1 Background

An engineer is interested in the effects of cutting speed A, tool geometry B, and cutting angle C on the life (in hours) of a machine tool. Two levels of each factor are chosen, and one replicate of a 23 factorial design is run.

2 Question 1

Assume that the design is run in two blocks.

2.1 Part a

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

I would confound the three factor interaction into the two blocks since it is the most likely to be insignificant to the results of the experiment.

  • Block 1 (ABC = +1): a,b,c,abc
  • Block 2 (ABC = -1): (1),ab,ac,bc

2.2 Part b

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

Block 1 (ABC -) Block 2 (ABC +)
(1) a
ab b
ac c
bc abc

3 Question 2

Assume the design is to be run in four blocks.

3.1 Part a

Which effects would you confound with the blocks?

I would confound AC and AC with the blocks which would also confound AB within the blocks

3.2 Part b

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

Block 1 Block 2 Block 3 Block 4
(1) a b c
abc bc ac ab

4 Question 3

The following data was collected for the design that you proposed in question 1(b):

run = c("(1)", "a", "b", "ab", "c", "ac", "bc", "abc")
design <- expand.grid(A = c(-1,1), B = c(-1,1), C = c(-1,1))
life <- c(22, 32, 35, 55, 44, 40, 60, 39)
dat<-data.frame(run,design,life)
dat$AC <- dat$A * dat$C
dat$BC <- dat$B * dat$C
dat$block <- factor(interaction(dat$AC, dat$BC))
levels(dat$block) <- paste("Block", seq_along(levels(dat$block)))
dat
##   run  A  B  C life AC BC   block
## 1 (1) -1 -1 -1   22  1  1 Block 4
## 2   a  1 -1 -1   32 -1  1 Block 3
## 3   b -1  1 -1   35  1 -1 Block 2
## 4  ab  1  1 -1   55 -1 -1 Block 1
## 5   c -1 -1  1   44 -1 -1 Block 1
## 6  ac  1 -1  1   40  1 -1 Block 2
## 7  bc -1  1  1   60 -1  1 Block 3
## 8 abc  1  1  1   39  1  1 Block 4

Generate a half normal plot for the collected data

library(DoE.base)
model <- lm(life ~ block + A + B + C, data = dat)
halfnormal(model)

4.1 Part a

It doesn’t appear that any main effects are significant based on the results from the half normal plot.

4.2 Part b

Based on the results from the half normal plot no blocks were identified as outliers which would identify them as being a significant factor in the model equation.

5 Complete R Code

run = c("(1)", "a", "b", "ab", "c", "ac", "bc", "abc")
design <- expand.grid(A = c(-1,1), B = c(-1,1), C = c(-1,1))
life <- c(22, 32, 35, 55, 44, 40, 60, 39)
dat<-data.frame(run,design,life)
dat$AC <- dat$A * dat$C
dat$BC <- dat$B * dat$C
dat$block <- factor(interaction(dat$AC, dat$BC))
levels(dat$block) <- paste("Block", seq_along(levels(dat$block)))
dat

library(DoE.base)
model <- lm(life ~ block + A + B + C, data = dat)
halfnormal(model)