HW13
Ayodeji, Md Ariful Haque
2022-11-25
Question 8.2
We are experimenting on \(2^{4-1}\) where k=4 and p=1
I=ABCD and I=-ABCD
library(FrF2)
res3<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
res3## 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)
response1 <- add.response(res3,response)
aliasprint(response1)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCWe can conclude that since the two factor interaction is aliased with each other, they are in the \(4^{th}\) resolution.
Running the summary of our design we have
summary(response1)## 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(response1)
MEPlot(response1,show.alias=TRUE)
We can see from the Daniel plot that none of the factors are significant.
Because of this we can say that none of the factors affect the Crack length.
Question 8.24
We are experimenting on \(2^{5-1}\) where k=5 and p=1
I=ABCDE and I=-ABCDE
res1<- FrF2(nfactors = 5, resolution = 5 ,randomize = FALSE)
aliasprint(res1)## $legend
## [1] A=A B=B C=C D=D E=E
##
## [[2]]
## [1] no aliasing among main effects and 2fisThe summary of our design is
summary(res1)## Call:
## FrF2(nfactors = 5, resolution = 5, randomize = FALSE)
##
## Experimental design of type FrF2
## 16 runs
##
## 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
## [1] E=ABCD
##
##
## Alias structure:
## [[1]]
## [1] no aliasing among main effects and 2fis
##
##
## The design itself:
## A B C D E
## 1 -1 -1 -1 -1 1
## 2 1 -1 -1 -1 -1
## 3 -1 1 -1 -1 -1
## 4 1 1 -1 -1 1
## 5 -1 -1 1 -1 -1
## 6 1 -1 1 -1 1
## 7 -1 1 1 -1 1
## 8 1 1 1 -1 -1
## 9 -1 -1 -1 1 -1
## 10 1 -1 -1 1 1
## 11 -1 1 -1 1 1
## 12 1 1 -1 1 -1
## 13 -1 -1 1 1 1
## 14 1 -1 1 1 -1
## 15 -1 1 1 1 -1
## 16 1 1 1 1 1
## class=design, type= FrF2Confounding the AB factor combination with blocks
AB<- c("+","-","-","+","+","-","-","+","+","-","-","+","+","-","-","+")
block<- c(1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1)
dat<- data.frame(res1,AB,block)
dat## A B C D E AB block
## 1 -1 -1 -1 -1 1 + 1
## 2 1 -1 -1 -1 -1 - 2
## 3 -1 1 -1 -1 -1 - 2
## 4 1 1 -1 -1 1 + 1
## 5 -1 -1 1 -1 -1 + 1
## 6 1 -1 1 -1 1 - 2
## 7 -1 1 1 -1 1 - 2
## 8 1 1 1 -1 -1 + 1
## 9 -1 -1 -1 1 -1 + 1
## 10 1 -1 -1 1 1 - 2
## 11 -1 1 -1 1 1 - 2
## 12 1 1 -1 1 -1 + 1
## 13 -1 -1 1 1 1 + 1
## 14 1 -1 1 1 -1 - 2
## 15 -1 1 1 1 -1 - 2
## 16 1 1 1 1 1 + 1We can see that the AB and CDE factor interactions are confounded with blocks
Also, we see that No Two factor interaction is confounded in block 1 and block 2
Overall we noticed that,
Main effects confounded in block 1- E, C,D
Main effects confounded in block 2- A,B
Question 8.25
We are experimenting on \(2^{7-2}\) where k=7 and p=2
I=ABCDEF and I=-ABCDEF ;
I=ABDEG and I=-ABDEG
With a generalized interaction of I=CEFG, I=-CEFG
design1<- FrF2(nruns = 32,nfactors=7,blocks = 4,randomize=TRUE)
design1## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 20.1.5 1 1 -1 -1 1 1 1 -1
## 2 2 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 3 3 10.1.3 1 -1 1 -1 -1 1 1 1
## 4 4 27.1.7 1 1 1 -1 1 -1 -1 1
## 5 5 16.1.4 1 -1 1 1 1 1 -1 -1
## 6 6 7.1.2 1 -1 -1 1 1 -1 1 1
## 7 7 29.1.8 1 1 1 1 -1 -1 1 -1
## 8 8 22.1.6 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 28.2.7 2 1 1 -1 1 1 -1 1
## 11 11 30.2.8 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 8.2.2 2 -1 -1 1 1 1 1 1
## 15 15 9.2.3 2 -1 1 -1 -1 -1 1 1
## 16 16 21.2.6 2 1 -1 1 -1 -1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 31.3.8 3 1 1 1 1 -1 1 1
## 18 18 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 19 19 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 20 20 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 21 21 18.3.5 3 1 -1 -1 -1 1 1 1
## 22 22 12.3.3 3 -1 1 -1 1 1 1 -1
## 23 23 24.3.6 3 1 -1 1 1 1 -1 -1
## 24 24 14.3.4 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 13.4.4 4 -1 1 1 -1 -1 -1 1
## 28 28 4.4.1 4 -1 -1 -1 1 1 -1 1
## 29 29 6.4.2 4 -1 -1 1 -1 1 1 -1
## 30 30 11.4.3 4 -1 1 -1 1 -1 1 -1
## 31 31 23.4.6 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 framesummary(design1)## Call:
## FrF2(nruns = 32, nfactors = 7, 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 20.1.5 1 1 -1 -1 1 1 1 -1
## 2 2 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 3 3 10.1.3 1 -1 1 -1 -1 1 1 1
## 4 4 27.1.7 1 1 1 -1 1 -1 -1 1
## 5 5 16.1.4 1 -1 1 1 1 1 -1 -1
## 6 6 7.1.2 1 -1 -1 1 1 -1 1 1
## 7 7 29.1.8 1 1 1 1 -1 -1 1 -1
## 8 8 22.1.6 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 28.2.7 2 1 1 -1 1 1 -1 1
## 11 11 30.2.8 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 8.2.2 2 -1 -1 1 1 1 1 1
## 15 15 9.2.3 2 -1 1 -1 -1 -1 1 1
## 16 16 21.2.6 2 1 -1 1 -1 -1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 31.3.8 3 1 1 1 1 -1 1 1
## 18 18 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 19 19 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 20 20 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 21 21 18.3.5 3 1 -1 -1 -1 1 1 1
## 22 22 12.3.3 3 -1 1 -1 1 1 1 -1
## 23 23 24.3.6 3 1 -1 1 1 1 -1 -1
## 24 24 14.3.4 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 13.4.4 4 -1 1 1 -1 -1 -1 1
## 28 28 4.4.1 4 -1 -1 -1 1 1 -1 1
## 29 29 6.4.2 4 -1 -1 1 -1 1 1 -1
## 30 30 11.4.3 4 -1 1 -1 1 -1 1 -1
## 31 31 23.4.6 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 frameWe observe that only ACE, BFG, and ABCEFG are confounded with blocks
Question 8.28
Lamtemp <- c(rep(c("-1","1"),8))
Lamtime <- c(rep(c("-1","-1","1","1"),4))
LamPres<- c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
Firtemp <- c(rep(c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1"),1))
Fircytime <- c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
Firdewpoint <- c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
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)
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
)
dat<-cbind(Lamtemp,Lamtime,LamPres,Firtemp,Fircytime,Firdewpoint,response,total,mean,std)
dat <-as.data.frame(dat)
dat## Lamtemp Lamtime LamPres Firtemp Fircytime Firdewpoint response total mean
## 1 -1 -1 -1 -1 -1 -1 0.0167 629 157.25
## 2 1 -1 -1 -1 1 1 0.0062 192 48
## 3 -1 1 -1 -1 1 -1 0.0041 176 44
## 4 1 1 -1 -1 -1 1 0.0073 223 55.75
## 5 -1 -1 1 -1 1 1 0.0047 223 55.75
## 6 1 -1 1 -1 -1 -1 0.0219 920 230
## 7 -1 1 1 -1 -1 1 0.0121 389 97.25
## 8 1 1 1 -1 1 -1 0.0255 900 225
## 9 -1 -1 -1 1 -1 -1 0.0032 201 50.25
## 10 1 -1 -1 1 1 1 0.0078 341 85.25
## 11 -1 1 -1 1 1 -1 0.0043 126 31.5
## 12 1 1 -1 1 -1 1 0.0186 640 160
## 13 -1 -1 1 1 1 1 0.011 455 113.75
## 14 1 -1 1 1 -1 -1 0.0065 371 92.75
## 15 -1 1 1 1 -1 1 0.0155 603 150.75
## 16 1 1 1 1 1 -1 0.0093 460 115
## 17 -1 -1 -1 -1 -1 -1 0.0128 629 157.25
## 18 1 -1 -1 -1 1 1 0.0066 192 48
## 19 -1 1 -1 -1 1 -1 0.0043 176 44
## 20 1 1 -1 -1 -1 1 0.0081 223 55.75
## 21 -1 -1 1 -1 1 1 0.0047 223 55.75
## 22 1 -1 1 -1 -1 -1 0.0258 920 230
## 23 -1 1 1 -1 -1 1 0.009 389 97.25
## 24 1 1 1 -1 1 -1 0.025 900 225
## 25 -1 -1 -1 1 -1 -1 0.0023 201 50.25
## 26 1 -1 -1 1 1 1 0.0158 341 85.25
## 27 -1 1 -1 1 1 -1 0.0027 126 31.5
## 28 1 1 -1 1 -1 1 0.0137 640 160
## 29 -1 -1 1 1 1 1 0.0086 455 113.75
## 30 1 -1 1 1 -1 -1 0.0109 371 92.75
## 31 -1 1 1 1 -1 1 0.0158 603 150.75
## 32 1 1 1 1 1 -1 0.0124 460 115
## 33 -1 -1 -1 -1 -1 -1 0.0149 629 157.25
## 34 1 -1 -1 -1 1 1 0.0044 192 48
## 35 -1 1 -1 -1 1 -1 0.0042 176 44
## 36 1 1 -1 -1 -1 1 0.0039 223 55.75
## 37 -1 -1 1 -1 1 1 0.004 223 55.75
## 38 1 -1 1 -1 -1 -1 0.0147 920 230
## 39 -1 1 1 -1 -1 1 0.0092 389 97.25
## 40 1 1 1 -1 1 -1 0.0226 900 225
## 41 -1 -1 -1 1 -1 -1 0.0077 201 50.25
## 42 1 -1 -1 1 1 1 0.006 341 85.25
## 43 -1 1 -1 1 1 -1 0.0028 126 31.5
## 44 1 1 -1 1 -1 1 0.0158 640 160
## 45 -1 -1 1 1 1 1 0.0101 455 113.75
## 46 1 -1 1 1 -1 -1 0.0126 371 92.75
## 47 -1 1 1 1 -1 1 0.0145 603 150.75
## 48 1 1 1 1 1 -1 0.011 460 115
## 49 -1 -1 -1 -1 -1 -1 0.0185 629 157.25
## 50 1 -1 -1 -1 1 1 0.002 192 48
## 51 -1 1 -1 -1 1 -1 0.005 176 44
## 52 1 1 -1 -1 -1 1 0.003 223 55.75
## 53 -1 -1 1 -1 1 1 0.0089 223 55.75
## 54 1 -1 1 -1 -1 -1 0.0296 920 230
## 55 -1 1 1 -1 -1 1 0.0086 389 97.25
## 56 1 1 1 -1 1 -1 0.0169 900 225
## 57 -1 -1 -1 1 -1 -1 0.0069 201 50.25
## 58 1 -1 -1 1 1 1 0.0045 341 85.25
## 59 -1 1 -1 1 1 -1 0.0028 126 31.5
## 60 1 1 -1 1 -1 1 0.0159 640 160
## 61 -1 -1 1 1 1 1 0.0158 455 113.75
## 62 1 -1 1 1 -1 -1 0.0071 371 92.75
## 63 -1 1 1 1 -1 1 0.0145 603 150.75
## 64 1 1 1 1 1 -1 0.0133 460 115
## std
## 1 24.418
## 2 20.976
## 3 4.083
## 4 25.025
## 5 22.41
## 6 63.639
## 7 16.029
## 8 39.42
## 9 26.725
## 10 50.341
## 11 7.681
## 12 20.083
## 13 31.12
## 14 29.51
## 15 6.75
## 16 17.45
## 17 24.418
## 18 20.976
## 19 4.083
## 20 25.025
## 21 22.41
## 22 63.639
## 23 16.029
## 24 39.42
## 25 26.725
## 26 50.341
## 27 7.681
## 28 20.083
## 29 31.12
## 30 29.51
## 31 6.75
## 32 17.45
## 33 24.418
## 34 20.976
## 35 4.083
## 36 25.025
## 37 22.41
## 38 63.639
## 39 16.029
## 40 39.42
## 41 26.725
## 42 50.341
## 43 7.681
## 44 20.083
## 45 31.12
## 46 29.51
## 47 6.75
## 48 17.45
## 49 24.418
## 50 20.976
## 51 4.083
## 52 25.025
## 53 22.41
## 54 63.639
## 55 16.029
## 56 39.42
## 57 26.725
## 58 50.341
## 59 7.681
## 60 20.083
## 61 31.12
## 62 29.51
## 63 6.75
## 64 17.45Question a.
The above design uses a \(2^{6-2}\) with Resolution IIII and with 16 runs.
b.)
the following which are alias relationship are displayed below
res4<- FrF2(nfactors = 6,resolution = 4 , randomize = TRUE)
aliasprint(res4)## $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=DEUsing table 8.14 from the Montgomery textbook,
The design generators is
E=(+-ABC) and F=(+-BCD) that is I=ABCE , I=BCDF
The generalized interaction is I=ADEF
C.
Lamtemp <- c(rep(c(-1,1),8))
Lamtime <- c(rep(c(-1,-1,1,1),4))
LamPres <- c(rep(c(-1,-1,-1,-1,1,1,1,1),2))
Firtemp <- c(rep(c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1),1))
Fircytime <- c(rep(c(-1,1,1,-1,1,-1,-1,1),2))
Firdewpoint <- c(rep(c(-1,1,-1,1,1,-1,1,-1),2))
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)
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
)
dat <- cbind(Lamtemp,Lamtime,LamPres,Firtemp,Fircytime,Firdewpoint,response,total,mean,std)
dat <- as.data.frame(dat)
dat## Lamtemp Lamtime LamPres Firtemp Fircytime Firdewpoint response total mean
## 1 -1 -1 -1 -1 -1 -1 0.0167 629 157.25
## 2 1 -1 -1 -1 1 1 0.0062 192 48.00
## 3 -1 1 -1 -1 1 -1 0.0041 176 44.00
## 4 1 1 -1 -1 -1 1 0.0073 223 55.75
## 5 -1 -1 1 -1 1 1 0.0047 223 55.75
## 6 1 -1 1 -1 -1 -1 0.0219 920 230.00
## 7 -1 1 1 -1 -1 1 0.0121 389 97.25
## 8 1 1 1 -1 1 -1 0.0255 900 225.00
## 9 -1 -1 -1 1 -1 -1 0.0032 201 50.25
## 10 1 -1 -1 1 1 1 0.0078 341 85.25
## 11 -1 1 -1 1 1 -1 0.0043 126 31.50
## 12 1 1 -1 1 -1 1 0.0186 640 160.00
## 13 -1 -1 1 1 1 1 0.0110 455 113.75
## 14 1 -1 1 1 -1 -1 0.0065 371 92.75
## 15 -1 1 1 1 -1 1 0.0155 603 150.75
## 16 1 1 1 1 1 -1 0.0093 460 115.00
## 17 -1 -1 -1 -1 -1 -1 0.0128 629 157.25
## 18 1 -1 -1 -1 1 1 0.0066 192 48.00
## 19 -1 1 -1 -1 1 -1 0.0043 176 44.00
## 20 1 1 -1 -1 -1 1 0.0081 223 55.75
## 21 -1 -1 1 -1 1 1 0.0047 223 55.75
## 22 1 -1 1 -1 -1 -1 0.0258 920 230.00
## 23 -1 1 1 -1 -1 1 0.0090 389 97.25
## 24 1 1 1 -1 1 -1 0.0250 900 225.00
## 25 -1 -1 -1 1 -1 -1 0.0023 201 50.25
## 26 1 -1 -1 1 1 1 0.0158 341 85.25
## 27 -1 1 -1 1 1 -1 0.0027 126 31.50
## 28 1 1 -1 1 -1 1 0.0137 640 160.00
## 29 -1 -1 1 1 1 1 0.0086 455 113.75
## 30 1 -1 1 1 -1 -1 0.0109 371 92.75
## 31 -1 1 1 1 -1 1 0.0158 603 150.75
## 32 1 1 1 1 1 -1 0.0124 460 115.00
## 33 -1 -1 -1 -1 -1 -1 0.0149 629 157.25
## 34 1 -1 -1 -1 1 1 0.0044 192 48.00
## 35 -1 1 -1 -1 1 -1 0.0042 176 44.00
## 36 1 1 -1 -1 -1 1 0.0039 223 55.75
## 37 -1 -1 1 -1 1 1 0.0040 223 55.75
## 38 1 -1 1 -1 -1 -1 0.0147 920 230.00
## 39 -1 1 1 -1 -1 1 0.0092 389 97.25
## 40 1 1 1 -1 1 -1 0.0226 900 225.00
## 41 -1 -1 -1 1 -1 -1 0.0077 201 50.25
## 42 1 -1 -1 1 1 1 0.0060 341 85.25
## 43 -1 1 -1 1 1 -1 0.0028 126 31.50
## 44 1 1 -1 1 -1 1 0.0158 640 160.00
## 45 -1 -1 1 1 1 1 0.0101 455 113.75
## 46 1 -1 1 1 -1 -1 0.0126 371 92.75
## 47 -1 1 1 1 -1 1 0.0145 603 150.75
## 48 1 1 1 1 1 -1 0.0110 460 115.00
## 49 -1 -1 -1 -1 -1 -1 0.0185 629 157.25
## 50 1 -1 -1 -1 1 1 0.0020 192 48.00
## 51 -1 1 -1 -1 1 -1 0.0050 176 44.00
## 52 1 1 -1 -1 -1 1 0.0030 223 55.75
## 53 -1 -1 1 -1 1 1 0.0089 223 55.75
## 54 1 -1 1 -1 -1 -1 0.0296 920 230.00
## 55 -1 1 1 -1 -1 1 0.0086 389 97.25
## 56 1 1 1 -1 1 -1 0.0169 900 225.00
## 57 -1 -1 -1 1 -1 -1 0.0069 201 50.25
## 58 1 -1 -1 1 1 1 0.0045 341 85.25
## 59 -1 1 -1 1 1 -1 0.0028 126 31.50
## 60 1 1 -1 1 -1 1 0.0159 640 160.00
## 61 -1 -1 1 1 1 1 0.0158 455 113.75
## 62 1 -1 1 1 -1 -1 0.0071 371 92.75
## 63 -1 1 1 1 -1 1 0.0145 603 150.75
## 64 1 1 1 1 1 -1 0.0133 460 115.00
## std
## 1 24.418
## 2 20.976
## 3 4.083
## 4 25.025
## 5 22.410
## 6 63.639
## 7 16.029
## 8 39.420
## 9 26.725
## 10 50.341
## 11 7.681
## 12 20.083
## 13 31.120
## 14 29.510
## 15 6.750
## 16 17.450
## 17 24.418
## 18 20.976
## 19 4.083
## 20 25.025
## 21 22.410
## 22 63.639
## 23 16.029
## 24 39.420
## 25 26.725
## 26 50.341
## 27 7.681
## 28 20.083
## 29 31.120
## 30 29.510
## 31 6.750
## 32 17.450
## 33 24.418
## 34 20.976
## 35 4.083
## 36 25.025
## 37 22.410
## 38 63.639
## 39 16.029
## 40 39.420
## 41 26.725
## 42 50.341
## 43 7.681
## 44 20.083
## 45 31.120
## 46 29.510
## 47 6.750
## 48 17.450
## 49 24.418
## 50 20.976
## 51 4.083
## 52 25.025
## 53 22.410
## 54 63.639
## 55 16.029
## 56 39.420
## 57 26.725
## 58 50.341
## 59 7.681
## 60 20.083
## 61 31.120
## 62 29.510
## 63 6.750
## 64 17.450library(GAD)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.dat$Lamtemp <- as.fixed(dat$Lamtemp)
dat$Lamtime <- as.fixed(dat$Lamtime)
dat$LamPres <- as.fixed(dat$LamPres)
dat$Firtemp <- as.fixed(dat$Firtemp)
dat$Firdewpoint <- as.fixed(dat$Firdewpoint)
dat$Fircytime <- as.fixed(dat$Fircytime)
model<- aov(response~Lamtemp*Lamtime*LamPres*Firtemp*Fircytime*Firdewpoint,data = dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## Lamtemp 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## Lamtime 1 0.0000053 0.0000053 0.614 0.43725
## LamPres 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## Firtemp 1 0.0000323 0.0000323 3.712 0.05995 .
## Fircytime 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## Firdewpoint 1 0.0000803 0.0000803 9.218 0.00387 **
## Lamtemp:Lamtime 1 0.0000587 0.0000587 6.738 0.01249 *
## Lamtime:LamPres 1 0.0000527 0.0000527 6.053 0.01754 *
## Lamtemp:Firtemp 1 0.0000239 0.0000239 2.741 0.10431
## Lamtime:Firtemp 1 0.0000849 0.0000849 9.739 0.00305 **
## LamPres:Firtemp 1 0.0000622 0.0000622 7.139 0.01027 *
## Firtemp:Fircytime 1 0.0000088 0.0000088 1.007 0.32062
## Firtemp:Firdewpoint 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## Lamtemp:Lamtime:Firtemp 1 0.0000000 0.0000000 0.005 0.94291
## Lamtime:LamPres:Firtemp 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 that would affect average camber is A,C,E,F also some interaction effects that can be seen above.
D.
camber<- 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)
var<-c(camber^2)
A<- c(rep(c("-1","1"),8))
B<- c(rep(c("-1","-1","1","1"),4))
C<- c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
D<- c(rep(c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1"),1))
E<- c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
F<- c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
dat2<- data.frame(A,B,C,D,E,F,var)
model2<-lm(camber~A*B*C*D*E*F,data=dat2)
DanielPlot(model2)
We conclude that based on the Daniel’s plot that
Laminating temperature and Laminating time are factors that are significantly affecting the standard deviation.
Running the model of the significant factors above we have
model5<-aov(camber~A+B,data = dat2)
summary(model5)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 1012 1012 8.505 0.01202 *
## B 1 1099 1099 9.241 0.00948 **
## Residuals 13 1546 119
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the ANOVA analysis, we can see that since the pvalue of Laminating temperature(0.01202) and Laminating time(0.00948) are less than the reference level of significance(0.05).
We ultimately conclude that Laminating temperature and Laminating time are affecting standard deviation of camber.
E.
Lamtemp <- c(rep(c("-1","1"),8))
Lamtime <- c(rep(c("-1","-1","1","1"),4))
LamPres <- c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
Firtemp <- c(rep(c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1"),1))
Fircytime <- c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
Firdewpoint <- c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
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)
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
)
dat3<- cbind(Lamtemp,Lamtime,LamPres,Firtemp,Fircytime,Firdewpoint,response,total,mean,std)
dat3<- as.data.frame(dat3)
model5<-lm(dat$response~dat$Lamtemp*dat$Lamtime*dat$LamPres*dat$Firtemp*dat$Fircytime*dat$Firdewpoint)
coef(model5)## (Intercept)
## 0.01572500
## dat$Lamtemp1
## 0.00325000
## dat$Lamtime1
## -0.00713750
## dat$LamPres1
## 0.00402500
## dat$Firtemp1
## -0.01070000
## dat$Fircytime1
## -0.00418750
## dat$Firdewpoint1
## -0.00998750
## dat$Lamtemp1:dat$Lamtime1
## 0.00372500
## dat$Lamtemp1:dat$LamPres1
## NA
## dat$Lamtime1:dat$LamPres1
## 0.00710000
## dat$Lamtemp1:dat$Firtemp1
## -0.00255000
## dat$Lamtime1:dat$Firtemp1
## 0.00796875
## dat$LamPres1:dat$Firtemp1
## -0.00047500
## dat$Lamtemp1:dat$Fircytime1
## NA
## dat$Lamtime1:dat$Fircytime1
## NA
## dat$LamPres1:dat$Fircytime1
## NA
## dat$Firtemp1:dat$Fircytime1
## 0.00148125
## dat$Lamtemp1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$Firdewpoint1
## NA
## dat$LamPres1:dat$Firdewpoint1
## NA
## dat$Firtemp1:dat$Firdewpoint1
## 0.01549375
## dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1
## 0.00021250
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1
## -0.00693750
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$LamPres1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## NAsummary(model5)##
## Call:
## lm.default(formula = dat$response ~ dat$Lamtemp * dat$Lamtime *
## dat$LamPres * dat$Firtemp * dat$Fircytime * dat$Firdewpoint)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.008300 -0.001350 -0.000350 0.001744 0.007275
##
## Coefficients: (48 not defined because of singularities)
## Estimate
## (Intercept) 0.0157250
## dat$Lamtemp1 0.0032500
## dat$Lamtime1 -0.0071375
## dat$LamPres1 0.0040250
## dat$Firtemp1 -0.0107000
## dat$Fircytime1 -0.0041875
## dat$Firdewpoint1 -0.0099875
## dat$Lamtemp1:dat$Lamtime1 0.0037250
## dat$Lamtemp1:dat$LamPres1 NA
## dat$Lamtime1:dat$LamPres1 0.0071000
## dat$Lamtemp1:dat$Firtemp1 -0.0025500
## dat$Lamtime1:dat$Firtemp1 0.0079688
## dat$LamPres1:dat$Firtemp1 -0.0004750
## dat$Lamtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Fircytime1 NA
## dat$LamPres1:dat$Fircytime1 NA
## dat$Firtemp1:dat$Fircytime1 0.0014813
## dat$Lamtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Firdewpoint1 0.0154937
## dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1 0.0002125
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1 -0.0069375
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## Std. Error
## (Intercept) 0.0014760
## dat$Lamtemp1 0.0014760
## dat$Lamtime1 0.0018077
## dat$LamPres1 0.0014760
## dat$Firtemp1 0.0020874
## dat$Fircytime1 0.0010437
## dat$Firdewpoint1 0.0010437
## dat$Lamtemp1:dat$Lamtime1 0.0020874
## dat$Lamtemp1:dat$LamPres1 NA
## dat$Lamtime1:dat$LamPres1 0.0020874
## dat$Lamtemp1:dat$Firtemp1 0.0020874
## dat$Lamtime1:dat$Firtemp1 0.0025565
## dat$LamPres1:dat$Firtemp1 0.0020874
## dat$Lamtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Fircytime1 NA
## dat$LamPres1:dat$Fircytime1 NA
## dat$Firtemp1:dat$Fircytime1 0.0014760
## dat$Lamtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Firdewpoint1 0.0014760
## dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1 0.0029520
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1 0.0029520
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## t value
## (Intercept) 10.654
## dat$Lamtemp1 2.202
## dat$Lamtime1 -3.948
## dat$LamPres1 2.727
## dat$Firtemp1 -5.126
## dat$Fircytime1 -4.012
## dat$Firdewpoint1 -9.570
## dat$Lamtemp1:dat$Lamtime1 1.785
## dat$Lamtemp1:dat$LamPres1 NA
## dat$Lamtime1:dat$LamPres1 3.401
## dat$Lamtemp1:dat$Firtemp1 -1.222
## dat$Lamtime1:dat$Firtemp1 3.117
## dat$LamPres1:dat$Firtemp1 -0.228
## dat$Lamtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Fircytime1 NA
## dat$LamPres1:dat$Fircytime1 NA
## dat$Firtemp1:dat$Fircytime1 1.004
## dat$Lamtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Firdewpoint1 10.497
## dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1 0.072
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1 -2.350
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## Pr(>|t|)
## (Intercept) 3.06e-14
## dat$Lamtemp1 0.032509
## dat$Lamtime1 0.000257
## dat$LamPres1 0.008899
## dat$Firtemp1 5.24e-06
## dat$Fircytime1 0.000210
## dat$Firdewpoint1 1.05e-12
## dat$Lamtemp1:dat$Lamtime1 0.080655
## dat$Lamtemp1:dat$LamPres1 NA
## dat$Lamtime1:dat$LamPres1 0.001359
## dat$Lamtemp1:dat$Firtemp1 0.227809
## dat$Lamtime1:dat$Firtemp1 0.003083
## dat$LamPres1:dat$Firtemp1 0.820954
## dat$Lamtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Fircytime1 NA
## dat$LamPres1:dat$Fircytime1 NA
## dat$Firtemp1:dat$Fircytime1 0.320619
## dat$Lamtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Firdewpoint1 5.05e-14
## dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1 0.942912
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1 0.022926
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1 NA
##
## (Intercept) ***
## dat$Lamtemp1 *
## dat$Lamtime1 ***
## dat$LamPres1 **
## dat$Firtemp1 ***
## dat$Fircytime1 ***
## dat$Firdewpoint1 ***
## dat$Lamtemp1:dat$Lamtime1 .
## dat$Lamtemp1:dat$LamPres1
## dat$Lamtime1:dat$LamPres1 **
## dat$Lamtemp1:dat$Firtemp1
## dat$Lamtime1:dat$Firtemp1 **
## dat$LamPres1:dat$Firtemp1
## dat$Lamtemp1:dat$Fircytime1
## dat$Lamtime1:dat$Fircytime1
## dat$LamPres1:dat$Fircytime1
## dat$Firtemp1:dat$Fircytime1
## dat$Lamtemp1:dat$Firdewpoint1
## dat$Lamtime1:dat$Firdewpoint1
## dat$LamPres1:dat$Firdewpoint1
## dat$Firtemp1:dat$Firdewpoint1 ***
## dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1 *
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## dat$Lamtemp1:dat$Lamtime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$LamPres1:dat$Firdewpoint1
## dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Firtemp1:dat$Firdewpoint1
## dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1
## dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1
## dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Firdewpoint1
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## dat$Lamtemp1:dat$Lamtime1:dat$LamPres1:dat$Firtemp1:dat$Fircytime1:dat$Firdewpoint1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.002952 on 48 degrees of freedom
## Multiple R-squared: 0.849, Adjusted R-squared: 0.8018
## F-statistic: 18 on 15 and 48 DF, p-value: 9.012e-15The expression for reducing camber(y) is
y= 0.015735 + 0.0010094A +0.0017844C -0.0041875E - 0.0077469F
In conclusion we have to let A, C be at low levels [-1,-1] and E,F high [+1,+1].
Question 8.40
Part A,
the experiment investigated A,B,C,D which are four factors.
Part B,
the resolution of the design used is FOUR
Part C,
O = 8
AD =10
BD =12
AB =7
CD =13
AC = 6
BC =5
ABCD =11
EffectofA<-(2*(AD+AB+AC+ABCD-O-BD-CD-BC))/(8)
EffectofA## [1] -1EffectofB<-(2*(BD+AB+BC+ABCD-O-AD-CD-AC))/(8)
EffectofB## [1] -0.5EffectofC <- (2*(CD+AC+BC+ABCD-O-AD-BD-AB))/(8)
EffectofC## [1] -0.5EffectofD <- (2*(AD+BD+CD+ABCD-O-AB-AC-BC))/(8)
EffectofD## [1] 5We can see that
Effect of A is -1
Effect of B is -0.5
Effect of C is -0.5
Effect of D is 5
QUESTION 8.48.
design4<- FrF2(nfactors = 5,nruns = 8,generators = c("-ABC","BC"), randomize = FALSE)
design4## A B C D E
## 1 -1 -1 -1 1 1
## 2 1 -1 -1 -1 1
## 3 -1 1 -1 -1 -1
## 4 1 1 -1 1 -1
## 5 -1 -1 1 -1 -1
## 6 1 -1 1 1 -1
## 7 -1 1 1 1 1
## 8 1 1 1 -1 1
## class=design, type= FrF2.generatorssummary(design4)## Call:
## FrF2(nfactors = 5, nruns = 8, generators = c("-ABC", "BC"), randomize = FALSE)
##
## Experimental design of type FrF2.generators
## 8 runs
##
## 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
## [1] D=-ABC E=BC
##
##
## Alias structure:
## $main
## [1] A=-DE B=CE C=BE D=-AE E=-AD=BC
##
## $fi2
## [1] AB=-CD AC=-BD
##
##
## The design itself:
## A B C D E
## 1 -1 -1 -1 1 1
## 2 1 -1 -1 -1 1
## 3 -1 1 -1 -1 -1
## 4 1 1 -1 1 -1
## 5 -1 -1 1 -1 -1
## 6 1 -1 1 1 -1
## 7 -1 1 1 1 1
## 8 1 1 1 -1 1
## class=design, type= FrF2.generatorsdesign5<-fold.design(design4)
aliasprint(design5)## $legend
## [1] A=A B=B C=C D=fold E=D F=E
##
## $main
## character(0)
##
## $fi2
## [1] AB=-CE AC=-BE AD=EF AE=-BC=DF AF=DE BD=-CF BF=-CDa. the design generator for column D is -ABC
b. the design generator for column E is BC
c. From the Montgometry textbook the resolution of the design is FOUR.
Question 8.60
library(FrF2)
design<- FrF2(nfactors=7,resolution=3,randomize=FALSE)
design## A B C D E F G
## 1 -1 -1 -1 1 1 1 -1
## 2 1 -1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 -1 1
## 4 1 1 -1 1 -1 -1 -1
## 5 -1 -1 1 1 -1 -1 1
## 6 1 -1 1 -1 1 -1 -1
## 7 -1 1 1 -1 -1 1 -1
## 8 1 1 1 1 1 1 1
## class=design, type= FrF2design2<-fold.design(design,column=1)
design2## A B C fold D E F G
## 1 -1 -1 -1 original 1 1 1 -1
## 2 1 -1 -1 original -1 -1 1 1
## 3 -1 1 -1 original -1 1 -1 1
## 4 1 1 -1 original 1 -1 -1 -1
## 5 -1 -1 1 original 1 -1 -1 1
## 6 1 -1 1 original -1 1 -1 -1
## 7 -1 1 1 original -1 -1 1 -1
## 8 1 1 1 original 1 1 1 1
## 9 1 -1 -1 mirror 1 1 1 -1
## 10 -1 -1 -1 mirror -1 -1 1 1
## 11 1 1 -1 mirror -1 1 -1 1
## 12 -1 1 -1 mirror 1 -1 -1 -1
## 13 1 -1 1 mirror 1 -1 -1 1
## 14 -1 -1 1 mirror -1 1 -1 -1
## 15 1 1 1 mirror -1 -1 1 -1
## 16 -1 1 1 mirror 1 1 1 1
## class=design, type= FrF2.foldeddesign.info(design2)$aliased$main## [1] "B=CG=FH" "C=BG=EH" "E=CH=FG" "F=BH=EG" "G=BC=EF" "H=BF=CE"aliasprint(design2)## $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