Homework 13
Eghorieta /
11/24/2021
Question 8.2
Suppose that in Problem 6.15, only a one-half fraction of the 24 design could be run. Construct the design and per- form the analysis, using the data from replicate I.
Design model
## 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':
##
## lengths## Call:
## FrF2(nfactors = 4, resolution = 3, 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= FrF2Then we get factors that appear to be large in the model design
##
## Call:
## lm.default(formula = response ~ A * B * C * D, data = des.res)
##
## Residuals:
## ALL 8 residuals are 0: no residual degrees of freedom!
##
## Coefficients: (8 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.0350 NA NA NA
## A1 1.5053 NA NA NA
## B1 2.0055 NA NA NA
## C1 -1.7282 NA NA NA
## D1 2.5255 NA NA NA
## A1:B1 0.9172 NA NA NA
## A1:C1 -1.8015 NA NA NA
## B1:C1 0.1943 NA NA NA
## A1:D1 NA NA NA NA
## B1:D1 NA NA NA NA
## C1:D1 NA NA NA NA
## A1:B1:C1 NA NA NA NA
## A1:B1:D1 NA NA NA NA
## A1:C1:D1 NA NA NA NA
## B1:C1:D1 NA NA NA NA
## A1:B1:C1:D1 NA NA NA NA
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 7 and 0 DF, p-value: NAAB=CD AC=BD AD=BC The following model terms A,B,C,D,AB,AC and BD appear to be large. BD because its alias AC=BD, since BD is large due to main effects B and D. However we conduct halfnormal and main effects plot for further screening.
From the plots, halfnormal doesn’t depict a clear representation of significant model terms, but main effect plots indicate model terms A,B,C,D are significant.
Analysis of variance
## Df Sum Sq Mean Sq
## A 1 18.13 18.13
## B 1 32.18 32.18
## C 1 23.89 23.89
## D 1 51.03 51.03
## A:B 1 6.73 6.73
## A:C 1 25.96 25.96
## B:C 1 0.30 0.30The model is not significant and will require additional runs to determine anova test. Perhaps a one-quarter fraction of the design to carry out the experiment
Question 8.24
Construct a \(2^{5-1}\) design. Show how the design may be run in two blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
For 2 blocks of 8 obs each, we would require a one quarter design to get 16 runs to place ‘+’ signs in block1 and ‘-’ signs in block2. Also note main effects cannot be used for blocks.
## $legend
## [1] A=A B=B C=C D=D E=E
##
## [[2]]
## [1] no aliasing among main effects and 2fis## Call:
## FrF2(nfactors = 5, resolution = 4, 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= FrF2## [1] 16## 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= FrF2## t AB blocks
## 1 e 1 1
## 2 a -1 2
## 3 b -1 2
## 4 abe 1 1
## 5 c 1 1
## 6 ace -1 2
## 7 bce -1 2
## 8 abc 1 1
## 9 d 1 1
## 10 ade -1 2
## 11 bde -1 2
## 12 abd 1 1
## 13 cde 1 1
## 14 acd -1 2
## 15 bcd -1 2
## 16 abcde 1 1Factors AB and CDE are confounded in the blocks.
Question 8.25
Construct a \(2^{7−2}\) design. Show how the design may be run in four blocks of eight observations each. Are any main effects or two-factor interactions confounded with blocks?
## [1] 32## t ACE BFG blocks
## 1 (I) -1 -1 1
## 2 ag 1 1 4
## 3 bg -1 -1 1
## 4 ab 1 1 4
## 5 cfg 1 -1 3
## 6 acf -1 1 2
## 7 bcf 1 -1 3
## 8 abcfg -1 1 2
## 9 df -1 1 2
## 10 adfg 1 -1 3
## 11 bdfg -1 1 2
## 12 abdf 1 -1 3
## 13 cdg 1 1 4
## 14 acd -1 -1 1
## 15 bcd 1 1 4
## 16 abcdg -1 -1 1
## 17 ef 1 1 4
## 18 aefg -1 -1 1
## 19 befg 1 1 4
## 20 abef -1 -1 1
## 21 ceg -1 1 2
## 22 ace 1 -1 3
## 23 bce -1 1 2
## 24 abceg 1 -1 3
## 25 de 1 -1 3
## 26 adeg -1 1 2
## 27 bdeg 1 -1 3
## 28 abde -1 1 2
## 29 cdefg -1 -1 1
## 30 acdef 1 1 4
## 31 bcdef -1 -1 1
## 32 abcdefg 1 1 4Question 8.28
A 16-run experiment was performed in a semiconductor manufacturing plant to study the effects of six factors on the curvature or camber of the substrate devices produced
## 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) The experimenter used a \(2^{6-2}\) design and a resolution 4 to get 16 runs
## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DEb) The aliases are model terms AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DE
c) From the camber data presented in the question, the experimenter changed the coded levels for model term F. Hence we manually create the designs to fit the experiment
Using the original design generated
## Number of observations used: 16
## Formula:
## camber ~ (A + B + C + D + E + F)^2
##
## Call:
## lm.default(formula = fo, data = model.frame(fo, data = formula))
##
## Residuals:
## 1 2 3 4 5 6 7 8 9 10 11
## 189.6 -120.3 120.3 -189.6 -189.6 120.2 -120.3 189.6 -189.6 120.3 -120.3
## 12 13 14 15 16
## 189.6 189.6 -120.3 120.3 -189.6
##
## Coefficients: (8 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 428.062 112.269 3.813 0.0624 .
## A1 77.812 112.269 0.693 0.5599
## B1 11.562 112.269 0.103 0.9274
## C1 112.063 112.269 0.998 0.4234
## D1 -28.437 112.269 -0.253 0.8237
## E1 -68.938 112.269 -0.614 0.6017
## F1 1.063 112.269 0.009 0.9933
## A1:B1 38.312 112.269 0.341 0.7654
## A1:C1 44.813 112.269 0.399 0.7284
## A1:D1 -24.438 112.269 -0.218 0.8479
## A1:E1 36.312 112.269 0.323 0.7770
## A1:F1 46.063 112.269 0.410 0.7214
## B1:C1 NA NA NA NA
## B1:D1 NA NA NA NA
## B1:E1 NA NA NA NA
## B1:F1 NA NA NA NA
## C1:D1 -39.437 112.269 -0.351 0.7589
## C1:E1 NA NA NA NA
## C1:F1 14.813 112.269 0.132 0.9071
## D1:E1 NA NA NA NA
## D1:F1 NA NA NA NA
## E1:F1 NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 449.1 on 2 degrees of freedom
## Multiple R-squared: 0.5713, Adjusted R-squared: -2.215
## F-statistic: 0.205 on 13 and 2 DF, p-value: 0.9737Using the experimenter’s design by changing the coding levels for factor F to correspond with the experiment
##
## Call:
## lm.default(formula = camber ~ A * B * C * D * E * FF, data = dat)
##
## Residuals:
## ALL 16 residuals are 0: no residual degrees of freedom!
##
## Coefficients: (48 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 428.062 NA NA NA
## A 77.813 NA NA NA
## B 11.562 NA NA NA
## C 112.063 NA NA NA
## D -28.437 NA NA NA
## E -68.938 NA NA NA
## FF -154.938 NA NA NA
## A:B 38.312 NA NA NA
## A:C 44.813 NA NA NA
## B:C 36.313 NA NA NA
## A:D -24.438 NA NA NA
## B:D 46.063 NA NA NA
## C:D -39.437 NA NA NA
## A:E NA NA NA NA
## B:E NA NA NA NA
## C:E NA NA NA NA
## D:E 14.812 NA NA NA
## A:FF NA NA NA NA
## B:FF NA NA NA NA
## C:FF NA NA NA NA
## D:FF NA NA NA NA
## E:FF NA NA NA NA
## A:B:C NA NA NA NA
## A:B:D 1.063 NA NA NA
## A:C:D NA NA NA NA
## B:C:D -34.688 NA NA NA
## A:B:E NA NA NA NA
## A:C:E NA NA NA NA
## B:C:E NA NA NA NA
## A:D:E NA NA NA NA
## B:D:E NA NA NA NA
## C:D:E NA NA NA NA
## A:B:FF NA NA NA NA
## A:C:FF NA NA NA NA
## B:C:FF NA NA NA NA
## A:D:FF NA NA NA NA
## B:D:FF NA NA NA NA
## C:D:FF NA NA NA NA
## A:E:FF NA NA NA NA
## B:E:FF NA NA NA NA
## C:E:FF NA NA NA NA
## D:E:FF NA NA NA NA
## A:B:C:D NA NA NA NA
## A:B:C:E NA NA NA NA
## A:B:D:E NA NA NA NA
## A:C:D:E NA NA NA NA
## B:C:D:E NA NA NA NA
## A:B:C:FF NA NA NA NA
## A:B:D:FF NA NA NA NA
## A:C:D:FF NA NA NA NA
## B:C:D:FF NA NA NA NA
## A:B:E:FF NA NA NA NA
## A:C:E:FF NA NA NA NA
## B:C:E:FF NA NA NA NA
## A:D:E:FF NA NA NA NA
## B:D:E:FF NA NA NA NA
## C:D:E:FF NA NA NA NA
## A:B:C:D:E NA NA NA NA
## A:B:C:D:FF NA NA NA NA
## A:B:C:E:FF NA NA NA NA
## A:B:D:E:FF NA NA NA NA
## A:C:D:E:FF NA NA NA NA
## B:C:D:E:FF NA NA NA NA
## A:B:C:D:E:FF NA NA NA NA
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 15 and 0 DF, p-value: NAHalf normal plot of design generated and experimenter’s design
Its not still clear from both plots as to which model terms are significant as this is merely a screening test to identify significant model terms. We would construct main effects plot for both designs
MEPlot(des.res,show.alias=TRUE,main ="Main Effects plot for design generated ")
MEPlot(u,show.alias=TRUE, main = "Main Effects plot for experimenter's design")
From the plots the design generated has main effects A,C,D,E to be significant. While the experimenter’s design has main effects A,C,E and F to be significant.
Anova for design generated
model<-aov(camber~A+C+D+E, data = des.res)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 96877 96877 1.923 0.1930
## C 1 200928 200928 3.989 0.0711 .
## D 1 12939 12939 0.257 0.6223
## E 1 76038 76038 1.509 0.2449
## Residuals 11 554111 50374
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the result, values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case none of the model terms appear to be significant
model<-aov(camber~A+C+E+FF, data = dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 96877 96877 5.824 0.034407 *
## C 1 200928 200928 12.080 0.005188 **
## E 1 76038 76038 4.572 0.055787 .
## FF 1 384090 384090 23.092 0.000549 ***
## Residuals 11 182960 16633
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the result, values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case model terms A,C,E and F are significant, hence they affect average camber.
d) Do any of the process variables affect the variability in camber measurements?
##
## Call:
## lm.default(formula = camber2 ~ A * B * C * D * E * FF, data = dat2)
##
## Residuals:
## ALL 16 residuals are 0: no residual degrees of freedom!
##
## Coefficients: (48 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.35375 NA NA NA
## A 7.95175 NA NA NA
## B -8.28862 NA NA NA
## C 2.93725 NA NA NA
## D -1.64625 NA NA NA
## E -1.16862 NA NA NA
## FF -4.62800 NA NA NA
## A:B 0.47763 NA NA NA
## A:C 1.26200 NA NA NA
## B:C -0.09012 NA NA NA
## A:D -2.31325 NA NA NA
## B:D -2.42788 NA NA NA
## C:D -5.43725 NA NA NA
## A:E NA NA NA NA
## B:E NA NA NA NA
## C:E NA NA NA NA
## D:E 4.10913 NA NA NA
## A:FF NA NA NA NA
## B:FF NA NA NA NA
## C:FF NA NA NA NA
## D:FF NA NA NA NA
## E:FF NA NA NA NA
## A:B:C NA NA NA NA
## A:B:D -0.34062 NA NA NA
## A:C:D NA NA NA NA
## B:C:D 1.69912 NA NA NA
## A:B:E NA NA NA NA
## A:C:E NA NA NA NA
## B:C:E NA NA NA NA
## A:D:E NA NA NA NA
## B:D:E NA NA NA NA
## C:D:E NA NA NA NA
## A:B:FF NA NA NA NA
## A:C:FF NA NA NA NA
## B:C:FF NA NA NA NA
## A:D:FF NA NA NA NA
## B:D:FF NA NA NA NA
## C:D:FF NA NA NA NA
## A:E:FF NA NA NA NA
## B:E:FF NA NA NA NA
## C:E:FF NA NA NA NA
## D:E:FF NA NA NA NA
## A:B:C:D NA NA NA NA
## A:B:C:E NA NA NA NA
## A:B:D:E NA NA NA NA
## A:C:D:E NA NA NA NA
## B:C:D:E NA NA NA NA
## A:B:C:FF NA NA NA NA
## A:B:D:FF NA NA NA NA
## A:C:D:FF NA NA NA NA
## B:C:D:FF NA NA NA NA
## A:B:E:FF NA NA NA NA
## A:C:E:FF NA NA NA NA
## B:C:E:FF NA NA NA NA
## A:D:E:FF NA NA NA NA
## B:D:E:FF NA NA NA NA
## C:D:E:FF NA NA NA NA
## A:B:C:D:E NA NA NA NA
## A:B:C:D:FF NA NA NA NA
## A:B:C:E:FF NA NA NA NA
## A:B:D:E:FF NA NA NA NA
## A:C:D:E:FF NA NA NA NA
## B:C:D:E:FF NA NA NA NA
## A:B:C:D:E:FF NA NA NA NA
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 15 and 0 DF, p-value: NAModel terms that appear large are A,B,FF,CD,DE, however we would run other screening test to determine significant model terms
From the plots A,B,C and FF are identified to be main effects that are significant
CD<-C*D
DE<-D*E
CD<-as.factor(CD)
DE<-as.factor(DE)
model2<-aov(camber2~A+B+FF+CD, data = dat2)
summary(model2)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 1011.7 1011.7 15.231 0.00247 **
## B 1 1099.2 1099.2 16.549 0.00186 **
## FF 1 342.7 342.7 5.159 0.04420 *
## CD 1 473.0 473.0 7.121 0.02185 *
## Residuals 11 730.7 66.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the result, values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case model terms A,B,FF and CD are significant hence they affect average camber.
e)If it is important to reduce camber as much as possible, what recommendations would you make?
Run A and C at the low level and E and F at the high level. B at the low level enables a lower variation without affecting the average camber
Question 8.40
Consider the following experiment in DOE text, 8.10 Problems Pg. 387:
## 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= FrF2## 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
##
## 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
## 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= FrF2a) How many factors did this experiment investigate? The answer is 4
b) What is the resolution of this design? The answer is resolution 4
c) Calculate the estimates of the main effects.
d) What is the complete defining relation for this design? (I) = ABCD
Question 8.48
Consider the following design in DOE text, 8.10 Problems Pg. 389:
a) What is the generator for column D? D = -ABC b) What is the generator for column E? E = BC c) If this design were folded over, what is the resolution of the combined design? IV
Question 8.60
Consider a partial fold over for the \(2^{7-4}\) resolution III design. Suppose that the partial fold over of this design is constructed using column A ( + signs only). Determine the alias relationships in the combined design.
The alias relationships are shown below
## $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## Question 8.2
library(FrF2)
des.res<-FrF2(nfactors = 4,resolution =3,randomize = FALSE)
#aliasprint(des.res)
response<-c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
des.res<-add.response(des.res,response)
summary(des.res)
summary(lm(response~A*B*C*D,des.res))
DanielPlot(des.res,half=TRUE)
MEPlot(des.res,show.alias=TRUE)
#str(des.res)
model<-aov(response~A*B*C*D, data = des.res)
summary(model)
## Question 8.24
des.res<-FrF2(nfactors = 5,resolution =4,randomize = FALSE)
aliasprint(des.res)
summary(des.res)
## Question 8.25
t<-c("(I)","ag","bg","ab","cfg","acf","bcf","abcfg","df","adfg","bdfg","abdf","cdg","acd","bcd","abcdg","ef","aefg","befg","abef","ceg","ace","bce","abceg","de","adeg","bdeg","abde","cdefg","acdef","bcdef","abcdefg")
ACE<-c(-1,1,-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1,-1,1)
BFG<-c(-1,1,-1,1,-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1,-1,1,-1,1)
blocks<-c(1,4,1,4,3,2,3,2,2,3,2,3,4,1,4,1,4,1,4,1,2,3,2,3,3,2,3,2,1,4,1,4)
length(BFG)
data.frame(t,ACE,BFG,blocks)
## Question 8.28
des.res<-FrF2(nfactors = 6,resolution =4,randomize = FALSE)
summary(des.res)
aliasprint(des.res)
camber<-c(629,192,176,223,223,920,389,900,201,341,126,640,455,371,603,460)
des.res<-add.response(des.res,camber)
summary(lm(des.res))
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))
FF<-c(rep(c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1),1))
dat<-data.frame(A,B,C,D,E,FF,camber)
summary(u<-lm(camber~A*B*C*D*E*FF,data=dat))
DanielPlot(des.res,half=FALSE, main = "Design Generated Normal Plot for Camber")
DanielPlot(u,half=FALSE, main = "Experimenter Design Normal Plot for Camber")
`
MEPlot(des.res,show.alias=TRUE,main ="Main Effects plot for design generated ")
MEPlot(u,show.alias=TRUE, main = "Main Effects plot for experimenter's design")
model<-aov(camber~A+C+D+E, data = des.res)
summary(model)
model<-aov(camber~A+C+E+FF, data = dat)
summary(model)
camber2<-c(24.418, 20.976, 4.083, 25.025, 22.410, 63.639, 16.029, 39.420, 26.725, 50.341, 7.681, 20.083, 31.120, 29.510, 6.750, 17.450)
dat2<-(A,B,C,D,E,FF)
summary(v<-lm(camber2~A*B*C*D*E*FF, data = dat2))
DanielPlot(v,half=FALSE)
MEPlot(v,show.alias=TRUE)
CD<-C*D
CD<-as.factor(CD)
DE<-as.factor(DE)
DE<-D*E
model2<-aov(camber2~A+B+FF+CD, data = dat2)
summary(model2)
## Question 8.40
des.res<-FrF2(nfactors = 4,resolution =4,randomize = FALSE)
#aliasprint(des.res)
des.res
summary(des.res)
Y<-C(8,10,12,7,13,6,5,11)
## Question 8.60
des.res<-FrF2(nfactors = 7,resolution =3,randomize = FALSE)
#summary(des.res)
newdesign<-fold.design(des.res,column=1)
aliasprint(newdesign)
summary(newdesign)