designs from Jones and Montgomery[49]. The
HierAFS()function in the same package can be used to fit a model using a forward stepwise regression that enforces model hierarchy.
5.5.1 Example
Montgomery[73] presented an example of a 16-run 26IV−2 fractional factorial experiment for studying factors that affected the thickness of a photoresist layer applied to a silicon wafer in a spin-coater. The factors and levels are shown inTable 5.10
TABLE 5.10: Factors and Levels in Spin-Coater Experiment
Factors Level (−) Level (+)
A–spin speed − +
B–acceleration − +
C–volume of resist applied − +
D–spin time − +
E–resist viscosity − +
F–exhaust rate − +
In this example, the analysis showed that main effects A, B, C, and E, along with the confounded string of two-factor interactions AB=CE, were significant. However, since both main effects A and B and main effects C and E were significant, the effect-heredity principle did not help in trying to determine whether the CE interaction or the AB interaction was causing the significance of the confounded string. This situation is different from the last example. In that example, main effects C, F, and D were significant along with the confounded string of two-factor interaction AD=CF=EG. Since C and F were the largest main effects, the effect-heredity principle would lead one to believe that CF was causing the significance of the confounded string of interactions.
Since neither the hierarchical ordering principle nor the effect-heredity principle made it clear what interaction was causing the significance of the confounded string AB=CE in the present example, 16 additional follow-up experiments were run to determine which interaction was causing the signifi- cance and to identify the optimal factor settings. The follow-up experiments were selected according to a fold-over plan Lawson[59], and the analysis of the combined set of 32 experiments showed that the four main effects found in the analysis of the original 16 experiments were significant along with the CE interaction.
To demonstrate the value of an Alternative Screening Design, Johnson and Jones[47] considered the same situation as this example. They used the model found after analysis of the combined 32 experiments and simulated data for the photo-resist thickness. In the code below, theAltscreen()function in the daewrpackage was used to generate the same six-factor alternative screening
design used in Johnson and Jones’ article, and the thickness data was their simulated data. TheHierAFS()function identified the correct model for the simulated data in three steps, and the variable that would have entered in the fourth step,BE, was not significant (p-value=0.11327). Had the original experiment been conducted with this design, no follow-up experiments would have been required, and the conclusions would have been reached with half the number of experiments.
R>library(daewr)
R>Design<-Altscreen(nfac=6,randomize=FALSE)
R>Thickness<-c(4494,4592,4357,4489,4513,4483,4288, 4448,4691,4671,4219,4271,4530,4632, 4337,4391)
R>cat("Table of Design and Response") R>cbind(Design,Thickness)
R>HierAFS(Thickness,Design,m=0,c=6,step=4)
formula R2
y~A 0.660
y~B+A 0.791
y~C+E+C:E+B+A 0.953 y~B+E+B:E+C+C:E+A 0.965
The R code below shows the calculated effects (from the lm() function) for the model found by Johnson and Jones[47] and by using three steps of the HierAFS()function. The assumptions of the least-squares fit can be checked using theplot.lm()function, as shown in the block of code aboveFigure 5.7.
R> mod<-lm(Thickness~A+B+C+E+C:E,data=Design) R> summary(mod)
Call:
lm(formula = Thickness ~ A + B + C + E + C:E, data = Design) Residuals:
Min 1Q Median 3Q Max
-56.625 -19.625 2.625 23.125 53.875 Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 4462.875 9.494 470.073 < 2e-16 ***
A 85.500 11.628 7.353 2.44e-05 ***
B -77.750 11.628 -6.687 5.45e-05 ***
C -34.250 9.494 -3.608 0.00479 **
E 21.500 9.494 2.265 0.04700 *
C:E 54.750 13.427 4.078 0.00222 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 37.98 on 10 degrees of freedom
Multiple R-squared: 0.9533,Adjusted R-squared: 0.93 F-statistic: 40.85 on 5 and 10 DF, p-value: 2.447e-06
A=spin speed had a positive effect on thickness, and increasing spin speed increased thickness. B=acceleration had a negative effect on thickness, and increasing acceleration decreased thickness. Because of the significant interac- tion, it is difficult to say from the main effects what combinations of C=volume of resist applied, and E=resist viscosity would increase or decrease the thick- ness. The code below produces the interaction plot shown inFigure 5.18. This plot makes it clear.
R>resist_viscosity<-Design$E
R>interaction.plot(Design$C,resist_viscosity,Thickness, type="b",xlab="volume of resist applied",
pch=c(1,2),col=c(1,2))
FIGURE 5.18: CE Interaction Plot
The greatest thickness results with a low C=volume of resist applied and a low E=resist viscosity, and the smallest thickness is produced using low E=resist viscosity but a high C=volume of resist applied.