The Annals of Applied Statistics

Modeling temporal gradients in regionally aggregated California asthma hospitalization data

Harrison Quick, Sudipto Banerjee, and Bradley P. Carlin

Full-text: Open access

Abstract

Advances in Geographical Information Systems (GIS) have led to the enormous recent burgeoning of spatial-temporal databases and associated statistical modeling. Here we depart from the rather rich literature in space–time modeling by considering the setting where space is discrete (e.g., aggregated data over regions), but time is continuous. Our major objective in this application is to carry out inference on gradients of a temporal process in our data set of monthly county level asthma hospitalization rates in the state of California, while at the same time accounting for spatial similarities of the temporal process across neighboring counties. Use of continuous time models here allows inference at a finer resolution than at which the data are sampled. Rather than use parametric forms to model time, we opt for a more flexible stochastic process embedded within a dynamic Markov random field framework. Through the matrix-valued covariance function we can ensure that the temporal process realizations are mean square differentiable, and may thus carry out inference on temporal gradients in a posterior predictive fashion. We use this approach to evaluate temporal gradients where we are concerned with temporal changes in the residual and fitted rate curves after accounting for seasonality, spatiotemporal ozone levels and several spatially-resolved important sociodemographic covariates.

Article information

Source
Ann. Appl. Stat., Volume 7, Number 1 (2013), 154-176.

Dates
First available in Project Euclid: 9 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1365527194

Digital Object Identifier
doi:10.1214/12-AOAS600

Mathematical Reviews number (MathSciNet)
MR3086414

Zentralblatt MATH identifier
06171267

Keywords
Gaussian process gradients Markov chain Monte Carlo spatial process models spatially associated functional data

Citation

Quick, Harrison; Banerjee, Sudipto; Carlin, Bradley P. Modeling temporal gradients in regionally aggregated California asthma hospitalization data. Ann. Appl. Stat. 7 (2013), no. 1, 154--176. doi:10.1214/12-AOAS600. https://projecteuclid.org/euclid.aoas/1365527194


Export citation

References

  • Adler, R. J. (2009). The Geometry of Random Fields. SIAM, Philadelphia, PA.
  • Baladandayuthapani, V., Mallick, B. K., Hong, M. Y., Lupton, J. R., Turner, N. D. and Carroll, R. J. (2008). Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis. Biometrics 64 64–73, 321–322.
  • Banerjee, S. (2010). Spatial gradients and wombling. In Handbook of Spatial Statistics (Gelfand A. E., Diggle P., Guttorp P. and Fuentes M., eds.) 559–575. CRC Press, Boca Raton, FL.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Banerjee, S. and Gelfand, A. E. (2003). On smoothness properties of spatial processes. J. Multivariate Anal. 84 85–100.
  • Banerjee, S., Gelfand, A. E. and Sirmans, C. F. (2003). Directional rates of change under spatial process models. J. Amer. Statist. Assoc. 98 946–954.
  • Besag, J. (1986). On the statistical analysis of dirty pictures. J. Roy. Statist. Soc. Ser. B 48 259–302.
  • California Department of Health Services. (2003). California asthma facts. Available at http://www.ehib.org/papers/CaliforniaAsthmaFacts010503.pdf.
  • Carlin, B. P. and Louis, T. A. (2009). Bayesian Methods for Data Analysis, 3rd ed. CRC Press, Boca Raton, FL.
  • Cressie, N. A. C. (1993). Statistics for Spatial Data, 2nd ed. Wiley, New York.
  • Cressie, N. and Chan, N. H. (1989). Spatial modeling of regional variables. J. Amer. Statist. Assoc. 84 393–401.
  • Cressie, N. and Huang, H.-C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. J. Amer. Statist. Assoc. 94 1330–1340.
  • Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-temporal Data, 1st ed. Wiley, Hoboken, NJ.
  • Czado, C., Gneiting, T. and Held, L. (2009). Predictive model assessment for count data. Biometrics 65 1254–1261.
  • Dawid, A. P. and Sebastiani, P. (1999). Coherent dispersion criteria for optimal experimental design. Ann. Statist. 27 65–81.
  • Delicado, P., Giraldo, R., Comas, C. and Mateu, J. (2010). Statistics for spatial functional data: Some recent contributions. Environmetrics 21 224–239.
  • English, P. B., Behren, J. V., Harnly, M. and Neutra, R. R. (1998). Childhood asthma along the United States/Mexico border: Hospitalizations and air quality in two California counties. Rev. Panam. Salud Publica 3 392–399.
  • Freeman, M. F. and Tukey, J. W. (1950). Transformations related to the angular and the square root. Ann. Math. Statistics 21 607–611.
  • Gelfand, A. E., Banerjee, S. and Gamerman, D. (2005). Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics 16 465–479.
  • Gelfand, A. E. and Banerjee, S. (2010). Multivariate spatial process models. In Handbook of Spatial Statistics (Gelfand A. E., Diggle P., Guttorp P. and Fuentes M., eds.) 495–515. CRC Press, Boca Raton, FL.
  • Gelfand, A. E., Ghosh, S. K., Knight, J. R. and Sirmans, C. F. (1998). Spatio-temporal modeling of residential sales data. J. Bus. Econom. Statist. 16 312–321.
  • Gneiting, T. (2002). Nonseparable, stationary covariance functions for space–time data. J. Amer. Statist. Assoc. 97 590–600.
  • Gneiting, T. and Guttorp, P. (2010). Continuous parameter spatio-temporal processes. In Handbook of Spatial Statistics (Gelfand A. E., Diggle P., Guttorp P. and Fuentes M., eds.) 427–436. CRC Press, Boca Raton, FL.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • Handcock, M. S. and Wallis, J. R. (1994). An approach to statistical spatial-temporal modeling of meteorological fields. J. Amer. Statist. Assoc. 89 368–390.
  • Lawson, A. B., Song, H.-R., Cai, B., Hossain, M. M. and Huang, K. (2010). Space–time latent component modeling of geo-referenced health data. Stat. Med. 29 2012–2027.
  • MacNab, Y. C. and Gustafson, P. (2007). Regression B-spline smoothing in Bayesian disease mapping: With an application to patient safety surveillance. Stat. Med. 26 4455–4474.
  • Mardia, K. V., Kent, J. T., Goodall, C. R. and Little, J. A. (1996). Kriging and splines with derivative information. Biometrika 83 207–221.
  • Martínez-Beneito, M. A., López-Quilez, A. and Botella-Rocamora, P. (2008). An autoregressive approach to spatio-temporal disease mapping. Stat. Med. 27 2874–2889.
  • Pace, R. K., Barry, R., Gilley, O. W. and Sirmans, C. F. (2000). A method for spatiotemporal forecasting with an application to real estate and financial economics. J. Forecast. 16 229–240.
  • Pfeifer, P. E. and Deutsch, S. J. (1980a). Independence and sphericity tests for the residuals of space–time ARMA models. Comm. Statist. Simulation Comput. 9 533–549.
  • Pfeiffer, P. E. and Deutsch, S. J. (1980b). Stationarity and invertibility regions for low order STARMA models. Comm. Statist. Simulation Comput. 9 551–562.
  • Quick, H., Banerjee, S. and Carlin, B. P. (2013). Supplement to “Modeling temporal gradients in regionally aggregated California asthma hospitalization data.” DOI:10.1214/12-AOAS600SUPP.
  • Ramsay, J. O. and Silverman, B. W. (1997). Functional Data Analysis, 1st ed. Springer, New York.
  • Reich, B. J. and Hodges, J. S. (2008). Modeling longitudinal spatial periodontal data: A spatially adaptive model with tools for specifying priors and checking fit. Biometrics 64 790–799.
  • Schmid, V. and Held, L. (2004). Bayesian extrapolation of space–time trends in cancer registry data. Biometrics 60 1034–1042.
  • Short, M., Carlin, B. P. and Bushhouse, S. (2002). Using hierarchical spatial models for cancer control planning in Minnesota (United States). Cancer Causes Control 13 903–916.
  • 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.
  • Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • Stein, M. L. (2005). Space–time covariance functions. J. Amer. Statist. Assoc. 100 310–321.
  • Stoffer, D. S. (1986). Estimation and identification of space–time ARMAX models in the presence of missing data. J. Amer. Statist. Assoc. 81 762–772.
  • Stroud, J. R., Müller, P. and Sansó, B. (2001). Dynamic models for spatiotemporal data. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 673–689.
  • Ugarte, M. D., Goicoa, T. and Militino, A. F. (2010). Spatio-temporal modeling of mortality risks using penalized splines. Environmetrics 21 270–289.
  • Vivar, J. C. and Ferreira, M. A. R. (2009). Spatiotemporal models for Gaussian areal data. J. Comput. Graph. Statist. 18 658–674.
  • Waller, L., Carlin, B. P., Xia, H. and Gelfand, A. E. (1997). Hierarchical spatio-temporal mapping of disease rates. J. Amer. Statist. Assoc. 92 607–617.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.
  • Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250–261.

Supplemental materials

  • Supplementary material: Imputation of missing daily hospitalization counts, MCMC details, alternative models and comparison with discrete-time models. As data for days with between one and four asthma hospitalizations are missing, we impute county-specific values for these days using a method similar to Besag’s iterated conditional modes method [Besag (1986)] but with means. We also lay out the details for the MCMC implementation, discuss more general versions of our model and compare our gradient estimates to finite differences from a simple discrete-time model.