# Spatial and Spatio-Temporal Models for Modeling Epidemiological Data with Excess Zeros

*Int. J. Environ. Res. Public Health*

**2015**,

*12*(9), 10536-10548; https://doi.org/10.3390/ijerph120910536

## Abstract

**:**

## 1. Introduction

## 2. Models for Data with Excess Zeros

#### 2.1. Hurdle Models

#### 2.2. Zero-Inflated Models

#### 2.3. Model Choice between a Hurdle Model and a Zero-Inflated Model

#### 2.4. Spatial and Spatio-Temporal Models with Excess Zeros

#### 2.5. Software Tools and Implementation

## 3. Case Study: Lyme disease in Illinois

**Figure 1.**Histogram (

**a**) and map (

**b**) of the total number of confirmed cases of Lyme disease in Illinois by county for the 5–year interval 2007–2011.

**Figure 2.**INLA mesh for the study region, blue line represents the border for the state of Illinois, and red dots represent the coordinates for the county seats.

## 4. Results and Discussion

Model | DIC | Effective p |
---|---|---|

Spatial Poisson Hurdle | 404 | 42.94 |

Spatial Zero-Inflated Poisson | 360 | 48.54 |

Spatial Poisson Hurdle with Probability Model | 380 | 44.84 |

Spatial Negative Binomial Hurdle | 459 | 11.85 |

Spatial Zero-Inflated Negative Binomial | 420 | 11.53 |

Spatial Neg. Bin. Hurdle with Probability Model | 435 | 13.84 |

Coefficient | Mean | Standard Deviation | 95% CI |
---|---|---|---|

Truncated Poisson | |||

Intercept | −3.2931 | 1.6830 | (−6.6478, −0.0008) |

Elevation | 0.0051 | 0.0019 | (0.0014, 0.0089) |

Population per square mile | −0.0007 | 0.0056 | (−0.0120, 0.0102) |

Zero-Inflation Probability | |||

Intercept | 7.4643 | 1.8093 | (4.1338, 11.2494) |

Elevation | −0.0097 | 0.0022 | (−0.0143, −0.0056) |

Population per square mile | −0.0025 | 0.0086 | (−0.0196, 0.0143) |

## 5. Conclusions

## Supplementary Files

Supplementary File 1## Conflicts of Interest

## References

- Lawson, A.B. Statistical Methods in Spatial Epidemiology; John Wiley & Sons: West Sussex, UK, 2013. [Google Scholar]
- Cohen, A. Estimation in mixtures of discrete distributions. In Proceedings of the International Symposium on Discrete Distributions, Montreal, QC, Canada; Pergamon Press: New York, NY, USA, 1963; pp. 373–378. [Google Scholar]
- Lambert, D. Zero-inflated Poisson regression with an application to defects in manufacturing. Technometrics
**1992**, 34, 1–14. [Google Scholar] - Hilbe, J.M. Modeling Count Data; Cambridge University Press: New York, NY, USA, 2014. [Google Scholar]
- Cragg, J.G. Some statistical models for limited dependent variables with application to the demand for durable goods. Econometrica
**1971**, 39, 829–844. [Google Scholar] [CrossRef] - Mullahy, J. Specification and testing of some modified count data models. J. Econom.
**1986**, 33, 341–365. [Google Scholar] [CrossRef] - Cameron, A.C.; Trivedi, P.K. Regression Analysis of Count Data; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Elliott, P.; Wartenberg, D. Spatial epidemiology: Current approaches and future challenges. Environ. Health Perspect.
**2004**, 112, 998–1006. [Google Scholar] [CrossRef] [PubMed] - Berliner, L.M. Hierarchical Bayesian time series models. In Maximum Entropy and Bayesian Methods; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996; pp. 15–22. [Google Scholar]
- Arab, A.; Hooten, M.B.; Wikle, C.K. Hierarchical spatial models. In Encyclopedia of GIS; Springer Science & Business Media: New York, NY, USA, 2008; pp. 425–431. [Google Scholar]
- Cressie, N.; Wikle, C.K. Statistics for Spatio-temporal Data; John Wiley & Sons: New Jersey, NJ, USA, 2011. [Google Scholar]
- Banerjee, S.; Carlin, B.P.; Gelfand, A.E. Hierarchical Modeling and Analysis for Spatial Data, 2nd ed.; CRC Press: Florida, FL, USA, 2015. [Google Scholar]
- Wikle, C.K.; Anderson, C.J. Climatological analysis of tornado report counts using a hierarchical Bayesian spatiotemporal model. J. Geophys. Res. Atmos.
**2003**, 108. [Google Scholar] [CrossRef] - Agarwal, D.K.; Gelfand, A.E.; Citron-Pousty, S. Zero-inflated models with application to spatial count data. Environ. Ecol. Stat.
**2002**, 9, 341–355. [Google Scholar] [CrossRef] - Neelon, B.; Gosh, P.; Loebs, P. A spatial Poisson hurdle model for exploring geographic variation in emergency department visits. J. Roy. Stat. Soc. A
**2013**, 176, 389–413. [Google Scholar] [CrossRef] [PubMed] - Oleson, J.J.; Wikle, C.K. Predicting infectious disease outbreak risk via migratory waterfowl vectors. J. Appl. Stat.
**2013**, 40, 656–673. [Google Scholar] [CrossRef] - Amek, N.; Bayoh, N.; Hamel, M.; Lindblade, K.A.; Gimnig, J.; Laserson, K.F.; Slutsker, L.; Smith, T.; Vounatsou, P. Spatio-temporal modeling of sparse geostatistical malaria sporozoite rate data using a zero inflated binomial model. Spat. Spatiotemporal Epidemiol.
**2011**, 2, 283–290. [Google Scholar] [CrossRef] [PubMed] - Musenge, E.; Chirwa, T.F.; Kahn, K.; Vounatsou, P. Bayesian analysis of zero inflated spatiotemporal HIV/TB child mortality data through the INLA and SPDE approaches: Applied to data observed between 1992 and 2010 in rural North East South Africa. Int. J. Appl. Earth Obs.
**2013**, 22, 86–98. [Google Scholar] [CrossRef] [PubMed] - Cressie, N. Statistics for Spatial Data: Wiley Series in Probability and Statistics; John Wiley & Sons: New Jersey, NJ, USA, 1993. [Google Scholar]
- Arab, A.; Holan, S.H.; Wikle, C.K.; Wildhaber, M.L. Semiparametric bivariate zero-inflated Poisson models with application to studies of abundance for multiple species. Environmetrics
**2012**, 23, 183–196. [Google Scholar] [CrossRef] - Rue, H.; Martino, S.; Chopin, N. Approximate Bayesian inference for latent Gaussian models by using integratednested Laplace approximations. J. Roy. Stat. Soc. B
**2009**, 71, 319–392. [Google Scholar] [CrossRef] - Blangiardo, M.; Cameletti, M.; Baio, G.; Rue, H. Spatial and spatio-temporal models with R-INLA. Spat. Spatiotemporal Epidemiol.
**2013**, 7, 39–55. [Google Scholar] [CrossRef] [PubMed] - Quiroz, Z.C.; Prates, M.O.; Rue, H. A Bayesian approach to estimate the biomass of anchovies off the coast of Perú. Biometrics
**2015**, 71, 208–217. [Google Scholar] [CrossRef] [PubMed] - Spiegelhalter, D.; Best, N.; Carlin, B.; Van Der Linde, A. Bayesian measures of model complexity and fit. J. Roy. Stat. Soc. B
**2002**, 64, 583–639. [Google Scholar] [CrossRef] - Radolf, J.D.; Caimano, M.J.; Stevenson, B.; Hu, L.T. Of ticks, mice and men: understanding the dual-host lifestyle of Lyme disease spirochaetes. Nat. Rev. Microbial.
**2012**, 10, 87–99. [Google Scholar] [CrossRef] [PubMed] - Mead, P.S. Epidemiology of Lyme disease. Infect. Dis. Clin. North Am.
**2015**, 29, 187–210. [Google Scholar] [CrossRef] [PubMed] - Center for Disease Control and Prevention: Reported Cases of Lyme Disease by State or Locality, 2004–2013. Available online: http://www.cdc.gov/lyme/stats/chartstables/reportedcases_statelocality.html (accessed on 20 June 2015).
- Tran, P.; Waller, L. Variability in results from negative binomial models for lyme disease measured at different spatial scales. Environ. Res.
**2015**, 136, 373–380. [Google Scholar] [CrossRef] [PubMed] - Lindgren, F.; Rue, H. Bayesian Spatial and Spatiotemporal Modelling with R-INLA. Available online: http://inla.googlecode.com/hg-history/5cba347753615e2bfdab62141acd4d6e858136bd/r-inla.org/papers/jss/lindgren.pdf (accessed on 4 July 2015).
- Diuk-Wasser, M.A.; Hoen, A.G.; Cislo, P.; Brinkerhoff, R.; Hamer, S.A.; Rowland, M.; Cortinas, R.; Vourc’h, G.; Melton, F.; Hickling, G.J.; et al. Human risk of infection with Borrelia burgdorferi, the Lyme disease agent, in Eastern United States. Am. J. Trop. Med. Hyg.
**2012**, 86, 320–327. [Google Scholar] [CrossRef] [PubMed]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Arab, A.
Spatial and Spatio-Temporal Models for Modeling Epidemiological Data with Excess Zeros. *Int. J. Environ. Res. Public Health* **2015**, *12*, 10536-10548.
https://doi.org/10.3390/ijerph120910536

**AMA Style**

Arab A.
Spatial and Spatio-Temporal Models for Modeling Epidemiological Data with Excess Zeros. *International Journal of Environmental Research and Public Health*. 2015; 12(9):10536-10548.
https://doi.org/10.3390/ijerph120910536

**Chicago/Turabian Style**

Arab, Ali.
2015. "Spatial and Spatio-Temporal Models for Modeling Epidemiological Data with Excess Zeros" *International Journal of Environmental Research and Public Health* 12, no. 9: 10536-10548.
https://doi.org/10.3390/ijerph120910536