R help - Feb 2010 - Interpretation of high order interaction terms.

I have difficulties in interpreting high order interaction terms in
high-way ANOVA.

According to Introductory Statistics with R by Peter Dalgaard (Section 12.5),

"The exact definition of the interaction terms and the interpretation of
their
associated regression coefficients can be elusive. Some peculiar things
happen if an interaction term is present but one or more of the main
effects are missing. The full details are probably best revealed through
experimentation."


However, I do want to understand the full details. I have checked a
few books, but I haven't found a book that describes high order
interactions in a satisfactory way. Besides learning interactions
through experimentation, could somebody let me know if there are any
detailed documentation on interactions?

## Artificial data with all interactions significant.

## The interaction2wt plot shows all main effects and all pairwise
## interactions.  We see in the "Y ~ A|B" panel (or in the
## interaction.plot) that Y goes uphill for levels 1 and 2 of B and
## goes down and then up for level 3 of B.  This is the two-way A:B
## interaction.  At each level of B, the main effect of A differs.

## From the xyplot, we see that at level 1 of C, Y goes down for level
## 3 of B.  At level 2 of C, Y goes down a lot and then up for level 3
## of B.  This is the three-way A:B:C interaction.  At each level of C
## the two-way interaction of A and B differs.


set.seed(1)

require(HH)  ## needed for interaction2wt()

threeway <-
data.frame(matrix(c(
1, 1, 1, 1,
2, 2, 1, 1,
3, 3, 1, 1,
1, 1, 2, 1,
2, 2, 2, 1,
3, 3, 2, 1,
3, 1, 3, 1,
2, 2, 3, 1,
1, 3, 3, 1,
1, 1, 1, 2,
2, 2, 1, 2,
3, 3, 1, 2,
1, 1, 2, 2,
2, 2, 2, 2,
3, 3, 2, 2,
3, 1, 3, 2,
-4, 2, 3, 2,
1, 3, 3, 2),
byrow=TRUE, 18, 4,
dimnames=list(1:18,c("Y","A","B","C"))))
for (i in 2:4) threeway[[i]] <- factor(threeway[[i]])
threeway <- rbind(threeway, threeway)
threeway$Y <- threeway$Y + rnorm(36, s=.5)


anova(aov(Y ~ A * B * C, data=threeway))

with(threeway, interaction.plot(A, B, Y))  ## shows just the "Y ~ A|B"
panel

## all two-way interactions and main effects
interaction2wt(Y ~ A + B + C, data=threeway)  ## library(HH) required

xyplot(x ~ A | C, groups=B,
       data=aggregate(threeway[,1], threeway[,-1], mean), type="l",
       auto.key=list(title="B", space="left",
         border=TRUE, lines=TRUE, points=FALSE),
       strip=strip.custom(strip.names=c(TRUE,TRUE)))