(a):

Data collection layout is listed as in the results.

library(agricolae)
trts<-c(2,2,2,2)
#Specify the K number of replications (r=1)
design <- design.ab(trt=trts, r=1, design="crd",seed=879879)
design$book
##    plots r A B C D
## 1    101 1 2 1 2 1
## 2    102 1 2 1 1 1
## 3    103 1 1 1 1 1
## 4    104 1 2 1 2 2
## 5    105 1 1 1 2 2
## 6    106 1 2 2 1 1
## 7    107 1 1 2 1 1
## 8    108 1 1 2 1 2
## 9    109 1 1 2 2 1
## 10   110 1 1 1 2 1
## 11   111 1 2 2 1 2
## 12   112 1 2 2 2 2
## 13   113 1 2 1 1 2
## 14   114 1 1 1 1 2
## 15   115 1 1 2 2 2
## 16   116 1 2 2 2 1

(b):

Data are collected as listed in code below.

obs <- c(60,36,29,62,41,37,35,29,48,42,45,71,40,31,53,61)
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)
library(DoE.base)
## 载入需要的程辑包：grid
## 载入需要的程辑包：conf.design
## Registered S3 method overwritten by 'DoE.base':
##   method           from       
##   factorize.factor conf.design
## 
## 载入程辑包：'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':
## 
##     lengths
dat1 <- cbind(A,B,C,D,obs)
dat1 <- data.frame(dat1)
mod <- lm(obs~A*B*C*D,data = dat1)
coef(mod)
## (Intercept)           A           B           C           D         A:B 
##      45.000       6.500       2.375       9.750       1.500      -0.375 
##         A:C         B:C         A:D         B:D         C:D       A:B:C 
##       2.250       1.125       1.500       0.625       0.500      -0.625 
##       A:B:D       A:C:D       B:C:D     A:B:C:D 
##       0.875      -0.500       1.125      -0.625
halfnormal(mod)
## 
## Significant effects (alpha=0.05， Lenth method):
## [1] C A

(c):

\(\overline{y_{ijklm}}=\mu+\alpha_{i}+\beta_{j}+\gamma_{k}+\delta_{l}+\alpha\beta_{ij}+\alpha\gamma_{ik}+\alpha\delta_{il}+\beta\gamma_{jk}+\beta\delta_{jl}+\gamma\delta_{kl}+\alpha\beta\gamma_{ijk}+\alpha\beta\delta_{ijl}+\alpha\gamma\delta_{ikl}+\beta\gamma\delta_{jkl}+\alpha\beta\gamma\delta_{ijkl}+\epsilon_{ijklm}\)

From the halfnorm plot, we know that A and C has the significance.

dat1$A <- as.factor(dat1$A)
dat1$B <- as.factor(dat1$B)
dat1$C <- as.factor(dat1$C)
dat1$D <- as.factor(dat1$D)

model<-aov(obs~A+C+A*C,data=dat1)
summary(model)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1    676   676.0  33.246 8.94e-05 ***
## C            1   1521  1521.0  74.803 1.68e-06 ***
## A:C          1     81    81.0   3.984   0.0692 .  
## Residuals   12    244    20.3                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(d):

\(\overline{y_{ikm}}=\mu+\alpha_{i}+\gamma_{k}+\alpha\gamma_{ik}+\epsilon_{ikm}\)

From ANOVA, p(a:c)=0.0692>0.05, so interaction between A and C is not significant. So we investigate the main effect A and C. Since P(A)=8.94e-05<0.05, P(B)=1.68e-06<0.05, they are significant. Both A and C do affect the means.

\(\overline{y_{ikm}}=\mu+\alpha_{i}+\gamma_{k}+\epsilon_{ikm}\)

Code display:

library(agricolae)
trts<-c(2,2,2,2)
#Specify the K number of replications (r=1)
design <- design.ab(trt=trts, r=1, design="crd",seed=879879)
design$book
obs <- c(60,36,29,62,41,37,35,29,48,42,45,71,40,31,53,61)
A <- c(1,1,-1,1,-1,1,-1,-1,-1,-1,1,1,1,-1,-1,1)
B <- c(-1,-1,-1,-1,-1,1,1,1,1,-1,1,1,-1,-1,1,1)
C <- c(1,-1,-1,1,1,-1,-1,-1,1,1,-1,1,-1,-1,1,1)
D <- c(-1,-1,-1,1,1,-1,-1,1,-1,-1,1,1,1,1,1,-1)
library(DoE.base)
dat1 <- cbind(A,B,C,D,obs)
dat1 <- data.frame(dat1)
mod <- lm(obs~A*B*C*D,data = dat1)
coef(mod)
halfnormal(mod)
dat1$A <- as.factor(dat1$A)
dat1$B <- as.factor(dat1$B)
dat1$C <- as.factor(dat1$C)
dat1$D <- as.factor(dat1$D)

model<-aov(obs~A+C+A*C,data=dat1)
summary(model)