HW13
Sahil Gaikwad
11/28/2021
Question 8.2
As only a one-half fraction of the 2^4 design could be run, hence the required design will have 2^(4-1) observations with I = ABCD.
#install.packages("FrF2")
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':
##
## lengthsdes.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= FrF2aliasprint(des.res4)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCresponse<-c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
des.resp<-add.response(des.res4,response)
summary(des.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= FrF2DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)
In resolution IV designs, we know that the main effects are not aliased with other main effects or two factor interaction
From the Daniel plot results we can conclude that no factor or interaction is significant that is affecting the cracking length
Question 8.24
To construct a 2^(5-1) design and to show how the design may be run in two blocks of eight observations each and also to determine any main effects or two-factor interactions confounded with blocks
des.res5<-FrF2(nfactors=5,blocks=2,nruns=16,randomize=FALSE,alias.block.2fis = TRUE)
des.res5## run.no run.no.std.rp Blocks A B C D E
## 1 1 5.1.1 1 -1 1 -1 -1 -1
## 2 2 6.1.2 1 -1 1 -1 1 1
## 3 3 7.1.3 1 -1 1 1 -1 1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 9.1.5 1 1 -1 -1 -1 -1
## 6 6 10.1.6 1 1 -1 -1 1 1
## 7 7 11.1.7 1 1 -1 1 -1 1
## 8 8 12.1.8 1 1 -1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 1.2.1 2 -1 -1 -1 -1 1
## 10 10 2.2.2 2 -1 -1 -1 1 -1
## 11 11 3.2.3 2 -1 -1 1 -1 -1
## 12 12 4.2.4 2 -1 -1 1 1 1
## 13 13 13.2.5 2 1 1 -1 -1 1
## 14 14 14.2.6 2 1 1 -1 1 -1
## 15 15 15.2.7 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=Esummary(des.res5)## Call:
## FrF2(nfactors = 5, blocks = 2, nruns = 16, randomize = FALSE,
## alias.block.2fis = 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=ABCD
##
## $`block generators`
## [1] AB
##
##
## no aliasing of main effects or 2fis among experimental factors
##
## Aliased with block main effects:
## [1] AB
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E
## 1 1 5.1.1 1 -1 1 -1 -1 -1
## 2 2 6.1.2 1 -1 1 -1 1 1
## 3 3 7.1.3 1 -1 1 1 -1 1
## 4 4 8.1.4 1 -1 1 1 1 -1
## 5 5 9.1.5 1 1 -1 -1 -1 -1
## 6 6 10.1.6 1 1 -1 -1 1 1
## 7 7 11.1.7 1 1 -1 1 -1 1
## 8 8 12.1.8 1 1 -1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E
## 9 9 1.2.1 2 -1 -1 -1 -1 1
## 10 10 2.2.2 2 -1 -1 -1 1 -1
## 11 11 3.2.3 2 -1 -1 1 -1 -1
## 12 12 4.2.4 2 -1 -1 1 1 1
## 13 13 13.2.5 2 1 1 -1 -1 1
## 14 14 14.2.6 2 1 1 -1 1 -1
## 15 15 15.2.7 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 frameIn this case we have 2 blocks and from the results we can conclude that no main effects are confounded with the blocks but the interaction AB and CDE are confounded with the blocks
Question 8.25
To construct a 2^(7-2) design and to show how the design may be run in four blocks of eight observations each and also to determine any main effects or two-factor interactions confounded with blocks
des.res7<-FrF2(nfactors=7,blocks=4,nruns=32,randomize=FALSE,alias.block.2fis = TRUE)
des.res7## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 13.1.1 1 -1 1 1 -1 -1 -1 -1
## 2 2 14.1.2 1 -1 1 1 -1 1 -1 1
## 3 3 15.1.3 1 -1 1 1 1 -1 -1 1
## 4 4 16.1.4 1 -1 1 1 1 1 -1 -1
## 5 5 17.1.5 1 1 -1 -1 -1 -1 1 -1
## 6 6 18.1.6 1 1 -1 -1 -1 1 1 1
## 7 7 19.1.7 1 1 -1 -1 1 -1 1 1
## 8 8 20.1.8 1 1 -1 -1 1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.1 2 -1 1 -1 -1 -1 1 -1
## 10 10 10.2.2 2 -1 1 -1 -1 1 1 1
## 11 11 11.2.3 2 -1 1 -1 1 -1 1 1
## 12 12 12.2.4 2 -1 1 -1 1 1 1 -1
## 13 13 21.2.5 2 1 -1 1 -1 -1 -1 -1
## 14 14 22.2.6 2 1 -1 1 -1 1 -1 1
## 15 15 23.2.7 2 1 -1 1 1 -1 -1 1
## 16 16 24.2.8 2 1 -1 1 1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 5.3.1 3 -1 -1 1 -1 -1 1 1
## 18 18 6.3.2 3 -1 -1 1 -1 1 1 -1
## 19 19 7.3.3 3 -1 -1 1 1 -1 1 -1
## 20 20 8.3.4 3 -1 -1 1 1 1 1 1
## 21 21 25.3.5 3 1 1 -1 -1 -1 -1 1
## 22 22 26.3.6 3 1 1 -1 -1 1 -1 -1
## 23 23 27.3.7 3 1 1 -1 1 -1 -1 -1
## 24 24 28.3.8 3 1 1 -1 1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 1.4.1 4 -1 -1 -1 -1 -1 -1 1
## 26 26 2.4.2 4 -1 -1 -1 -1 1 -1 -1
## 27 27 3.4.3 4 -1 -1 -1 1 -1 -1 -1
## 28 28 4.4.4 4 -1 -1 -1 1 1 -1 1
## 29 29 29.4.5 4 1 1 1 -1 -1 1 1
## 30 30 30.4.6 4 1 1 1 -1 1 1 -1
## 31 31 31.4.7 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=Gsummary(des.res7)## Call:
## FrF2(nfactors = 7, blocks = 4, nruns = 32, randomize = FALSE,
## alias.block.2fis = 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=ABDE
##
## $`block generators`
## [1] AB AC
##
##
## no aliasing of main effects or 2fis among experimental factors
##
## Aliased with block main effects:
## [1] AB AC AF BC BF CF
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 13.1.1 1 -1 1 1 -1 -1 -1 -1
## 2 2 14.1.2 1 -1 1 1 -1 1 -1 1
## 3 3 15.1.3 1 -1 1 1 1 -1 -1 1
## 4 4 16.1.4 1 -1 1 1 1 1 -1 -1
## 5 5 17.1.5 1 1 -1 -1 -1 -1 1 -1
## 6 6 18.1.6 1 1 -1 -1 -1 1 1 1
## 7 7 19.1.7 1 1 -1 -1 1 -1 1 1
## 8 8 20.1.8 1 1 -1 -1 1 1 1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.1 2 -1 1 -1 -1 -1 1 -1
## 10 10 10.2.2 2 -1 1 -1 -1 1 1 1
## 11 11 11.2.3 2 -1 1 -1 1 -1 1 1
## 12 12 12.2.4 2 -1 1 -1 1 1 1 -1
## 13 13 21.2.5 2 1 -1 1 -1 -1 -1 -1
## 14 14 22.2.6 2 1 -1 1 -1 1 -1 1
## 15 15 23.2.7 2 1 -1 1 1 -1 -1 1
## 16 16 24.2.8 2 1 -1 1 1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 5.3.1 3 -1 -1 1 -1 -1 1 1
## 18 18 6.3.2 3 -1 -1 1 -1 1 1 -1
## 19 19 7.3.3 3 -1 -1 1 1 -1 1 -1
## 20 20 8.3.4 3 -1 -1 1 1 1 1 1
## 21 21 25.3.5 3 1 1 -1 -1 -1 -1 1
## 22 22 26.3.6 3 1 1 -1 -1 1 -1 -1
## 23 23 27.3.7 3 1 1 -1 1 -1 -1 -1
## 24 24 28.3.8 3 1 1 -1 1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 1.4.1 4 -1 -1 -1 -1 -1 -1 1
## 26 26 2.4.2 4 -1 -1 -1 -1 1 -1 -1
## 27 27 3.4.3 4 -1 -1 -1 1 -1 -1 -1
## 28 28 4.4.4 4 -1 -1 -1 1 1 -1 1
## 29 29 29.4.5 4 1 1 1 -1 -1 1 1
## 30 30 30.4.6 4 1 1 1 -1 1 1 -1
## 31 31 31.4.7 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 frameNo main effects or two factor interactions are confounded with the blocks
Blocks are confounded with ACE, BFG, and ABCEFG
Question 8.28
des.res28<-FrF2(nfactors=6,resolution=4,randomize = FALSE)
print (des.res28)## 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= FrF2a) Given experiment is a 2^(6-2) design of 16 runs with resolution IV, hence here K = 6 and P = 2
aliasprint(des.res28)## $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=DEsummary(des.res28)## 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= FrF2b) The alias relationships were shown in results of previous chunk
c)
obs<-c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.011,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.009,0.025,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.004,0.0147,0.0092,0.0226,0.0077,0.006,0.0028,0.0158,0.0101,0.0126,0.0145,0.011,0.0185,0.002,0.005,0.003,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
temp<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
time<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
press<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
ftemp<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
cycle<-c(-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1)
dew<-c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1)
temp<- as.factor(temp)
time<- as.factor(time)
press<- as.factor(press)
ftemp<- as.factor(ftemp)
cycle<- as.factor(cycle)
dew<- as.factor(dew)
df<-data.frame(obs,temp,time,press,ftemp,cycle,dew)
aov.model <- aov(obs~temp*time*press*ftemp*cycle*dew,data = df)
summary(aov.model)## Df Sum Sq Mean Sq F value Pr(>F)
## temp 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## time 1 0.0000053 0.0000053 0.614 0.43725
## press 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## ftemp 1 0.0000323 0.0000323 3.712 0.05995 .
## cycle 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## dew 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## temp:time 1 0.0000587 0.0000587 6.738 0.01249 *
## temp:press 1 0.0000803 0.0000803 9.218 0.00387 **
## time:press 1 0.0000527 0.0000527 6.053 0.01754 *
## temp:ftemp 1 0.0000239 0.0000239 2.741 0.10431
## time:ftemp 1 0.0000849 0.0000849 9.739 0.00305 **
## press:ftemp 1 0.0000622 0.0000622 7.139 0.01027 *
## ftemp:cycle 1 0.0000088 0.0000088 1.007 0.32062
## temp:time:ftemp 1 0.0000000 0.0000000 0.005 0.94291
## time:press:ftemp 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 ' ' 1From the results of ANOVA test, we can conclude that the some of the main effcets and higher interactions are significant and can affect the avaerage camber. Hence factors that will affect the average camber are temp,press, cycle and dew
d)
sdeviation<-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)
df1<-data.frame(sdeviation,temp,time,press,ftemp,cycle,dew)
df1## sdeviation temp time press ftemp cycle dew
## 1 24.418 -1 -1 -1 -1 -1 -1
## 2 20.976 1 -1 -1 -1 1 1
## 3 4.083 -1 1 -1 -1 1 -1
## 4 25.025 1 1 -1 -1 -1 1
## 5 22.410 -1 -1 1 -1 1 1
## 6 63.639 1 -1 1 -1 -1 -1
## 7 16.029 -1 1 1 -1 -1 1
## 8 39.420 1 1 1 -1 1 -1
## 9 26.725 -1 -1 -1 1 -1 1
## 10 50.341 1 -1 -1 1 1 -1
## 11 7.681 -1 1 -1 1 1 1
## 12 20.083 1 1 -1 1 -1 -1
## 13 31.120 -1 -1 1 1 1 -1
## 14 29.510 1 -1 1 1 -1 1
## 15 6.750 -1 1 1 1 -1 -1
## 16 17.450 1 1 1 1 1 1aovmodel<-aov(sdeviation ~ temp*time*press*ftemp*cycle*dew,data = df1)
DanielPlot(aovmodel)
From the result of Daniel plot, we could conclude that the variables that affect variability in camber measurements are temp and time
aovmodel2<-aov(sdeviation~time+temp,data = df1)
summary(aovmodel2)## Df Sum Sq Mean Sq F value Pr(>F)
## time 1 1099 1099 9.241 0.00948 **
## temp 1 1012 1012 8.505 0.01202 *
## Residuals 13 1546 119
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the result of ANOVA analysis, we could conclude that the variables that affect variability in camber measurements are temp and time
e)
modelnew<-lm(obs~temp*time*press*ftemp*cycle*dew,data = df)
coef(modelnew)## (Intercept) temp1
## 0.015725000 0.001009375
## time1 press1
## -0.007137500 0.001784375
## ftemp1 cycle1
## -0.002953125 -0.004187500
## dew1 temp1:time1
## -0.007746875 0.003725000
## temp1:press1 time1:press1
## 0.004481250 0.007100000
## temp1:ftemp1 time1:ftemp1
## -0.002550000 0.007968750
## press1:ftemp1 temp1:cycle1
## -0.000475000 NA
## time1:cycle1 press1:cycle1
## NA NA
## ftemp1:cycle1 temp1:dew1
## 0.001481250 NA
## time1:dew1 press1:dew1
## NA NA
## ftemp1:dew1 cycle1:dew1
## NA NA
## temp1:time1:press1 temp1:time1:ftemp1
## NA 0.000212500
## temp1:press1:ftemp1 time1:press1:ftemp1
## NA -0.006937500
## temp1:time1:cycle1 temp1:press1:cycle1
## NA NA
## time1:press1:cycle1 temp1:ftemp1:cycle1
## NA NA
## time1:ftemp1:cycle1 press1:ftemp1:cycle1
## NA NA
## temp1:time1:dew1 temp1:press1:dew1
## NA NA
## time1:press1:dew1 temp1:ftemp1:dew1
## NA NA
## time1:ftemp1:dew1 press1:ftemp1:dew1
## NA NA
## temp1:cycle1:dew1 time1:cycle1:dew1
## NA NA
## press1:cycle1:dew1 ftemp1:cycle1:dew1
## NA NA
## temp1:time1:press1:ftemp1 temp1:time1:press1:cycle1
## NA NA
## temp1:time1:ftemp1:cycle1 temp1:press1:ftemp1:cycle1
## NA NA
## time1:press1:ftemp1:cycle1 temp1:time1:press1:dew1
## NA NA
## temp1:time1:ftemp1:dew1 temp1:press1:ftemp1:dew1
## NA NA
## time1:press1:ftemp1:dew1 temp1:time1:cycle1:dew1
## NA NA
## temp1:press1:cycle1:dew1 time1:press1:cycle1:dew1
## NA NA
## temp1:ftemp1:cycle1:dew1 time1:ftemp1:cycle1:dew1
## NA NA
## press1:ftemp1:cycle1:dew1 temp1:time1:press1:ftemp1:cycle1
## NA NA
## temp1:time1:press1:ftemp1:dew1 temp1:time1:press1:cycle1:dew1
## NA NA
## temp1:time1:ftemp1:cycle1:dew1 temp1:press1:ftemp1:cycle1:dew1
## NA NA
## time1:press1:ftemp1:cycle1:dew1 temp1:time1:press1:ftemp1:cycle1:dew1
## NA NAsummary(modelnew)##
## Call:
## lm.default(formula = obs ~ temp * time * press * ftemp * cycle *
## dew, data = df)
##
## 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
## temp1 0.0010094 0.0016502 0.612 0.543644
## time1 -0.0071375 0.0018077 -3.948 0.000257
## press1 0.0017844 0.0016502 1.081 0.284963
## ftemp1 -0.0029531 0.0019525 -1.512 0.136976
## cycle1 -0.0041875 0.0010437 -4.012 0.000210
## dew1 -0.0077469 0.0007380 -10.497 5.05e-14
## temp1:time1 0.0037250 0.0020874 1.785 0.080655
## temp1:press1 0.0044812 0.0014760 3.036 0.003866
## time1:press1 0.0071000 0.0020874 3.401 0.001359
## temp1:ftemp1 -0.0025500 0.0020874 -1.222 0.227809
## time1:ftemp1 0.0079688 0.0025565 3.117 0.003083
## press1:ftemp1 -0.0004750 0.0020874 -0.228 0.820954
## temp1:cycle1 NA NA NA NA
## time1:cycle1 NA NA NA NA
## press1:cycle1 NA NA NA NA
## ftemp1:cycle1 0.0014813 0.0014760 1.004 0.320619
## temp1:dew1 NA NA NA NA
## time1:dew1 NA NA NA NA
## press1:dew1 NA NA NA NA
## ftemp1:dew1 NA NA NA NA
## cycle1:dew1 NA NA NA NA
## temp1:time1:press1 NA NA NA NA
## temp1:time1:ftemp1 0.0002125 0.0029520 0.072 0.942912
## temp1:press1:ftemp1 NA NA NA NA
## time1:press1:ftemp1 -0.0069375 0.0029520 -2.350 0.022926
## temp1:time1:cycle1 NA NA NA NA
## temp1:press1:cycle1 NA NA NA NA
## time1:press1:cycle1 NA NA NA NA
## temp1:ftemp1:cycle1 NA NA NA NA
## time1:ftemp1:cycle1 NA NA NA NA
## press1:ftemp1:cycle1 NA NA NA NA
## temp1:time1:dew1 NA NA NA NA
## temp1:press1:dew1 NA NA NA NA
## time1:press1:dew1 NA NA NA NA
## temp1:ftemp1:dew1 NA NA NA NA
## time1:ftemp1:dew1 NA NA NA NA
## press1:ftemp1:dew1 NA NA NA NA
## temp1:cycle1:dew1 NA NA NA NA
## time1:cycle1:dew1 NA NA NA NA
## press1:cycle1:dew1 NA NA NA NA
## ftemp1:cycle1:dew1 NA NA NA NA
## temp1:time1:press1:ftemp1 NA NA NA NA
## temp1:time1:press1:cycle1 NA NA NA NA
## temp1:time1:ftemp1:cycle1 NA NA NA NA
## temp1:press1:ftemp1:cycle1 NA NA NA NA
## time1:press1:ftemp1:cycle1 NA NA NA NA
## temp1:time1:press1:dew1 NA NA NA NA
## temp1:time1:ftemp1:dew1 NA NA NA NA
## temp1:press1:ftemp1:dew1 NA NA NA NA
## time1:press1:ftemp1:dew1 NA NA NA NA
## temp1:time1:cycle1:dew1 NA NA NA NA
## temp1:press1:cycle1:dew1 NA NA NA NA
## time1:press1:cycle1:dew1 NA NA NA NA
## temp1:ftemp1:cycle1:dew1 NA NA NA NA
## time1:ftemp1:cycle1:dew1 NA NA NA NA
## press1:ftemp1:cycle1:dew1 NA NA NA NA
## temp1:time1:press1:ftemp1:cycle1 NA NA NA NA
## temp1:time1:press1:ftemp1:dew1 NA NA NA NA
## temp1:time1:press1:cycle1:dew1 NA NA NA NA
## temp1:time1:ftemp1:cycle1:dew1 NA NA NA NA
## temp1:press1:ftemp1:cycle1:dew1 NA NA NA NA
## time1:press1:ftemp1:cycle1:dew1 NA NA NA NA
## temp1:time1:press1:ftemp1:cycle1:dew1 NA NA NA NA
##
## (Intercept) ***
## temp1
## time1 ***
## press1
## ftemp1
## cycle1 ***
## dew1 ***
## temp1:time1 .
## temp1:press1 **
## time1:press1 **
## temp1:ftemp1
## time1:ftemp1 **
## press1:ftemp1
## temp1:cycle1
## time1:cycle1
## press1:cycle1
## ftemp1:cycle1
## temp1:dew1
## time1:dew1
## press1:dew1
## ftemp1:dew1
## cycle1:dew1
## temp1:time1:press1
## temp1:time1:ftemp1
## temp1:press1:ftemp1
## time1:press1:ftemp1 *
## temp1:time1:cycle1
## temp1:press1:cycle1
## time1:press1:cycle1
## temp1:ftemp1:cycle1
## time1:ftemp1:cycle1
## press1:ftemp1:cycle1
## temp1:time1:dew1
## temp1:press1:dew1
## time1:press1:dew1
## temp1:ftemp1:dew1
## time1:ftemp1:dew1
## press1:ftemp1:dew1
## temp1:cycle1:dew1
## time1:cycle1:dew1
## press1:cycle1:dew1
## ftemp1:cycle1:dew1
## temp1:time1:press1:ftemp1
## temp1:time1:press1:cycle1
## temp1:time1:ftemp1:cycle1
## temp1:press1:ftemp1:cycle1
## time1:press1:ftemp1:cycle1
## temp1:time1:press1:dew1
## temp1:time1:ftemp1:dew1
## temp1:press1:ftemp1:dew1
## time1:press1:ftemp1:dew1
## temp1:time1:cycle1:dew1
## temp1:press1:cycle1:dew1
## time1:press1:cycle1:dew1
## temp1:ftemp1:cycle1:dew1
## time1:ftemp1:cycle1:dew1
## press1:ftemp1:cycle1:dew1
## temp1:time1:press1:ftemp1:cycle1
## temp1:time1:press1:ftemp1:dew1
## temp1:time1:press1:cycle1:dew1
## temp1:time1:ftemp1:cycle1:dew1
## temp1:press1:ftemp1:cycle1:dew1
## time1:press1:ftemp1:cycle1:dew1
## temp1:time1:press1:ftemp1:cycle1:dew1
## ---
## 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-15From the above analysis, we conclude that the factors temp and press should be run at low level and the factors cycle and dew should be run at high level
Question 8.40
a) The experiment investigate the 4 factors (a,b,c,d)
b) The resolution of the design is 4
c)
one<-8
AD<-10
BD<-12
AB<-7
CD<-13
AC<-6
BC<-5
ABCD<-11
factA_eff<-(2*(AD+AB+AC+ABCD-one-BD-CD-BC))/(16)
print(factA_eff)## [1] -0.5factB_eff<-(2*(BD+AB+BC+ABCD-one-AD-CD-AC))/(16)
print(factB_eff)## [1] -0.25factB_eff<-(2*(CD+AC+BC+ABCD-one-AD-BD-AB))/(16)
print(factB_eff)## [1] -0.25factB_eff<-(2*(AD+BD+CD+ABCD-one-AB-AC-BC))/(16)
print(factB_eff)## [1] 2.5Estimate of main effects is as follows
A: -0.5 B: -0.25 C: -0.25 D: 2.5
d)
The complete defining relation for given design is I = ABCD
Question 8.48
The given design has 5 factors and 8 observations, hence design is of form 2^(5-2) and the resolution of the given design is 3.
a) The generator of column D is -ABC because column D will have same signs as of -ABC
b) The generator of column E is BC because E has similar signs of BC
c) If this design was folded over, the resolution of the combined design is 4, as the resolution of original design was III and folding will add one more
Question 8.60
Considering partial fold over for 2^(7-4) design with resolution III and constructing partial fold over of this design using column A
library(FrF2)
des.res7<-FrF2(nfactors=7,resolution=3,randomize=FALSE)
des.res7## 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.res7n<-fold.design(des.res7,column=1)
des.res7n## 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.res7n)## $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.res7n)## 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.foldedThe alias relationships for main effects aliased with 2factor interaction are as follows due to resolution III
B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE
The alias relationships for 2 factor interaction effects aliased with other 2factor interaction are as follows
AB=-DE AC=-DF AD=-BE=-CF=-GH AE=-BD AF=-CD
AG=-DH AH=-DG
All Source Code
#install.packages("FrF2")
library(FrF2)
des.res4<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res4
aliasprint(des.res4)
response<-c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
des.resp<-add.response(des.res4,response)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)
des.res5<-FrF2(nfactors=5,blocks=2,nruns=16,randomize=FALSE,alias.block.2fis = TRUE)
des.res5
aliasprint(des.res5)
summary(des.res5)
des.res7<-FrF2(nfactors=7,blocks=4,nruns=32,randomize=FALSE,alias.block.2fis = TRUE)
des.res7
aliasprint(des.res7)
summary(des.res7)
des.res28<-FrF2(nfactors=6,resolution=4,randomize = FALSE)
print (des.res28)
aliasprint(des.res28)
summary(des.res28)
obs<-c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.011,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.009,0.025,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.004,0.0147,0.0092,0.0226,0.0077,0.006,0.0028,0.0158,0.0101,0.0126,0.0145,0.011,0.0185,0.002,0.005,0.003,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
temp<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
time<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
press<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
ftemp<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
cycle<-c(-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1)
dew<-c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1)
temp<- as.factor(temp)
time<- as.factor(time)
press<- as.factor(press)
ftemp<- as.factor(ftemp)
cycle<- as.factor(cycle)
dew<- as.factor(dew)
df<-data.frame(obs,temp,time,press,ftemp,cycle,dew)
aov.model <- aov(obs~temp*time*press*ftemp*cycle*dew,data = df)
summary(aov.model)
sdeviation<-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)
df1<-data.frame(sdeviation,temp,time,press,ftemp,cycle,dew)
df1
aovmodel<-aov(sdeviation ~ temp*time*press*ftemp*cycle*dew,data = df1)
DanielPlot(aovmodel)
aovmodel2<-aov(sdeviation~time+temp,data = df1)
summary(aovmodel2)
modelnew<-lm(obs~temp*time*press*ftemp*cycle*dew,data = df)
coef(modelnew)
summary(modelnew)
one<-8
AD<-10
BD<-12
AB<-7
CD<-13
AC<-6
BC<-5
ABCD<-11
factA_eff<-(2*(AD+AB+AC+ABCD-one-BD-CD-BC))/(16)
print(factA_eff)
factB_eff<-(2*(BD+AB+BC+ABCD-one-AD-CD-AC))/(16)
print(factB_eff)
factB_eff<-(2*(CD+AC+BC+ABCD-one-AD-BD-AB))/(16)
print(factB_eff)
factB_eff<-(2*(AD+BD+CD+ABCD-one-AB-AC-BC))/(16)
print(factB_eff)
library(FrF2)
des.res7<-FrF2(nfactors=7,resolution=3,randomize=FALSE)
des.res7
des.res7n<-fold.design(des.res7,column=1)
des.res7n
aliasprint(des.res7n)
summary(des.res7n)