# A Risk Assessment for Ozone Regulation Based on Statistical Rollback
## 5 Concluding Remarks  
As the rollback approach provides an adjusted ozone process in a deterministic manner, we introduce a statistical rollback approach based on the quantile matching method to allow for uncertainty and thus provide flexible rollback sets. The proposed methods are applied to epidemiologic data (NMMAPs) and we assess the impact of new ozone regulation standards on public health under various settings. Another possible model is to consider different time lags in the Poisson process model with constrained or unconstrained distributed lags together. Using respiratory deaths rather than nonaccidental deaths may also provide useful results. One of the issues that need to be addressed is collinearity between ozone and temperature, for example. Other studies revealed strong effects in the Northeast and Industrial Midwest, less strong but still significant effects in the Southeast and possibly Southern California. Thus, we can also consider a regional or spatial structure in the method. Finally, the parametric rollback approach using quantile matching may allow uncertainty in ozone process itself, which can be extended to a fully Bayesian framework.  
[PERSON] concieved the idea and developed the method presented herein and wrote the paper. [PERSON] performed data analysis. Both authors have read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.  
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2018R1D1A1B07043352).  
**Acknowledgments:** We are grateful to the Editor-in-chief, Associate Editor and the anonymous referees for their helpful comments.  
**Conflicts of Interest:** There are no potential conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Covariate** & **Reg.** & **Prop.** & **Prop. w/ BG** & **Quadratic** & **Weibull** & **Stat.W** & **Stat.LN** \\ \hline Bayesian & level 75 & 1.96 & 1.20 & 0.98 & -0.52 & 1.02 & 0.99 \\  & (1.06, 2.92) & (0.61, 1.79) & (0.51, 1.46) & (-1.33, 0.25) & (0.54, 1.51) & (0.50, 1.48) \\  & level 60 & 4.62 & 2.76 & 2.40 & 2.67 & 2.56 & 2.49 \\  & (2.64, 6.69) & (1.58, 4.06) & (1.36, 3.52) & (1.38, 3.99) & (1.52, 3.70) & (1.45, 3.62) \\ \hline MLE & level 75 & 2.23 & 1.31 & 1.09 & -0.30 & 1.21 & 1.19 \\  & (1.14, 3.21) & (0.62, 1.91) & (0.54, 1.58) & (-1.15, 0.48) & (0.66, 1.70) & (0.64, 1.69) \\  & level 60 & 5.07 & 2.93 & 2.52 & 3.03 & 2.56 & 2.52 \\  & (3.00, 7.28) & (1.65, 4.29) & (1.42, 3.68) & (1.69, 4.37) & (1.49, 3.73) & (1.39, 3.69) \\ \hline Pooled & level 75 & 1.92 & 1.16 & 0.95 & -0.50 & 1.15 & 1.10 \\ MLE & (1.79, 2.05) & (1.09, 1.24) & (0.89, 1.02) & (-0.61, -0.38) & (1.08, 1.22) & (1.04, 1.18) \\  & level 60 & 4.65 & 2.78 & 2.40 & 2.69 & 2.49 & 2.44 \\  & (4.37, 4.92) & (2.62, 2.93) & (2.27, 2.54) & (2.48, 2.88) & (2.38, 2.62) & (2.32, 2.59) \\ \hline \end{tabular}
\end{table}
Table 4: Posterior means and 95% confidence intervals of total mortality reduction per 1000 deaths (1998–2000): city-specific rollback and CDL model.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Prop.** & **Prop. w/ BG** & **Quadratic** & **Weibull** & **Stat.W** & **Stat.LN** \\ \hline With High Temp & 1.96 & 1.20 & 0.98 & -0.52 & 1.02 & 0.99 \\  & (1.06, 2.92) & (0.61, 1.79) & (0.51, 1.46) & (-1.33, 0.25) & (0.54, 1.51) & (0.50, 1.48) \\ \hline Without High Temp & 1.39 & 0.83 & 0.67 & -0.47 & 0.75 & 0.71 \\  & (0.33, 2.41) & (0.15, 1.46) & (0.15, 1.17) & (-1.36, 0.45) & (0.22, 1.26) & (0.18, 1.22) \\ \hline \end{tabular}
\end{table}
Table 5: Posterior means and 95% confidence intervals of total mortality reduction per 1000 Deaths (1998–2000): without high temperature (>85).