Homework 13
Rishidhar, Hemanth
2022-12-01
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':
##
## lengthsdesign_1<- FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
design_1## 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(design_1)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCObservation_1<- c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
design_2<- add.response(design_1,Observation_1)
design_2## A B C D Observation_1
## 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(design_2,half = TRUE)
MEPlot(design_2,show.alias = TRUE)
Answer:From the daniel plot we can say that no factor is Significant.
Question 8.24
library(FrF2)
design_3<- FrF2(nfactors = 5, nruns = 16,blocks = 2, randomize = TRUE)
summary(design_3)## 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 8.1.4 1 -1 1 1 1 -1
## 2 2 1.1.1 1 -1 -1 -1 -1 -1
## 3 3 15.1.8 1 1 1 1 -1 1
## 4 4 3.1.2 1 -1 -1 1 -1 1
## 5 5 6.1.3 1 -1 1 -1 1 1
## 6 6 13.1.7 1 1 1 -1 -1 -1
## 7 7 12.1.6 1 1 -1 1 1 -1
## 8 8 10.1.5 1 1 -1 -1 1 1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 9.2.5 2 1 -1 -1 -1 1
## 10 10 4.2.2 2 -1 -1 1 1 1
## 11 11 11.2.6 2 1 -1 1 -1 -1
## 12 12 16.2.8 2 1 1 1 1 1
## 13 13 2.2.1 2 -1 -1 -1 1 -1
## 14 14 14.2.7 2 1 1 -1 1 -1
## 15 15 5.2.3 2 -1 1 -1 -1 1
## 16 16 7.2.4 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(design_3)## $legend
## [1] A=A B=B C=C D=D E=E
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE AC=BE AE=BCdesign_3## run.no run.no.std.rp Blocks A B C D E
## 1 1 8.1.4 1 -1 1 1 1 -1
## 2 2 1.1.1 1 -1 -1 -1 -1 -1
## 3 3 15.1.8 1 1 1 1 -1 1
## 4 4 3.1.2 1 -1 -1 1 -1 1
## 5 5 6.1.3 1 -1 1 -1 1 1
## 6 6 13.1.7 1 1 1 -1 -1 -1
## 7 7 12.1.6 1 1 -1 1 1 -1
## 8 8 10.1.5 1 1 -1 -1 1 1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 9.2.5 2 1 -1 -1 -1 1
## 10 10 4.2.2 2 -1 -1 1 1 1
## 11 11 11.2.6 2 1 -1 1 -1 -1
## 12 12 16.2.8 2 1 1 1 1 1
## 13 13 2.2.1 2 -1 -1 -1 1 -1
## 14 14 14.2.7 2 1 1 -1 1 -1
## 15 15 5.2.3 2 -1 1 -1 -1 1
## 16 16 7.2.4 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 this we can see The block does not interfere with any of the main effects or two-factor interactions.
we can see the block is confounded with ABD
Question 8.25
library(FrF2)
design_4<- FrF2(nfactors = 7, nruns = 32, blocks = 4, randomize = TRUE)
summary(design_4)## 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 20.1.5 1 1 -1 -1 1 1 1 -1
## 2 2 29.1.8 1 1 1 1 -1 -1 1 -1
## 3 3 7.1.2 1 -1 -1 1 1 -1 1 1
## 4 4 22.1.6 1 1 -1 1 -1 1 -1 1
## 5 5 10.1.3 1 -1 1 -1 -1 1 1 1
## 6 6 27.1.7 1 1 1 -1 1 -1 -1 1
## 7 7 1.1.1 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 21.2.6 2 1 -1 1 -1 -1 -1 1
## 10 10 15.2.4 2 -1 1 1 1 -1 -1 -1
## 11 11 9.2.3 2 -1 1 -1 -1 -1 1 1
## 12 12 19.2.5 2 1 -1 -1 1 -1 1 -1
## 13 13 28.2.7 2 1 1 -1 1 1 -1 1
## 14 14 8.2.2 2 -1 -1 1 1 1 1 1
## 15 15 30.2.8 2 1 1 1 -1 1 1 -1
## 16 16 2.2.1 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 24.3.6 3 1 -1 1 1 1 -1 -1
## 19 19 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 20 20 18.3.5 3 1 -1 -1 -1 1 1 1
## 21 21 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 22 22 12.3.3 3 -1 1 -1 1 1 1 -1
## 23 23 14.3.4 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 13.4.4 4 -1 1 1 -1 -1 -1 1
## 26 26 23.4.6 4 1 -1 1 1 -1 -1 -1
## 27 27 26.4.7 4 1 1 -1 -1 1 -1 -1
## 28 28 17.4.5 4 1 -1 -1 -1 -1 1 1
## 29 29 32.4.8 4 1 1 1 1 1 1 1
## 30 30 6.4.2 4 -1 -1 1 -1 1 1 -1
## 31 31 11.4.3 4 -1 1 -1 1 -1 1 -1
## 32 32 4.4.1 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 framedesign_4## 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 29.1.8 1 1 1 1 -1 -1 1 -1
## 3 3 7.1.2 1 -1 -1 1 1 -1 1 1
## 4 4 22.1.6 1 1 -1 1 -1 1 -1 1
## 5 5 10.1.3 1 -1 1 -1 -1 1 1 1
## 6 6 27.1.7 1 1 1 -1 1 -1 -1 1
## 7 7 1.1.1 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 21.2.6 2 1 -1 1 -1 -1 -1 1
## 10 10 15.2.4 2 -1 1 1 1 -1 -1 -1
## 11 11 9.2.3 2 -1 1 -1 -1 -1 1 1
## 12 12 19.2.5 2 1 -1 -1 1 -1 1 -1
## 13 13 28.2.7 2 1 1 -1 1 1 -1 1
## 14 14 8.2.2 2 -1 -1 1 1 1 1 1
## 15 15 30.2.8 2 1 1 1 -1 1 1 -1
## 16 16 2.2.1 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 24.3.6 3 1 -1 1 1 1 -1 -1
## 19 19 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 20 20 18.3.5 3 1 -1 -1 -1 1 1 1
## 21 21 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 22 22 12.3.3 3 -1 1 -1 1 1 1 -1
## 23 23 14.3.4 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 13.4.4 4 -1 1 1 -1 -1 -1 1
## 26 26 23.4.6 4 1 -1 1 1 -1 -1 -1
## 27 27 26.4.7 4 1 1 -1 -1 1 -1 -1
## 28 28 17.4.5 4 1 -1 -1 -1 -1 1 1
## 29 29 32.4.8 4 1 1 1 1 1 1 1
## 30 30 6.4.2 4 -1 -1 1 -1 1 1 -1
## 31 31 11.4.3 4 -1 1 -1 1 -1 1 -1
## 32 32 4.4.1 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 frameAnswer:Here there are no two factor interactions and main effects confounded with blocks
Question 8.28
library(FrF2)
design_5 <- FrF2(nfactors = 6,nruns = 16,generators = c("ABC","ACD"), randomize = FALSE)
design_5## 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= FrF2.generatorssummary(design_5)## Call:
## FrF2(nfactors = 6, nruns = 16, generators = c("ABC", "ACD"),
## randomize = FALSE)
##
## Experimental design of type FrF2.generators
## 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=ACD
##
##
## Alias structure:
## $fi2
## [1] AB=CE AC=BE=DF AD=CF AE=BC AF=CD BD=EF BF=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= FrF2.generatorsaliasprint(design_5)## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE AC=BE=DF AD=CF AE=BC AF=CD BD=EF BF=DEObservation_2 <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.0110,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.0090,0.0250,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.0040,0.0147,0.0092,0.0226,0.0077,0.0060,0.0028,0.0158,0.0101,0.0126,0.0145,0.0110,0.0185,0.0020,0.0050,0.0030,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
A.a <- rep(design_5$A,4)
B.a <- rep(design_5$B,4)
C.a <- rep(design_5$C,4)
D.a <- rep(design_5$D,4)
E.a <- rep(design_5$E,4)
F.a <- rep(design_5$F,4)
design_6 <- aov(Observation_2~A.a*B.a*C.a*D.a*E.a*F.a)
summary(design_6)## Df Sum Sq Mean Sq F value Pr(>F)
## A.a 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## B.a 1 0.0000053 0.0000053 0.614 0.43725
## C.a 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## D.a 1 0.0000323 0.0000323 3.712 0.05995 .
## E.a 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## F.a 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## A.a:B.a 1 0.0000587 0.0000587 6.738 0.01249 *
## A.a:C.a 1 0.0000803 0.0000803 9.218 0.00387 **
## B.a:C.a 1 0.0000527 0.0000527 6.053 0.01754 *
## A.a:D.a 1 0.0000239 0.0000239 2.741 0.10431
## B.a:D.a 1 0.0000849 0.0000849 9.739 0.00305 **
## C.a:D.a 1 0.0000622 0.0000622 7.139 0.01027 *
## D.a:E.a 1 0.0000088 0.0000088 1.007 0.32062
## A.a:B.a:D.a 1 0.0000000 0.0000000 0.005 0.94291
## B.a:C.a:D.a 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 ' ' 1##c) The variables A,C,E,F lamination temperature,lamination pressure,firing cycle temperature,firing dew point
##d)
sd<- c(24.418,20.976,4.083,25.025,22.410,63.639,16.029,39.42,26.725,50.341,7.681,20.083,31.12,29.51,6.75,17.45)
value <- aov(sd~A*B*C*D*E*F,data = design_5)
halfnormal(value,ME.partial = TRUE)##
## The following effects are completely aliased:## [1] A:E B:E C:E A:F B:F C:F
## [7] D:F E:F A:B:C A:C:D A:B:E A:C:E
## [13] B:C:E A:D:E B:D:E C:D:E A:B:F A:C:F
## [19] B:C:F A:D:F B:D:F C:D:F A:E:F B:E:F
## [25] C:E:F D:E:F A:B:C:D A:B:C:E A:B:D:E A:C:D:E
## [31] B:C:D:E A:B:C:F A:B:D:F A:C:D:F B:C:D:F A:B:E:F
## [37] A:C:E:F B:C:E:F A:D:E:F B:D:E:F C:D:E:F A:B:C:D:E
## [43] A:B:C:D:F A:B:C:E:F A:B:D:E:F A:C:D:E:F B:C:D:E:F A:B:C:D:E:F##
## Significant effects (alpha=0.05, Lenth method):## [1] B1 A1
d) as we can see that the process variables A&B affect the variability in camber measurements.
Question 8.40
library(FrF2)
design_8<- FrF2(nfactors = 4, nruns = 8, randomize = FALSE)
y<- c(8,10,12,7,13,6,5,11)
Observation_3<- add.response(design_8,y)
summary(Observation_3)## Call:
## FrF2(nfactors = 4, nruns = 8, 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] y
##
## 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 y
## 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= FrF2A <- rep(c(-1,1),4)
B <- c(rep(-1,2),rep(1,2))
B <- rep(B,2)
C <- c(rep(-1,4),rep(1,4))
D<- A*B*C
system1<- aov(y~A*B*C*D)
coef(system1)## (Intercept) A B C D A:B
## 9.00 -0.50 -0.25 -0.25 2.50 0.75
## A:C B:C
## 0.25 -0.50Answers
A: 4 Factors
B: the resoultion is Four
C: A: -0.50 B: -0.25 C:-0.25 D:2.50 A:B 0.75 A:C 0.25 B:C -0.50
D : ABCD
Question 4.48
library(FrF2)
design_9<- FrF2(nfactors = 5,nruns = 8,generators = c("-ABC","BC"), randomize = FALSE)
design_9## 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(design_9)## 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.generatorsdesign.c <-fold.design(design_9)
aliasprint(design.c)## $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=-CDAnswer:
A: -ABC
B: BC
C: If this design is folded over, the resolution of the combined design could be FOUR
Question 8.60
library(FrF2)
design_10 <- FrF2(nfactors = 7,resolution = 3, randomize = TRUE)
design.c<- fold.design(design_10,column = 1)
design_10## 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= FrF2design.c## 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(design.c)## $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