Types of correlations between two observations when random effects are crossed On page 378 of Raudenbush & Bryk's (2002) book, they recognize 3 possible types of correlations among two observations given two (fully or partially) crossed random effects (i.e., neighborhood & school):
(1) two observations coming from the same school and same neighborhood.
(2) two observations coming from same neighborhood but different schools.
(3) two observations coming from same school but different neighborhoods.
Question: I wonder how to justify which of the 3 types of correlations among two observations are possible in the following toy, neighborhood-school dataset given its structure (below the data)?
m = "
neighbor school
1        2
1        2
2        1
2        1
2        1
3        2
3        2
4        1
"
### DATA STRUCTURE in `R` code:
dat <- read.table(text=m,h=T)
xtabs(~neighbor+school,data = dat)

       school
neighbor 1 2
       1 0 2
       2 3 0
       3 0 2
       4 1 0


I'm not really sure that I understand the question:

I wonder how to justify which of the 3 types of correlations among two observations are possible in the following toy, neighborhood-school dataset given its structure (below the data)?

I don't see how you can "justify" any correlation at all. Provided that the variance components can be estimated (ie the model converges to a non-singular result) then you simply just calculate the 3 intra-class correlations (ICCs) that you mentioned.
As for the comments on the OP regarding the interpretation of these correlations, you said:

And, in fact, how come only two observations can be correlated?(I think we are talking about imaginary correlations that come about if we repeat our exact experiment innumerable times then if we examine observations with in the same school at any two iterations, then they will be likely positively correlated). Is my understanding of the meaning of two observations being correlated in the context of clustered data correct?

We're not talking about imaginary correlations, we are talking about the correlation between 2 observations chosen at random, one from each group, the point being that observations in one cluster are likely to be similar to other observations in the same cluster, rather than to observations from other clusters. Take a simple 2 level model with one grouping variable and no random slopes; so there will be only 1 ICC, and this is the correlation between observations within the same cluster. For nested models with more than one level and for models with crossed random effects there will be several ICCs.
