Simplifying coxme models

Hello all,

I have some questions about specifying a coxme model and then simplifying
it after reading the coxme documentation and posts here. The situation is
this:

I glued 4 pieces of small coral fragments onto small ceramic tiles, which I
placed at 4 distances east and west of 10 large coral colonies (i.e. site).
Thus, each tile represents one distance-direction-site combination. I
checked the small coral fragments daily to see which had died overnight and
at the end of the experiment some were still alive (thus, censored). I
therefore had 4 fragments per tile*4 distances*2 directions*10 sites = 320
small fragments. Distance and direction are fixed effects, while the tile
that each fragment is on and the site are random effects. In addition, each
large colony is a different size, so the size of the large colonies should
be a random effect, too (SiteSize).

The model I wrote to express this is:
mefull<-coxme(Surv(death, censor) ~ Distance*Direction+(1|Site/Tile)
+(1|SiteSize))

First, can anyone tell me if this properly specifies the situation I
described above?

After running this model, I found that neither fixed effect nor their
interaction was significant. Also, the standard deviation for Site and
SiteSize are identical (~1.12), which seems strange to me. Is there a
reason for that? The fact that they are both greater than 1 indicates to me
that they contribute a lot of variation to survival. Is that correct?

My next major question is how to simplify this model. My instinct (and
based on reading Terry Therneau's manuals and other posts here) is to
remove each random effect in turn and compare the AICs of the integrated
log-likelihood of the resulting models; the higher AIC is the preferred
model in this formulation. Is that correct?

However, I'd also like to try to try to simplify the model through removal
of the non-significant fixed effects, starting with their interaction. How
can I do this while also removing random effects? What terms should I start
with removing, or does the order not matter as long as I start with
higher-order terms (i.e. interaction)? Can I try as many combinations as I
like or do issues with multiple tests come into play?

Some options are removing one of the random effects (me2) or removing the
interaction between the fixed effects but keeping the random effects in
place (me3).
me2<-coxme(surv ~ Distance*Direction+(1|Site/Tile))
me3<-coxme(surv ~ Distance+Direction+(1|Site/Tile)+(1|SiteSize))

When I run these and other combinations of factors, their AIC is always
lower than that of the full model, which suggests to me that the full model
is best. Any guidance on how to simplify this model would be greatly
appreciated.

Finally, to compare the model with random effects to one without, can I
compare the NULL log-likelihood with the integrated log-likelihood? From my
understanding of the coxme manual, the one closer to 0 is the better model,
so if the integrated one is closer to 0 then the model with random effects
is preferred over the one without random effects.

Thanks very much for your time and help.

David Gibbs
Georgia Institute of Technology

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