Homework 13
Leonardo Wei
2022-11-27
Question 8.2
Installing necessary libraries
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':
##
## lengthsdes.res4 <- FrF2(nfactors = 4,resolution = 4,randomize=TRUE)
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= FrF2obs <- c(17.273,4.368,13.876,7.037,16.867,15.653,11.846,9.360)
design <- add.response(des.res4,obs)
DanielPlot(design,half=TRUE)
No significant effects were found. Therefore, we may proceed to perform the ANOVA checking for the four interactions effects first:
A <- c("+","+","-","-","+","+","-","-")
A <- c("+","+","-","-","+","+","-","-")
B <- c("+","-","+","-","-","+","-","+")
C <- c("-","+","-","-","-","+","+","+")
D <- c("-","-","+","-","+","+","+","-")
dat <- data.frame(A,B,C,D,obs)
model.aov <- aov(obs~A*B*C*D,data=dat)
summary(model.aov)## 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 0.30Question 8.24
des.res4 <- FrF2(nfactors = 5,resolution = 4,randomize=TRUE)
des.res4## 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
By choosing AB as the blocking criteria, we can see that the effect CDE is also cofounded with blocks.
Question 8.25
des.res3 <- FrF2(nruns=32,nfactors = 7,randomize=FALSE)
des.res3## 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
## 9 -1 -1 -1 1 -1 -1 -1
## 10 1 -1 -1 1 -1 1 1
## 11 -1 1 -1 1 -1 1 1
## 12 1 1 -1 1 -1 -1 -1
## 13 -1 -1 1 1 -1 1 -1
## 14 1 -1 1 1 -1 -1 1
## 15 -1 1 1 1 -1 -1 1
## 16 1 1 1 1 -1 1 -1
## 17 -1 -1 -1 -1 1 -1 -1
## 18 1 -1 -1 -1 1 1 1
## 19 -1 1 -1 -1 1 1 1
## 20 1 1 -1 -1 1 -1 -1
## 21 -1 -1 1 -1 1 1 -1
## 22 1 -1 1 -1 1 -1 1
## 23 -1 1 1 -1 1 -1 1
## 24 1 1 1 -1 1 1 -1
## 25 -1 -1 -1 1 1 -1 1
## 26 1 -1 -1 1 1 1 -1
## 27 -1 1 -1 1 1 1 -1
## 28 1 1 -1 1 1 -1 1
## 29 -1 -1 1 1 1 1 1
## 30 1 -1 1 1 1 -1 -1
## 31 -1 1 1 1 1 -1 -1
## 32 1 1 1 1 1 1 1
## class=design, type= FrF2
Question 8.28
Item A
They used a \(2^{6-2}_{IV}\) with 16 runs.
Item B
The alias relationship is as follows: \[ A=BCE=CDF=ABDEF\\ B=ACE=ABCDF=DEF\\ C=ABE=ADF=BCDEF\\ D=ABCDE=ACF=BEF\\ E=ABC=ABDEF=BDF\\ F=ABCEF=ACD=BDE\\ AB=CE=BCDF=ADEF\\ AC=BE=DF=ABCDEF\\ AD=BCDE=CF=ABEF\\ AE=BC=CDEF=ABDF\\ AF=BCEF=CD=ABDE\\ BD=ACDE=ABCF=EF\\ BF=ACEF=ABCD=DE\]
Item C
lamination_temperature <- rep(c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1),4)
lamination_time <-rep(c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1),4)
lamination_pressure <- rep(c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1),4)
firing_temperature <- rep(c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1),4)
firing_cycle_time <- rep(c(-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1),4)
firing_dew_point <- rep(c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1),4)
obs <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.0110,0.0065,0.0155,0.0093,
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,
0.0185,0.0020,0.005,0.0030,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
model <- aov(obs~lamination_temperature*lamination_time*lamination_pressure*firing_temperature*firing_cycle_time*firing_dew_point)
summary(model)## Df Sum Sq
## lamination_temperature 1 0.0002422
## lamination_time 1 0.0000053
## lamination_pressure 1 0.0005023
## firing_temperature 1 0.0000323
## firing_cycle_time 1 0.0001901
## firing_dew_point 1 0.0009602
## lamination_temperature:lamination_time 1 0.0000587
## lamination_temperature:lamination_pressure 1 0.0000803
## lamination_time:lamination_pressure 1 0.0000527
## lamination_temperature:firing_temperature 1 0.0000239
## lamination_time:firing_temperature 1 0.0000849
## lamination_pressure:firing_temperature 1 0.0000622
## firing_temperature:firing_cycle_time 1 0.0000088
## lamination_temperature:lamination_time:firing_temperature 1 0.0000000
## lamination_time:lamination_pressure:firing_temperature 1 0.0000481
## Residuals 48 0.0004183
## Mean Sq F value
## lamination_temperature 0.0002422 27.793
## lamination_time 0.0000053 0.614
## lamination_pressure 0.0005023 57.644
## firing_temperature 0.0000323 3.712
## firing_cycle_time 0.0001901 21.815
## firing_dew_point 0.0009602 110.192
## lamination_temperature:lamination_time 0.0000587 6.738
## lamination_temperature:lamination_pressure 0.0000803 9.218
## lamination_time:lamination_pressure 0.0000527 6.053
## lamination_temperature:firing_temperature 0.0000239 2.741
## lamination_time:firing_temperature 0.0000849 9.739
## lamination_pressure:firing_temperature 0.0000622 7.139
## firing_temperature:firing_cycle_time 0.0000088 1.007
## lamination_temperature:lamination_time:firing_temperature 0.0000000 0.005
## lamination_time:lamination_pressure:firing_temperature 0.0000481 5.523
## Residuals 0.0000087
## Pr(>F)
## lamination_temperature 3.17e-06 ***
## lamination_time 0.43725
## lamination_pressure 9.14e-10 ***
## firing_temperature 0.05995 .
## firing_cycle_time 2.45e-05 ***
## firing_dew_point 5.05e-14 ***
## lamination_temperature:lamination_time 0.01249 *
## lamination_temperature:lamination_pressure 0.00387 **
## lamination_time:lamination_pressure 0.01754 *
## lamination_temperature:firing_temperature 0.10431
## lamination_time:firing_temperature 0.00305 **
## lamination_pressure:firing_temperature 0.01027 *
## firing_temperature:firing_cycle_time 0.32062
## lamination_temperature:lamination_time:firing_temperature 0.94291
## lamination_time:lamination_pressure:firing_temperature 0.02293 *
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The analysis shows that lamination temperature, lamination pressure, firing cycle time, and firing dew point affects the average camber.
Item D
lamination_temperature_std <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
lamination_time_std <-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
lamination_pressure_std <- c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
firing_temperature_std <- c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
firing_cycle_time_std <- c(-1,1,1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1)
firing_dew_point_std <- c(-1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1,1)
obs <- 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)
dat <- data.frame(lamination_temperature_std,lamination_pressure_std,firing_cycle_time_std,firing_dew_point_std,obs)
model2 <- aov(obs~lamination_temperature_std*lamination_pressure_std*firing_cycle_time_std*firing_dew_point_std)
summary(model2)## Df
## lamination_temperature_std 1
## lamination_pressure_std 1
## firing_cycle_time_std 1
## firing_dew_point_std 1
## lamination_temperature_std:lamination_pressure_std 1
## lamination_temperature_std:firing_cycle_time_std 1
## lamination_pressure_std:firing_cycle_time_std 1
## lamination_temperature_std:firing_dew_point_std 1
## lamination_pressure_std:firing_dew_point_std 1
## firing_cycle_time_std:firing_dew_point_std 1
## lamination_temperature_std:lamination_pressure_std:firing_cycle_time_std 1
## lamination_temperature_std:lamination_pressure_std:firing_dew_point_std 1
## lamination_temperature_std:firing_cycle_time_std:firing_dew_point_std 1
## lamination_pressure_std:firing_cycle_time_std:firing_dew_point_std 1
## lamination_temperature_std:lamination_pressure_std:firing_cycle_time_std:firing_dew_point_std 1
## Sum Sq
## lamination_temperature_std 1011.7
## lamination_pressure_std 138.0
## firing_cycle_time_std 21.9
## firing_dew_point_std 342.7
## lamination_temperature_std:lamination_pressure_std 25.5
## lamination_temperature_std:firing_cycle_time_std 0.1
## lamination_pressure_std:firing_cycle_time_std 3.7
## lamination_temperature_std:firing_dew_point_std 473.0
## lamination_pressure_std:firing_dew_point_std 85.6
## firing_cycle_time_std:firing_dew_point_std 94.3
## lamination_temperature_std:lamination_pressure_std:firing_cycle_time_std 1099.2
## lamination_temperature_std:lamination_pressure_std:firing_dew_point_std 43.4
## lamination_temperature_std:firing_cycle_time_std:firing_dew_point_std 1.9
## lamination_pressure_std:firing_cycle_time_std:firing_dew_point_std 46.2
## lamination_temperature_std:lamination_pressure_std:firing_cycle_time_std:firing_dew_point_std 270.2
## Mean Sq
## lamination_temperature_std 1011.7
## lamination_pressure_std 138.0
## firing_cycle_time_std 21.9
## firing_dew_point_std 342.7
## lamination_temperature_std:lamination_pressure_std 25.5
## lamination_temperature_std:firing_cycle_time_std 0.1
## lamination_pressure_std:firing_cycle_time_std 3.7
## lamination_temperature_std:firing_dew_point_std 473.0
## lamination_pressure_std:firing_dew_point_std 85.6
## firing_cycle_time_std:firing_dew_point_std 94.3
## lamination_temperature_std:lamination_pressure_std:firing_cycle_time_std 1099.2
## lamination_temperature_std:lamination_pressure_std:firing_dew_point_std 43.4
## lamination_temperature_std:firing_cycle_time_std:firing_dew_point_std 1.9
## lamination_pressure_std:firing_cycle_time_std:firing_dew_point_std 46.2
## lamination_temperature_std:lamination_pressure_std:firing_cycle_time_std:firing_dew_point_std 270.2