Is crossed random-effect the only choice?

Dear Thierry,

Thank you for your interesting comment (H being nested in X). I read your
informative webpage as well which was in large part in line with this
comment: (https://stats.stackexchange.com/a/228814/140365).

I think a little context can help. Think of H as a group of studies (each
with one or more rows). And think of X as scientific formulas each of which
a study has used (for all its rows) to measure the same construct.

Given this context and the data below, do you think there is a "nesting"
or a "crossing" (full or partial) relationship between studies (H) and the
formulas (X) they used, why?

Thanks, Jack
H  X
1   2
1   2
2   1
2   1
2   1
3   2
4   1

On Mon, Jul 19, 2021 at 1:58 AM Thierry Onkelinx <thierry.onkelinx at inbo.be>
wrote:

Dear Jack,

In your example H is implicitly nested in X. See
https://www.muscardinus.be/2017/07/lme4-random-effects/ for
more information on nested vs crossed effects.

Best regards,

ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE
AND FOREST
Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
thierry.onkelinx at inbo.be
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be


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Op vr 16 jul. 2021 om 01:09 schreef Jack Solomon <kj.jsolomon at gmail.com>:

Dear Ben,

Just to make sure, the structure of my data is below. With this data
structure, I wonder why ~ (1|H) + (1|X) would indicate that H and X are
crossed random-effects?

Because theoretically every value of X is capable of meeting every value
of
H (Or because each value of X means the same thing across any given value
of H)?

Does this also mean each unique cluster (separately for H & X) is
considered correlated with another cluster?

Thank you, Jack

H  X
1   2
1   2
2   1
2   1
2   1
3   2
4   1

On Thu, Jul 15, 2021 at 8:46 AM Ben Bolker <bbolker at gmail.com> wrote:

On 7/15/21 9:44 AM, Jack Solomon wrote:

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

   Yes, and yes.

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

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

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