# Specification of nested random effects in lme4

Suppose we have repeated observations on pupils, where pupils are nested within schools. In lme4 I believe the correct way to specify the random effects part of the formula is

(1|School/Pupil)

which expands to

(1|School) + (1|School:Pupil)

My questions are:

1. What exactly is the : symbol doing in the formula ? Does it make this an interaction term, as per the usual way that R works or is it something else ?
2. What would be the interpretation of a formula which only contains (1|School:Pupil) ?
3. What would be the interpretation of a model specified as (1|School) + (1|Pupil) + (1|School:Pupil)

1. Yes, : is the interaction operator. For example it is specifically mentioned as such in the lme4: Mixed-effects modeling with R book draft by D. Bates (see Sect. 2.2.1.1 on Nested Factors)
2. It effectively says that instead of having an intercept varying among schools and among pupils within schools ((1|schools)+(1|school:pupil)), you have intercept variations only among pupils within schools. It is a rather restrictive model and I think is hard to justify because you would expect at least some minimal effect to be due to school as a factor.
3. The interpretation would be that you are having an intercept varying among schools, among pupils and among pupils within schools. I strongly suspect this would be an overdetermined model. It could potentially work if you have multiple individual students who attended multiple schools in your sample but even then it would be probably computationally intractable. Conceptually, you would have to say that you are able to differentiate between the variation due to school, pupil and school:pupil; which I think is a bit of a tall order for not exceptionally large and well-behaved samples.