Is crossed random-effect the only choice?

Dear Ben,

In the case of #3 in your response, if the researcher intends to 
generalize beyond the 3 levels of the categorical factor/ predictor X, 
then can s/he use: ~ (1|H) + (1|X)?

If yes, then H and X will be crossed?

Thanks,
Jack

On Sat, Jul 10, 2021, 10:36 PM Jack Solomon <kj.jsolomon at gmail.com 
<mailto:kj.jsolomon at gmail.com>> wrote:

    Dear?Ben,

    Thank you for your informative?response. I think # 4 is what matches
    my situation.

    Thanks again, Jack

    On Sat, Jul 10, 2021 at 8:30 PM Ben Bolker <bbolker at gmail.com
    <mailto:bbolker at gmail.com>> wrote:

         ? ?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
         >
         >? ? ? ?[[alternative HTML version deleted]]
         >
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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