DOE HW WEEK 13
Yashwanth, Palash Jariwala
2022-12-07
DOE HOMEWORK WEEK 13
Question-8.2
Construct the design and perform the analysis, using the data from replicate I
library(FrF2)## Warning: package 'FrF2' was built under R version 4.2.2## Loading required package: DoE.base## Warning: package 'DoE.base' was built under R version 4.2.2## Loading required package: grid## Loading required package: conf.design## Registered S3 method overwritten by 'DoE.base':
## method from
## factorize.factor conf.design##
## Attaching package: 'DoE.base'## The following objects are masked from 'package:stats':
##
## aov, lm## The following object is masked from 'package:graphics':
##
## plot.design## The following object is masked from 'package:base':
##
## lengthsOne half fraction design for 2^4 design which gives 2^(K-p) where K=4 and P=1 and I is the replicateI which gives relation between A,B,C,D
dsn <- FrF2(nfactors = 4,resolution = 4,randomize = FALSE)
aliasprint(dsn)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCTaking the data from problem 6.15
repI<-c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
repI## [1] 7.037 16.867 13.876 17.273 11.846 4.368 9.360 15.653dsn## A B C D
## 1 -1 -1 -1 -1
## 2 1 -1 -1 1
## 3 -1 1 -1 1
## 4 1 1 -1 -1
## 5 -1 -1 1 1
## 6 1 -1 1 -1
## 7 -1 1 1 -1
## 8 1 1 1 1
## class=design, type= FrF2dsnrepI<-add.response(dsn,repI)
dsnrepI## A B C D repI
## 1 -1 -1 -1 -1 7.037
## 2 1 -1 -1 1 16.867
## 3 -1 1 -1 1 13.876
## 4 1 1 -1 -1 17.273
## 5 -1 -1 1 1 11.846
## 6 1 -1 1 -1 4.368
## 7 -1 1 1 -1 9.360
## 8 1 1 1 1 15.653
## class=design, type= FrF2summary(dsnrepI)## Call:
## FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
##
## Experimental design of type FrF2
## 8 runs
##
## Factor settings (scale ends):
## A B C D
## 1 -1 -1 -1 -1
## 2 1 1 1 1
##
## Responses:
## [1] repI
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
##
## $generators
## [1] D=ABC
##
##
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
##
##
## The design itself:
## A B C D repI
## 1 -1 -1 -1 -1 7.037
## 2 1 -1 -1 1 16.867
## 3 -1 1 -1 1 13.876
## 4 1 1 -1 -1 17.273
## 5 -1 -1 1 1 11.846
## 6 1 -1 1 -1 4.368
## 7 -1 1 1 -1 9.360
## 8 1 1 1 1 15.653
## class=design, type= FrF2Doing a plot to check the factor significance
halfnormal(dsnrepI)##
## The following effects are completely aliased:## [1] B:C B:D C:D## no significant effects
MEPlot(dsnrepI)
From the halfnormal probability plot we can see The effects are aliased and no significant effects in the factors affecting cracklength The main affects of B and D are slightly different mean deviation than A & C
Question-8.24
Constructing a 2^(5-1) design to Show how the design may be run in two blocks of eight observations each. Testing whether there are any main affects of Two-factor interactions confounded with blocks Where K=5 and p=1
dgn<-FrF2(nfactors = 5,nruns = 16,blocks = 2,randomize = TRUE)
dgn## run.no run.no.std.rp Blocks A B C D E
## 1 1 15.1.8 1 1 1 1 -1 1
## 2 2 1.1.1 1 -1 -1 -1 -1 -1
## 3 3 13.1.7 1 1 1 -1 -1 -1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 6.1.3 1 -1 1 -1 1 1
## 6 6 10.1.5 1 1 -1 -1 1 1
## 7 7 12.1.6 1 1 -1 1 1 -1
## 8 8 3.1.2 1 -1 -1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 16.2.8 2 1 1 1 1 1
## 10 10 9.2.5 2 1 -1 -1 -1 1
## 11 11 11.2.6 2 1 -1 1 -1 -1
## 12 12 14.2.7 2 1 1 -1 1 -1
## 13 13 4.2.2 2 -1 -1 1 1 1
## 14 14 5.2.3 2 -1 1 -1 -1 1
## 15 15 7.2.4 2 -1 1 1 -1 -1
## 16 16 2.2.1 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 framealiasprint(dgn)## $legend
## [1] A=A B=B C=C D=D E=E
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE AC=BE AE=BCsummary(dgn)## Call:
## FrF2(nfactors = 5, nruns = 16, blocks = 2, randomize = 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=ABC
##
## $`block generators`
## [1] ABD
##
##
## Alias structure:
## $fi2
## [1] AB=CE AC=BE AE=BC
##
## Aliased with block main effects:
## [1] none
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E
## 1 1 15.1.8 1 1 1 1 -1 1
## 2 2 1.1.1 1 -1 -1 -1 -1 -1
## 3 3 13.1.7 1 1 1 -1 -1 -1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 6.1.3 1 -1 1 -1 1 1
## 6 6 10.1.5 1 1 -1 -1 1 1
## 7 7 12.1.6 1 1 -1 1 1 -1
## 8 8 3.1.2 1 -1 -1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 16.2.8 2 1 1 1 1 1
## 10 10 9.2.5 2 1 -1 -1 -1 1
## 11 11 11.2.6 2 1 -1 1 -1 -1
## 12 12 14.2.7 2 1 1 -1 1 -1
## 13 13 4.2.2 2 -1 -1 1 1 1
## 14 14 5.2.3 2 -1 1 -1 -1 1
## 15 15 7.2.4 2 -1 1 1 -1 -1
## 16 16 2.2.1 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 frameFrom the design we can see there are no main effects or two factor interactions confounded with the blocks The affect confounded with the block is ABD
Question-8.25
Constructing a 2^(7-2) design to show if design may be run in four blocks of eight observations each Where k=7 and p=2
dmn<-FrF2(nfactors = 7,nruns = 32,blocks = 4,randomize = TRUE)
dmn## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 2 2 20.1.5 1 1 -1 -1 1 1 1 -1
## 3 3 16.1.4 1 -1 1 1 1 1 -1 -1
## 4 4 27.1.7 1 1 1 -1 1 -1 -1 1
## 5 5 22.1.6 1 1 -1 1 -1 1 -1 1
## 6 6 10.1.3 1 -1 1 -1 -1 1 1 1
## 7 7 29.1.8 1 1 1 1 -1 -1 1 -1
## 8 8 7.1.2 1 -1 -1 1 1 -1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 30.2.8 2 1 1 1 -1 1 1 -1
## 10 10 28.2.7 2 1 1 -1 1 1 -1 1
## 11 11 8.2.2 2 -1 -1 1 1 1 1 1
## 12 12 9.2.3 2 -1 1 -1 -1 -1 1 1
## 13 13 2.2.1 2 -1 -1 -1 -1 1 -1 -1
## 14 14 21.2.6 2 1 -1 1 -1 -1 -1 1
## 15 15 15.2.4 2 -1 1 1 1 -1 -1 -1
## 16 16 19.2.5 2 1 -1 -1 1 -1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 14.3.4 3 -1 1 1 -1 1 -1 1
## 18 18 31.3.8 3 1 1 1 1 -1 1 1
## 19 19 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 20 20 24.3.6 3 1 -1 1 1 1 -1 -1
## 21 21 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 22 22 18.3.5 3 1 -1 -1 -1 1 1 1
## 23 23 12.3.3 3 -1 1 -1 1 1 1 -1
## 24 24 3.3.1 3 -1 -1 -1 1 -1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 17.4.5 4 1 -1 -1 -1 -1 1 1
## 26 26 26.4.7 4 1 1 -1 -1 1 -1 -1
## 27 27 6.4.2 4 -1 -1 1 -1 1 1 -1
## 28 28 11.4.3 4 -1 1 -1 1 -1 1 -1
## 29 29 23.4.6 4 1 -1 1 1 -1 -1 -1
## 30 30 4.4.1 4 -1 -1 -1 1 1 -1 1
## 31 31 32.4.8 4 1 1 1 1 1 1 1
## 32 32 13.4.4 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 framealiasprint(dmn)## $legend
## [1] A=A B=B C=C D=D E=E F=F G=G
##
## $main
## character(0)
##
## $fi2
## [1] AB=CF=DG AC=BF AD=BG AF=BC AG=BD CD=FG CG=DFsummary(dmn)## Call:
## FrF2(nfactors = 7, nruns = 32, blocks = 4, randomize = 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=ABD
##
## $`block generators`
## [1] ACD ABE
##
##
## Alias structure:
## $fi2
## [1] AB=CF=DG AC=BF AD=BG AF=BC AG=BD CD=FG CG=DF
##
## Aliased with block main effects:
## [1] none
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 2 2 20.1.5 1 1 -1 -1 1 1 1 -1
## 3 3 16.1.4 1 -1 1 1 1 1 -1 -1
## 4 4 27.1.7 1 1 1 -1 1 -1 -1 1
## 5 5 22.1.6 1 1 -1 1 -1 1 -1 1
## 6 6 10.1.3 1 -1 1 -1 -1 1 1 1
## 7 7 29.1.8 1 1 1 1 -1 -1 1 -1
## 8 8 7.1.2 1 -1 -1 1 1 -1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 30.2.8 2 1 1 1 -1 1 1 -1
## 10 10 28.2.7 2 1 1 -1 1 1 -1 1
## 11 11 8.2.2 2 -1 -1 1 1 1 1 1
## 12 12 9.2.3 2 -1 1 -1 -1 -1 1 1
## 13 13 2.2.1 2 -1 -1 -1 -1 1 -1 -1
## 14 14 21.2.6 2 1 -1 1 -1 -1 -1 1
## 15 15 15.2.4 2 -1 1 1 1 -1 -1 -1
## 16 16 19.2.5 2 1 -1 -1 1 -1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 14.3.4 3 -1 1 1 -1 1 -1 1
## 18 18 31.3.8 3 1 1 1 1 -1 1 1
## 19 19 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 20 20 24.3.6 3 1 -1 1 1 1 -1 -1
## 21 21 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 22 22 18.3.5 3 1 -1 -1 -1 1 1 1
## 23 23 12.3.3 3 -1 1 -1 1 1 1 -1
## 24 24 3.3.1 3 -1 -1 -1 1 -1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 17.4.5 4 1 -1 -1 -1 -1 1 1
## 26 26 26.4.7 4 1 1 -1 -1 1 -1 -1
## 27 27 6.4.2 4 -1 -1 1 -1 1 1 -1
## 28 28 11.4.3 4 -1 1 -1 1 -1 1 -1
## 29 29 23.4.6 4 1 -1 1 1 -1 -1 -1
## 30 30 4.4.1 4 -1 -1 -1 1 1 -1 1
## 31 31 32.4.8 4 1 1 1 1 1 1 1
## 32 32 13.4.4 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 frameThere are no main and two-factor interaction affects confounded with the block The affects confounded with block are ACD & ABE
Question-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.
Part-a
In this experiment a 2^(6-2) design is used by the experimenters Where k=6,p=2
drt<-FrF2(nfactors = 6,nruns = 16,randomize = FALSE)
drt## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 1
## 4 1 1 -1 -1 -1 -1
## 5 -1 -1 1 -1 1 -1
## 6 1 -1 1 -1 -1 1
## 7 -1 1 1 -1 -1 1
## 8 1 1 1 -1 1 -1
## 9 -1 -1 -1 1 -1 1
## 10 1 -1 -1 1 1 -1
## 11 -1 1 -1 1 1 -1
## 12 1 1 -1 1 -1 1
## 13 -1 -1 1 1 1 1
## 14 1 -1 1 1 -1 -1
## 15 -1 1 1 1 -1 -1
## 16 1 1 1 1 1 1
## class=design, type= FrF2summary(drt)## Call:
## FrF2(nfactors = 6, nruns = 16, randomize = FALSE)
##
## Experimental design of type FrF2
## 16 runs
##
## Factor settings (scale ends):
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $generators
## [1] E=ABC F=ABD
##
##
## Alias structure:
## $fi2
## [1] AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DE
##
##
## The design itself:
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 1
## 4 1 1 -1 -1 -1 -1
## 5 -1 -1 1 -1 1 -1
## 6 1 -1 1 -1 -1 1
## 7 -1 1 1 -1 -1 1
## 8 1 1 1 -1 1 -1
## 9 -1 -1 -1 1 -1 1
## 10 1 -1 -1 1 1 -1
## 11 -1 1 -1 1 1 -1
## 12 1 1 -1 1 -1 1
## 13 -1 -1 1 1 1 1
## 14 1 -1 1 1 -1 -1
## 15 -1 1 1 1 -1 -1
## 16 1 1 1 1 1 1
## class=design, type= FrF2Part-b
aliasprint(drt)## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DEPart-c
To check whether the process variables affects the average camber
Taking the data of the camber for replicate
obs<-c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.0110,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.0090,0.0250,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.0040,0.0147,0.0092,0.0226,0.0077,0.0060,0.0028,0.0158,0.0101,0.0126,0.0145,0.0110,0.0185,0.0020,0.0050,0.0030,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
R1 <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
R2 <- c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
R3 <- c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
R4 <- c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
R5 <- c(-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1)
R6 <- c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1)Taking the data as factors
R1 <- as.factor(R1)
R2 <- as.factor(R2)
R3 <- as.factor(R3)
R4 <- as.factor(R4)
R5 <- as.factor(R5)
R6 <- as.factor(R6)Making the data into data frame
RTD <- data.frame(R1,R2,R3,R4,R5,R6,obs)Doing the anova analysis to test whether the variables are affecting the average camber
av.mdl <- aov(obs~R1*R2*R3*R4*R5*R6,data = RTD)
summary(av.mdl)## Df Sum Sq Mean Sq F value Pr(>F)
## R1 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## R2 1 0.0000053 0.0000053 0.614 0.43725
## R3 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## R4 1 0.0000323 0.0000323 3.712 0.05995 .
## R5 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## R6 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## R1:R2 1 0.0000587 0.0000587 6.738 0.01249 *
## R1:R3 1 0.0000803 0.0000803 9.218 0.00387 **
## R2:R3 1 0.0000527 0.0000527 6.053 0.01754 *
## R1:R4 1 0.0000239 0.0000239 2.741 0.10431
## R2:R4 1 0.0000849 0.0000849 9.739 0.00305 **
## R3:R4 1 0.0000622 0.0000622 7.139 0.01027 *
## R4:R5 1 0.0000088 0.0000088 1.007 0.32062
## R1:R2:R4 1 0.0000000 0.0000000 0.005 0.94291
## R2:R3:R4 1 0.0000481 0.0000481 5.523 0.02293 *
## Residuals 48 0.0004183 0.0000087
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The variables affecting the Average camber are R1(Lamination temperature),R3(Lamination Pressure),R5(Firing cycle time),R6(Firing Dew Point)
Part-d
Process variables affecting the variability in camber measurements Taking the data
std <- c(24.418,20.976,4.083,25.025,22.41,63.639,16.029,39.42,26.725,50.341,7.681,20.083,31.12,29.51,6.75,17.45)
RTDSD <- data.frame(R1,R2,R3,R4,R5,R6,std)
RTDSD## R1 R2 R3 R4 R5 R6 std
## 1 -1 -1 -1 -1 -1 -1 24.418
## 2 1 -1 -1 -1 1 1 20.976
## 3 -1 1 -1 -1 1 -1 4.083
## 4 1 1 -1 -1 -1 1 25.025
## 5 -1 -1 1 -1 1 1 22.410
## 6 1 -1 1 -1 -1 -1 63.639
## 7 -1 1 1 -1 -1 1 16.029
## 8 1 1 1 -1 1 -1 39.420
## 9 -1 -1 -1 1 -1 1 26.725
## 10 1 -1 -1 1 1 -1 50.341
## 11 -1 1 -1 1 1 1 7.681
## 12 1 1 -1 1 -1 -1 20.083
## 13 -1 -1 1 1 1 -1 31.120
## 14 1 -1 1 1 -1 1 29.510
## 15 -1 1 1 1 -1 -1 6.750
## 16 1 1 1 1 1 1 17.450Doing an anova analysis
aovb<-aov(std~R1+R2+R3+R4+R5+R6)
summary(aovb)## Df Sum Sq Mean Sq F value Pr(>F)
## R1 1 1011.7 1011.7 9.101 0.0146 *
## R2 1 1099.2 1099.2 9.889 0.0118 *
## R3 1 138.0 138.0 1.242 0.2940
## R4 1 43.4 43.4 0.390 0.5478
## R5 1 21.9 21.9 0.197 0.6680
## R6 1 342.7 342.7 3.083 0.1130
## Residuals 9 1000.4 111.2
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1MEPlot(aovb)
From the anova analysis and main effect plot we can see that R1(Lamination temperature) & R2(Lamination time) affect the variability in camber measurements
Part-e
Recommendations to reduce camber as much as possible
coefficients(av.mdl)## (Intercept) R11 R21 R31 R41 R51
## 0.015725000 0.001009375 -0.007137500 0.001784375 -0.002953125 -0.004187500
## R61 R11:R21 R11:R31 R21:R31 R11:R41 R21:R41
## -0.007746875 0.003725000 0.004481250 0.007100000 -0.002550000 0.007968750
## R31:R41 R41:R51 R11:R21:R41 R21:R31:R41
## -0.000475000 0.001481250 0.000212500 -0.006937500The equation of this experiment \[Y_{ijkl}=0.015725000(Intercept)+0.001009375(Lamination temp)+0.001784375(Lamination Pressure)+(-0.004187500)(Firing Cycle)+(-0.007746875)(Firing Dew point)\]
From the experiment data we can say that the process variables affecting the camber should be be balanced in such a way that it does not affect the average camber.
QUESTION 8.40
Consider the following experiment:
Part A
How many factors did this experiment investigate? This experiment investigated “Four” Factors.
Part B
What is the resolution of this design? Resolution of the design employed in this experiment is “Four”.
Part C
Effects are as under:
One <- c(8)
AD <- c(10)
BD <- c(12)
AB <- c(7)
CD <- c(13)
AC <- c(6)
BC <- c(5)
ABCD <- c(11)EffA <- (2*(AD+AB+AC+ABCD-One-BD-CD-BC))/(16)
EffA## [1] -0.5EffB <- (2*(BD+AB+BC+ABCD-One-AD-CD-AC))/(16)
EffB## [1] -0.25EffC <- (2*(CD+AC+BC+ABCD-One-AD-BD-AB))/(16)
EffC## [1] -0.25EffD <- (2*(AD+BD+CD+ABCD-One-AB-AC-BC))/(16)
EffD## [1] 2.5Part D Defining relation for this design is I = ABCD.
Question 8.48
Consider the following design:
Part A
What is the generator for column D? If we tally the readings in column D we can see that the design generator for column D is -ABC.
Part B
What is the generator for column E? If we tally the readings in column E we can see that the design generator for column E is BC.
Part C
If this design were folded over, what is the resolution of the combined design? Resolution of folded over design is “Four”.
Question 8.60
Consider a partial fold over for the 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.
dsg <- FrF2(nfactors=7,resolution=3,randomize=FALSE)
dsg ## A B C D E F G
## 1 -1 -1 -1 1 1 1 -1
## 2 1 -1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 -1 1
## 4 1 1 -1 1 -1 -1 -1
## 5 -1 -1 1 1 -1 -1 1
## 6 1 -1 1 -1 1 -1 -1
## 7 -1 1 1 -1 -1 1 -1
## 8 1 1 1 1 1 1 1
## class=design, type= FrF2dsg2 <- fold.design(dsg,column=1)
dsg2## 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.foldeddsg3 <- dsg2[-c(1,3,5,7,10,12,14,16),]
dsg3## A B C fold D E F G
## 2 1 -1 -1 original -1 -1 1 1
## 4 1 1 -1 original 1 -1 -1 -1
## 6 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
## 11 1 1 -1 mirror -1 1 -1 1
## 13 1 -1 1 mirror 1 -1 -1 1
## 15 1 1 1 mirror -1 -1 1 -1aliasprint(dsg2)## $legend
## [1] A=A B=B C=C D=fold E=D F=E G=F H=G
##
## $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