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Methods: We applied a variant of the difference-in-differences approach, which serves to approximate random assignment of exposure across the population and hence estimate a causal effect. Specifically, we estimated the association between long-term exposure to PM2.5 and mortality while controlling for geographical differences using dummy variables for each census tract in New Jersey, a state-wide time trend using dummy variables for each year from 2004 to 2009, and mean summer and winter temperatures for each tract in each year. This approach assumed that no variable changing differentially over time across space other than seasonal temperatures confounded the association.

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Further, the causal modeling approach has not been used to estimate the effects of long-term exposure to PM2.5 on mortality. To estimate causal effects, we need a counterfactual framework. Causal modeling seeks to estimate the difference in value of the expected mortality in the population under the exposure they received versus what it would have been had they received an alternative exposure. Because that counterfactual cannot be observed, various methods seek legitimate surrogates for the unobserved potential outcome. Randomized trials are one approach but are not feasible for environmental exposures. Causal methods in observational epidemiology seek alternative ways to estimate a substitute for the counterfactual outcome (; ; ). One approach uses formal modeling techniques, such as inverse probability weighting and propensity scores, to make the exposure independent of all measured predictors and relies on the untestable assumption of no unmeasured confounding (; ). Another approach relies on natural experiments or “random shocks,” which are used as instrumental variables. The variation in such an instrumental variable is a subset of the variation in exposure that is believed to be independent of measured and unmeasured confounders. However, the assumption that exposure variations caused by the instrumental variable are randomly assigned with respect to all measured or unmeasured confounders is untestable and often relies on external information for justification. When using natural experiments or random shocks, some studies made use of the temporal variation in exposure caused by the random shock. For example, compared the mortality rates before (1984–1990) and after (1990–1996) the ban on coal sales in Dublin, Ireland (). The ban is an instrumental variable that was related to a substantial reduction in air pollution after its implementation. It is likely that the ban or a change in policy was independent of measured or unmeasured variables that confounded the association between air pollution and mortality. Other studies relied on the spatiotemporal variation in exposure caused by the instrumental variable, an example of which is the difference-in-differences approach. For example, Card and Krueger evaluated the difference in fast-food employment in New Jersey between February 1992 (2 months before an increase in the minimum wage) and November 1992 (5 months after the increase) and compared it with the difference in fast-food employment between February and November 1992 in Pennsylvania, a neighboring state that did not change its minimum wage (). The increase in the minimum wage was a random shock. In other words, the authors estimated the difference in the change (difference) in employment over time between the two states. Measured or unmeasured factors that might have confounded the association between the minimum wage and fast-food employment at each point in time (e.g., education) might have varied between the two states, but as long as any temporal variation in such factors was comparable between the states, they would not confound the difference in the change in employment over time between the states. Therefore, if the untestable assumption that the change in the minimum wage was the only factor influencing the difference in the rate of change in fast-food employment between New Jersey and Pennsylvania was true, the difference in differences was unconfounded.

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The difference-in-differences approach was applied to estimate the causal effects of long-term exposure to PM2.5 on mortality among people in New Jersey. We also estimated the association for people > 65 years old and people ≤ 65 years old by stratification. We tested if the association was modified by the mean summer temperature and by the mean winter temperature. We performed this test by adding into the model two sets of product terms: one set comprised the product terms between the spline of the mean summer temperature and PM2.5, and the other set comprised the product terms between the spline of the mean winter temperature and PM2.5. We also tested if the association was modified by ecological SES variables at the census tract level using Census 2000 data (the percentage of black residents, the median household income, and median home values) and by ecological health condition at the county level using BRFSS data from 2004–2009 (age-adjusted prevalence of diabetes and smoking). These effect modifications were tested by adding a product term between PM2.5 and the modifier into the model. Not only did we test these effect modifications among the whole population, we also tested them in a subgroup analysis by restricting the study population to the white residents (70% of the total population) to determine whether the results were consistent within a race group. Consistency could reflect whether the association estimated using the whole population was confounded by individual-level race group. We did not repeat the analysis for other race groups owing to insufficient power to detect effect modifications. In addition, because these effect modifiers all reflected the SES of a census tract and were potentially related to each other, we fitted a model with simultaneous interactions of PM2.5 with percent of black residents, home value, household income, smoking rate, and diabetes rate to determine the most robust modifiers. We used backward elimination to select the modifiers. Specifically, we started with a model with all five interaction terms. Then, the interaction term with the largest p-value was dropped, and a model without that interaction term was refitted. We repeated this procedure and stopped dropping variables until each of the remaining interaction terms had a p-value

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