[R-sig-ME] Under what conditions does it make sense to fit random intercepts for an interaction, but not the main effects?

[Robert Long]

I am actually re-posting an old question from Cross Validated that I am
interested in, but has not received any answers:
https://stats.stackexchange.com/questions/402641/under-what-conditions-does-it-make-sense-to-fit-random-intercepts-for-an-interac


I am aware that when specifying the random structure for one factor (B)
nested within another factor (A), we can use:

(1|A) + (1|A:B)

I am trying to understand section 2.3.1 in the online book chapter 2 by
Douglas Bates: http://lme4.r-forge.r-project.org/book/Ch2.pdf
which is using the InstEval dataset, which is an evaluation of lecturers by
students at the Swiss Federal Institute for Technology–Zurich (ETH–Zurich):

> str(InstEval)
'data.frame': 73421 obs. of 7 variables:
$ s : Factor w/ 2972 levels "1","2","3","4",..: 1 1 1 1 2 2 3 3 3 ..
$ d : Factor w/ 1128 levels "1","6","7","8",..: 525 560 832 1068 6..
$ studage: Ord.factor w/ 4 levels "2"<"4"<"6"<"8": 1 1 1 1 1 1 1 1 1 1 ..
$ lectage: Ord.factor w/ 6 levels "1"<"2"<"3"<"4"<..: 2 1 2 2 1 1 1 1 1..
$ service: Factor w/ 2 levels "0","1": 1 2 1 2 1 1 2 1 1 1 ...
$ dept : Factor w/ 14 levels "15","5","10",..: 14 5 14 12 2 2 13 3 3 ..
$ y : int 5 2 5 3 2 4 4 5 5 4 ...
Factor s designates the student and d the instructor. The dept factor is
the department for the course and service indicates whether the course was
a service course taught to students from other departments. Thus these data
are partially crossed.

The model fitted in the text is:

fm4 <- lmer(y ~ 1 + (1|s) + (1|d) + (1|dept:service), InstEval, REML=0)

My question is: why is the interaction fitted as a random intercept without
(or instead of) the main effect also being fitted in this case, and in
general: when would we fit random effects for an interaction but not for
either of the main effects ? These are not nested factors, so I guess that
has something to do with it, but why is dept not specified as a random
intercept instead ? The text goes on to say

We could pursue other mixed-effects models here, such as using the dept
factor and not the dept:service interaction to define random effects, but
we will revisit these data in the next chapter and follow up on some of
these variations there.

However, as far as I know, there is no Chapter 3 !!!!

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