# A Risk Assessment for Ozone Regulation Based on Statistical Rollback
## 3 Application to NMMAPS Data  
### Statistical Modeling  
We apply an overdispersed Poisson model in generalized linear models (see [8; 12]) to the NMMAPS data. Denote \(Y_{t}^{c}\) by the number of daily non-accidental deaths in community \(c\) on day \(t\). The Poisson process with intensity function \(\mu_{t}^{c}\) can be expressed as  
\[Y_{t}^{c}\sim Poisson(\mu_{t}^{c})\text{ with }Var(Y_{t}^{c})=\phi_{c}\mu_{t}^{c}, \tag{1}\]  
where the parameter \(\phi_{c}\) describes overdispersion for community \(c\). Note that all overdispersed Poisson models are assumed to be mutually independent over time. We can also model the intensity function \(\mu_{t}^{c}\) with some essential covariates such as ozone levels at different lags, seasonality, long-term trends, weather, and co-pollutants for three age groups (<65, 65-74, and \(\geq\)75 year). Natural cubic splines are useful tools for getting smoothing functions of time to account for seasonality and long-term trends in which influenza epidemics, for example, can affect mortality. The interaction term between smoothing functions of time and age-specific indicators (<65, 65-74, and \(\geq\)75 years) is considered as it can adjust the possible seasonal mortality patterns by age group. We also control for some potential covariates related to weather by smoothing functions of dew point, average dew points of the previous three days, temperature, the average temperature of the previous three days as follows:  
\[\log\mu_{t}^{c} = \beta^{c}x_{t}^{c}+\alpha^{c}DOW_{t}+\gamma_{1}^{c}ns(time,7/year )+\gamma_{2}^{c}ns(T_{t}^{c},6) \tag{2}\] \[+ \gamma_{3}^{c}ns(T_{t-1,t-3}^{c},6)+\gamma_{4}^{c}ns(D_{t}^{c},3) +\gamma_{5}^{c}ns(D_{t-1,t-3}^{c},3)\] (3) \[+ \text{ interaction terms for age groups and time}, \tag{4}\]  
where \(\mu_{t}^{c}\) is the expected number of deaths, \(x_{t}^{c}\) is the average daily O\({}_{3}\) concentrations of the current and the previous day in community \(c\) on the day \(t\), \(DOW_{t}\) is the days of the week (categorical) on day \(t\). We define \(ns(time,7/year)\), \(ns(T_{t}^{c},6)\), \(ns(T_{t-1,t-3}^{c},6)\), \(ns(D_{t}^{c},3)\) and \(ns(D_{t-1,t-3}^{c},3)\) as the natural cubic spline function of time with 7 degrees of freedom per year, temperature with 6 degrees of freedom, the average temperature of the previous three days with 6 degrees of freedom, dew point with 3 degrees of freedom and the average dew points of the previous three days with 3 degree of freedom, respectively. The last term in (4) indicates the interaction terms of age-specific indicators and natural cubic spline functions of time. Relative mortality rates associated with exposure to ozone over the past few days can be estimated in a specific community by constrained or unconstrained distributed-lag models as daily ozone levels are readily available. Those models can be more flexible in the sense that they can be more suitable for exploring the time lag between exposures to ozone and deaths than single-lag models (see [7]). For example, the constrained distributed-lag models (CDL) and unconstrained distributed-lag models (UDL) can be expressed as  
\[\beta^{c}\mathbf{x}_{t}^{c}=\beta_{0}^{c}x_{t}+\beta_{1}^{c}\mathbf{\bar{x} }_{t:t-3}^{c}+\beta_{2}^{c}\mathbf{\bar{x}}_{t:t-6}^{c}\quad\text{ or }\quad\beta^{c}\mathbf{x}_{t}^{c}=\sum_{j=0}^{6}\beta_{j}^{c}x_{t-j}^{c}, \tag{5}\]  
respectively. The national average relative mortality rate caused by ozone effects can be estimated through a Bayesian hierarchical model in which both variability within-community and across-community can be accounted. The national average relative rate will be estimated by integrating estimates for the relative rate from the distributed-lag models for each specific community. Through the two-stage model, first, variation across communities isconsidered over the short-term ozone effects. The national average relative rate is then estimated.