HW6
Yasir Iqbal
Last compiled on November 14, 2024 at 11:00 AM - CST
6.8:
Model equation is:
\[ y_{ij} =\mu +\alpha_i +\beta_j + \alpha \beta_{ij} +\epsilon_{ijk} \]
CultureMedium <- c(1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2)
Time <- c(rep(12,12),rep(18,12))
Values <- c(21,22,25,26,23,28,24,25,20,26,29,27,37,39,31,34,38,38,29,33,35,36,30,35)
CM <- as.factor(CultureMedium)
Time <- as.factor(Time)
Data <- data.frame(CM,Time,Values)
Model <- aov(Values~CM*Time,data = Data)
summary(Model)## Df Sum Sq Mean Sq F value Pr(>F)
## CM 1 9.4 9.4 1.835 0.190617
## Time 1 590.0 590.0 115.506 9.29e-10 ***
## CM:Time 1 92.0 92.0 18.018 0.000397 ***
## Residuals 20 102.2 5.1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Since the p-value of interaction is very small so we say that it is significant and reject null hypothesis and look at the interaction plot.
interaction.plot(CultureMedium,Time,Values,col = c("blue","red"))
library(ggfortify)## Loading required package: ggplot2library(ggplot2)
autoplot(Model)
The QQ norm plot is pretty normal but the plot of constant variation has some variation so the model might not be adequate.
6.12:
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)
Obs <- c(14.037,13.880,14.821,14.888,16.165,13.860,14.757,14.921,13.972,14.032,14.843,14.415,13.907,13.914,14.878,14.932)
A <- as.factor(A)
B <- as.factor(B)
Data <- data.frame(A,B,Obs)(a):
One <- c(14.037,16.165,13.972,13.907)
A <- c(13.88,13.86,14.032,13.914)
B <- c(14.821,14.757,14.843,14.878)
AB <- c(14.888,14.921,14.415,14.932)
S1 <- sum(One)
SA <- sum(A)
SB <- sum(B)
SAB <- sum(AB)
EffectA <- (2*(SA+SAB-S1-SB)/(4*4))
EffectB <- (2*(SB+SAB-S1-SA)/(4*4))
EffectAB <- (2*(SA+SB-S1-SAB)/(4*4))(b):
Model <- aov(Obs~A*B,data = Data)
summary(Model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 0.403 0.4026 1.262 0.2833
## B 1 1.374 1.3736 4.305 0.0602 .
## A:B 1 0.317 0.3170 0.994 0.3386
## Residuals 12 3.828 0.3190
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The interaction does not seem to be significant so we can check for main effects.
Model <- aov(Obs~A+B,data = Data)
summary(Model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 0.403 0.4026 1.263 0.2815
## B 1 1.374 1.3736 4.308 0.0584 .
## Residuals 13 4.145 0.3189
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Main effects are also not significant.
(c):
Model <- lm(Obs~A*B,data = Data)
coef(Model)## (Intercept) A1 B1 A1:B1
## 14.52025 -0.59875 0.30450 0.56300summary(Model)##
## Call:
## lm(formula = Obs ~ A * B, data = Data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.61325 -0.14431 -0.00563 0.10188 1.64475
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.5202 0.2824 51.414 1.93e-15 ***
## A1 -0.5987 0.3994 -1.499 0.160
## B1 0.3045 0.3994 0.762 0.461
## A1:B1 0.5630 0.5648 0.997 0.339
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5648 on 12 degrees of freedom
## Multiple R-squared: 0.3535, Adjusted R-squared: 0.1918
## F-statistic: 2.187 on 3 and 12 DF, p-value: 0.1425Residual equation is
\[ y_{ijk} = 14.52025-0.59874\alpha_i +0.304550\beta_j +0.5630 \alpha \beta \gamma_{ijk} +\epsilon_{ijk} \]
(d):
autoplot(Model)
From the plots, both the conditions of normality and constant variance does not seem to satisfy.
(e):
As we did earliar in the homeworks, we can do BoxCox and find the appropriate \(\lambda\) , and run the ANOVA to fix it.
6.21:
Typeofputter <- c(rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7))
LengthofPutt <- c(rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7))
Slopeofputt <- c(rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7))
Breakofputt <- c(rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7))
DistancefromCup <- c(10,18,14,12.5,19,16,18.5, 0,16.5,4.5,17.5,20.5,17.5,33, 4,6,1,14.5,12,14,5, 0,10,34,11,25.5,21.5,0, 0,0,18.5,19.5,16,15,11, 5,20.5,18,20,29.5,19,10, 6.5,18.5,7.5,6,0,10,0, 16.5,4.5,0,23.5,8,8,8, 4.5,18,14.5,10,0,17.5,6, 19.5,18,16,5.5,10,7,36, 15,16,8.5,0,0.5,9,3, 41.5,39,6.5,3.5,7,8.5,36, 8,4.5,6.5,10,13,41,14, 21.5,10.5,6.5,0,15.5,24,16, 0,0,0,4.5,1,4,6.5, 18,5,7,10,32.5,18.5,8)
library(GAD)
Typeofputter <- as.fixed(Typeofputter)
LengthofPutt <- as.fixed(LengthofPutt)
Slopeofputt <- as.fixed(Slopeofputt)
Breakofputt <- as.fixed(Breakofputt)
Dat3 <- data.frame(LengthofPutt, Typeofputter, Breakofputt, Slopeofputt, DistancefromCup)Model <- lm(DistancefromCup~LengthofPutt*Typeofputter*Breakofputt*Slopeofputt, data = Dat3)
coef(Model)## (Intercept)
## 15.4285714
## LengthofPutt1
## 0.2142857
## Typeofputter1
## -7.3571429
## Breakofputt1
## -4.0000000
## Slopeofputt1
## -5.3571429
## LengthofPutt1:Typeofputter1
## 6.2857143
## LengthofPutt1:Breakofputt1
## 5.7857143
## Typeofputter1:Breakofputt1
## 2.8571429
## LengthofPutt1:Slopeofputt1
## 5.7142857
## Typeofputter1:Slopeofputt1
## 4.7142857
## Breakofputt1:Slopeofputt1
## 7.7857143
## LengthofPutt1:Typeofputter1:Breakofputt1
## -9.4285714
## LengthofPutt1:Typeofputter1:Slopeofputt1
## 0.6428571
## LengthofPutt1:Breakofputt1:Slopeofputt1
## -12.1428571
## Typeofputter1:Breakofputt1:Slopeofputt1
## -11.7857143
## LengthofPutt1:Typeofputter1:Breakofputt1:Slopeofputt1
## 14.7857143Model <- aov(Model)
gad(Model)## $anova
## Analysis of Variance Table
##
## Response: DistancefromCup
## Df Sum Sq Mean Sq F value
## LengthofPutt 1 917.1 917.15 10.5878
## Typeofputter 1 388.1 388.15 4.4809
## Breakofputt 1 145.1 145.15 1.6756
## Slopeofputt 1 1.4 1.40 0.0161
## LengthofPutt:Typeofputter 1 218.7 218.68 2.5245
## LengthofPutt:Breakofputt 1 11.9 11.90 0.1373
## Typeofputter:Breakofputt 1 115.0 115.02 1.3278
## LengthofPutt:Slopeofputt 1 93.8 93.81 1.0829
## Typeofputter:Slopeofputt 1 56.4 56.43 0.6515
## Breakofputt:Slopeofputt 1 1.6 1.63 0.0188
## LengthofPutt:Typeofputter:Breakofputt 1 7.3 7.25 0.0837
## LengthofPutt:Typeofputter:Slopeofputt 1 113.0 113.00 1.3045
## LengthofPutt:Breakofputt:Slopeofputt 1 39.5 39.48 0.4558
## Typeofputter:Breakofputt:Slopeofputt 1 33.8 33.77 0.3899
## LengthofPutt:Typeofputter:Breakofputt:Slopeofputt 1 95.6 95.65 1.1042
## Residuals 96 8315.8 86.62
## Pr(>F)
## LengthofPutt 0.001572 **
## Typeofputter 0.036862 *
## Breakofputt 0.198615
## Slopeofputt 0.899280
## LengthofPutt:Typeofputter 0.115377
## LengthofPutt:Breakofputt 0.711776
## Typeofputter:Breakofputt 0.252054
## LengthofPutt:Slopeofputt 0.300658
## Typeofputter:Slopeofputt 0.421588
## Breakofputt:Slopeofputt 0.891271
## LengthofPutt:Typeofputter:Breakofputt 0.772939
## LengthofPutt:Typeofputter:Slopeofputt 0.256228
## LengthofPutt:Breakofputt:Slopeofputt 0.501207
## Typeofputter:Breakofputt:Slopeofputt 0.533858
## LengthofPutt:Typeofputter:Breakofputt:Slopeofputt 0.295994
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The p-valus of the factors are less than 0.05, so we can reject the null hypothesis.
autoplot(Model)
It does not seem to follow the normality and constant variation and hence I think that the model is not adequate.
6.36:
A<-rep(c(-1,1),8)
B<-rep(c(-1,-1,1,1),4)
C<-rep(c(rep(-1,4),rep(1,4)),2)
D<-c(rep(-1,8),rep(1,8))
Resistivity<-c(1.92,11.28,1.09,5.75,2.13,9.53,1.03,5.35,1.60,11.73,1.16,4.68,2.16,9.11,1.07,5.30)
Dat<-data.frame(A,B,C,D,Resistivity)
Dat## A B C D Resistivity
## 1 -1 -1 -1 -1 1.92
## 2 1 -1 -1 -1 11.28
## 3 -1 1 -1 -1 1.09
## 4 1 1 -1 -1 5.75
## 5 -1 -1 1 -1 2.13
## 6 1 -1 1 -1 9.53
## 7 -1 1 1 -1 1.03
## 8 1 1 1 -1 5.35
## 9 -1 -1 -1 1 1.60
## 10 1 -1 -1 1 11.73
## 11 -1 1 -1 1 1.16
## 12 1 1 -1 1 4.68
## 13 -1 -1 1 1 2.16
## 14 1 -1 1 1 9.11
## 15 -1 1 1 1 1.07
## 16 1 1 1 1 5.30(a):
ModeL <- lm(Resistivity~A*B*C*D, data=Dat)
coef(ModeL)## (Intercept) A B C D A:B
## 4.680625 3.160625 -1.501875 -0.220625 -0.079375 -1.069375
## A:C B:C A:D B:D C:D A:B:C
## -0.298125 0.229375 -0.056875 -0.046875 0.029375 0.344375
## A:B:D A:C:D B:C:D A:B:C:D
## -0.096875 -0.010625 0.094375 0.141875library(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':
##
## lengthshalfnormal(ModeL)##
## Significant effects (alpha=0.05, Lenth method):## [1] A B A:B A:B:C
The significant effects are A, B and A🅱️C.
(b):
AA <- as.fixed(A)
BA <- as.fixed(B)
Dat1 <- data.frame(AA,BA,Resistivity)
Model <- aov(Resistivity~AA*BA, data=Dat1)
GAD::gad(Model)## $anova
## Analysis of Variance Table
##
## Response: Resistivity
## Df Sum Sq Mean Sq F value Pr(>F)
## AA 1 159.833 159.833 333.088 4.049e-10 ***
## BA 1 36.090 36.090 75.211 1.630e-06 ***
## AA:BA 1 18.297 18.297 38.130 4.763e-05 ***
## Residuals 12 5.758 0.480
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1autoplot(Model)
Both the normality and constant variance does not seem to be true.
(c):
RRresistivity <- log(Resistivity)
Dat2<-data.frame(A,B,C,D,Resistivity)
Model <- lm(Resistivity~A*B*C*D, data=Dat2)
coef(Model)## (Intercept) A B C D A:B
## 4.680625 3.160625 -1.501875 -0.220625 -0.079375 -1.069375
## A:C B:C A:D B:D C:D A:B:C
## -0.298125 0.229375 -0.056875 -0.046875 0.029375 0.344375
## A:B:D A:C:D B:C:D A:B:C:D
## -0.096875 -0.010625 0.094375 0.141875halfnormal(Model)##
## Significant effects (alpha=0.05, Lenth method):## [1] A B A:B A:B:C
Ac <- as.fixed(A)
Bc <- as.fixed(B)
Dat3 <- data.frame(Ac,Bc,RRresistivity)
Model <- aov(RRresistivity~Ac+Bc, data=Dat3)
GAD::gad(Model)## $anova
## Analysis of Variance Table
##
## Response: RRresistivity
## Df Sum Sq Mean Sq F value Pr(>F)
## Ac 1 10.5721 10.5721 962.95 1.408e-13 ***
## Bc 1 1.5803 1.5803 143.94 2.095e-08 ***
## Residuals 13 0.1427 0.0110
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1autoplot(Model)
Now, the model seems to be adequate.
Dat4 <- data.frame(AA,BA,RRresistivity)
Model <- aov(RRresistivity~AA+BA, data=Dat4)
GAD::gad(Model)## $anova
## Analysis of Variance Table
##
## Response: RRresistivity
## Df Sum Sq Mean Sq F value Pr(>F)
## AA 1 10.5721 10.5721 962.95 1.408e-13 ***
## BA 1 1.5803 1.5803 143.94 2.095e-08 ***
## Residuals 13 0.1427 0.0110
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1autoplot(Model)
Both the conditions are satisfied so the model is again adequate.
Dat5 <- data.frame(RRresistivity,A,B)
Model <- lm(RRresistivity~A+B, data=Dat5)
coef(Model)## (Intercept) A B
## 1.1854171 0.8128703 -0.3142776(d):
log(Resistivity) = 1.1854171 + (0.8128703)A + (-0.3142776)B + \(\epsilon\)
6.39:
A <- 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)
B <- 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)
C <- 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)
D <- 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)
E <- 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)
Obs <- c(8.11,5.56,5.77,5.82,9.17,7.8,3.23,5.69,8.82,14.23,9.2,8.94,8.68,11.49,6.25,9.12,7.93,5,7.47,12,9.86,3.65,6.4,11.61,12.43,17.55,8.87,25.38,13.06,18.85,11.78,26.05)
Data <- data.frame(A,B,C,D,E,Obs)(a):
library(DoE.base)
Model <- lm(Obs~A*B*C*D*E,data = Data)
coef(Model)## (Intercept) A B C D E
## 10.1803125 1.6159375 0.0434375 -0.0121875 2.9884375 2.1878125
## A:B A:C B:C A:D B:D C:D
## 1.2365625 -0.0015625 -0.1953125 1.6665625 -0.0134375 0.0034375
## A:E B:E C:E D:E A:B:C A:B:D
## 1.0271875 1.2834375 0.3015625 1.3896875 0.2503125 -0.3453125
## A:C:D B:C:D A:B:E A:C:E B:C:E A:D:E
## -0.0634375 0.3053125 1.1853125 -0.2590625 0.1709375 0.9015625
## B:D:E C:D:E A:B:C:D A:B:C:E A:B:D:E A:C:D:E
## -0.0396875 0.3959375 -0.0740625 -0.1846875 0.4071875 0.1278125
## B:C:D:E A:B:C:D:E
## -0.0746875 -0.3553125halfnormal(Model)##
## Significant effects (alpha=0.05, Lenth method):## [1] D E A:D A D:E B:E A:B A:B:E A:E A:D:E
summary(Model)##
## Call:
## lm.default(formula = Obs ~ A * B * C * D * E, data = Data)
##
## Residuals:
## ALL 32 residuals are 0: no residual degrees of freedom!
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.180312 NaN NaN NaN
## A 1.615938 NaN NaN NaN
## B 0.043438 NaN NaN NaN
## C -0.012187 NaN NaN NaN
## D 2.988437 NaN NaN NaN
## E 2.187812 NaN NaN NaN
## A:B 1.236562 NaN NaN NaN
## A:C -0.001563 NaN NaN NaN
## B:C -0.195313 NaN NaN NaN
## A:D 1.666562 NaN NaN NaN
## B:D -0.013438 NaN NaN NaN
## C:D 0.003437 NaN NaN NaN
## A:E 1.027187 NaN NaN NaN
## B:E 1.283437 NaN NaN NaN
## C:E 0.301562 NaN NaN NaN
## D:E 1.389688 NaN NaN NaN
## A:B:C 0.250312 NaN NaN NaN
## A:B:D -0.345312 NaN NaN NaN
## A:C:D -0.063437 NaN NaN NaN
## B:C:D 0.305313 NaN NaN NaN
## A:B:E 1.185313 NaN NaN NaN
## A:C:E -0.259062 NaN NaN NaN
## B:C:E 0.170938 NaN NaN NaN
## A:D:E 0.901563 NaN NaN NaN
## B:D:E -0.039687 NaN NaN NaN
## C:D:E 0.395938 NaN NaN NaN
## A:B:C:D -0.074063 NaN NaN NaN
## A:B:C:E -0.184687 NaN NaN NaN
## A:B:D:E 0.407187 NaN NaN NaN
## A:C:D:E 0.127812 NaN NaN NaN
## B:C:D:E -0.074688 NaN NaN NaN
## A:B:C:D:E -0.355312 NaN NaN NaN
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 31 and 0 DF, p-value: NAModel2 <- aov(Obs~A+B+D+E+A*B+A*D+A*E+B*E+D*E+A*B*E+A*D*E,data = Data)
summary(Model2)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 83.56 83.56 51.362 6.10e-07 ***
## B 1 0.06 0.06 0.037 0.849178
## D 1 285.78 285.78 175.664 2.30e-11 ***
## E 1 153.17 153.17 94.149 5.24e-09 ***
## A:B 1 48.93 48.93 30.076 2.28e-05 ***
## A:D 1 88.88 88.88 54.631 3.87e-07 ***
## A:E 1 33.76 33.76 20.754 0.000192 ***
## B:E 1 52.71 52.71 32.400 1.43e-05 ***
## D:E 1 61.80 61.80 37.986 5.07e-06 ***
## A:B:E 1 44.96 44.96 27.635 3.82e-05 ***
## A:D:E 1 26.01 26.01 15.988 0.000706 ***
## Residuals 20 32.54 1.63
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(b):
autoplot(Model2)
The model does not seem to be adequate.
(c):
A <- 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)
B <- 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)
D <- 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)
E <- 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)
Obs <- c(8.11,5.56,5.77,5.82,9.17,7.8,3.23,5.69,8.82,14.23,9.2,8.94,8.68,11.49,6.25,9.12,7.93,5,7.47,12,9.86,3.65,6.4,11.61,12.43,17.55,8.87,25.38,13.06,18.85,11.78,26.05)
Data <- data.frame(A,B,D,E,Obs)
Model <- lm(Obs~A*B*D*E,data = Data)
coef(Model)## (Intercept) A B D E A:B
## 10.1803125 1.6159375 0.0434375 2.9884375 2.1878125 1.2365625
## A:D B:D A:E B:E D:E A:B:D
## 1.6665625 -0.0134375 1.0271875 1.2834375 1.3896875 -0.3453125
## A:B:E A:D:E B:D:E A:B:D:E
## 1.1853125 0.9015625 -0.0396875 0.4071875halfnormal(Model)##
## Significant effects (alpha=0.05, Lenth method):## [1] D E A:D A D:E B:E A:B A:B:E A:E A:D:E
summary(Model)##
## Call:
## lm.default(formula = Obs ~ A * B * D * E, data = Data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4750 -0.5637 0.0000 0.5637 1.4750
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.18031 0.21360 47.661 < 2e-16 ***
## A 1.61594 0.21360 7.565 1.14e-06 ***
## B 0.04344 0.21360 0.203 0.841418
## D 2.98844 0.21360 13.991 2.16e-10 ***
## E 2.18781 0.21360 10.243 1.97e-08 ***
## A:B 1.23656 0.21360 5.789 2.77e-05 ***
## A:D 1.66656 0.21360 7.802 7.66e-07 ***
## B:D -0.01344 0.21360 -0.063 0.950618
## A:E 1.02719 0.21360 4.809 0.000193 ***
## B:E 1.28344 0.21360 6.009 1.82e-05 ***
## D:E 1.38969 0.21360 6.506 7.24e-06 ***
## A:B:D -0.34531 0.21360 -1.617 0.125501
## A:B:E 1.18531 0.21360 5.549 4.40e-05 ***
## A:D:E 0.90156 0.21360 4.221 0.000650 ***
## B:D:E -0.03969 0.21360 -0.186 0.854935
## A:B:D:E 0.40719 0.21360 1.906 0.074735 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.208 on 16 degrees of freedom
## Multiple R-squared: 0.9744, Adjusted R-squared: 0.9504
## F-statistic: 40.58 on 15 and 16 DF, p-value: 7.07e-107.12:
library(GAD)
length<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
type<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
brk<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
sp<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
I<-c(10,0,4,0,0,5,6.5,16.5,4.5,19.5,15,41.5,8,21.5,0,18)
II<-c(18,16.5,6,10,0,20.5,18.5,4.5,18,18,16,39,4.5,10.5,0,5)
III<-c(14,4.5,1,34,18.5,18,7.5,0,14.5,16,8.5,6.5,6.5,6.5,0,7)
IV<-c(12.5,17.5,14.5,11,19.5,20,6,23.5,10,5.5,0,3.5,10,0,4.5,10)
V<-c(19,20.5,12,25.5,16,29.5,0,8,0,10,0.5,7,13,15.5,1,32.5)
VI<-c(16,17.5,14,21.5,15,19,10,8,17.5,7,9,8.5,41,24,4,18.5)
VII<-c(18.5,33,5,0,11,10,0,8,6,36,3,36,14,16,6.5,8)
Obs<-c(I,II,III,IV,V,VI,VII)
Blk <- c(rep(1,16),rep(2,16),rep(3,16),rep(4,16),rep(5,16),rep(6,16),rep(7,16))
block<-as.fixed(Blk)
length<-as.fixed(length)
type<-as.fixed(type)
brk<-as.fixed(brk)
sp<-as.fixed(sp)
data<-data.frame(length,type,brk,sp,Obs,block)model<-lm(Obs~length+type+brk+sp+block+length*type+length*brk+type*brk+length*sp+type*sp+brk*sp+length*type*brk+length*type*sp+length*brk*sp+type*brk*sp+length*type*brk*sp,data = data)
GAD::gad(model)## $anova
## Analysis of Variance Table
##
## Response: Obs
## Df Sum Sq Mean Sq F value Pr(>F)
## length 1 917.1 917.15 10.3962 0.00176 **
## type 1 388.1 388.15 4.3998 0.03875 *
## brk 1 145.1 145.15 1.6453 0.20290
## sp 1 1.4 1.40 0.0158 0.90021
## block 6 376.1 62.68 0.7105 0.64202
## length:type 1 218.7 218.68 2.4788 0.11890
## length:brk 1 11.9 11.90 0.1348 0.71433
## type:brk 1 115.0 115.02 1.3038 0.25655
## length:sp 1 93.8 93.81 1.0633 0.30522
## type:sp 1 56.4 56.43 0.6397 0.42594
## brk:sp 1 1.6 1.63 0.0184 0.89227
## length:type:brk 1 7.3 7.25 0.0822 0.77499
## length:type:sp 1 113.0 113.00 1.2809 0.26073
## length:brk:sp 1 39.5 39.48 0.4476 0.50520
## type:brk:sp 1 33.8 33.77 0.3828 0.53767
## length:type:brk:sp 1 95.6 95.65 1.0842 0.30055
## Residuals 90 7939.7 88.22
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Going backwards and removing the 4th level interaction.
model1<-lm(Obs~length+type+brk+sp+block+length*type+length*brk+type*brk+length*sp+type*sp+brk*sp+length*type*brk+length*type*sp+length*brk*sp+type*brk*sp,data = data)
GAD::gad(model1)## $anova
## Analysis of Variance Table
##
## Response: Obs
## Df Sum Sq Mean Sq F value Pr(>F)
## length 1 917.1 917.15 10.3866 0.001762 **
## type 1 388.1 388.15 4.3957 0.038804 *
## brk 1 145.1 145.15 1.6438 0.203067
## sp 1 1.4 1.40 0.0158 0.900250
## block 6 376.1 62.68 0.7098 0.642528
## length:type 1 218.7 218.68 2.4765 0.119026
## length:brk 1 11.9 11.90 0.1347 0.714449
## type:brk 1 115.0 115.02 1.3026 0.256734
## length:sp 1 93.8 93.81 1.0623 0.305413
## type:sp 1 56.4 56.43 0.6391 0.426128
## brk:sp 1 1.6 1.63 0.0184 0.892318
## length:type:brk 1 7.3 7.25 0.0821 0.775082
## length:type:sp 1 113.0 113.00 1.2797 0.260920
## length:brk:sp 1 39.5 39.48 0.4472 0.505381
## type:brk:sp 1 33.8 33.77 0.3824 0.537843
## Residuals 91 8035.4 88.30
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Getting rid of the 3rd level interaction.
model2<-lm(Obs~length+type+brk+sp+block+length*type+length*brk+type*brk+length*sp+type*sp+brk*sp,data = data)
GAD::gad(model2)## $anova
## Analysis of Variance Table
##
## Response: Obs
## Df Sum Sq Mean Sq F value Pr(>F)
## length 1 917.1 917.15 10.5882 0.001577 **
## type 1 388.1 388.15 4.4810 0.036886 *
## brk 1 145.1 145.15 1.6757 0.198640
## sp 1 1.4 1.40 0.0161 0.899281
## block 6 376.1 62.68 0.7236 0.631626
## length:type 1 218.7 218.68 2.5246 0.115406
## length:brk 1 11.9 11.90 0.1373 0.711780
## type:brk 1 115.0 115.02 1.3279 0.252075
## length:sp 1 93.8 93.81 1.0830 0.300678
## type:sp 1 56.4 56.43 0.6515 0.421601
## brk:sp 1 1.6 1.63 0.0188 0.891272
## Residuals 95 8228.9 86.62
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Removing the 2nd level interaction now.
model3<-lm(Obs~length+type+brk+sp+block,data = data)
GAD::gad(model3)## $anova
## Analysis of Variance Table
##
## Response: Obs
## Df Sum Sq Mean Sq F value Pr(>F)
## length 1 917.1 917.15 10.6152 0.001528 **
## type 1 388.1 388.15 4.4925 0.036496 *
## brk 1 145.1 145.15 1.6799 0.197888
## sp 1 1.4 1.40 0.0161 0.899137
## block 6 376.1 62.68 0.7254 0.630108
## Residuals 101 8726.3 86.40
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Only putterlength and puttertype are significant.
model3<-lm(Obs~length+type,data = data)
library(ggplot2)
library(ggfortify)
autoplot(model3)
The normality plot looks good but the constant variance is not.
7.20:
Block1 <- c('a','b','cd','abcd','ace','bce','de','abde','cf','abcf','adf','bdf','ef','abef','acdef','bcdef')
Block2 <- c('c','abc','ad','bd','e','abe','acde','bcde','af','bf','cdf','abcdf','acef','bcef','def','abdef')
Block3 <- c('ac','bc','d','abd','ae','be','cde','abcde','f','abf','acdf','bcdf','cef','abcef','adef','bdef')
Block4 <- c('(1)','ab','acd','bcd','ce','abce','ade','bde','acf','bcf','df','abdf','aef','bef','cdef','abcdef')
confounding_scheme<-cbind(Block1,Block2,Block3,Block4)
confounding_scheme## Block1 Block2 Block3 Block4
## [1,] "a" "c" "ac" "(1)"
## [2,] "b" "abc" "bc" "ab"
## [3,] "cd" "ad" "d" "acd"
## [4,] "abcd" "bd" "abd" "bcd"
## [5,] "ace" "e" "ae" "ce"
## [6,] "bce" "abe" "be" "abce"
## [7,] "de" "acde" "cde" "ade"
## [8,] "abde" "bcde" "abcde" "bde"
## [9,] "cf" "af" "f" "acf"
## [10,] "abcf" "bf" "abf" "bcf"
## [11,] "adf" "cdf" "acdf" "df"
## [12,] "bdf" "abcdf" "bcdf" "abdf"
## [13,] "ef" "acef" "cef" "aef"
## [14,] "abef" "bcef" "abcef" "bef"
## [15,] "acdef" "def" "adef" "cdef"
## [16,] "bcdef" "abdef" "bdef" "abcdef"7.21:
block1 <- c('b','acd','ce','abde','abcf','df','aef','bcdef')
block2 <- c('abc','d','ae','bcde','bf','acdf','cef','abdef')
block3 <- c('a','bcd','abce','de','cf','abdf','bef','acdef')
block4 <- c('c','abd','be','acde','af','bcdf','abcef','def')
block5 <- c('ac','bd','abe','cde','f','abcdf','bcef','adef')
block6 <- c('-1','abcd','bce','ade','acf','bdf','abef','cdef')
block7 <- c('bc','ad','e','abcde','abf','cdf','acef','bdef')
block8 <- c('ab','cd','ace','bde','bcf','adf','ef','abcdef')
confounding_scheme2<-cbind(block1,block2,block3,block4,block5,block6,block7,block8)
confounding_scheme2## block1 block2 block3 block4 block5 block6 block7 block8
## [1,] "b" "abc" "a" "c" "ac" "-1" "bc" "ab"
## [2,] "acd" "d" "bcd" "abd" "bd" "abcd" "ad" "cd"
## [3,] "ce" "ae" "abce" "be" "abe" "bce" "e" "ace"
## [4,] "abde" "bcde" "de" "acde" "cde" "ade" "abcde" "bde"
## [5,] "abcf" "bf" "cf" "af" "f" "acf" "abf" "bcf"
## [6,] "df" "acdf" "abdf" "bcdf" "abcdf" "bdf" "cdf" "adf"
## [7,] "aef" "cef" "bef" "abcef" "bcef" "abef" "acef" "ef"
## [8,] "bcdef" "abdef" "acdef" "def" "adef" "cdef" "bdef" "abcdef"8.2:
library(FrF2)## Warning: package 'FrF2' was built under R version 4.4.2design <- 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= FrF2aliasprint(design)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCNo Main effets are aliased but the two factors interactions are aliased with each other.
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= FrF2DanielPlot(design.resp,half=TRUE)
MEPlot(design.resp,show.alias=TRUE)
Daniel plot shows that none of the factors are significant.
8.24:
des.res <- FrF2(nfactors = 5, resolution = 5 ,randomize = FALSE)
aliasprint(des.res)## $legend
## [1] A=A B=B C=C D=D E=E
##
## [[2]]
## [1] no aliasing among main effects and 2fissummary(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 <- 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 + 1Main effects confounded in block 1=E, C,D and main effects in block 2= A, B.
8.25:
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 16.1.4 1 -1 1 1 1 1 -1 -1
## 2 2 29.1.8 1 1 1 1 -1 -1 1 -1
## 3 3 20.1.5 1 1 -1 -1 1 1 1 -1
## 4 4 27.1.7 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 22.1.6 1 1 -1 1 -1 1 -1 1
## 8 8 10.1.3 1 -1 1 -1 -1 1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.3 2 -1 1 -1 -1 -1 1 1
## 10 10 28.2.7 2 1 1 -1 1 1 -1 1
## 11 11 19.2.5 2 1 -1 -1 1 -1 1 -1
## 12 12 21.2.6 2 1 -1 1 -1 -1 -1 1
## 13 13 15.2.4 2 -1 1 1 1 -1 -1 -1
## 14 14 2.2.1 2 -1 -1 -1 -1 1 -1 -1
## 15 15 30.2.8 2 1 1 1 -1 1 1 -1
## 16 16 8.2.2 2 -1 -1 1 1 1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 18 18 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 19 19 12.3.3 3 -1 1 -1 1 1 1 -1
## 20 20 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 21 21 24.3.6 3 1 -1 1 1 1 -1 -1
## 22 22 14.3.4 3 -1 1 1 -1 1 -1 1
## 23 23 31.3.8 3 1 1 1 1 -1 1 1
## 24 24 18.3.5 3 1 -1 -1 -1 1 1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 11.4.3 4 -1 1 -1 1 -1 1 -1
## 26 26 23.4.6 4 1 -1 1 1 -1 -1 -1
## 27 27 17.4.5 4 1 -1 -1 -1 -1 1 1
## 28 28 6.4.2 4 -1 -1 1 -1 1 1 -1
## 29 29 26.4.7 4 1 1 -1 -1 1 -1 -1
## 30 30 4.4.1 4 -1 -1 -1 1 1 -1 1
## 31 31 13.4.4 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.
8.28:
(A):
k=6 , p=2 Fraction: $2_{i\nu}^{6-2}$
Runs=16, Design generatos: E=ABC and F=BCD
Defining relations: I=ABCE, I=BCDE, I=ADEF.
(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=DE(C):
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)
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 ' ' 1Factors A, C, E and F are significantly affecting the average number.
(D):
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)
Model2 <- lm(SD~A*B*C*D*E*F1,data = Data2)
DanielPlot(Model2)
(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 0.0014812 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-158.40:
Four factors.
Resolution is also four.
One <- c(8)
AD <- c(10)
BD <- c(12)
AB <- c(7)
CD <- c(13)
AC <- c(6)
BC <- c(5)
ABCD <- c(11)
EffectA <- (2*(AD+AB+AC+ABCD-One-BD-CD-BC))/(8)
EffectB <- (2*(BD+AB+BC+ABCD-One-AD-CD-AC))/(8)
EffectC <- (2*(CD+AC+BC+ABCD-One-AD-BD-AB))/(8)
EffectD <- (2*(AD+BD+CD+ABCD-One-AB-AC-BC))/(8)- Defining relation is I=ABCD
8.48:
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.generatorsThe design generator for column D is -ABC.
The design generator for colum E is BC.
Using table 8.14, resolution is four.
8.60:
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