Is crossed random-effect the only choice?

   The "crossed vs random" terminology is only relevant in models with
more than one grouping variable.  I would call (1|X) " a random effect
of X" or more precisely "a random-intercept model with grouping variable X"

   However, your question is a little unclear to me.  Is X a grouping
variable or a predictor variable (numeric or categorical) that varies
across groups?

   I can think of four possibilities.

  1. X is the grouping variable (e.g. "hospital"). Then ~ (1|X) is a
model that describes variation in the model intercept / baseline value,
across hospitals.

  2. X is a continuous covariate (e.g. annual hospital budget).  Then if
H is the factor designating hospitals, we want  ~ X + (1|H) (plus any
other fixed effects of interest. (It doesn't make sense / isn't
identifiable to fit a random-slopes model ~ (H | X) because budgets
don't vary within hospitals.

3. X is a categorical / factor predictor (e.g. hospital size class
{small, medium, large} with multiple hospitals measured in each size
class:  ~ X + (1|H) (the same as #2).

4. X is a categorical predictor with unique values for each hospital
(e.g. postal code).  Then X is redundant with H, you shouldn't try to
include them both in the same model.

On 7/10/21 4:55 PM, Jack Solomon wrote:

Hello Allo,

In my two-level data structure, I have a cluster-level variable (called
"X"; one that doesn't vary in any cluster). If I intend to generalize
beyond X's current possible levels, then, I should take X as a random
effect.

However, because "X" doesn't vary in any cluster, therefore, such a

random

effect necessarily must be a crossed random effect (e.g., "~ 1 | X"),
correct?

If yes, then what is "X" crossed with?

Thank you,
Jack

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--
Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Director, School of Computational Science and Engineering
Graduate chair, Mathematics & Statistics