[R-sig-ME] Is crossed random-effect the only choice?

[Ben Bolker]

   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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> 
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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