# A Bayesian Quantile Modeling for Spatiotemporal Relative Risk: An Application to Adverse Risk Detection of Respiratory Diseases in South Carolina, USA

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Quantile Modeling for Count Data

#### 2.2. Quantile Modeling for Spatiotemporal Relative Risk

#### 2.2.1. Log-Laplace Distribution

#### 2.2.2. Spatiotemporal Relative Risk Quantile Modeling

#### 2.3. Simulation Study

## 3. A Case Study and Results

#### 3.1. Posterior Estimates in Adverse Risk Detection

#### 3.2. A Case Study of Adverse Risk Detection of Respiratory Diseases in South Carolina, USA

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Maps of the estimated 90th quantile relative risks from the proposed quantile (

**top**row) and mean (

**middle**row) regression models versus the simulated relative risks (

**bottom**row) during years 2 (left) to 4 (right).

**Figure 2.**Maps of the estimated median quantile relative risks from the proposed quantile (

**top**row) and mean (

**middle**row) regression models versus the simulated relative risks (

**bottom**row) during years 2 (left) to 4 (right).

**Figure 3.**Maps of the estimated 10th quantile relative risks from the proposed quantile (

**top**row) and mean (

**middle**row) regression models versus the simulated relative risks (

**bottom**row) during years 2 (left) to 4 (right).

**Figure 4.**Scatterplots of simulated and estimated from quantile (

**upper**) versus mean (

**lower**) regression models at 10th (left), 50th (middle), and 90th (right) quantiles on the log scale. The red line represents the equality of the predicted and true quantiles.

**Figure 5.**Weekly emergency room discharges of respiratory diseases in South Carolina 2009. The red line represents the mean number of cases averaged over all counties.

**Figure 6.**Maps of exceedence probability under mean regression for each area with cut-off points of 1 (c = 1; first row) and 2 (c = 2; third row). Maps of exceedance probability under quantile modeling with cut-off points of 1 (c = 1; second row) and 2 (c = 2; forth row) compared to the posterior estimated relative risk (fifth row) during weeks of 34 (left column), 35 (middle), and 36 (right column).

**Figure 7.**Plots of crude relative risks (standardized incidence rate; blue), posterior mean relative risks (solid black) with corresponding 95% credible intervals (dash black), and estimated 5th, 50th and 95th quantiles from log-Laplace regression (solid red) for Beaufort (upper left), Greenwood (upper right), Charleston (lower left), and Richland (lower right) counties.

Level | 0.9 | 0.5 | 0.1 |
---|---|---|---|

Quantile model | 689.54 | 236.64 | 742.08 |

Mean model | 2446.15 | 1245.37 | 2734.15 |

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**MDPI and ACS Style**

Rotejanaprasert, C.; Lawson, A.B.
A Bayesian Quantile Modeling for Spatiotemporal Relative Risk: An Application to Adverse Risk Detection of Respiratory Diseases in South Carolina, USA. *Int. J. Environ. Res. Public Health* **2018**, *15*, 2042.
https://doi.org/10.3390/ijerph15092042

**AMA Style**

Rotejanaprasert C, Lawson AB.
A Bayesian Quantile Modeling for Spatiotemporal Relative Risk: An Application to Adverse Risk Detection of Respiratory Diseases in South Carolina, USA. *International Journal of Environmental Research and Public Health*. 2018; 15(9):2042.
https://doi.org/10.3390/ijerph15092042

**Chicago/Turabian Style**

Rotejanaprasert, Chawarat, and Andrew B. Lawson.
2018. "A Bayesian Quantile Modeling for Spatiotemporal Relative Risk: An Application to Adverse Risk Detection of Respiratory Diseases in South Carolina, USA" *International Journal of Environmental Research and Public Health* 15, no. 9: 2042.
https://doi.org/10.3390/ijerph15092042