The Annals of Applied Statistics

Measuring the vulnerability of the Uruguayan population to vector-borne diseases via spatially hierarchical factor models

Hedibert F. Lopes, Alexandra M. Schmidt, Esther Salazar, Mariana Gómez, and Marcel Achkar

Full-text: Open access


We propose a model-based vulnerability index of the population from Uruguay to vector-borne diseases. We have available measurements of a set of variables in the census tract level of the 19 Departmental capitals of Uruguay. In particular, we propose an index that combines different sources of information via a set of micro-environmental indicators and geographical location in the country. Our index is based on a new class of spatially hierarchical factor models that explicitly account for the different levels of hierarchy in the country, such as census tracts within the city level, and cities in the country level. We compare our approach with that obtained when data are aggregated in the city level. We show that our proposal outperforms current and standard approaches, which fail to properly account for discrepancies in the region sizes, for example, number of census tracts. We also show that data aggregation can seriously affect the estimation of the cities vulnerability rankings under benchmark models.

Article information

Ann. Appl. Stat., Volume 6, Number 1 (2012), 284-303.

First available in Project Euclid: 6 March 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Areal data Bayesian inference model comparison spatial interpolation spatial smoothing


Lopes, Hedibert F.; Schmidt, Alexandra M.; Salazar, Esther; Gómez, Mariana; Achkar, Marcel. Measuring the vulnerability of the Uruguayan population to vector-borne diseases via spatially hierarchical factor models. Ann. Appl. Stat. 6 (2012), no. 1, 284--303. doi:10.1214/11-AOAS497.

Export citation


  • Adger, W. N. (2006). Vulnerability. Global Environmental Change 16 268–281.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRC, London.
  • Beltrami, M. (2008). Evolución de la pobreza em Uruguay por el método del ingreso. Período 1986–2001 (in Spanish). Technical report, Instituto Nacional de Estadística, República Oriental del Uruguay.
  • Besag, J. and Kooperberg, C. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733–746.
  • Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics (with discussion). Ann. Inst. Statist. Math. 43 1–59.
  • Blaikie, P., Cannon, T., Davis, I. and Wisner, B. (1994). At Risk, Natural Hazards, People’s Vulnerability and Disasters. Routledge, London.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • Clark, W. C. et al. (2000). Assessing vulnerability to global environmental risks. Technical Report 2000-12, Belfer Center for Science and International Affairs, John F. Kennedy School of Government, Harvard Univ.
  • Cutter, S. L., Boruff, B. J. and Shirley, W. L. (2003). Social vulnerability to environmental hazards. Social Science Quarterly 84 242–261.
  • Eakin, H. and Luers, A. L. (2006). Assessing the vulnerability of social–environmental systems. Annu. Rev. Environ. Resour. 31 365–394.
  • Ferreira, M. A. R. and De Oliveira, V. (2007). Bayesian reference analysis for Gaussian Markov random fields. J. Multivariate Anal. 98 789–812.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd ed. Chapman and Hall/CRC, Boca Raton, FL.
  • Gelfand, A. E. and Ghosh, S. K. (1998). Model choice: A minimum posterior predictive loss approach. Biometrika 85 1–11.
  • Gneiting, T., Balabdaoui, F. and Raftery, A. E. (2007). Probabilistic forecasts, calibration and sharpness. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 243–268.
  • Hahn, M., Riederer, A. and Foster, S. (2009). The livelihood vulnerability index: A pragmatic approach to assessing risks from climate variability and change—a case study in Mozambique. Global Environmental Change 19 74–88.
  • Harville, D. A. (1997). Matrix Algebra from a Statistician’s Perspective. Springer, New York.
  • Hogan, J. W. and Tchernis, R. (2004). Bayesian factor analysis for spatially correlated data, with application to summarizing area-level material deprivation from census data. J. Amer. Statist. Assoc. 99 314–324.
  • Lopes, H. F., Salazar, E. and Gamerman, D. (2008). Spatial dynamic factor models. Bayesian Anal. 3 759–792.
  • Lopes, H. F. and West, M. (2004). Bayesian model assessment in factor analysis. Statist. Sinica 14 41–67.
  • Lopes, H. F., Schmidt, A. M., Salazar, E., Goméz, M. and Achkar, M. (2011). Supplement to “Measuring the vulnerability of the Uruguayan population to vector-borne diseases via spatially hierarchical factor models.” DOI:10.1214/11-AOAS497SUPPA, DOI:10.1214/11-AOAS497SUPPB.
  • Lyth, A., Holbrook, N. and Beggs, P. (2005). Climate, urbanization and vulnerability to vector-borne disease in subtropical coastal Australia: Sustainable policy for a changing environment. Global Environmental Change Part B: Environmental Hazards 6 189–200.
  • O’Brien, K. L., Leichenko, R., Kelkarc, U., Venemad, H., Aandahl, G., Tompkins, H., Javed, A., Bhadwal, S., Nygaard, S. B. L. and West, J. (2004). Mapping vulnerability to multiple stressors: Climate change and globalization in India. Global Environmental Change 14 303–313.
  • Reid, C. E., O’Neill, M. S., Gronlund, C. J., Briness, S. J., Brown, D. G., Diez-Roux, A. V. and Schwartz, J. (2009). Mapping community determinants of heat vulnerability. Environmental Health Perspectives 117 1730–1735.
  • Rue, H., Follestad, T., Wist, H. T. and Martino, S. (2007). GMRFLib: A C-library for fast and exact simulation of Gaussian Markov random fields. Technical report, Dept. Mathematical Sciences, The Norwegian Institute of Technology, Trondheim.
  • Rygel, L., O’Sullivan, D. and Yarnal, B. (2006). A method for constructing a social vulnerability index: An application to hurricane storm surges in a developed country. Mitigation and Adaptation Strategies for Global Change 11 741–764.
  • Sarewitz, D., Pielke, R. and Keykhah, M. (2003). Vulnerability and risk: Some thoughts from a political and policy perspective. Risk Anal. 23 805–810.
  • Schmidt, A. M. and Gelfand, A. E. (2003). A Bayesian coregionalization model for multivariate pollutant data. Journal of Geophysics Research 108 8783.
  • Schmidtlein, M. C., Deutsh, R. C., Piegorsch, W. W. and Cutter, S. L. (2008). A sensitivity analysis of the social vulnerability index. Risk Anal. 28 1099–1114.
  • Sen, A. K. (1981). Poverty and Famines: An Essay on Entitlement and Deprivation. Clarendon, Oxford.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583–639.
  • Sun, D., Tsutakawa, R. K. and Speckman, P. L. (1999). Posterior distribution of hierarchical models using CAR(1) distributions. Biometrika 86 341–350.
  • van Leishout, M., Kovats, R., Livermore, M. and Martens, P. (2004). Climate change and malaria: Analysis of the SRES climate and socio-economic scenarios. Global Environmental Change 14 87–99.
  • Wang, F. and Wall, M. M. (2003). Generalized common spatial factor model. Biostatistics 4 569–582.
  • Whittle, P. (1954). On stationary processes in the plane. Biometrika 41 434–449.

Supplemental materials

  • Supplementary material A: MCMC scheme and Model Selection. The full conditional distributions for both the spatially hierarchical factor model (SHFM) and the unstructured hierarchical factor model (UHFM) are presented in this supplement. We also provide a brief overview of the model comparison criteria used in the paper, namely, (i) expected posterior deviation (EPD), (ii) deviance information criterion (DIC), (iii) continuous ranked probability score (CRPS), (iv) mean absolute error (MAE), and (v) mean square error (MSE).
  • Supplementary material B: Ox Code for SHFM. The folder data contains files with the 11 socio-economic indicators (the columns of the files) observed at the census tract level (the rows of the file) for each one of the 19 Uruguayan capitals (montevideo.txt, for instance, has 1,031 rows and 11 columns). The folder neigmat contains 19 files with the neighborhood matrices for each one of the 19 capitals after rearranging the numbering of the census tract using the GMRFLib-library of Rue et al. (2007). The files shfm.ox and functions.ox contain the Ox code to perform MCMC-based posterior inference for our spatially hierarchical factor model (SHFM).