Adding a random effect for a categorical variable I'm following up on this great answer. Given the structure of my data (below), is it possible to add a random-effect for H (a cluster ID variable) and X (a categorical variable not varying in H) as represented by the lme4 formula: ~ (1|H) + (1|X)?
If yes, would that indicate that H and X are crossed random-effects even though NO value of X is capable of meeting every value of H as suggested by this answer?
Finally, under nested random-effects, we say correlations exist among observations coming from the same cluster, but how do correlations come about under crossed random-effects as in my case?
H   X
1   2
1   2
2   1
2   1
2   1
3   2
4   1


Yes, (1|H) + (1|X) would specify crossed random effects, when there is no nesting - you said that X doesn't vary within H and presumably the reverse is also true.

Does this also mean each unique H cluster is correlated with another H cluster AND each unique X cluster is correlated with another X cluster?

I'm not sure if I've understood that question, but the correlation is between observations within each cluster (of each variable), not between them.
