specifying/interpreting random effects with near-zero variance in glmer()

Margaret Metz <mrmetz at ...> writes:

[snip]

I am using glmer() and a logit link for the survival model,
including fixed factors of 3 topographic models ("topo1", "topo2",
and "topo3" for simplicity), starting height ("ht") I have 130+
species ("sp") found at 200 census stations ("station"). ?Not all
species are found at all stations, and the sample size per species
ranges from 10 - 1200 individuals (and I could restrict these
further to ones with a sample size greater than some threshold).

?The topo variables are continuous, right?

?You probably don't need to -- this is one of the strengths of
the mixed modeling approach.

?I would like to know whether the topographic variables are
significant predictors of mortality while including the random
factors of census station to account for non-independence of
seedlings at the same location (which have the same topo
measurements) and species to allow for variation in species'
responses. ?I expect that both the slope and intercept of species'
responses to each variable could be quite different. ?To allow for
different slopes/intercepts among species, I have centered the
continuous variables and specified the model as:

glmer(survival ~ topo1 + topo2 + topo3 + ht +
(0 + topo1 | sp) + (0 + topo2 | sp) + (0 + topo3 | sp) + (1 | sp) +
(1 | station), data=seedlingdata, family=binomial)

?This looks reasonable, you might want to check for overdispersion.

Questions: When I do this, there is a random intercept for station,
a random intercept for species, and then random slopes among species
for the relationship with the topographic variables as follows in
the model output. ?I believe this is allowing for the variation
among species that I intend, but would like confirmation of this
specification vs. something like (topo1 | sp) or (1 + topo1 | sp) as
someone else has suggested to me.

(topo1 | sp) is equivalent to (1 | topo1 | sp) (as
(0 + topo1 | sp) is equivalent to (topo1 - 1 | sp)

?If you have enough data you could try

(topo1 + topo2 + topo3 | sp )

which allows for correlation among the effects of the topographic
variables -- although you can run out of data pretty quickly in
some cases, and it sounds from stuff below as though you're running
low on signal anyway. ?(This model has (n+1)*(n+2)/2 = 10 parameters --
4 variances (topo[1-3] plus intercept) and 6 covariances -- as opposed
to the 4 variances of the model you are using.) (I'm not counting
the station variable in these totals.)

Any version of these models that I have run results in significant
fixed factors and zero or near-zero variances for the random
effects. ?I interpret this to mean that the topographic variables
are important predictors of seedling mortality, but that the
relationship does not vary among species groups nor census
locations. ?Is this your interpretation too or need I worry about
model specification or the sample size or variance structure of my
variables?

? This is a reasonable interpretation. ?However, be aware that this
is signal-to-noise / sample-size dependent. ?There could be (is, by
definition, in an ecological system) some among-species and
among-station variance that you just can't detect with this data set.
(In a classical model with a balanced, nested, etc. design you would
probably just find a small (non-significant) variance in this case,
rather than a practically-zero one -- on the other hand, there are
other classical models where you would actually estimate a *negative*
variance.)

?A suggestion was made to confirm a lack of spatial autocorrelation
in the residuals of this model, but I am not sure that is
appropriate given the inclusion of the random effect of census
station and the fixed effects of topography, which are shared by
seedlings at the same station. ?Can anyone suggest an appropriate
reference to support or refute this suggestion?

?I don't have a reference but I would suggest that checking for
spatial autocorrelation might be worthwhile. Spatial autocorrelation
would detect the effects of _unmeasured_ covariates that were more
similar among nearby stations.

Finally, if the response to topography DID significantly vary among
species, where in this model would I see it? ?In a large variance
for the species slopes or intercept?

?Exactly (variance among species in responses to topo1, topo2, topo3)

Or would I need to include

species as a fixed factor crossed with the topographic variables?

?(topo1 | sp) is effectively crossing topo with species.

?I would consider looking (at least graphically) for evidence
of nonlinearity in the responses to the continuous variables ...
you could fit a GAM without *too* much extra effort, and with
this size dataset it might produce interesting results.