Relating abundance and cover data

Dear list

This seems like something I really should know by now, but I'm getting so 
confused, I'd really appreciate a little help!

I am trying to model the relationship between relative abundance (%) and 
relative cover (%) data for plant species. I want to know to 
what extent the 2 measures correlate, and to compare the extent of this 
correlation at different sites. Obviously, both sets of data are 
zero-inflated and highly skewed.

The "traditional" thing to do would be to log-transform both of them and 
use lm(). However, a recent paper (O'Hara & Kotze, 2010) argues that a 
much better approach is to use glm() and to specify Poisson or negative 
binomial models, rather than using transformations. This does make a lot 
of sense, I think!

I have tried using "quasipoisson" and "quasibinomial" families in glm(), 
but I am left with a number of questions: 

1) Should relative abundance and relative cover be treated as "count" 
data, given that the values are not actually integers but rather 
percentages?

2) Which parts of the output of glm(...family=quasipoisson(link=log)) do I 
use to evaluate the fit? Just residual deviance and the p value?

3) How do I plot the data so as to graphically represent the model? If I 
am using a log link should I use log axes for x and y?

Thanks so much for any help!
Karen

Interesting paper by O'Hara and Kotze, but it does not refer to cover
(compositional) data, but rather to count data.  Cover data is actually
a considerably harder problem to handle in the generalized linear model
case (alas), *unless* the data come from a point count of some sort
(i.e., where you know the 'denominator', or the total number of counts
that would correspond to 100% cover, in which case you can use a
binomial GLM: see e.g. [Seavy, N. E, S. Quader, John D. Alexander, and
C. John Ralph. 2002. Generalized linear models and point count data:
statistical considerations for the design and analysis of monitoring
studies. In Bird Conservation Implementation and Integration in the
Americas: Proceedings of the Third International Partners in Flight
Conference, ed. C. John Ralph and Terrell D. Rich, 2:744-753. Asilomar,
CA: U.S. Dept. of Agriculture, Forest Service, Pacific Southwest
Research Station, March 20.
http://www.fs.fed.us/psw/publications/documents/psw_gtr191/psw_gtr191_0744-0753_seavy.pdf.]

   The natural (to a statistician) way to deal with this would be via
beta regression [Smithson, Michael, and Jay Verkuilen. 2006. A better
lemon squeezer? Maximum-likelihood regression with beta-distributed
dependent variables. Psychological Methods 11, no. 1 (March): 54-71.
doi:2006-03820-004.]  Beta distributions are a natural description of
cover -- they are distributions defined on [0,1] with a simple
mathematical description, that can be fitted similarly to GLMs [see the
'betareg' package on CRAN].  I think I heard a talk at ESA a few years
ago that used beta regression (or maybe I just thought it should have
used beta regression).  There's one big problem, though -- zeros do not
naturally fit into the statistical framework, so you have to do some
kind of ad hoc fix for this (this is discussed, briefly, in Smithson and
Verkuilen 2006).  I looked for ecology papers that used beta regression
or cited SV2006, but didn't find very many (see below).

   If you have a point count statistic, I would analyze your data in
terms of 'number of points occupied out of total census points', with a
binomial or quasibinomial model.  If they are assessed in some other way
where there is no natural denominator, I would either (sigh) use
transformations or look into beta regression.

  I'd be interested to hear other opinions.

  good luck,
    Ben Bolker




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