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/ ------------------------------------------------------------------- This list is for discussion of modelling issues and the BUGS software. For help with crashes and error messages, first mail [log in to unmask] To mail the BUGS list, mail to [log in to unmask] Before mailing, please check the archive at www.jiscmail.ac.uk/lists/bugs.html Please do not mail attachments to the list. To leave the BUGS list, send LEAVE BUGS to [log in to unmask] If this fails, mail [log in to unmask], NOT the whole list