1 1. Two Blocks

1.1 (a)

For two blocks, confound the highest-order interaction with blocks:

\[ \boxed{ABC} \]

1.2 (b)

Using \(ABC\) as the block generator:

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

2 2. Four Blocks

2.1 (a)

Use two independent block generators, for example \(AB\) and \(AC\).

Their generalized interaction is

\[ (AB)(AC)=BC \]

Therefore, the effects confounded with blocks are

\[ \boxed{AB,\ AC,\ BC} \]

2.2 (b)

  • Block 1 \((AB+,AC+)\): \((1), abc\)
  • Block 2 \((AB+,AC-)\): \(ab, c\)
  • Block 3 \((AB-,AC+)\): \(b, ac\)
  • Block 4 \((AB-,AC-)\): \(a, bc\)

3 3. Half-Normal Plot

A <- rep(c(-1,1), 4)
B <- rep(rep(c(-1,1), each=2), 2)
C <- rep(c(-1,1), each=4)

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

dat <- data.frame(A,B,C,Life)

model <- lm(Life ~ A*B*C, data=dat)

effects <- 2*coef(model)[-1]
effects
##      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
abs_eff <- sort(abs(effects))
n <- length(abs_eff)

p <- ((1:n)-0.5)/n
half_q <- qnorm((1+p)/2)

plot(half_q, abs_eff,
     xlab="Half-Normal Quantiles",
     ylab="Absolute Effects",
     main="Half-Normal Plot of Effects")

text(half_q, abs_eff,
     labels=names(abs_eff),
     pos=4, cex=.8)

3.1 (a)

The half-normal plot suggests that B, C, and AC are the important effects.

3.2 (b)

The block effect is confounded with ABC. Since ABC does not appear to be one of the dominant effects on the half-normal plot, the block effect does not appear to be significant.