Problem 1

a.

The highest-order interaction, ABC, should be confounded with blocks because higher-order interactions are generally less likely to be significant than main effects and lower-order interactions.

The block generator is ABC.

b.

Block 1 (ABC = +1): a, b, c, abc

Block 2 (ABC = -1): (1), ab, ac, bc

Each block contains four treatment combinations.

Problem 2

a.

For four blocks, two independent block generators are needed, AB and AC will be chosen.

Generalized interaction:

\[AB \times AC=A^2BC=BC\]

Therefore, AB, AC, and BC are confounded with blocks.

b.

Block 1 (AB = +, AC = +): (1), abc

Block 2 (AB = +, AC = -): ab, c

Block 3 (AB = -, AC = +): ac, b

Block 4 (AB = -, AC = -): a, bc

Each block contains two treatment combinations.

Problem 3

Load Data

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
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)

obs<-c(22,32,35,55,44,40,60,39)

dat<-data.frame(A,B,C,obs)
dat
##    A  B  C obs
## 1 -1 -1 -1  22
## 2  1 -1 -1  32
## 3 -1  1 -1  35
## 4  1  1 -1  55
## 5 -1 -1  1  44
## 6  1 -1  1  40
## 7 -1  1  1  60
## 8  1  1  1  39

Half-Normal Plot

mod<-lm(obs~A*B*C,data=dat)

coef(mod)
## (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(mod)

## no significant effects

a.

The effects AC, B, and C appear to be the most significant because they have the largest estimated effects and depart from the straight-line pattern on the half-normal plot. BC and ABC have smaller effects and do not clearly separate from the other points, so there is insufficient evidence to conclude they are significant.

b.

The block effect is confounded with the ABC interaction. Since ABC has an estimated effect of -6.75, there may be some block variation, but it is not among the largest effects. The block effect does not appear clearly significant because ABC does not show a strong departure from the general pattern in the half-normal plot.

R Script

library(DoE.base)

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)

obs<-c(22,32,35,55,44,40,60,39)

dat<-data.frame(A,B,C,obs)
dat
mod<-lm(obs~A*B*C,data=dat)

coef(mod)

halfnormal(mod)