Homework week 12
Dowthyaksai Lagadapati
2022-11-21
7.12
Hypothesis :
\(H_{0}\) : \(\tau\beta_{ij}\) = 0 \(\forall\) ij ,
\(H_{a}\) : \(\tau\beta_{ij}\) \(\neq\) 0 \(\exists\) ij .
library(GAD)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.A <- rep(rep(c(-1,1),8),7)
B <- rep(rep(c(1,1,-1,-1),4),7)
C <- rep(rep(c(rep(1,4),rep(-1,4)),2),7)
D <- rep(c(rep(1,8),rep(-1,8)),7)
obs7.12 <- c(10.0,0.0,4.0,0.0,0.0,5.0,6.5,16.5,4.5,19.5,15.0,41.5,8.0,21.5,0.0,18.0,18.0,16.5,6.0,10.0,0.0,20.5,18.5,4.5,18.0,18.0,16.0,39.0,4.5,10.5,0.0,5.0,14.0,4.5,1.0,34.0,18.5,18.0,7.5,0.0,14.5,16.0,8.5,6.5,6.5,6.5,0.0,7.0,12.5,17.5,14.5,11.0,19.5,20.0,6.0,23.5,10.0,5.5,0.0,3.5,10.0,0.0,4.5,10.0,19.0,20.5,12.0,25.5,16.0,29.5,0.0,8.0,0.0,10.0,0.5,7.0,13.0,15.5,1.0,32.5,16.0,17.5,14.0,21.5,15.0,19.0,10.0,8.0,17.5,7.0,9.0,8.5,41.0,24.0,4.0,18.5,18.5,33.0,5.0,0.0,11.0,10.0,0.0,8.0,6.0,36.0,3.0,36.0,14.0,16.0,6.5,8.0)Performing anova :
dat7.12 <- data.frame(A,B,C,D,obs7.12)
model7.12 <- aov(obs7.12 ~ A*B*C*D, data = dat7.12)
summary(model7.12)## 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 ' ' 1Interaction plot between factors
interaction.plot(A,B,obs7.12)
interaction.plot(A,C,obs7.12)
interaction.plot(A,D,obs7.12)
interaction.plot(B,C,obs7.12)
interaction.plot(B,D,obs7.12)
interaction.plot(C,D,obs7.12)
Manipulating data :
block <- as.fixed(rep(seq(1,7),16))
A <- as.fixed(A)
B <- as.fixed(B)
C <- as.fixed(C)
D <- as.fixed(D)
model7.12 <- lm(obs7.12 ~ block + A*B*C*D)
gad(model7.12)## Analysis of Variance Table
##
## Response: obs7.12
## Df Sum Sq Mean Sq F value Pr(>F)
## block 6 397.2 66.21 0.7525 0.609049
## A 1 917.1 917.15 10.4240 0.001736 **
## B 1 388.1 388.15 4.4116 0.038494 *
## C 1 145.1 145.15 1.6497 0.202299
## D 1 1.4 1.40 0.0159 0.900075
## A:B 1 218.7 218.68 2.4855 0.118411
## A:C 1 11.9 11.90 0.1352 0.713967
## B:C 1 115.0 115.02 1.3073 0.255919
## A:D 1 93.8 93.81 1.0662 0.304579
## B:D 1 56.4 56.43 0.6414 0.425322
## C:D 1 1.6 1.63 0.0185 0.892129
## A:B:C 1 7.3 7.25 0.0824 0.774695
## A:B:D 1 113.0 113.00 1.2844 0.260101
## A:C:D 1 39.5 39.48 0.4488 0.504635
## B:C:D 1 33.8 33.77 0.3838 0.537130
## A:B:C:D 1 95.6 95.65 1.0871 0.299912
## Residual 90 7918.5 87.98
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Interaction plots between factors after blocking
interaction.plot(A,B,obs7.12)
interaction.plot(A,C,obs7.12)
interaction.plot(A,D,obs7.12)
interaction.plot(B,C,obs7.12)
interaction.plot(B,D,obs7.12)
interaction.plot(C,D,obs7.12)
The p-values of the main effects and interactions do not significantly change before and after blocking. The blocking does not affect the interactions observed from the interaction plots before and after blocking.
7.20
Selecting ABCE and ABDF effects also confounds CDEF.
ABCE * ABDF = A^2B^2 CDEF
Below is self manual randomly generated design
B1 <- c("A","B","CD","ABCD","ACE","BCE","DE","ABDE","CF","ABCF","ADF","BDF","EF","ABEF","ACDEF","BCDEF")
B2 <- c("C","ABC","AD","BD","E","ABE","ACDE","BCDE","AF","BF","CDF","ABCDF","ACEF","BCEF","DEF","ABDEF")
B3 <- c("AC","BC","D","ABD","AE","BE","CDE","ABCDE","F","ABF","ACDF","BCDF","CEF","ABCEF","ADEF","BDEF")
B4 <- c("(1)","AB","ACD","BCD","CE","ABCE","ADE","BDE","ACF","BCF","DF","ABDF","AEF","BEF","CDEF","ABCDEF")
design <- cbind(B1,B2,B3,B4)
design## B1 B2 B3 B4
## [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
In addition to the three confounded effects in the problem statement, four other effects are confounded with the design (BDE, CDEF, BCF, and ADF). This is a self generated random design.
Bl4 <- c("B","ACD","CE","ABDE","ABCF","DE","AEF","BCDEF")
Bl2 <- c("ABC","D","AE","BCDE","BF","ACDF","CEF","ABDEF")
Bl3 <- c("A","BCD","ABCE","DE","CF","ABDF","DEF","ACDEF")
Bl1 <- c("C","ABD","BE","ACDE","AF","BCDF","ABCEF","DEF")
Bl5 <- c("AC","BD","ABE","CDE","F","ABCDF","BCEF","ADEF")
Bl6 <- c("(1)","ABCD","BCE","ADE","ACF","BDF","ABEF","CDEF")
Bl7 <- c("BC","AD","E","ABCDE","ABF","CDF","ACEF","BDEF")
Bl8 <- c("AB","CD","ACE","BDE","BCF","ADF","EF","ABCDEF")
design7.21 <- cbind(Bl1,Bl2,Bl3,Bl4,Bl5,Bl6,Bl7,Bl8)
design7.21## Bl1 Bl2 Bl3 Bl4 Bl5 Bl6 Bl7 Bl8
## [1,] "C" "ABC" "A" "B" "AC" "(1)" "BC" "AB"
## [2,] "ABD" "D" "BCD" "ACD" "BD" "ABCD" "AD" "CD"
## [3,] "BE" "AE" "ABCE" "CE" "ABE" "BCE" "E" "ACE"
## [4,] "ACDE" "BCDE" "DE" "ABDE" "CDE" "ADE" "ABCDE" "BDE"
## [5,] "AF" "BF" "CF" "ABCF" "F" "ACF" "ABF" "BCF"
## [6,] "BCDF" "ACDF" "ABDF" "DE" "ABCDF" "BDF" "CDF" "ADF"
## [7,] "ABCEF" "CEF" "DEF" "AEF" "BCEF" "ABEF" "ACEF" "EF"
## [8,] "DEF" "ABDEF" "ACDEF" "BCDEF" "ADEF" "CDEF" "BDEF" "ABCDEF"