Homework Week - 13
Saad Mirza
Harshitha
Last compiled on November 29, 2022 at 4:41 AM - CST
1 Question 8.2:
1.1 Solution:
Given only a one half fraction of the \(2^4\) design could be run, we have following
k=4 and p=1
Fraction: \(2_{iv}^{4-1}\)
No of Runs: 8
Design Generators: D= \(\pm\) ABC
Defining Relation: I=ABCD and I=-ABCD
Set up resolution IV design with 4 Factors:
library(FrF2)
design <- FrF2(nfactors=4,resolution=4,randomize=FALSE)
design## 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= FrF2Print aliased relationships:
aliasprint(design)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCCollect data and add response:
response<-c(7.037, 16.867, 13.876, 17.273, 11.846, 4.368, 9.36, 15.653)
design.resp <- add.response(design,response)
summary(design.resp)## 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= FrF2Generate Half Normal Plot:
DanielPlot(design.resp,half=TRUE)
Generate Main Effects Plot:
MEPlot(design.resp,show.alias=TRUE)
2 Question 8.24:
2.1 Solution:
Given only a one half fraction of the \(2^5\) design could be run, we have following
k=5 and p=1
Fraction: \(2_{v}^{5-1}\)
No of Runs: 16
Design Generators: E= \(\pm\) ABCD
Defining Relation: I=ABCDE and I=-ABCDE
Set up resolution V design with 5 Factors:
des.res <- FrF2(nfactors = 5, resolution = 5 ,randomize = FALSE)
des.res## 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= FrF2Print aliased relationships:
aliasprint(des.res)## $legend
## [1] A=A B=B C=C D=D E=E
##
## [[2]]
## [1] no aliasing among main effects and 2fisSummary:
summary(des.res)## 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= FrF2AB factor combinations are confounded with blocks:
AB <- c("+","-","-","+","+","-","-","+","+","-","-","+","+","-","-","+")
Block <- c(1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1)
Data <- data.frame(des.res,AB,Block)
Data## 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 + 1Apart from Main Effects:
--> AB & CDE factor combinations are confounded with the blocks3 Question 8.25:
3.1 Solution:
Given Quater fraction of the \(2^7\) design to be run, we have following
k=7 and p=2
Fraction: \(2_{iv}^{7-2}\)
No of Runs: 32
Design Generators: F= \(\pm\) ABCD & G=\(\pm\) ABDE
Defining Relations: I=ABCDF, I= -ABCDF, I=ABDEG, I= -ABDEG, I=CEFG, I= -CEFG
Set up resolution IV design with 7 Factors & 4 Blocks:
design <- FrF2(nruns = 32,nfactors=7,blocks = 4,randomize=TRUE)
summary(design)## 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 27.1.7 1 1 1 -1 1 -1 -1 1
## 2 2 20.1.5 1 1 -1 -1 1 1 1 -1
## 3 3 22.1.6 1 1 -1 1 -1 1 -1 1
## 4 4 10.1.3 1 -1 1 -1 -1 1 1 1
## 5 5 7.1.2 1 -1 -1 1 1 -1 1 1
## 6 6 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 7 7 29.1.8 1 1 1 1 -1 -1 1 -1
## 8 8 16.1.4 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 15.2.4 2 -1 1 1 1 -1 -1 -1
## 11 11 30.2.8 2 1 1 1 -1 1 1 -1
## 12 12 28.2.7 2 1 1 -1 1 1 -1 1
## 13 13 21.2.6 2 1 -1 1 -1 -1 -1 1
## 14 14 19.2.5 2 1 -1 -1 1 -1 1 -1
## 15 15 8.2.2 2 -1 -1 1 1 1 1 1
## 16 16 9.2.3 2 -1 1 -1 -1 -1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 18.3.5 3 1 -1 -1 -1 1 1 1
## 18 18 31.3.8 3 1 1 1 1 -1 1 1
## 19 19 14.3.4 3 -1 1 1 -1 1 -1 1
## 20 20 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 21 21 12.3.3 3 -1 1 -1 1 1 1 -1
## 22 22 24.3.6 3 1 -1 1 1 1 -1 -1
## 23 23 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 24 24 25.3.7 3 1 1 -1 -1 -1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 6.4.2 4 -1 -1 1 -1 1 1 -1
## 26 26 32.4.8 4 1 1 1 1 1 1 1
## 27 27 26.4.7 4 1 1 -1 -1 1 -1 -1
## 28 28 23.4.6 4 1 -1 1 1 -1 -1 -1
## 29 29 13.4.4 4 -1 1 1 -1 -1 -1 1
## 30 30 4.4.1 4 -1 -1 -1 1 1 -1 1
## 31 31 11.4.3 4 -1 1 -1 1 -1 1 -1
## 32 32 17.4.5 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 frame4 Question 8.28:
What type of design did the experimenters use?
What are the alias relationships in this design?
Do any of the process variables affect average camber?
Do any of the process variables affect the variability in camber measurements?
If it is important to reduce camber as much as possible, what recommendations would you make?
4.1 Solution:
PART A:
Given in the table above, we can infer that that there are 6 Factors and 16 Runs made. From Table 8.14 in Douglas C. Montgomery we can conclude following:
k=6 and p=2
Fraction: \(2_{iv}^{6-2}\)
No of Runs: 16
Design Generators: E= \(\pm\) ABC & F=\(\pm\) BCD
Defining Relations: I=ABCE, I=BCDF, I=ADEF
--> The above design uses 2^(6−2) design with 16 runs at resolution 4PART B:
des.res <- FrF2(nfactors = 6,resolution = 4 , randomize = TRUE)
aliasprint(des.res)## $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:
Reading the Data from Question:
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)
F1 <- c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1)
A <- as.factor(A)
B <- as.factor(B)
C <- as.factor(C)
D <- as.factor(D)
E <- as.factor(E)
F1 <- as.factor(F1)
response<-c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.0110,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.0090,0.0250,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.0040,0.0147,0.0092,0.0226,0.0077,0.0060,0.0028,0.0158,0.0101,0.0126,0.0145,0.0110,0.0185,0.0020,0.0050,0.0030,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
Data <- data.frame(A,B,C,D,E,F1,response)Running AOV model to check significant Factors:
Model <- aov(response~A*B*C*D*E*F1,data = Data)
summary(Model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## B 1 0.0000053 0.0000053 0.614 0.43725
## C 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## D 1 0.0000323 0.0000323 3.712 0.05995 .
## E 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## F1 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## A:B 1 0.0000587 0.0000587 6.738 0.01249 *
## A:C 1 0.0000803 0.0000803 9.218 0.00387 **
## B:C 1 0.0000527 0.0000527 6.053 0.01754 *
## A:D 1 0.0000239 0.0000239 2.741 0.10431
## B:D 1 0.0000849 0.0000849 9.739 0.00305 **
## C:D 1 0.0000622 0.0000622 7.139 0.01027 *
## D:E 1 0.0000088 0.0000088 1.007 0.32062
## A:B:D 1 0.0000000 0.0000000 0.005 0.94291
## B:C:D 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 ' ' 1Note: Half Normal plot in Part C is unable to generate with error code “Error in halfnormal.lm(Model) : partially aliased main effect”
PART D:
Reading the Variability Data:
SD <- 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 <- SD^2
Data2 <- data.frame(A,B,C,D,E,F1,var)Generating Half Normal Plot:
Model2 <- lm(SD~A*B*C*D*E*F1,data = Data2)
DanielPlot(Model2)
Running the Model:
Model3 <- aov(SD~A+B,data = Data2)
summary(Model3)## 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 ' ' 1PART E:
Model2 <- lm(response~A*B*C*D*E*F1,data = Data)
coef(Model2)## (Intercept) A1 B1 C1
## 0.015725000 0.001009375 -0.007137500 0.001784375
## D1 E1 F11 A1:B1
## -0.002953125 -0.004187500 -0.007746875 0.003725000
## A1:C1 B1:C1 A1:D1 B1:D1
## 0.004481250 0.007100000 -0.002550000 0.007968750
## C1:D1 A1:E1 B1:E1 C1:E1
## -0.000475000 NA NA NA
## D1:E1 A1:F11 B1:F11 C1:F11
## 0.001481250 NA NA NA
## D1:F11 E1:F11 A1:B1:C1 A1:B1:D1
## NA NA NA 0.000212500
## A1:C1:D1 B1:C1:D1 A1:B1:E1 A1:C1:E1
## NA -0.006937500 NA NA
## B1:C1:E1 A1:D1:E1 B1:D1:E1 C1:D1:E1
## NA NA NA NA
## A1:B1:F11 A1:C1:F11 B1:C1:F11 A1:D1:F11
## NA NA NA NA
## B1:D1:F11 C1:D1:F11 A1:E1:F11 B1:E1:F11
## NA NA NA NA
## C1:E1:F11 D1:E1:F11 A1:B1:C1:D1 A1:B1:C1:E1
## NA NA NA NA
## A1:B1:D1:E1 A1:C1:D1:E1 B1:C1:D1:E1 A1:B1:C1:F11
## NA NA NA NA
## A1:B1:D1:F11 A1:C1:D1:F11 B1:C1:D1:F11 A1:B1:E1:F11
## NA NA NA NA
## A1:C1:E1:F11 B1:C1:E1:F11 A1:D1:E1:F11 B1:D1:E1:F11
## NA NA NA NA
## C1:D1:E1:F11 A1:B1:C1:D1:E1 A1:B1:C1:D1:F11 A1:B1:C1:E1:F11
## NA NA NA NA
## A1:B1:D1:E1:F11 A1:C1:D1:E1:F11 B1:C1:D1:E1:F11 A1:B1:C1:D1:E1:F11
## NA NA NA NAsummary(Model2)##
## Call:
## lm.default(formula = response ~ A * B * C * D * E * F1, data = Data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.008300 -0.001350 -0.000350 0.001744 0.007275
##
## Coefficients: (48 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0157250 0.0014760 10.654 3.06e-14 ***
## A1 0.0010094 0.0016502 0.612 0.543644
## B1 -0.0071375 0.0018077 -3.948 0.000257 ***
## C1 0.0017844 0.0016502 1.081 0.284963
## D1 -0.0029531 0.0019525 -1.512 0.136976
## E1 -0.0041875 0.0010437 -4.012 0.000210 ***
## F11 -0.0077469 0.0007380 -10.497 5.05e-14 ***
## A1:B1 0.0037250 0.0020874 1.785 0.080655 .
## A1:C1 0.0044812 0.0014760 3.036 0.003866 **
## B1:C1 0.0071000 0.0020874 3.401 0.001359 **
## A1:D1 -0.0025500 0.0020874 -1.222 0.227809
## B1:D1 0.0079688 0.0025565 3.117 0.003083 **
## C1:D1 -0.0004750 0.0020874 -0.228 0.820954
## A1:E1 NA NA NA NA
## B1:E1 NA NA NA NA
## C1:E1 NA NA NA NA
## D1:E1 0.0014813 0.0014760 1.004 0.320619
## A1:F11 NA NA NA NA
## B1:F11 NA NA NA NA
## C1:F11 NA NA NA NA
## D1:F11 NA NA NA NA
## E1:F11 NA NA NA NA
## A1:B1:C1 NA NA NA NA
## A1:B1:D1 0.0002125 0.0029520 0.072 0.942912
## A1:C1:D1 NA NA NA NA
## B1:C1:D1 -0.0069375 0.0029520 -2.350 0.022926 *
## A1:B1:E1 NA NA NA NA
## A1:C1:E1 NA NA NA NA
## B1:C1:E1 NA NA NA NA
## A1:D1:E1 NA NA NA NA
## B1:D1:E1 NA NA NA NA
## C1:D1:E1 NA NA NA NA
## A1:B1:F11 NA NA NA NA
## A1:C1:F11 NA NA NA NA
## B1:C1:F11 NA NA NA NA
## A1:D1:F11 NA NA NA NA
## B1:D1:F11 NA NA NA NA
## C1:D1:F11 NA NA NA NA
## A1:E1:F11 NA NA NA NA
## B1:E1:F11 NA NA NA NA
## C1:E1:F11 NA NA NA NA
## D1:E1:F11 NA NA NA NA
## A1:B1:C1:D1 NA NA NA NA
## A1:B1:C1:E1 NA NA NA NA
## A1:B1:D1:E1 NA NA NA NA
## A1:C1:D1:E1 NA NA NA NA
## B1:C1:D1:E1 NA NA NA NA
## A1:B1:C1:F11 NA NA NA NA
## A1:B1:D1:F11 NA NA NA NA
## A1:C1:D1:F11 NA NA NA NA
## B1:C1:D1:F11 NA NA NA NA
## A1:B1:E1:F11 NA NA NA NA
## A1:C1:E1:F11 NA NA NA NA
## B1:C1:E1:F11 NA NA NA NA
## A1:D1:E1:F11 NA NA NA NA
## B1:D1:E1:F11 NA NA NA NA
## C1:D1:E1:F11 NA NA NA NA
## A1:B1:C1:D1:E1 NA NA NA NA
## A1:B1:C1:D1:F11 NA NA NA NA
## A1:B1:C1:E1:F11 NA NA NA NA
## A1:B1:D1:E1:F11 NA NA NA NA
## A1:C1:D1:E1:F11 NA NA NA NA
## B1:C1:D1:E1:F11 NA NA NA NA
## A1:B1:C1:D1:E1:F11 NA NA NA NA
## ---
## 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 equation for this experiment in terms of coded variables is as under
--> Yi,j,k,l = 0.0157250 + 0.0010094(A) + 0.0017844(C) - 0.0041875(E) - 0.0077469(F)Therefore,
-->It would be best to keep A and C at low level and E and F at high level to reduce camber as much as possible5 Question 8.40:
5.1 Solution:
PART A:
--> The Experiment investigated “Four” FactorsPART B:
--> Resolution of the design employed in this experiment is “Four”PART C:
Effects are as follows:
One <- c(8)
AD <- c(10)
BD <- c(12)
AB <- c(7)
CD <- c(13)
AC <- c(6)
BC <- c(5)
ABCD <- c(11)Effect A:
EffectA <- (2*(AD+AB+AC+ABCD-One-BD-CD-BC))/(8)
EffectA## [1] -1Effect B:
EffectB <- (2*(BD+AB+BC+ABCD-One-AD-CD-AC))/(8)
EffectB## [1] -0.5Effect C:
EffectC <- (2*(CD+AC+BC+ABCD-One-AD-BD-AB))/(8)
EffectC## [1] -0.5Effect D:
EffectD <- (2*(AD+BD+CD+ABCD-One-AB-AC-BC))/(8)
EffectD## [1] 5PART D:
--> Defining relation for this design is I = ABCD6 Question 8.48:
What is the generator for column D?
What is the generator for column E?
If this design were folded over, what is the resolution of the combined design?
6.1 Solution:
design<- FrF2(nfactors = 5,nruns = 8,generators = c("-ABC","BC"), randomize = FALSE)
summary(design)## 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.generatorsPART A:
--> The design generator for column D is -ABC.PART B:
--> The design generator for column E is BC..PART C:
--> From table 8.14 from Douglas Montgomery Resolution of folded over design for such scenrio is “Four”.7 Question 8.60:
7.1 Solution:
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.foldeddesign3 <- design2[-c(1,3,5,7,10,12,14,16),]
design3## 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(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--> The relations are presented above