I'd like to take this opportunity both to thank those who sent me
suggestions on the above subject and to share those suggestions with the
rest of the BUGS list. Following are my letter and the responses.
Hi BUGS users,
I have data consisting of a bivariate response (an overdispersed
Poisson variable and an overdispersed Binomial variable), together with
6 explanatory variables, measured at 90 irregularly spaced points in a
geographical region. There's no problem setting up an appropriate
hierarchical generalized linear model for each response separately, with
error terms {e_i, i = 1, ..., 90} to reflect spatial dependence.
However, I have the following two questions:
(1) The nature of the problem is such that a geostatistical
covariance structure, e.g.,
corr(e_i, e_j) = exp{ -gamma * dist(i, j) }
seems more appropriate than a conditional autoregressive model.
However, I've read that such models can take a VERY long time to run in
BUGS. Has anyone had experience with a problem of this size?
(2) Since the response is bivariate, I should really model both
components simultaneously. Has someone had experience modeling
bivariate spatially distributed responses in BUGS, with either
autoregressive or explicitly defined covariance structures?
Thanks,
Sam Oman
--
Professor Samuel D. Oman
Department of Statistics
Hebrew University of Jerusalem
Mount Scopus, Jerusalem
91905 Israel
telephone: +972 2 5883 442
facsimile: +972 2 5883 549
Subject:
Re: bivariate response & geostatistical cov structure
From:
"Lance Waller" <[log in to unmask]>
Date:
Tue, 18 Nov 2003 08:42:59 -0500
To:
"Samuel Oman" <[log in to unmask]>
Dear Professor Oman:
I've had some experience with question 1. As it turns out, the CAR
structure is almost custom-made for MCMC since we define a conditional
distribution for each error component given the values of the others. As a
result, we work with the precision matrix (inverse of the var-cov matrix)
the entire time. For the geostatistical model of the elements of the
var-cov matrix, we need to invert the var-cov matrix for each set of
updates, resulting in much longer run times. While there are some
parameterizations that work more efficiently than others, there is still a
substantial computational cost of using MCMC directly with the var-cov
matrix. In my opinion, the trade-off is between a fairly efficient MCMC
spatial smoothing without the nicety of readily interpretable covariance
parameters (indeed, many CAR formulations have singular precision matrices,
see Besag and Kooperberg 1995, Biometrika for a nice discussion), and a slow
MCMC providing direct estimates of covariance parameters. This is somewhat
of an oversimplicification, but I think is the crux of the slowness.
(Best et al 1999, Bayesian Statistics 6 also consider exponentially decaying
spatial weights coupled with a CAR structure...it runs quickly but again,
the induced correlation structure is not readily apparent).
For question 2, see related papers:
A bivariate Bayes method for improving the estimates of mortality rates with
a twofold conditional autoregressive model
Kim H, Sun DC, Tsutakawa RK
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
96 (456): 1506-1521 DEC 2001
Late detection of breast and colorectal cancer in Minnesota counties: an
application of spatial smoothing and clustering
Thomas AJ, Carlin BP
STATISTICS IN MEDICINE
22 (1): 113-127 JAN 15 2003
I hope you find this helpful,
Lance Waller
Subject:
RE: bivariate response & geostatistical cov structure
From:
"Best, Nicky G" <[log in to unmask]>
Date:
Tue, 18 Nov 2003 12:39:29 -0000
To:
"'Samuel Oman'" <[log in to unmask]>
Dear Samuel,
In answer to (1), with just 90 areas, the spatial.exp model (i.e. the
geostatistical covariance srtucutre model) in winbugs should be OK so long
as you have a reasonably powerful PC. A post-doc of mine is currently
running this model with 170 areas and it takes in the order of 10's on
minutes to run.
In answer to (2), we have implemented a multivariate version of the CAR
model in WinBUGS (which basically allows the conditional distributions of a
vector of random effects in each area to be multivariate normal with
arbitrary correlation between effects in the same area, and the usual
autoregressive dependence between the same effect in neighbouring areas).
This is not available in the cirrent 'public' version on WinBUGS 1.4, but I
can send you a patch to implement it if you are interested. I am not sure
how you could extend the geostatistical covariance model in WinBUGS though,
as it is not really possible to specify explicit covariance structures other
than the 'inbuilt' exponential one in spatial.exp, which will only work for
one response.
Subject:
Re: [BUGS] bivariate response & geostatistical cov structure
From:
Andrew Lawson <[log in to unmask]>
Date:
Thu, 20 Nov 2003 12:25:45 -0500
To:
Samuel Oman <[log in to unmask]>
Hello Samuel
For the Book ' Disease Mapping with WinBUGS and MLwiN' we examined the
use of the 'spatialpred' and 'unipred'
functions. In general there are two problems with them. First, they are
very slow with even reasonably small data sets (due I guess to the need
to carry out inversion). Second, they are extremely sensitive to the
spatial configuration of the data. In the second case we examined very
small data sets with non-regular lattices and found that even with #
nodes <10 you could have singularities due to the topology of the
lattice. When this happens WinBUGS crashes.
Sudipto Banerjee at UMN also has found difficulties with the size problem.
best wishes
Andrew
Professor Andrew B. Lawson
Department of Epidemiology and Biostatistics
Arnold School of Public Health
University of South Carolina
Columbia SC 29208
USA
ph: 803-777-6647
fax:803-777-2524
email:[log in to unmask]
web site:http://www.sph.sc.edu/alawson/
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