Problem 7.12
Model Equation:
\(Y_{ijklmn} = \mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\gamma_k+(\alpha\gamma)_{ik}+(\beta\gamma)_{jk}+\delta_l+.........+(\alpha\beta\gamma\delta)_{ijkl}+\tau_m+\epsilon_{ijklmn}\)
Where, \(\tau_m\) = Block effect and n = replicate
ANOVA with Blocking
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 <- c(rep(-1,7),rep(1,7))
B <- c(rep(-1,14),rep(1,14))
C <- c(rep(-1,28),rep(1,28))
D <- c(rep(-1,56),rep(1,56))
Block <- c(rep(seq(1,7),16))
obs <- 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)
A <- as.fixed(A)
B <- as.fixed(B)
C <- as.fixed(C)
D <- as.fixed(D)
Block <- as.fixed(Block)
dat <- data.frame(A,B,C,D,Block,obs)
datmodel <- aov(obs~(A*B*C*D)+Block,data=dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 917 917.1 10.396 0.00176 **
## B 1 388 388.1 4.400 0.03875 *
## C 1 145 145.1 1.645 0.20290
## D 1 1 1.4 0.016 0.90021
## Block 6 376 62.7 0.710 0.64202
## A:B 1 219 218.7 2.479 0.11890
## A:C 1 12 11.9 0.135 0.71433
## B:C 1 115 115.0 1.304 0.25655
## A:D 1 94 93.8 1.063 0.30522
## B:D 1 56 56.4 0.640 0.42594
## C:D 1 2 1.6 0.018 0.89227
## A:B:C 1 7 7.3 0.082 0.77499
## A:B:D 1 113 113.0 1.281 0.26073
## A:C:D 1 39 39.5 0.448 0.50520
## B:C:D 1 34 33.8 0.383 0.53767
## A:B:C:D 1 96 95.6 1.084 0.30055
## Residuals 90 7940 88.2
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1When considering Block, P value’s of Factor A and Factor B are less than \(\alpha=0.05\) level of significance and hence the length of Putt (Factor A) and the type of putter (Factor B) are significant. All other remaining factors, their interaction terms and the Block are not significant based on their respective P value’s.
ANOVA without Blocking
library(GAD)
A <- c(rep(-1,7),rep(1,7))
B <- c(rep(-1,14),rep(1,14))
C <- c(rep(-1,28),rep(1,28))
D <- c(rep(-1,56),rep(1,56))
Block <- c(rep(seq(1,7),16))
obs <- 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)
A <- as.fixed(A)
B <- as.fixed(B)
C <- as.fixed(C)
D <- as.fixed(D)
Block <- as.fixed(Block)
dat <- data.frame(A,B,C,D,Block,obs)
datmodel <- 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 ' ' 1When considering not to Block, P value’s of Factor A and Factor B are less than \(\alpha=0.05\) level of significance and hence the length of Putt (Factor A) and the type of putter (Factor B) are significant. All other remaining factors and their interaction terms are not significant based on their respective P value’s.
If Comparing between Block and not to Block- On both cases, the length of Putt (Factor A) and the type of putter (Factor B) are significant. Also P-value’s on both cases (Block/not to Block) are pretty much similar. Also, when Blocking, value of SSE is smaller than the value of SSE (when not Blocking).
Problem 7.20
A <- c(-1,1)
B <- c(rep(-1,2),rep(1,2))
C <- c(rep(-1,4),rep(1,4))
D <- c(rep(-1,8),rep(1,8))
E <- c(rep(-1,16),rep(1,16))
F <- c(rep(-1,32),rep(1,32))
ABCF <- c(A*B*C*F)
CDEF <- c(C*D*E*F)
ABDE <- c(A*B*D*E)
dat <- data.frame(A,B,C,D,E,F,ABCF,CDEF,ABDE)
datProblem 7.21
A <- c(-1,1)
B <- c(rep(-1,2),rep(1,2))
C <- c(rep(-1,4),rep(1,4))
D <- c(rep(-1,8),rep(1,8))
E <- c(rep(-1,16),rep(1,16))
F <- c(rep(-1,32),rep(1,32))
ABEF <- c(A*B*E*F)
ABCD <- c(A*B*C*D)
ACE <- c(A*C*E)
BCF <- c(B*C*F)
BDE <- c(B*D*E)
CDEF <- c(C*D*E*F)
ADF <- c(A*D*F)
dat <- data.frame(A,B,C,D,E,F,ABEF,ABCD,ACE,BCF,BDE,CDEF,ADF)
dat