Many thanks to Brian Gray, Darla Munroe, Carlos Carroll, Wayne
Thogmartin, who replied to the question below...I've pasted in
Basically, it seems as if there is no off-the-peg solution to this
problem. I'm going to look into the Gotway & Stroup paper, and also
look at transforming the data to utilise linear regression instead. The
response variable is counts of deaths, so I reckon I might get away
with age-sex specific/standardised mortality rates to use as a linear
> I'm running poisson regressions for a large number of small areas
> (several thousand contiguous polygons) - predicting counts of events
> with several predictor variables for each small area. I'd like to be
> able to adjust these models to account for spatial autocorrelation.
> Does anyone know of software (ideally free/cheap) that will do this in
> a reasonably straightforward way? Either stand-alone or as an add-on to
> Arc/info or arcview. I can also use Stata, SAS, SPSS etc.
How are you adjusting your p-values to account for the multiple
regressions--each with a potential for a Type I error/s? And, how do
you determine which points are in which polygon: if they are
spatially-correlated, could information associated with points be
shared across polygons? sorry for the questions, but my interest is in
modeling spatially-correlated nonnormal data. frankly, I haven't seen
extensions to multiple, practically-simultaneous regressions.
depending on the answers to the above questions, you might enjoy
reading Gotway, C.A. and W.W. Stroup. 1997. A generalized linear model
approach to spatial data analysis and prediction. Journal of
Agricultural, Biological, and Environmental Statistics 2: 157-178..
they examine issues pertaining to the analysis of nonnormal data under a
generalized linear model context.
You might want to contact Dan Griffith, Dept of Geography at Syracuse
University - he is working on an estimator for this exact case.
As far as I know, there is no built-in model for spatial
autocorrelation in a poisson regression (though there may be some code
out there - probably for GAUSS or something - you'd have to code the
autocorrelation into the maximum likelihood estimator - pretty sticky
Cressie indicates in his book on spatial statistics that an
"auto-Poisson" procedure (a Poisson regression incorporating spatial
autocorrelation) is infeasible. There are linear methods available in
Splus with the Spatial Statistics add-on that allow you to include
spatial autocorrelation in your models, but obviously a transformation
of the data would first be required.
You may be able to implement this in BUGS. You could ask the BUGS
or check the bugs WWW site
Just to be a little more clear: spatial effects in qualitative data
regression models are UGLY UGLY things...and no one has many good
solutions yet (though a few people are working furiously on it).
Basically, in any sort of qualitative data model, such as a possion
model - where your observed dependent variable is a count of a
occurrence/nonoccurence of some event - the observed process is not
where the spatial effect would/should be modeled. These regressions
are called latent, because there is some underlying process (that we do
not observe) that is generating the qualitative outcome.
For this reason, any spatial autocorrelation would be part of this
latent, unobserved process, not necessarily corresponding one-to-one to
the observed outcome.
Kurt Beron and Wim Vijverberg of U Texas, Dallas, have a chapter coming
out in the new Anselin spatial econometrics book (should come out this
year), New Advances in Spatial Econometrics, that has a really good and
careful review of spatial effects in probit models, and how difficult it
is to specify a full covariance structure taking these into account.
As I mentioned, Dan Griffith of Syracuse is working on poisson models.
I think Harry Kelejian (Dept of Economics, Maryland) has developed a
TEST for autocorrelation in possion models (but no correction).
You say you have thousands of polygons? YIKES. Beron and Vijverberg
developed a spatial probit estimator for 48 observations (or something
like that), and it takes several hours to run. The nXn weighting
structure/incidental parameter problem makes it very hard to identify
anything that big.
In response to 5:
I wonder if probit and Poisson are here confused? Continuous outcomes
are typically categorized using categories rather than counts. This
approach doesn't appear to describe Ben's case. Further, I am not sure
why a latent process must be assumed.
Counts are theoretically Poisson only if they meet a certain number of
assumptions/postulates. Autocorrelation is a violation, as I recall,
of these postulates. However, over- or underdispersion arising from
spatial autocorrelation may, in an estimation context, be handled from
a number of perspectives, including generalized estimating equations
and generalized linear mixed models. The negative binomial distribution
may also be used to model count data. I recommend Gotway, C.A. and
W.W. Stroup. 1997. A generalized linear model approach to spatial data
analysis and prediction. Journal of Agricultural, Biological, and
Environmental Statistics 2: 157-178.. they examine issues pertaining
to the analysis of nonnormal data under a generalized linear model
MRC Research Student
Department of Social Medicine
University of Bristol
Tel. (0117) 928 7288
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