# A Risk Assessment for Ozone Regulation Based on Statistical Rollback
## 3 Application to NMMAPS Data
### Inferences  
Estimating the impact of new ozone regulatory standards on the total nonaccidental deaths is not achieved through the national average relative rate \(\mathbf{\mu}\) as the rollback adjustment is applied to each community separately. Instead, we directly calculate the expected total nonaccidental deaths of the original and adjusted (rollbacked) ozone process, respectively. Denote \(g_{c}\) by a rollback function for community \(c\), which adjusts observed ozone concentrations to meet new ozone regulation standards. Our focus is then on the difference in expected nonaccidental deaths between before and after rollback transformation. Let \(E[\log(\mu_{t}^{c})]=\mathbf{x}_{t}^{c}\hat{\beta}^{c}+\mathbf{M}_{t}^{c}\hat{\theta} ^{c}\), where \(\mathbf{M}_{t}^{c}\) is the design matrix of covariates except for ozone (i.e., \(DOW_{t}\), \(ns(time,7/year)\), \(ns(T_{t}^{c},6)\), and \(ns(D_{t}^{c},6)\), etc. ), and \(\mathbf{x}_{t}^{c}\) is the design matrix of unconstrained (or constrained) distributed-lag ozones for community \(c\) (see [7]). If the amount of ozone reduction \(r\) is the same for each community \(c\), then the reduction ratio of the expected total death is \(1-\exp(-\mathbf{r}\hat{\beta}^{c})\) for each community \(c\) at time \(t\). Thus, the reduction rate of the national expected total death is \(1-\exp(-\mathbf{r}\hat{\mathbf{\mu}})\) at time \(t\). As the \(\mathbf{\mu}\) is obscure, the reduction rate of the national expected total death can be better estimated by weighting mean effects across the 98 cities, where weights are proportional to populations. For simplicity, the unconstrained distributed-lag model is only considered here although the constrained distributed-lag models can be also readily applied via reparameterization. Let \(z_{t}\) be rollbacked ozone concentrations of \(x_{t}\) for community \(c\) (i.e., \(z_{t}^{c}=g_{c}(x_{t}^{c})\)). After the rollback transformation, the expected total deaths for community \(c\) during the ozone season is as follows  
\[\sum_{t\in T_{\text{O}_{3}}}\exp\left(\sum_{j=0}^{6}\beta_{j}g_{c}(x_{t-j}^{c })+\mathbf{M}_{t}^{c}\hat{\theta}^{c}\right), \tag{8}\]  
where \(T_{\text{O}_{3}}\) is the set of time indices for the ozone season. The log ratio of national expected death before and after the rollback transformation during ozone season can be expressed as  
\[\log\sum_{c\in C}\sum_{t\in T_{\text{O}_{3}}}w_{c,t}\exp\left(\sum_{j=0}^{6} \beta_{j}(z_{t-j}^{c}-x_{t-j}^{c})\right), \tag{9}\]where \(C\) is the set of indices of communities in NMMAPS data and \(w_{c,t}\) is the weight of community \(c\) at time \(t\),  
\[w_{c,t}=\frac{\exp\left(\sum_{j=0}^{6}\beta_{j}x_{t-j}^{c}+\mathbf{M}_{t}^{c} \hat{\boldsymbol{\theta}}^{c}\right)}{\sum_{c\in C}\sum_{l\in T_{\text{O}_{3}}} \exp\left(\sum_{j=0}^{6}\beta_{j}x_{t-j}^{c}+\mathbf{M}_{t}^{c}\hat{ \boldsymbol{\theta}}^{c}\right)}. \tag{10}\]  
We can estimate the posterior distribution of the log ratio in (9) through the MCMC approach. As \(\pi(\boldsymbol{\beta}|\mathbf{Y})\) can be approximated by \(\pi(\boldsymbol{\beta}|\hat{\boldsymbol{\beta}})\) and a prior distribution of \((\boldsymbol{\mu},\Omega)\) well, we can generate \(\boldsymbol{\beta}\) from the posterior distribution \(\pi(\boldsymbol{\beta}|\mathbf{Y})\). One of the efficient algorithms in application is TLNISE (two level Normal independent sampling estimation) algorithm (see [13]). Without a Bayesian approach, restricted maximum likelihood (REML) method can provide similar estimates.