Problem

Three factors at two levels each: cutting speed A, tool geometry B and cutting angle C. One replicate of a 2^3 factorial design (8 runs).

1. Two blocks

a) Effect confounded with the blocks

With 2 blocks, one effect is confounded with the blocks. The highest-order interaction, ABC, is chosen, because it is the effect least likely to be important (hierarchy principle). The block effect is then the same as the ABC effect.

b) Design (corner points)

The treatment combinations with an even number of letters in common with ABC (an even number of high levels) go in one block and the others go in the other block (sign of ABC = A x B x C):

Block ABC sign Treatment combinations (corner points)
1 - (1), ab, ac, bc
2 + a, b, c, abc

2. Four blocks

a) Effects confounded with the blocks

With 4 blocks, 2 independent effects are chosen and their generalized interaction is also confounded, so 3 effects are confounded with the blocks. Choosing AB and AC, the generalized interaction is (AB)(AC) = A^2 BC = BC. Confounded effects: AB, AC and BC. The main effects A, B, C and the three-factor interaction ABC are not confounded, so they can still be estimated. (ABC and AB would not be a good choice, because their generalized interaction is C, a main effect.)

b) Design (corner points)

Each treatment combination is assigned using L1 = x1 + x2 (for AB) and L2 = x1 + x3 (for AC), mod 2, where x = 1 if the letter is in the treatment combination:

Block (L1, L2) Treatment combinations (corner points)
1 (0, 0) (1), abc
2 (1, 1) a, bc
3 (1, 0) b, ac
4 (0, 1) ab, c

3. Half normal plot (data from the design with 2 blocks, ABC confounded)

The data are entered in standard order. The block effect is the ABC effect, so it appears as A:B:C in the model.

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)
life <- c(22,32,35,55,44,40,60,39)
dat <- data.frame(A,B,C,life)
dat
##    A  B  C life
## 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
mod <- lm(life ~ 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
2*coef(mod)[-1]
##      A      B      C    A:B    A:C    B:C  A:B:C 
##   1.25  12.75   9.75  -1.75 -13.75  -5.25  -6.75
halfnormal(mod)

## no significant effects

Each coefficient is half of the corresponding effect (effect = 2 x coefficient). The effects are:

Effect Estimate
A 1.25
B 12.75
C 9.75
AB -1.75
AC -13.75
BC -5.25
ABC (block) -6.75

a) Significant effects: the largest effects are AC (-13.75), B (12.75) and C (9.75), but with only 8 runs and 7 effects none of them is clearly separated from the straight line: the half normal plot (Lenth method, alpha = 0.05) does not label any effect as significant. So, at alpha = 0.05, no effect appears to be significant. AC, B and C are the candidates that stand out the most.

b) Block: the block effect is the ABC effect (-6.75). It is in the middle of the other effects and lies on the straight line, close to the effects that behave like random error. So the block does not appear to be significant.