spatial auto-correlation or more complicated pseudo-replication?

Dear Thomas,

Extract the residuals from the model. Then use gstat::variogram() to 
calculate the empirical variogram of the residuals.? If there is 
spatial autocorrelation, you'll see an increase in the variance as the 
distance between observations increases.

I would expect that the birds have a stronger effect than the nests. 
Hence I'd use Pair ID. If the dataset would span more than 2 years you 
could try both a Pair and Nest random effect.

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 <mailto:thierry.onkelinx at inbo.be>
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be <http://www.inbo.be>

///////////////////////////////////////////////////////////////////////////////////////////
To call in the statistician after the experiment is done may be no 
more than asking him to perform a post-mortem examination: he may be 
able to say what the experiment died of. ~ Sir Ronald Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer does 
not ensure that a reasonable answer can be extracted from a given body 
of data. ~ John Tukey
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<https://www.inbo.be>


Op wo 22 apr. 2020 om 14:58 schreef Thomas Merkling 
<thomasmerkling00 at gmail.com <mailto:thomasmerkling00 at gmail.com>>:

    Dear Thierry,

    Thanks for reply. We used a sample of the population for our
    experiment, but for this sample we have information (treatment and
    Prop variable at each scale for all the nests.
    How would you suggest to test/check is there is spatial
    autocorrelation?
    I tried with the DHARMa package (which makes a Moran's I test
    adapted to mixed models), but it doesn't show if autocorrelation
    changes with distance, it just gives a p-value. I tried with a
    model with PairID as random effect (p = 0.33), but if I include
    nest as a random effect (some pairs changed in between the 2 years
    of the experiment, so there are less Nest IDs than Pair IDs) the
    p-value becomes 0.054 ...

    Kind regards,
    Thomas

    On 22/04/2020 13:57, Thierry Onkelinx wrote:

    Dear Thomas,

    Do you have information on all the nests or only on a sample of
    the nests? In case you have data on every nest, then I would look
    at a simple model with only treatment?and an iid nest effect.
    Then see if there is spatial autocorrelation. Variation at small
    ranges would indicate an effect of the treatment of the
    neighbouring nests.

    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 <mailto:thierry.onkelinx at inbo.be>
    Havenlaan 88 bus 73, 1000 Brussel
    www.inbo.be <http://www.inbo.be>

    ///////////////////////////////////////////////////////////////////////////////////////////
    To call in the statistician after the experiment is done may be
    no more than asking him to perform a post-mortem examination: he
    may be able to say what the experiment died of. ~ Sir Ronald
    Aylmer Fisher
    The plural of anecdote is not data. ~ Roger Brinner
    The combination of some data and an aching desire for an answer
    does not ensure that a reasonable answer can be extracted from a
    given body of data. ~ John Tukey
    ///////////////////////////////////////////////////////////////////////////////////////////

    <https://www.inbo.be>


    Op wo 22 apr. 2020 om 11:44 schreef Thomas Merkling
    <thomasmerkling00 at gmail.com <mailto:thomasmerkling00 at gmail.com>>:

        Hi all,

        I'm wondering how to best model data from an experimental design
        involving a spatial component. This is a study on seabirds
        nesting on
        artificial cliffs: each nest has been attributed an experimental
        treatment (supplemented or not), while making sure that there
        was a
        variable proportion of surrounding nests of the opposite
        treatment. Our
        main goal was to investigate if laying date of a focal pair was
        influenced by its treatment and/or by the proportion of
        surrounding
        nests of the opposite treatment (hereafter, "Prop"), which we
        calculated
        at 3 different spatial scale (local, panel and global, see
        https://drive.google.com/open?id=1OrJQCkNfBO6KOBHSlkOoQyAdTrqtIdY8
        for a
        visual representation).

        Hence, the treatment information of a focal pair is used in the
        "Treatment" predictor variable, but also in the calculation
        of "Prop"
        for the surrounding pairs (the number of pairs affected
        depending on the
        spatial scale considered), thereby leading to some
        pseudo-replication.
        Since this is dependent on the distance (i.e. "Prop" of pairs
        closer to
        a focal one are more influenced than pairs further away), we
        thought
        that accounting for spatial auto-correlation for be
        sufficient. We used
        the spaMM package to do so, and our models look something like:

        Laying ~ Treatment * Prop + Year + (1|PairID) +
        Matern(Y2011|x + y) +
        Matern(Y2012|x + y)

        with two Mat?rn correlation random effects (one for each year
        of the
        study) being included (x and y being the spatial coordinates
        of the nests).

        My question is: Is this random effect structure taking into
        account the
        fact that "Prop" of a focal pair depends on the "Treatment"
        of the
        surrounding pairs or not ? If not, how can we account for that?

        Thanks in advance for your help!
        Thomas


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