Random slope specification with interactions in fixed effects

Hello,

I am attempting to evaluate the influence of two main continuous
independent variables (?winter severity? aka WinterSeverity and ?% habitat
alteration? aka Alteration), and their interaction, on a continuous
dependent variable (?density?, measured as #animals/km2). I have reason to
suspect the impact of these two variables also depends on the underlying
habitat context (measured using a continuous variable, let?s say the
Normalized Difference Vegetation Index ?NDVI?). From this, my base model
is:

Density ~ NDVI*Alteration + NDVI*WinterSeverity + Alteration*WinterSeverity

Density is sampled in quadrats, with multiple quadrats clumped together in
?clusters?; such that the ?cluster? is the true sample unit, if you will,
and the quadrats are replicated samples of each ?cluster?.

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.

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
--
Melanie Dickie
PhD Candidate

        [[alternative HTML version deleted]]

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.

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