Problem

(1) Assume that the design is run in two blocks.

(a) Which effect(s) would you confound with the blocks?

ABC is confound with the blocks

(b) Propose a design (i.e. assign corner points to blocks)

Treatment A B C AB AC BC ABC
(1) − − − + + + −
a + − − − − + +
b − + − − + − +
ab + + − + − − −
c − − + + − − +
ac + − + − + − −
bc − + + − − + −
abc + + + + + + +

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

Block 2 (ABC = −1): (1), ab, ac, bc

(2) Assume the design is to be run in four blocks.

(a) Which effects would you confound with the blocks?

\[ 2^p=4 \]

So p = 2. G1=AB, G2=AC, G1G2= BC

AB, AC, BC are confound with the blocks

(b) Propose a design (i.e. assign corner points to blocks)

Treatment AB AC BC Block
(1) + + + 1
a − − + 2
b − + − 3
ab + − − 4
c + − − 4
ac − + − 3
bc − − + 2
abc + + + 1

Block 1: (1), abc

Block 2: a, bc

Block 3: b, ac

Block 4: c, ab

(3) Generate a half normal plot for the collected data

library(FrF2)
## Loading required package: 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
design <- FrF2(nfactors = 3, nruns = 8, randomize = FALSE)
## creating full factorial with 8 runs ...
response <- c(22, 32, 35, 55, 44, 40, 60, 39)
dat <- as.data.frame(design)
dat$ToolLife <- response
model <- aov(ToolLife ~ A * B * C, data = dat)
summary(model)
##             Df Sum Sq Mean Sq
## A            1    3.1     3.1
## B            1  325.1   325.1
## C            1  190.1   190.1
## A:B          1    6.1     6.1
## A:C          1  378.1   378.1
## B:C          1   55.1    55.1
## A:B:C        1   91.1    91.1
DanielPlot(model, half = TRUE)

(a) Which effects appear to be significant?

No effects appear to be significant at the 0.05 significance level.

(b)Does it appear that the block was significant?

dat <- data.frame(
  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),
  ToolLife = response)
dat$Block <- factor(
  ifelse(dat$A * dat$B * dat$C == 1,
         "Block 1", "Block 2"))
model.b <- aov(ToolLife ~ Block, data = dat)
summary(model.b)
##             Df Sum Sq Mean Sq F value Pr(>F)
## Block        1   91.1   91.12   0.571  0.479
## Residuals    6  957.8  159.63

No, the block does not appear to be significant. The block effect is confounded with ABC, and the corresponding point does not show a significant deviation in the half-normal plot.