Homework 13
Gabriel Farias Cacao
28/11/2021
Question 8.2
Suppose that in Problem 6.15, only a one-half fraction of the 2^4 design could be run. Construct the design and perform the analysis, using the data from replicate I.
The half normal plot below shows that all factors and interactions are not significant.
Question 8.24
Construct a 2^(5-1) design. Show how the design may be run in two blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
Yes, blocks are confounded with the interactions AB and CDE. C <- FrF2(nfactors=5, blocks=2, nruns=16, randomize=FALSE, alias.block.2fis=TRUE)
summary(C)## Call:
## FrF2(nfactors = 5, blocks = 2, nruns = 16, randomize = FALSE,
## alias.block.2fis = TRUE)
##
## Experimental design of type FrF2.blocked
## 16 runs
## blocked design with 2 blocks of size 8
##
## Factor settings (scale ends):
## A B C D E
## 1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E
##
## $`generators for design itself`
## [1] E=ABCD
##
## $`block generators`
## [1] AB
##
##
## no aliasing of main effects or 2fis among experimental factors
##
## Aliased with block main effects:
## [1] AB
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E
## 1 1 5.1.1 1 -1 1 -1 -1 -1
## 2 2 6.1.2 1 -1 1 -1 1 1
## 3 3 7.1.3 1 -1 1 1 -1 1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 9.1.5 1 1 -1 -1 -1 -1
## 6 6 10.1.6 1 1 -1 -1 1 1
## 7 7 11.1.7 1 1 -1 1 -1 1
## 8 8 12.1.8 1 1 -1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 1.2.1 2 -1 -1 -1 -1 1
## 10 10 2.2.2 2 -1 -1 -1 1 -1
## 11 11 3.2.3 2 -1 -1 1 -1 -1
## 12 12 4.2.4 2 -1 -1 1 1 1
## 13 13 13.2.5 2 1 1 -1 -1 1
## 14 14 14.2.6 2 1 1 -1 1 -1
## 15 15 15.2.7 2 1 1 1 -1 -1
## 16 16 16.2.8 2 1 1 1 1 1
## class=design, type= FrF2.blocked
## NOTE: columns run.no and run.no.std.rp are annotation,
## not part of the data frameQuestion 8.25
Construct a 2^(7-2) design. Show how the design may be run in two blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
Yes, blocks are confounded with the interactions ACE, BFG, ABCEFG. D <- FrF2(nfactors=7, blocks=4, nruns=32, randomize=FALSE, alias.block.2fis=TRUE)
summary(D)## Call:
## FrF2(nfactors = 7, blocks = 4, nruns = 32, randomize = FALSE,
## alias.block.2fis = TRUE)
##
## Experimental design of type FrF2.blocked
## 32 runs
## blocked design with 4 blocks of size 8
##
## Factor settings (scale ends):
## A B C D E F G
## 1 -1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F G=G
##
## $`generators for design itself`
## [1] F=ABC G=ABDE
##
## $`block generators`
## [1] AB AC
##
##
## no aliasing of main effects or 2fis among experimental factors
##
## Aliased with block main effects:
## [1] AB AC AF BC BF CF
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 13.1.1 1 -1 1 1 -1 -1 -1 -1
## 2 2 14.1.2 1 -1 1 1 -1 1 -1 1
## 3 3 15.1.3 1 -1 1 1 1 -1 -1 1
## 4 4 16.1.4 1 -1 1 1 1 1 -1 -1
## 5 5 17.1.5 1 1 -1 -1 -1 -1 1 -1
## 6 6 18.1.6 1 1 -1 -1 -1 1 1 1
## 7 7 19.1.7 1 1 -1 -1 1 -1 1 1
## 8 8 20.1.8 1 1 -1 -1 1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.1 2 -1 1 -1 -1 -1 1 -1
## 10 10 10.2.2 2 -1 1 -1 -1 1 1 1
## 11 11 11.2.3 2 -1 1 -1 1 -1 1 1
## 12 12 12.2.4 2 -1 1 -1 1 1 1 -1
## 13 13 21.2.5 2 1 -1 1 -1 -1 -1 -1
## 14 14 22.2.6 2 1 -1 1 -1 1 -1 1
## 15 15 23.2.7 2 1 -1 1 1 -1 -1 1
## 16 16 24.2.8 2 1 -1 1 1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 5.3.1 3 -1 -1 1 -1 -1 1 1
## 18 18 6.3.2 3 -1 -1 1 -1 1 1 -1
## 19 19 7.3.3 3 -1 -1 1 1 -1 1 -1
## 20 20 8.3.4 3 -1 -1 1 1 1 1 1
## 21 21 25.3.5 3 1 1 -1 -1 -1 -1 1
## 22 22 26.3.6 3 1 1 -1 -1 1 -1 -1
## 23 23 27.3.7 3 1 1 -1 1 -1 -1 -1
## 24 24 28.3.8 3 1 1 -1 1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 1.4.1 4 -1 -1 -1 -1 -1 -1 1
## 26 26 2.4.2 4 -1 -1 -1 -1 1 -1 -1
## 27 27 3.4.3 4 -1 -1 -1 1 -1 -1 -1
## 28 28 4.4.4 4 -1 -1 -1 1 1 -1 1
## 29 29 29.4.5 4 1 1 1 -1 -1 1 1
## 30 30 30.4.6 4 1 1 1 -1 1 1 -1
## 31 31 31.4.7 4 1 1 1 1 -1 1 -1
## 32 32 32.4.8 4 1 1 1 1 1 1 1
## class=design, type= FrF2.blocked
## NOTE: columns run.no and run.no.std.rp are annotation,
## not part of the data frameQuestion 8.28
A 16-run experiment was performed in a semiconductor manufacturing plant to study the effects of six factors on the curvature or camber of the substrate devices produced. The six variables and their levels are shown in Table P8.2
Each run was replicated four times, and a camber measurement was taken on the substrate. The data are shown in Table P8.3.
A<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
E<-c(-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1)
F<-c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1)
obs<-c(629,192,176,223,223,920,389,900,201,341,126,640,455,371,603,460)
df<-data.frame(A,B,C,D,E,F,obs)
model<-lm(obs~A*B*C*D*E*F, data = df)
aliases(model)##
## A = B:C:E = C:D:F = A:B:D:E:F
## B = A:C:E = D:E:F = A:B:C:D:F
## C = A:B:E = A:D:F = B:C:D:E:F
## D = A:C:F = B:E:F = A:B:C:D:E
## E = B:D:F = A:C:D:E:F = A:B:C
## F = B:D:E = A:B:C:E:F = A:C:D
## A:B = B:C:D:F = A:D:E:F = C:E
## A:C = A:B:C:D:E:F = B:E = D:F
## B:C = A:B:D:F = C:D:E:F = A:E
## A:D = B:C:D:E = A:B:E:F = C:F
## B:D = A:C:D:E = A:B:C:F = E:F
## C:D = A:B:D:E = B:C:E:F = A:F
## D:E = A:B:C:D = A:C:E:F = B:F
## A:B:D = C:D:E = B:C:F = A:E:F
## B:C:D = A:D:E = A:B:F = C:E:FWhat type of design did the experimenters use?
The 2^(6-2), 16 run design at resolution IV.
What are the alias relationships in this design?
The defining relation is I=ABCE=ACDF=BDEF
A (ABCE)= BCE A (ACDF)= CDF A (BDEF)= ABCDEF A=BCE=CDF=ABDEF B (ABCE)= ACE B (ACDF)= ABCDF B (BDEF)= DEF B=ACE=ABCDF=DEF C (ABCE)= ABE C (ACDF)= ADF C (BDEF)= BCDEF C=ABE=ADF=BCDEF D (ABCE)= ABCDE D (ACDF)= ACF D (BDEF)= BEF D=ABCDE=ACF=BEF E (ABCE)= ABC E (ACDF)= ACDEF E (BDEF)= BDF E=ABC=ABDEF=BDF F (ABCE)= ABCEF F (ACDF)= ACD F (BDEF)= BDE F=ABCEF=ACD=BDE AB (ABCE)= CE AB (ACDF)= BCDF AB (BDEF)= ADEF AB=CE=BCDF=ADEF AC (ABCE)= BE AC (ACDF)= DF AC (BDEF)= ABCDEF AC=BE=DF=ABCDEF AD (ABCE)= BCDE AD (ACDF)= CF AD (BDEF)= ABEF AD=BCDE=CF=ABEF AE (ABCE)= BC AE (ACDF)= CDEF AE (BDEF)= ABDF AE=BC=CDEF=ABDF AF (ABCE)= BCEF AF (ACDF)= CD AF (BDEF)= ABDE AF=BCEF=CD=ABDE BD (ABCE)= ACDE BD (ACDF)= ABCF BD (BDEF)= EF BD=ACDE=ABCF=EF BF (ABCE)= ACEF BF (ACDF)= ABCD BF (BDEF)= DE BF=ACEF=ABCD=DE
Do any of the process variables affect average camber?
Yes, variables A = Lamination Temperature, C = Lamination Pressure, E = Firing Cycle Time and F = Firing Dew Point.
Do any of the process variables affect the variability in camber measurements?
Yes, A, B, F, and AF interaction affect the variability in camber measurements.
If it is important to reduce camber as much as possible, what recommendations would you make?
Lower the level of A, B (Low levels of B enables a lower variation and does not affect the camber) and C and higher the level of E and F.
Question 8.40
Consider the following experiment:
Answer the following questions about this experiment: (a) How many factors did this experiment investigate?
Four Factors are investigated.What is the resolution of this design?
Resolution 4.
Calculate the estimates of the main effects.
D <- FrF2(nfactors=4, resolution=4,randomize=FALSE)
responseD<-c(8,10,12,7,13,6,5,11)
E <- add.response(D,responseD)
modelD <- coef(lm(E))[-1]*2
modelD[1:4]## A1 B1 C1 D1
## -1.0 -0.5 -0.5 5.0What is the complete defining relation for this design?
ABCD
Question 8.48
Consider the following design.
What is the generator for column D?
It is ABC.
What is the generator for column E?
It is BC.
If the design were folded over, what is the resolution of the combined design?
Folding adds one resolutions, so the design becomes resolution IV.
Question 8.60
Consider a partial fold over for the 2^(7-4)III design. Suppose that the partial fold over of this design is constructed using column A (+ signs only). Determine the alias relationships in the combined design.
f <- FrF2(nfactors=7, resolution=3,randomize=FALSE)
G <- fold.design(f,column=1)
summary(G)## Multi-step-call:
## [[1]]
## FrF2(nfactors = 7, resolution = 3, randomize = FALSE)
##
## $fold
## [1] 1
##
##
## Experimental design of type FrF2.folded
## 16 runs
##
## Factor settings (scale ends):
## A B C fold D E F G
## 1 -1 -1 -1 original -1 -1 -1 -1
## 2 1 1 1 mirror 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=fold E=D F=E G=F H=G
##
##
## Alias structure:
## $main
## [1] B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE
##
## $fi2
## [1] AB=-DE AC=-DF AD=-BE=-CF=-GH AE=-BD AF=-CD
## [6] AG=-DH AH=-DG
##
##
## The design itself:
## A B C fold D E F G
## 1 -1 -1 -1 original 1 1 1 -1
## 2 1 -1 -1 original -1 -1 1 1
## 3 -1 1 -1 original -1 1 -1 1
## 4 1 1 -1 original 1 -1 -1 -1
## 5 -1 -1 1 original 1 -1 -1 1
## 6 1 -1 1 original -1 1 -1 -1
## 7 -1 1 1 original -1 -1 1 -1
## 8 1 1 1 original 1 1 1 1
## 9 1 -1 -1 mirror 1 1 1 -1
## 10 -1 -1 -1 mirror -1 -1 1 1
## 11 1 1 -1 mirror -1 1 -1 1
## 12 -1 1 -1 mirror 1 -1 -1 -1
## 13 1 -1 1 mirror 1 -1 -1 1
## 14 -1 -1 1 mirror -1 1 -1 -1
## 15 1 1 1 mirror -1 -1 1 -1
## 16 -1 1 1 mirror 1 1 1 1
## class=design, type= FrF2.folded