Homework13
Jestin Job Sibychan
28/11/2021
library(FrF2)## Warning: package 'FrF2' was built under R version 4.1.2## Loading required package: DoE.base## 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':
##
## lengthslibrary(DoE.base)
library(GAD)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.8.2)
des.res4<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res4## 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= FrF2response<-c(7.037,16.867,13.876,17.273,11.846,4.368,9.36,15.653)
des.resp4<-add.response(des.res4,response)
des.resp4## A B C D response
## 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= FrF2aliasprint(des.resp4)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCsummary(des.resp4)## 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] response
##
## 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 response
## 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= FrF2DanielPlot(des.resp4)
Here we can see that the resolution 4 is used and so the main effects are not aliased with other maineffects and also not alliased with two factor interactions
From the plot we can say that no factors are significant
8.24
des.res5<-FrF2(nruns = 16, nfactors=5,resolution=5,randomize=FALSE,blocks = 2)## Warning in FrF2(nruns = 16, nfactors = 5, resolution = 5, randomize = FALSE, :
## resolution is ignored, if nruns is given.des.res5## run.no run.no.std.rp Blocks A B C D E
## 1 1 1.1.1 1 -1 -1 -1 -1 -1
## 2 2 3.1.2 1 -1 -1 1 -1 1
## 3 3 6.1.3 1 -1 1 -1 1 1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 10.1.5 1 1 -1 -1 1 1
## 6 6 12.1.6 1 1 -1 1 1 -1
## 7 7 13.1.7 1 1 1 -1 -1 -1
## 8 8 15.1.8 1 1 1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 2.2.1 2 -1 -1 -1 1 -1
## 10 10 4.2.2 2 -1 -1 1 1 1
## 11 11 5.2.3 2 -1 1 -1 -1 1
## 12 12 7.2.4 2 -1 1 1 -1 -1
## 13 13 9.2.5 2 1 -1 -1 -1 1
## 14 14 11.2.6 2 1 -1 1 -1 -1
## 15 15 14.2.7 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 framealiasprint(des.res5)## $legend
## [1] A=A B=B C=C D=D E=E
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE AC=BE AE=BCsummary(des.res5)## Call:
## FrF2(nruns = 16, nfactors = 5, resolution = 5, randomize = FALSE,
## blocks = 2)
##
## 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 1.1.1 1 -1 -1 -1 -1 -1
## 2 2 3.1.2 1 -1 -1 1 -1 1
## 3 3 6.1.3 1 -1 1 -1 1 1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 10.1.5 1 1 -1 -1 1 1
## 6 6 12.1.6 1 1 -1 1 1 -1
## 7 7 13.1.7 1 1 1 -1 -1 -1
## 8 8 15.1.8 1 1 1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 2.2.1 2 -1 -1 -1 1 -1
## 10 10 4.2.2 2 -1 -1 1 1 1
## 11 11 5.2.3 2 -1 1 -1 -1 1
## 12 12 7.2.4 2 -1 1 1 -1 -1
## 13 13 9.2.5 2 1 -1 -1 -1 1
## 14 14 11.2.6 2 1 -1 1 -1 -1
## 15 15 14.2.7 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 frameAlias Relationships for 2^(k-p) Fractional Factorial Designs with k less than or equal to 15 and n is less than or equal to 64
Here all three factor interactions are confounded with two factor interaction.
Hence no main effect or two factor interaction can be confounded with the block
#8.25
From the question we can see that it has 32 runs and it is a quarter fraction and hence we use resoludion 4 design
des.res7<-FrF2(nruns = 32, nfactors=7,resolution=4,randomize=FALSE,blocks = 4)## Warning in FrF2(nruns = 32, nfactors = 7, resolution = 4, randomize = FALSE, :
## resolution is ignored, if nruns is given.des.res7## 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 7.1.2 1 -1 -1 1 1 -1 1 1
## 3 3 10.1.3 1 -1 1 -1 -1 1 1 1
## 4 4 16.1.4 1 -1 1 1 1 1 -1 -1
## 5 5 20.1.5 1 1 -1 -1 1 1 1 -1
## 6 6 22.1.6 1 1 -1 1 -1 1 -1 1
## 7 7 27.1.7 1 1 1 -1 1 -1 -1 1
## 8 8 29.1.8 1 1 1 1 -1 -1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 2.2.1 2 -1 -1 -1 -1 1 -1 -1
## 10 10 8.2.2 2 -1 -1 1 1 1 1 1
## 11 11 9.2.3 2 -1 1 -1 -1 -1 1 1
## 12 12 15.2.4 2 -1 1 1 1 -1 -1 -1
## 13 13 19.2.5 2 1 -1 -1 1 -1 1 -1
## 14 14 21.2.6 2 1 -1 1 -1 -1 -1 1
## 15 15 28.2.7 2 1 1 -1 1 1 -1 1
## 16 16 30.2.8 2 1 1 1 -1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 18 18 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 19 19 12.3.3 3 -1 1 -1 1 1 1 -1
## 20 20 14.3.4 3 -1 1 1 -1 1 -1 1
## 21 21 18.3.5 3 1 -1 -1 -1 1 1 1
## 22 22 24.3.6 3 1 -1 1 1 1 -1 -1
## 23 23 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 24 24 31.3.8 3 1 1 1 1 -1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 4.4.1 4 -1 -1 -1 1 1 -1 1
## 26 26 6.4.2 4 -1 -1 1 -1 1 1 -1
## 27 27 11.4.3 4 -1 1 -1 1 -1 1 -1
## 28 28 13.4.4 4 -1 1 1 -1 -1 -1 1
## 29 29 17.4.5 4 1 -1 -1 -1 -1 1 1
## 30 30 23.4.6 4 1 -1 1 1 -1 -1 -1
## 31 31 26.4.7 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 framealiasprint(des.res7)## $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=DFHence no main effect or two factor effects can be confounded with blocks
8.28
library(FrF2)
library(DoE.base)
library(GAD)
des.res6<-FrF2(nfactors=6,resolution=4,randomize=FALSE)A)The design Generators can be seen as E= ABC & F= BCD
aliasprint(des.res6) # Here we can see the alias relationship## $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=DEB)Here we can see the alias relationship
summary(des.res6)## Call:
## FrF2(nfactors = 6, resolution = 4, 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= FrF2Lamtemp<-c("-1","1","-1","1","-1","1","-1","1","-1","1","-1","1","-1","1","-1","1")
Lamtemp1<-as.fixed(Lamtemp)
Lamtime<-c(rep(c("-1","-1","1","1"),4))
Lamtime1<-as.fixed(Lamtime)
LamPressure<-c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
LamPressure1<-as.fixed(LamPressure)
Firetemp<-c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1")
Firetemp1<-as.fixed(Firetemp)
Firetime<-c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
Firetime1<-as.fixed(Firetime)
Firedewpoint<-c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
Firedewpoint1<-as.fixed(Firedewpoint)
Response<-c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.011,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.009,0.025,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.004,0.0147,0.0092,0.0226,0.0077,0.006,0.0028,0.0158,0.0101,0.0126,0.0145,0.011,0.0185,0.002,0.005,0.003,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
Total<-c(629,192,176,223,223,920,389,900,201,341,126,640,455,371,603,460)
Mean<-c(157.25,48,44,55.75,55.75,230,97.25,225,50.25,85.25,31.5,160,113.75,92.75,150.75,115)
Stddeviation<-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
)
dat2<-data.frame(Lamtemp1,Lamtime1,LamPressure1,Firetemp1,Firetime1,Firedewpoint1,Response,Total,Mean,Stddeviation)
model<-aov(Response~Lamtemp1*Lamtime1*LamPressure1*Firetemp1*Firetime1*Firedewpoint1, data = dat2)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## Lamtemp1 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## Lamtime1 1 0.0000053 0.0000053 0.614 0.43725
## LamPressure1 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## Firetemp1 1 0.0000323 0.0000323 3.712 0.05995 .
## Firetime1 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## Firedewpoint1 1 0.0000803 0.0000803 9.218 0.00387 **
## Lamtemp1:Lamtime1 1 0.0000587 0.0000587 6.738 0.01249 *
## Lamtime1:LamPressure1 1 0.0000527 0.0000527 6.053 0.01754 *
## Lamtemp1:Firetemp1 1 0.0000239 0.0000239 2.741 0.10431
## Lamtime1:Firetemp1 1 0.0000849 0.0000849 9.739 0.00305 **
## LamPressure1:Firetemp1 1 0.0000622 0.0000622 7.139 0.01027 *
## Firetemp1:Firetime1 1 0.0000088 0.0000088 1.007 0.32062
## Firetemp1:Firedewpoint1 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## Lamtemp1:Lamtime1:Firetemp1 1 0.0000000 0.0000000 0.005 0.94291
## Lamtime1:LamPressure1:Firetemp1 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 ' ' 1C)From the aov table the process variable that affect camber are Lamination Temperature, Lamination Pressure, Firing cycle time,Firing Dew point temp
dat3<-data.frame(Lamtemp1,Lamtime1,LamPressure1,Firetemp1,Firetime1,Firedewpoint1,Stddeviation)
model1<-aov(Stddeviation~Lamtemp1*Lamtime1*LamPressure1*Firetemp1*Firetime1*Firedewpoint1, data = dat3)
DanielPlot(model1)
MEPlot(model1,show.alias=TRUE)
D)From the above plot we can say that the variables which affect variability in camber measurements are Lamination temperature and Lamination Time
E)To reduce the camber as much as possible the significant factors should be taken care of i.e Lamination Temperature, Lamination Pressure, Firing cycle time,Firing Dew point temp
8.40)
A)The experimentinvestigated 4 factors i.e. a,b,c,d
B)The resolution of the design is 4
C)The main effect
library(FrF2)
des.res8.4<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res8.4## 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= FrF2aliasprint(des.res8.4)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCresponse<- c(8,10,12,7,13,6,5,11)
des.res8.41<-add.response(des.res8.4,response)
summary(des.res8.41)## 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] response
##
## 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 response
## 1 -1 -1 -1 -1 8
## 2 1 -1 -1 1 10
## 3 -1 1 -1 1 12
## 4 1 1 -1 -1 7
## 5 -1 -1 1 1 13
## 6 1 -1 1 -1 6
## 7 -1 1 1 -1 5
## 8 1 1 1 1 11
## class=design, type= FrF2D)From above table we can see the generators
C)
Effect A= 1/4*(-8+10-12+7-13+6-5+11)=-1
Effect B= 1/4*(8-10+12+7-13-6+5+11)=-0.5
Effect c= 1/4*(-8-10-12-7+13+6+5+11)=-0.5
Effect D= 1/4*(-8+10+12-7+13-6-5+11)=5
8.48)
A) The design generator for column D is -ABC
B) The design generator for column E is BC
C) By referring to the table 8.14 we can say that the resolution for the combined design is 4
8.60)
library(FrF2)
des.res860<-FrF2(nfactors=7,resolution=3,randomize=FALSE)
des.res860## 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= FrF2des.res861<-fold.design(des.res860,column=1)
des.res861## 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.foldedaliasprint(des.res861)## $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=-DGsummary(des.res861)## 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