Comparing abundances at fixed locations in space - Syrjala test

On 2008-February-11  , at 10:19 , Barry Rowlingson wrote:

jiho wrote:

Thank you very much for this reference. However the problem it is   
dealing with is not really similar to the one I target. In this  
paper  the authors assess the differences in positions of neurones  
in a 2D  plane between three groups of patients, with replicates  
in each group.  So the data of interest are the coordinates.
In my case, the positions of sampling stations are fixed (and on  
a  grid if that helps [1]) and I want to assess the differences  
in  abundances of two groups at these positions. So the data of  
interest  are the abundances (normalized to remove the effect of  
total  population sizes), and more specifically, the way the  
abundances are  distributed on these points. Maybe the subject of  
this email is not  correctly stated then. I am not a native  
english speaker and when it  comes to technical terms, it is even  
worse.

"Spatial Point Pattern Analysis" only refers to cases where the  
locations of the points are 'interesting', which usually means they  
are generated by a stochastic process - like tree locations in a  
natural forest rather than rows of trees in a plantation.

Thanks for clarifying these terms. Indeed I am _not_ after spatial  
point pattern techniques. I changed the subject accordingly.

Analysis of data that comes from spatial locations that are  
'uninteresting' are another branch of statistics altogether. It  
will probably end up being generalised linear modelling with  
spatially-correlated errors, and how you deal with the correlations  
is the interesting part.

See if you can write down a model for your data and include a  
smoothly-varying spatial error term.... Then maybe we can find some  
R code to solve it. I don't think we'll find it in Spatstat, which  
I think is still exclusively spatial point pattern analysis. Have a  
look at geoRglm maybe...

Thank you for the pointer. The vignette of geoRglm seems promising,  
though much is about prediction from a given model while I am most  
interested in which terms are in the model, i.e. which variables  
have a notable influence on the repartition of the organisms. My  
scenario seems simpler than those presented however, since the data  
are standardized by the sampling effort, meaning that the same  
Poisson law applies to all points.

A continuous variable than would represent the spatiality in this  
dataset could simply be the distance from the lower-left corner of  
the sampling grid for example, or the distance from the island  
around which the sampling grid is designed (such a distance would  
have a biological meaning since we expect the abundances to be  
inversely proportional to it). Is that something that could fit your  
definition of a "smoothly-varying spatial error term" or am I  
completely mistaken?

Your answer and the vignette of geoRglm highlight how little I know  
about all this (I am just a young biologist after all) and how much  
reading I need to do. The page of geoRglm has a nice list of  
publications:
	http://www.daimi.au.dk/~olefc/geoRglm/Intro/books.html
Could you (or someone else) direct me towards the best introductory  
text(s) on this matter please?

Thank you very much for your help.

Jean-Olivier Irisson wrote:

Thank you for the pointer. The vignette of geoRglm seems promising,
though much is about prediction from a given model while I am most
interested in which terms are in the model, i.e. which variables  
have a
notable influence on the repartition of the organisms. My scenario  
seems
simpler than those presented however, since the data are  
standardized by
the sampling effort, meaning that the same Poisson law applies to all
points.

 I think you still need to fit a model, and then you can test how
useful your covariates are with standard techniques.

A continuous variable than would represent the spatiality in this
dataset could simply be the distance from the lower-left corner of  
the
sampling grid for example, or the distance from the island around  
which
the sampling grid is designed (such a distance would have a  
biological
meaning since we expect the abundances to be inversely proportional  
to
it). Is that something that could fit your definition of a
"smoothly-varying spatial error term" or am I completely mistaken?

 Think about fitting a straight line through some points. You find the
line that best fits your points. Then you look at the residual
differences between the line and your points. All the usual linear  
model
theory about predictions and significance depends on those residuals
being uncorrelated and independent. If you are fitting a straight line
to a curve then that won't be true, and if you then say something  
about
your straight line based on the linear model theory you'll be wrong.

 Now, you could fit a non-spatial generalised linear model to your  
data
using glm() in R and then map the residuals. If the residual map shows
structure, then there's something else going on that your model hasn't
accounted for. Perhaps there is an obvious trend due to a covariate
you've not included, such as elevation above sea level. You could then
add this to your model. If the residual surface looks like random  
noise
then you can use standard linear model theory to make conclusions  
about
your covariate parameters.

 If the residual surface doesn't look like random noise then that's
when you get into geoRglm functions which (I think) fit a GLM where  
the
error surface (that's your residuals) is defined by a gaussian random
field with a fitted covariance structure. Once that's done, the  
geoRglm
code will tell you about your covariate parameter significance (I  
think!
It's been a while since I've used it. Maybe Paulo and Ole can expand  
on
this).

 So what I'd do is:

 * fit a simple GLM using glm.
 * Look at parameter estimates and significance.
 * Draw a map of residuals.
 * Then worry about spatial correlation.

 Oh, I'd also, if I were you, try and find a local statistician  
expert!