Bayesian Analysis

Restricted Covariance Priors with Applications in Spatial Statistics

Theresa R. Smith, Jon Wakefield, and Adrian Dobra

Full-text: Open access

Abstract

We present a Bayesian model for area-level count data that uses Gaussian random effects with a novel type of G-Wishart prior on the inverse variance–covariance matrix. Specifically, we introduce a new distribution called the truncated G-Wishart distribution that has support over precision matrices that lead to positive associations between the random effects of neighboring regions while preserving conditional independence of non-neighboring regions. We describe Markov chain Monte Carlo sampling algorithms for the truncated G-Wishart prior in a disease mapping context and compare our results to Bayesian hierarchical models based on intrinsic autoregression priors. A simulation study illustrates that using the truncated G-Wishart prior improves over the intrinsic autoregressive priors when there are discontinuities in the disease risk surface. The new model is applied to an analysis of cancer incidence data in Washington State.

Article information

Source
Bayesian Anal., Volume 10, Number 4 (2015), 965-990.

Dates
First available in Project Euclid: 4 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.ba/1423083635

Digital Object Identifier
doi:10.1214/14-BA927

Mathematical Reviews number (MathSciNet)
MR3432246

Zentralblatt MATH identifier
1335.62064

Keywords
G-Wishart distribution Markov chain Monte Carlo (MCMC) spatial statistics disease mapping

Citation

Smith, Theresa R.; Wakefield, Jon; Dobra, Adrian. Restricted Covariance Priors with Applications in Spatial Statistics. Bayesian Anal. 10 (2015), no. 4, 965--990. doi:10.1214/14-BA927. https://projecteuclid.org/euclid.ba/1423083635


Export citation

References

  • Atay-Kayis, A. and Massam, H. (2005). “A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models.” Biometrika, 92: 317–335.
  • Banerjee, S., Carlin, B., and Gelfand, A. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC.
  • Besag, J. (1974). “Spatial interaction and the statistical analysis of lattice systems.” Journal of the Royal Statistical Society: Series B (Methodological), 36: 192–236.
  • Besag, J. and Kooperberg, C. (1995). “On conditional and intrinsic autogregressions.” Biometrika, 82: 733–746.
  • Besag, J., York, J., and Mollié, A. (1991). “Bayesian image restoration, with two applications in spatial statistics.” Annals of the Institute of Statistical Mathematics, 43: 1–59.
  • Cancer Research UK (2013). “Cancer statistics by type.” http://www.cancerresearchuk.org/cancer-info/cancerstats/types/. Last visited on 01/05/2013.
  • Carlin, B. and Banerjee, S. (2003). “Hierarchical multivariate CAR models for spatially correlated survival data.” In: Bayesian Statistics 7, 45–65. Oxford University Press.
  • Chen, M.-H., Shao, Q.-M., and Ibrahim, J. G. (2000). Monte Carlo Methods in Bayesian Computation. Springer: New York.
  • Dawid, A. (1981). “Some matrix-variate distribution theory: notational considerations and a Bayesian application.” Biometrika, 68: 265–274.
  • Dawid, A. P. and Lauritzen, S. L. (1993). “Hyper Markov laws in the statistical analysis of decomposable graphical models.” The Annals of Statistics, 21: 1272–1317.
  • Dempster, A. P. (1972). “Covariance selection.” Biometrics, 28: 157–175.
  • Diggle, P. and Ribeiro, P. J. (2007). Model-based Geostatistics. Springer.
  • Diggle, P., Tawn, J., and Moyeed, R. (1998). “Model-based geostatistics.” Journal of the Royal Statistical Society: Series C (Applied Statistics), 47: 299–350.
  • Dobra, A., Lenkoski, A., and Rodriguez, A. (2011). “Bayesian inference for general Gaussian graphical models with applications to multivariate lattice data.” Journal of the American Statistical Association, 106: 1418–1433.
  • Dutilleul, P. (1999). “The MLE algorithm for the matrix normal distribution.” Journal of Statistical Computation and Simulation, 64: 105–123.
  • Fong, Y., Rue, H., and Wakefield, J. (2009). “Bayesian inference for generalized linear mixed models.” Biostatistics, 11: 397–412.
  • Fosdick, B. K. and Hoff, P. (2014). “Separable factor analysis with applications to mortality data.” The Annals of Applied Statistics, 8: 120–147.
  • Fuentes, M. (2006). “Testing for separability of spatial–temporal covariance functions.” Journal of Statistical Planning and Inference, 136: 447–466.
  • Gelfand, A. and Vounatsou, P. (2003). “Proper multivariate conditional autoregressive models for spatial data analysis.” Biostatistics, 4: 11–25.
  • Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., and Rubin, D. (2013). Bayesian data analysis. CRC press.
  • Gneiting, T. (2002). “Nonseparable, stationary covariance functions for space–time data.” Journal of the American Statistical Association, 97: 590–600.
  • Gneiting, T. and Guttorp, P. (2010). “Continuous parameter spatio-temporal processes.” In: Gelfand, A., Diggle, P., Guttorp, P., and Fuentes, M. (eds.), Handbook of Spatial Statistics, 427–436. CRC Press.
  • Green, P. and Richardson, S. (2002). “Hidden Markov models and disease mapping.” Journal of the American Statistical Association, 97: 1055–1070.
  • Hoff, P. D. (2011). “Separable covariance arrays via the Tucker product, with applications to multivariate relational data.” Bayesian Analysis, 6: 179–196.
  • Hughes, J. and Haran, M. (2013). “Dimension reduction and alleviation of confounding for spatial generalized linear mixed models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75: 139–159.
  • Jin, X., Banerjee, S., and Carlin, B. P. (2007). “Order-free co-regionalized areal data models with application to multiple-disease mapping.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69: 817–838.
  • Jin, X., Carlin, B., and Banerjee, S. (2005). “Generalized hierarchical multivariate CAR models for areal data.” Biometrics, 61: 950–961.
  • Knorr-Held, L. (2000). “Bayesian modelling of inseparable space–time variation in disease risk.” Statistics in Medicine, 19: 2555–2568.
  • Knorr-Held, L. and Best, N. (2001). “A shared component model for detecting joint and selective clustering of two diseases.” Journal of the Royal Statistical Society: Series A (Statistics in Society), 164: 73–85.
  • Knorr-Held, L. and Raßer, G. (2000). “Bayesian detection of clusters and discontinuities in disease maps.” Biometrics, 56: 13–21.
  • Lauritzen, S. L. (1996). Graphical Models. Oxford University Press.
  • Lee, D. and Mitchell, R. (2013). “Locally adaptive spatial smoothing using conditional auto-regressive models.” Journal of the Royal Statistical Society: Series C (Applied Statistics), 62: 593–608.
  • Lee, D., Rushworth, A., and Sahu, S. (2014). “A Bayesian localized conditional autoregressive model for estimating the health effects of air pollution.” Biometrics, 70: 419–429.
  • Lunn, D., Thomas, A., Best, N., and Spiegelhalter, D. (2000). “WinBUGS – a Bayesian modelling framework: concepts, structure, and extensibility.” Statistics and Computing, 10: 325–337.
  • Mardia, K. and Goodall, C. (1993). “Spatial-temporal analysis of multivariate environmental monitoring data.” In: Patil, G. and Rao, C. (eds.), Multivariate Environmental Statistics, 347–385. Elsevier.
  • Quick, H., Banerjee, S., and Carlin, B. (2013). “Modeling temporal gradients in regionally aggregated California asthma hospitalization data.” The Annals of Applied Statistics, 7: 154–176.
  • Roverato, A. (2002). “Hyper inverse Wishart distribution for non-decomposable graphs and its application to Bayesian inference for Gaussian graphical models.” Scandinavian Journal of Statistics, 29: 391–411.
  • Rue, H. and Held, L. (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC.
  • Rue, H., Martino, S., and Chopin, N. (2009). “Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71: 319–392.
  • Smith, T. R., Wakefield, J., and Dobra, A. (2015). “Supplement to “Restricted Covariance Priors with Applications in Spatial Statistics”.”
  • Sørbye, S. H. and Rue, H. (2014). “Scaling intrinsic Gaussian Markov random field priors in spatial modelling.” Spatial Statistics, 8: 39–51.
  • Spiegelhalter, D., Best, N., Carlin, B., and Van Der Linde, A. (2002). “Bayesian measures of model complexity and fit.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64: 583–639.
  • Stein, M. L. (1999). Interpolation of spatial data: some theory for Kriging. Springer.
  • — (2005). “Space–time covariance functions.” Journal of the American Statistical Association, 100: 310–321.
  • Tobler, W. R. (1970). “A computer movie simulating urban growth in the Detroit region.” Economic Geography, 46: 234–240.
  • Wall, M. (2004). “A close look at the spatial structure implied by the CAR and SAR models.” Journal of Statistical Planning and Inference, 121: 311–324.
  • Wang, H. and Pillai, N. S. (2013). “On a class of shrinkage priors for covariance matrix estimation.” Journal of Computational and Graphical Statistics, 22: 689–707.
  • White, G. and Ghosh, S. K. (2009). “A stochastic neighborhood conditional autoregressive model for spatial data.” Computational Statistics & Data Analysis, 53: 3033–3046.

Supplemental materials