Question 1:

a) Design of 2^3 Factorial design

Answer: Block ABC

b) Proposing a design

Block 1: 1, ab, ac, bc

Block 2: a,b,c,abc

Question 2:

a) Assuming design run in 4 blocks

Answer: Effects AB, AC and BC.

b)

Block 1: 1, abc

Block 2: a,bc

Block 3: b,ac

Block 4: c, ac

Question 3:

Reading the data:

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)
AB<-c(1,-1,-1,1,1,-1,-1,1)
AC<-c(1,-1,1,-1,-1,1,-1,1)
BC<-c(1,1,-1,-1,-1,-1,1,1)
ABC<-c(-1,1,1,-1,1,-1,-1,1)
obs<-c(22,32,35,55,44,40,60,39)
Data<-data.frame(A,B,AB,C,AC,BC,ABC)

Half normal plot:

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':
## 
##     lengths
?halfnormal

Model <- lm(obs~A*B*C,data = Data)
coef(Model)
## (Intercept)           A           B           C         A:B         A:C 
##      40.875       0.625       6.375       4.875      -0.875      -6.875 
##         B:C       A:B:C 
##      -2.625      -3.375
halfnormal(Model)

## no significant effects

3 a) As per the half normal plot, none of the factors are significant at alpha=0.05.

  1. The coefficient for interaction for ABC is -3.375. Since ABC is not significant, the block is also not significant.

Complete code:

#Reading the data:
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)
AB<-c(1,-1,-1,1,1,-1,-1,1)
AC<-c(1,-1,1,-1,-1,1,-1,1)
BC<-c(1,1,-1,-1,-1,-1,1,1)
ABC<-c(-1,1,1,-1,1,-1,-1,1)
obs<-c(22,32,35,55,44,40,60,39)
Data<-data.frame(A,B,AB,C,AC,BC,ABC)

#Half normal plot:
library(DoE.base)
?halfnormal

Model <- lm(obs~A*B*C,data = Data)
coef(Model)

halfnormal(Model)