# A Risk Assessment for Ozone Regulation Based on Statistical Rollback
## 4 Results  
We investigate how the rollback transformation predicts that ozone series change under new ozone regulation standards possibly affecting public health. The transformation is based on the current AQI and new regulation standards in the attainment years. Figure 2 shows Q-Q plots of ozone levles under new regulation standards against the current AQI using various rallback functions and statistical rollbacks. A common AQI is applied to all cities (common rollback) or a different AQI is applied to each city (city-specific rollback). The regular meteorological model in the NMMAPS is based on nonlinear functions mainly consisting of temperature and dewpoint at lag 0 and the average of them at lags 13. However, meteorological confounding in a distributed-lag model for ozone may exist at lags 46 so that we account for temperature and dewpoint at lag 0 through 6 in the \"distributed-lag\" meteorological model through nonlinear splines with 4 and 3 df, respectively. We use the reduction rate of the expected total death to assess the effect of the proposed new ozone regulatory. Posterior means and 95% credible intervals for total mortality per 1000 deaths are shown in Tables 1 and 2. They are based on unconstrained distributed-lag models with a common rollback and city-specific rollback approach, respectively. Most credible intervals have positive lower limits in the common rollback approach except for Weibull rollback at level 75 regularization, providing good evidence that there are reduced rates. As expected, the mortality rate decreases as lower regulation increases. Similar results are observed in CDL. The ranges of the estimated reduction rates are approximately 0.94-1.9 for 75 ppb, 1.47-2.9 for 70 ppb and 4.9-5.0 for 60 ppb for all models and rollback functions. Statistical rollbacks tend to show similar results as quadratic rollbacks for overall regularization levels. The ranges of the estimated reduction rates are roughly 1.0-2.0 for 75 ppb, 1.4-2.8 for 70 ppb and 2.4-4.8 for 60 ppb for all rollback functions in the city-specific rollback approach. Posterior variances of the relative risks for each specific city vary with rollback functions. Rather than we include daily average ozone concentration as a covariate, we include daily maximum ozone concentration and daily 8-h maximum ozone concentration so that they reduce the mortality rate more in Table 3. We also conduct statistical inference with simple MLE and pooled MLE and then compare them with Bayesian inference. Both MLE and pooled MLE give a slightly higher reduction rate than one based on the Bayesian inference in Table 4. Several issues associated with high \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Reg.** & **Prop.** & **Prop. w/ BG** & **Quadratic** & **Weibull** & **Stat.W** & **Stat.LN** \\ \hline \multirow{4}{*}{CDL} & level 75 & 1.87 & 1.09 & 0.95 & –0.62 & 0.98 & 0.94 \\  & (1.07, 2.62) & (0.62, 1.53) & (0.54, 1.35) & (–1.18, –0.11) & (0.52, 1.37) & (0.49, 1.33) \\  & level 70 & 2.86 & 1.66 & 1.46 & 0.60 & 1.51 & 1.47 \\  & (1.65, 4.09) & (0.95, 2.39) & (0.84, 2.08) & (0.10, 1.09) & (0.88, 2.14) & (0.83, 2.10) \\  & level 60 & 4.92 & 2.86 & 2.51 & 2.98 & 2.61 & 2.54 \\  & (2.84, 7.02) & (1.64, 4.09) & (1.43, 3.61) & (1.53, 4.30) & (1.48, 3.75) & (1.46, 3.65) \\ \hline \multirow{4}{*}{UDL} & level 75 & 1.91 & 1.11 & 0.98 & –0.62 & 1.03 & 0.98 \\  & (1.15, 2.72) & (0.66, 1.59) & (0.58, 1.39) & (–1.17, –0.07) & (0.63, 1.44) & (0.59, 1.40) \\  & level 70 & 2.92 & 1.70 & 1.49 & 0.59 & 1.58 & 1.52 \\  & (1.66, 4.14) & (0.97, 2.41) & (0.84, 2.10) & (0.10, 1.09) & (0.92, 2.18) & (0.85, 2.13) \\  & level 60 & 5.01 & 2.92 & 2.56 & 3.05 & 2.79 & 2.60 \\  & (2.93, 7.19) & (1.66, 4.23) & (1.49, 3.70) & (1.73, 4.38) & (1.72, 3.93) & (1.56, 3.78) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Posterior means and 95% confidence intervals of total mortality reduction per 1000 deaths (1998–2000): common rollback.