Homwork 6
Peihang Li
2024-11-22
6.8
time <- rep(c(12, 18), each = 6)
medium <- c(rep(1,6),rep(1,6),rep(2,6),rep(2,6))
growth <- c(21,22,23,28,20,26,37,39,38,38,35,36,25,26,24,25,29,27,31,34,29,33,30,35)
time<-as.factor(time)
medium<-as.factor(medium)
data <- data.frame(time, medium, growth)
data## time medium growth
## 1 12 1 21
## 2 12 1 22
## 3 12 1 23
## 4 12 1 28
## 5 12 1 20
## 6 12 1 26
## 7 18 1 37
## 8 18 1 39
## 9 18 1 38
## 10 18 1 38
## 11 18 1 35
## 12 18 1 36
## 13 12 2 25
## 14 12 2 26
## 15 12 2 24
## 16 12 2 25
## 17 12 2 29
## 18 12 2 27
## 19 18 2 31
## 20 18 2 34
## 21 18 2 29
## 22 18 2 33
## 23 18 2 30
## 24 18 2 35model <- aov(growth~time+medium+time*medium, data = data)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## time 1 590.0 590.0 115.506 9.29e-10 ***
## medium 1 9.4 9.4 1.835 0.190617
## time:medium 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 ' ' 1plot(model,1)
plot(model,2)
Time and the interaction between time and medium appear to have significant effects on growth, at the 0.05 significance level.
The residual plots shows a fairly normal distribution and the residuals looks to be constant.
6.12
A <- rep(c(-1, 1, -1, 1), each = 4)
B <- rep(c(-1, -1, 1, 1), each = 4)
thickness <- c(14.037, 16.165, 13.972, 13.907,
13.880, 13.860, 14.032, 13.914,
14.821, 14.757, 14.843, 14.878,
14.888, 14.921, 14.415, 14.932)
A<-as.factor(A)
B<-as.factor(B)
dat<-data.frame(A,B,thickness)
dat## A B thickness
## 1 -1 -1 14.037
## 2 -1 -1 16.165
## 3 -1 -1 13.972
## 4 -1 -1 13.907
## 5 1 -1 13.880
## 6 1 -1 13.860
## 7 1 -1 14.032
## 8 1 -1 13.914
## 9 -1 1 14.821
## 10 -1 1 14.757
## 11 -1 1 14.843
## 12 -1 1 14.878
## 13 1 1 14.888
## 14 1 1 14.921
## 15 1 1 14.415
## 16 1 1 14.932n <- 4 # Number of replicates
total_ab <- sum(thickness[A == 1 & B == 1])
total_a <- sum(thickness[A == 1 & B == -1])
total_b <- sum(thickness[A == -1 & B == 1])
total_1 <- sum(thickness[A == -1 & B == -1])
# Factor effects
effect_A <- (total_ab + total_a - total_b - total_1) / (2 * n)
effect_B <- (total_ab + total_b - total_a - total_1) / (2 * n)
interaction_AB <- (total_ab - total_a - total_b + total_1) / (2 * n)
effect_A## [1] -0.31725effect_B## [1] 0.586interaction_AB## [1] 0.2815model <- aov(thickness ~ 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 ' ' 1plot(model,1)
plot(model,2)
Factor B appears to be the most significant factor, even though it still doesn’t meet the 0.05 sig. level.
regression_model <- lm(thickness ~ A * B, data = data)
coef(regression_model)## (Intercept) A1 B1 A1:B1
## 14.52025 -0.59875 0.30450 0.56300Observation 2 is an outlier in the normal probability plot and the plot of residual versus predicted.
Investigate the cause (e.g., experimental error, data recording issue); replace it with the average of the observations from that group.
6.21
obs <- c(10,0,4,0,0,5,6.5,16.5,4.5,19.5,15,41.5,8,21.5,0,18,18,16.5,6,10,0,20.5,18.5,4.5,18,18,16,39,4.5,10.5,0,5,14,4.5,1,34,18.5,18,7.5,0,14.5,16,8.5,6.5,6.5,6.5,0,7,12.5,17.5,14.5,11,19.5,20,6,23.5,10,5.5,0,3.5,10,0,4.5,10,19,20.5,12,25.5,16,29.5,0,8,0,10,0.5,7,13,15.5,1,32.5,16,17.5,14,21.5,15,19,10,8,17.5,7,9,8.5,41,24,4,18.5,18.5,33,5,0,11,10,0,8,6,36,3,36,14,16,6.5,8)
A <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B <- c(rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2))
C <- c(rep(1,4),rep(-1,4),rep(1,4),rep(-1,4))
D <- c(rep(1,8),rep(-1,8))
dat <- data.frame(A,B,C,D,obs)
model <- aov(obs~A*B*C*D,data = dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 917 917.1 10.588 0.00157 **
## B 1 388 388.1 4.481 0.03686 *
## C 1 145 145.1 1.676 0.19862
## D 1 1 1.4 0.016 0.89928
## A:B 1 219 218.7 2.525 0.11538
## A:C 1 12 11.9 0.137 0.71178
## B:C 1 115 115.0 1.328 0.25205
## A:D 1 94 93.8 1.083 0.30066
## B:D 1 56 56.4 0.651 0.42159
## C:D 1 2 1.6 0.019 0.89127
## A:B:C 1 7 7.3 0.084 0.77294
## A:B:D 1 113 113.0 1.305 0.25623
## A:C:D 1 39 39.5 0.456 0.50121
## B:C:D 1 34 33.8 0.390 0.53386
## A:B:C:D 1 96 95.6 1.104 0.29599
## Residuals 96 8316 86.6
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Length of the put and the type of putter have a significant effect at 0.05 sig level.
plot(model,1)
plot(model,2)
From the plots we can see the assumptions of constant variance and normality are violated to some extent.
6.36
obs1 <- 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)
A1 <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B1 <- c(rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2))
C1 <- c(rep(-1,4),rep(1,4),rep(-1,4),rep(1,4))
D1 <- c(rep(-1,8),rep(1,8))
dat1 <- data.frame(A1,B1,C1,D1,obs1)
dat1## A1 B1 C1 D1 obs1
## 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 5.30a.
library(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':
##
## lengthsmodel1 <- lm(obs1~A1*B1*C1*D1,data = dat1)
coef(model1)## (Intercept) A1 B1 C1 D1 A1:B1
## 4.680625 3.160625 -1.501875 -0.220625 -0.079375 -1.069375
## A1:C1 B1:C1 A1:D1 B1:D1 C1:D1 A1:B1:C1
## -0.298125 0.229375 -0.056875 -0.046875 0.029375 0.344375
## A1:B1:D1 A1:C1:D1 B1:C1:D1 A1:B1:C1:D1
## -0.096875 -0.010625 0.094375 0.141875halfnormal(model1)##
## Significant effects (alpha=0.05, Lenth method):## [1] A1 B1 A1:B1 A1:B1:C1
qqnorm(coef(model1))
qqline(coef(model1))
aov_mod <- aov(obs1~A1+B1+C1+A1*B1+A1*B1*C1,data = dat1)
plot(aov_mod)
## hat values (leverages) are all = 0.5
## and there are no factor predictors; no plot no. 5
The model does not seem to satisfy the assumption of constant variance or normal distribution.
c.
obs_ln <- log(obs1)
mod <- lm(obs_ln~A1+B1+C1+A1*B1+A1*B1*C1)
coef(mod)## (Intercept) A1 B1 C1 A1:B1 A1:C1
## 1.185417116 0.812870345 -0.314277554 -0.006408558 -0.024684570 -0.039723700
## B1:C1 A1:B1:C1
## -0.004225796 0.063434408halfnormal(mod)##
## Significant effects (alpha=0.05, Lenth method):## [1] A1 B1 A1:B1:C1 A1:C1 e7 e6
qqnorm(coef(mod))
qqline(coef(mod))
The log transformation has improved the results.
d.
mod <- lm(obs_ln~A1+B1+C1+A1*B1*C1+A1*C1)
coef(mod)## (Intercept) A1 B1 C1 A1:B1 A1:C1
## 1.185417116 0.812870345 -0.314277554 -0.006408558 -0.024684570 -0.039723700
## B1:C1 A1:B1:C1
## -0.004225796 0.0634344086.39
y <- 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)
A2 <- 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)
B2 <- c(rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2))
C2 <- c(rep(-1,4),rep(1,4),rep(-1,4),rep(1,4),rep(-1,4),rep(1,4),rep(-1,4),rep(1,4))
D2 <- c(rep(-1,8),rep(1,8),rep(-1,8),rep(1,8))
E2 <- c(rep(-1,16),rep(1,16))
dat2 <- data.frame(A2,B2,C2,D2,E2,y)a.
model3 <- aov(y~A2*B2*C2*D2*E2,data = dat2)
halfnormal(model3)##
## Significant effects (alpha=0.05, Lenth method):## [1] D2 E2 A2:D2 A2 D2:E2 B2:E2 A2:B2 A2:B2:E2
##
## [9] A2:E2 A2:D2:E2
b.
model4 <- aov(y~A2*B2*D2*E2,data = dat2)
plot(model4)
## hat values (leverages) are all = 0.5
## and there are no factor predictors; no plot no. 5
The assumption of constant variance seems to be violated.
c.
summary(model4)## Df Sum Sq Mean Sq F value Pr(>F)
## A2 1 83.56 83.56 57.233 1.14e-06 ***
## B2 1 0.06 0.06 0.041 0.841418
## D2 1 285.78 285.78 195.742 2.16e-10 ***
## E2 1 153.17 153.17 104.910 1.97e-08 ***
## A2:B2 1 48.93 48.93 33.514 2.77e-05 ***
## A2:D2 1 88.88 88.88 60.875 7.66e-07 ***
## B2:D2 1 0.01 0.01 0.004 0.950618
## A2:E2 1 33.76 33.76 23.126 0.000193 ***
## B2:E2 1 52.71 52.71 36.103 1.82e-05 ***
## D2:E2 1 61.80 61.80 42.328 7.24e-06 ***
## A2:B2:D2 1 3.82 3.82 2.613 0.125501
## A2:B2:E2 1 44.96 44.96 30.794 4.40e-05 ***
## A2:D2:E2 1 26.01 26.01 17.815 0.000650 ***
## B2:D2:E2 1 0.05 0.05 0.035 0.854935
## A2:B2:D2:E2 1 5.31 5.31 3.634 0.074735 .
## Residuals 16 23.36 1.46
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We can see that factors A, D, E, and interactions AB, AD, AE, BE, DE, ABE, and ADE are significant at 0.05 level.
d.
lm_mod <- lm(y~A2+D2+E2+A2*B2+A2*E2+A2*D2+B2*E2+D2*E2+A2*B2*E2+A2*D2*E2,data = dat2)
coef(lm_mod)## (Intercept) A2 D2 E2 B2 A2:B2
## 10.1803125 1.6159375 2.9884375 2.1878125 0.0434375 1.2365625
## A2:E2 A2:D2 E2:B2 D2:E2 A2:E2:B2 A2:D2:E2
## 1.0271875 1.6665625 1.2834375 1.3896875 1.1853125 0.90156257.12
obs <- c(10,0,4,0,0,5,6.5,16.5,4.5,19.5,15,41.5,8,21.5,0,18,18,16.5,6,10,0,20.5,18.5,4.5,18,18,16,39,4.5,10.5,0,5,14,4.5,1,34,18.5,18,7.5,0,14.5,16,8.5,6.5,6.5,6.5,0,7,12.5,17.5,14.5,11,19.5,20,6,23.5,10,5.5,0,3.5,10,0,4.5,10,19,20.5,12,25.5,16,29.5,0,8,0,10,0.5,7,13,15.5,1,32.5,16,17.5,14,21.5,15,19,10,8,17.5,7,9,8.5,41,24,4,18.5,18.5,33,5,0,11,10,0,8,6,36,3,36,14,16,6.5,8)
A <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B <- c(rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2),rep(1,2),rep(-1,2))
C <- c(rep(1,4),rep(-1,4),rep(1,4),rep(-1,4))
D <- c(rep(1,8),rep(-1,8))
block <- c(rep(1,16),rep(2,16),rep(3,16),rep(4,16),rep(5,16),rep(6,16),rep(7,16))
dat <- data.frame(A,B,C,D,block,obs)
lm_mod <- lm(obs~A*B*C*D+block,data = dat)
library(DoE.base)
halfnormal(lm_mod)## Warning in halfnormal.lm(lm_mod): halfnormal not recommended for models with
## more residual df than model df##
## Significant effects (alpha=0.05, Lenth method):## [1] A lof10 block lof65 lof9 B lof69
aov_mod <- aov(obs~A*B*C*D+block,data = dat)
summary(aov_mod)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 917 917.1 10.673 0.00151 **
## B 1 388 388.1 4.517 0.03616 *
## C 1 145 145.1 1.689 0.19687
## D 1 1 1.4 0.016 0.89888
## block 1 152 152.1 1.769 0.18663
## A:B 1 219 218.7 2.545 0.11398
## A:C 1 12 11.9 0.138 0.71068
## B:C 1 115 115.0 1.338 0.25020
## A:D 1 94 93.8 1.092 0.29877
## B:D 1 56 56.4 0.657 0.41976
## C:D 1 2 1.6 0.019 0.89084
## A:B:C 1 7 7.3 0.084 0.77206
## A:B:D 1 113 113.0 1.315 0.25437
## A:C:D 1 39 39.5 0.459 0.49952
## B:C:D 1 34 33.8 0.393 0.53224
## A:B:C:D 1 96 95.6 1.113 0.29411
## Residuals 95 8164 85.9
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 17.20
library(FrF2)
design <- FrF2(nruns = 64, nfactors = 6, factor.names = c("A", "B", "C", "D", "E", "F"))## creating full factorial with 64 runs ...Choosing ABCE and ABDF as the block generators, confound with CDEF
Block 1: ABCE = -1, ABDF = -1
{EF, DE, CF, CD, B, BDF, BCE, BCDEF, A, ADF, ACE, ACDEF, ABEF, ABDE, ABCF, ABCD}
Block 2: ABCE = -1, ABDF = +1
{E, DEF, C, CDF, BF, BD, BCEF, BCDE, AF, AD, ACEF, ACDE, ABE, ABDEF, ABC, ABCDF}
Block 3: ABCE = +1, ABDF = -1
{F, D, CEF, CDE, BE, BDEF, BC, BCDF, AE, ADEF, AC, ACDF, ABF, ABD, ABCEF, ABCDE}
Block 4: ABCE = +1, ABDF = +1
{1, DF, CE, CDEF, BEF, BDE, BCF, BCD, AEF, ADE, ACF, ACD, AB, ABDF, ABCE, ABCDEF}
7.21
Block 1 (ABCD = -1, ACE = -1, ABEF = -1)
{DF, CE, B, BCDEF, AEF, ACD, ABDE, ABCF}
Block 2 (ABCD = -1, ACE = -1, ABEF = +1)
{D, CEF, BF, BCDE, AE, ACDF, ABDEF, ABC}
Block 3 (ABCD = -1, ACE = +1, ABEF = -1)
{DE, CF, BEF, BCD, A, ACDEF, ABDF, ABCE}
Block 4 (ABCD = -1, ACE = +1, ABEF = +1)
{DEF, C, BE, BCDF, AF, ACDE, ABD, ABCEF}
Block 5 (ABCD = +1, ACE = -1, ABEF = -1)
{F, CDE, BD, BCEF, ADEF, AC, ABE, ABCDF}
Block 6 (ABCD = +1, ACE = -1, ABEF = +1)
{1, CDEF, BDF, BCE, ADE, ACF, ABEF, ABCD}
Block 7 (ABCD = +1, ACE = +1, ABEF = -1)
{E, CDF, BDEF, BC, AD, ACEF, ABF, ABCDE}
Block 8 (ABCD = +1, ACE = +1, ABEF = +1)
{EF, CD, BDE, BCF, ADF, ACE, AB, ABCDEF}
8.2
design <- data.frame(
A = c(-1, 1, -1, 1, -1, 1, -1, 1),
B = c(-1, -1, 1, 1, -1, -1, 1, 1),
C = c(-1, -1, -1, -1, 1, 1, 1, 1),
D = c(-1, 1, 1, -1, 1, -1, -1, 1), # D = ABC
Response = c(7.037, 14.707, 11.635, 17.273, 10.403, 4.368, 9.360, 13.440) # Replicate I
)
print(design)## A B C D Response
## 1 -1 -1 -1 -1 7.037
## 2 1 -1 -1 1 14.707
## 3 -1 1 -1 1 11.635
## 4 1 1 -1 -1 17.273
## 5 -1 -1 1 1 10.403
## 6 1 -1 1 -1 4.368
## 7 -1 1 1 -1 9.360
## 8 1 1 1 1 13.440model <- lm(Response ~ A + B + C + A:B + A:C + B:C, data = design)
summary(model) ##
## Call:
## lm.default(formula = Response ~ A + B + C + A:B + A:C + B:C,
## data = design)
##
## Residuals:
## 1 2 3 4 5 6 7 8
## -1.518 1.518 1.518 -1.518 1.518 -1.518 -1.518 1.518
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.0279 1.5184 7.263 0.0871 .
## A 1.4191 1.5184 0.935 0.5215
## B 1.8991 1.5184 1.251 0.4294
## C -1.6351 1.5184 -1.077 0.4764
## A:B 1.0104 1.5184 0.665 0.6262
## A:C -1.9079 1.5184 -1.257 0.4279
## B:C 0.1081 1.5184 0.071 0.9547
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.295 on 1 degrees of freedom
## Multiple R-squared: 0.849, Adjusted R-squared: -0.05671
## F-statistic: 0.9374 on 6 and 1 DF, p-value: 0.6585anova_results <- anova(model)
print(anova_results)## Analysis of Variance Table
##
## Response: Response
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 16.1113 16.1113 0.8735 0.5215
## B 1 28.8534 28.8534 1.5644 0.4294
## C 1 21.3891 21.3891 1.1597 0.4764
## A:B 1 8.1669 8.1669 0.4428 0.6262
## A:C 1 29.1199 29.1199 1.5789 0.4279
## B:C 1 0.0935 0.0935 0.0051 0.9547
## Residuals 1 18.4437 18.4437plot(model)
## hat values (leverages) are all = 0.875
## and there are no factor predictors; no plot no. 5
8.24
A B C D E Block
1 - - - - + 1
2 + - - - - 2
3 - + - - - 2
4 + + - - + 1
5 - - + - - 2
6 + - + - + 1
7 - + + - + 1
8 + + + - - 2
9 - - - + - 2
10 + - - + + 1
11 - + - + + 1
12 + + - + - 2
13 - - + + + 1
14 + - + + - 2
15 - + + + - 2
16 + + + + + 1
Main effects
A
B
C
D
E = A:B:C:D
Two-factor interactions
A:B = C:D*Block
A:C = B:D*Block
A:D = B:C*Block
A:E = B:C:D
B:C = A:D*Block
B:D = A:C*Block
B:E = A:C:D
C:D = A:B*Block
C:E = A:B:D
D:E = A:B:C
Block defining relation
Block = A:B:C:D
No main effects confounded with blocks.
8.25
main
A
B
C
D
E
F = A:B:C
G = B:C:D
interaction
A:B = F:G*Block
A:C = B:D*Block
A:D = B:C:G*Block
A:E = B:C:D:F*Block
Block
Block = F:G
No main effects confounded with blocks.
8.28
It’s a Fractional Factorial Design, with 6 factors and 16 runs. The resolution is IV. Fraction is 2 to the power of (6-2).
E = +/- ABC
F = +/- ACD
library(FrF2)
?FrF2
camber <- c(157.25, 48, 44, 55.75, 55.75, 230, 97.25, 225, 50.25, 85.25, 31.5, 160, 113.75, 92.75, 150.75, 115)
des.res4 <- FrF2(nruns = 16, nfactors = 6, generators = list(c(1, 2, 3), c(1, 3, 4)), randomize = FALSE)
des.res4## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 -1
## 4 1 1 -1 -1 -1 1
## 5 -1 -1 1 -1 1 1
## 6 1 -1 1 -1 -1 -1
## 7 -1 1 1 -1 -1 1
## 8 1 1 1 -1 1 -1
## 9 -1 -1 -1 1 -1 1
## 10 1 -1 -1 1 1 -1
## 11 -1 1 -1 1 1 1
## 12 1 1 -1 1 -1 -1
## 13 -1 -1 1 1 1 -1
## 14 1 -1 1 1 -1 1
## 15 -1 1 1 1 -1 -1
## 16 1 1 1 1 1 1
## class=design, type= FrF2.generatorsaliasprint(des.res4)## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE AC=BE=DF AD=CF AE=BC AF=CD BD=EF BF=DEdes.res<-add.response(des.res4, camber)
summary(des.res)## Call:
## FrF2(nruns = 16, nfactors = 6, generators = list(c(1, 2, 3),
## c(1, 3, 4)), randomize = FALSE)
##
## Experimental design of type FrF2.generators
## 16 runs
##
## Factor settings (scale ends):
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1
##
## Responses:
## [1] camber
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $generators
## [1] E=ABC F=ACD
##
##
## Alias structure:
## $fi2
## [1] AB=CE AC=BE=DF AD=CF AE=BC AF=CD BD=EF BF=DE
##
##
## The design itself:
## A B C D E F camber
## 1 -1 -1 -1 -1 -1 -1 157.25
## 2 1 -1 -1 -1 1 1 48.00
## 3 -1 1 -1 -1 1 -1 44.00
## 4 1 1 -1 -1 -1 1 55.75
## 5 -1 -1 1 -1 1 1 55.75
## 6 1 -1 1 -1 -1 -1 230.00
## 7 -1 1 1 -1 -1 1 97.25
## 8 1 1 1 -1 1 -1 225.00
## 9 -1 -1 -1 1 -1 1 50.25
## 10 1 -1 -1 1 1 -1 85.25
## 11 -1 1 -1 1 1 1 31.50
## 12 1 1 -1 1 -1 -1 160.00
## 13 -1 -1 1 1 1 -1 113.75
## 14 1 -1 1 1 -1 1 92.75
## 15 -1 1 1 1 -1 -1 150.75
## 16 1 1 1 1 1 1 115.00
## class=design, type= FrF2.generatorsDanielPlot(des.res, half=TRUE)
MEPlot(des.res, show.alias=TRUE)
camber_sd <- c(24.418, 20.976, 4.083, 25.025, 25.405, 63.635,
16.029, 40.289, 26.725, 25.921, 7.681, 31.122,
29.151, 6.753, 29.521, 17.455)
des.res4 <- FrF2(nruns = 16, nfactors = 6, generators = list(c(1, 2, 3), c(1, 3, 4)), randomize = FALSE)
des.res4## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 -1
## 4 1 1 -1 -1 -1 1
## 5 -1 -1 1 -1 1 1
## 6 1 -1 1 -1 -1 -1
## 7 -1 1 1 -1 -1 1
## 8 1 1 1 -1 1 -1
## 9 -1 -1 -1 1 -1 1
## 10 1 -1 -1 1 1 -1
## 11 -1 1 -1 1 1 1
## 12 1 1 -1 1 -1 -1
## 13 -1 -1 1 1 1 -1
## 14 1 -1 1 1 -1 1
## 15 -1 1 1 1 -1 -1
## 16 1 1 1 1 1 1
## class=design, type= FrF2.generatorsaliasprint(des.res4)## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE AC=BE=DF AD=CF AE=BC AF=CD BD=EF BF=DEdes.res<-add.response(des.res4, camber_sd)
summary(des.res)## Call:
## FrF2(nruns = 16, nfactors = 6, generators = list(c(1, 2, 3),
## c(1, 3, 4)), randomize = FALSE)
##
## Experimental design of type FrF2.generators
## 16 runs
##
## Factor settings (scale ends):
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1
##
## Responses:
## [1] camber_sd
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $generators
## [1] E=ABC F=ACD
##
##
## Alias structure:
## $fi2
## [1] AB=CE AC=BE=DF AD=CF AE=BC AF=CD BD=EF BF=DE
##
##
## The design itself:
## A B C D E F camber_sd
## 1 -1 -1 -1 -1 -1 -1 24.418
## 2 1 -1 -1 -1 1 1 20.976
## 3 -1 1 -1 -1 1 -1 4.083
## 4 1 1 -1 -1 -1 1 25.025
## 5 -1 -1 1 -1 1 1 25.405
## 6 1 -1 1 -1 -1 -1 63.635
## 7 -1 1 1 -1 -1 1 16.029
## 8 1 1 1 -1 1 -1 40.289
## 9 -1 -1 -1 1 -1 1 26.725
## 10 1 -1 -1 1 1 -1 25.921
## 11 -1 1 -1 1 1 1 7.681
## 12 1 1 -1 1 -1 -1 31.122
## 13 -1 -1 1 1 1 -1 29.151
## 14 1 -1 1 1 -1 1 6.753
## 15 -1 1 1 1 -1 -1 29.521
## 16 1 1 1 1 1 1 17.455
## class=design, type= FrF2.generatorsDanielPlot(des.res, half=TRUE)
# From the main effects plot for camber:
# A=-1, B=Not significant, C=-1, D=+1, E=+1, F=-1F has the largest and statistically significant effect on the average camber.
No factor has a significant effect on the variability
According to the Main Effects Plot, we can see that Factor F and Factor C have the most obvious effect on observation, even though only Factor F is significant. To reduce camber as much as possible, we can use lower level of Factor C and higher level of Factor F.
8.40
Each treatment combination is represented by factors labeled a, b, c, and d . This implies 4 factors were investigated.
2^{4-1} fractional factorial design.
resolution IV
Main effect of a: -0.5
Main effect of b: -0.25
Main effect of c: -0.25
Main effect of d: 2.5
The defining relation for this design is I = ABCD.
8.48
D=AB
E=BC
Resolution IV
8.60
D ~ BC
E ~ BCD
F ~ CD
G ~ BD