[R-sig-ME] Random slope specification with interactions in fixed effects

[João Veríssimo]

On 22/08/2023 00:52, Melanie Dickie wrote:
> One complicating twist is that the independent variables are all measured
> at each “cluster”, such that there is no variation in the independent
> variables within each cluster, but there is variation among clusters. Each
> of the quadrats within each cluster have different density values, but the
> same value for winter severity, % habitat alteration, and NDVI.
This means that by-cluster random slopes should not be included, since 
you cannot estimate their within-cluster effect.

If I'm seeing this correctly, the model that you're looking for is simply:
Density ~ NDVI*Alteration + NDVI*WinterSeverity + 
Alteration*WinterSeverity + (1|Cluster)

João

> Putting the twist aside for a moment, my understanding is that I should be
> using a random slope model to gain inference at the cluster level. The
> random effects would allow the covariate effects to vary among clusters and
> the fixed effects would capture the average effect of each covariate across
> clusters. From this, the full specification of the model is (with
> appropriate specification of the family and link):
>
> Density ~ NDVI*Alteration + NDVI*WinterSeverity + Alteration*WinterSeverity
> + ( NDVI*Alteration  |Cluster) + ( NDVI*WinterSeverity  |Cluster) + (
> NDVI*WinterSeverity  |Cluster)
>
> This model, however, appears to be too complex for my data and will not
> converge. To that end, I have also considered the following reduced model
> that does not include the interactions in the random effect structure (this
> model converges):
>
> Density ~ NDVI*Alteration + NDVI*WinterSeverity + Alteration*WinterSeverity
> + (0+NDVI|Cluster) + (0+Alteration|Cluster) + (0+WinterSeverity|Cluster)
>
> I have two main questions:
>
> 1.       For the simplified model, I am unsure of the interpretation of the
> beta coefficients for the fixed-effect interactions if only the independent
> variables with no interactions are specified as random slopes. Do the fixed
> coefficients still yield inferences at the cluster level?
>
> 2.       Is the lack of variation in independent variables within each
> cluster problematic? Is there an alternative way to model this that I am
> missing?
> Thank you for your help,
> Melanie