The Effects of Air Pollution on Health:
A Study of Los Angeles County
October, 2024

Air pollution poses a significant public health threat, with a growing body of research illustrating its links to various adverse health outcomes. Fine particulate matter (PM2.5 and PM10) has been particularly studied for its ability to penetrate deeply into the respiratory system and bloodstream, increasing the risk of respiratory and cardiovascular diseases. Research has consistently shown that both long-term and short-term exposure to these particles correlates with heightened mortality rates from conditions such as lung cancer and cardiopulmonary diseases. Additionally, pollutants like carbon monoxide (CO), sulfur dioxide (SO2), nitrogen dioxide (NO2), and ozone (O3) contribute to the public health burden, leading to increased hospital admissions and premature deaths.

The health impacts of air pollution are driven by complex mechanisms, including inflammation and oxidative stress. Pollutants can generate reactive oxygen species, causing cellular damage and systemic inflammation, which exacerbate chronic conditions like chronic obstructive pulmonary disease (COPD) and heart failure. Traditional studies have often relied on long-term exposure data, aggregated over annual or monthly averages. However, emerging research highlights that short-term fluctuations in air pollution levels can also have significant health implications. Weekly data, as opposed to daily or monthly measurements, can offer insights into intermediate-term effects of air pollution, capturing both short-term variations and longer-term trends.

Research has consistently shown that air pollution significantly affects various diseases, including respiratory and cardiovascular conditions. For instance, studies have demonstrated associations between short-term exposure to PM2.5 and increased exacerbations of COPD, while air pollution exposure has been linked to worsening respiratory symptoms and increased hospital admissions for pneumonia. The relationship between air pollution and health outcomes can vary by region, influenced by factors such as traffic density and meteorological conditions. For example, Los Angeles County experiences high pollution levels due to traffic and temperature inversions, which trap pollutants close to the ground.

Meteorological factors, such as temperature, air pressure, and humidity, also play a role in influencing pollution levels and health outcomes. High temperatures can enhance ground-level ozone formation, leading to increased respiratory symptoms.

Given the complexities of air pollution data, advanced statistical methods are crucial for analyzing its impacts on health outcomes. Generalized Additive Models (GAMs) are useful for capturing non-linear relationships and interactions between air pollution, meteorological covariates, and health outcomes. Meanwhile, ARIMA models help address temporal autocorrelation, making them valuable for forecasting pollutant levels and assessing their health impacts. Bayesian hierarchical models further enhance the analysis by allowing the incorporation of multiple levels of variation and uncertainty, which is essential for studying spatial and temporal variability in air pollution data.

Measurement error is a critical concern in air pollution studies, as it can lead to biased estimates of health effects. Approaches such as regression calibration and simulation-extrapolation (SIMEX) methods can help account for measurement error, providing more accurate estimates. Additionally, time series analysis using Poisson log-linear regression is a foundational tool for studying the effects of air pollution on health outcomes, effectively estimating the relative risk associated with exposure and addressing issues like overdispersion. Harmonic terms, such as sine and cosine functions, can be included in these models to capture seasonal patterns, which are particularly relevant in environmental epidemiology due to the seasonal variation in exposures and outcomes. State-space models offer additional flexibility in analyzing time series data by allowing for decomposition into underlying components, which is valuable for studies with noisy or incomplete data.

While substantial evidence exists regarding the adverse health effects of air pollution, particularly concerning respiratory and cardiovascular diseases, challenges remain. Many studies focus on long-term exposure or daily data, with fewer investigations addressing the intermediate-term impacts captured by weekly data. Furthermore, there is a need for region-specific analyses that account for localized pollution sources and meteorological conditions.

To address these gaps, this study employs advanced time series modeling techniques, including Poisson log-linear regression and ARIMA models, tailored to weekly air pollution and health outcome data. By integrating region-specific environmental variables and addressing potential measurement errors, this research aims to provide more accurate estimates of the short-to-intermediate-term effects of air pollution on public health, specifically focusing on respiratory diseases and cardiovascular diseases such as COPD, pneumonia, neoplasms, and heart failure in Los Angeles County.

The objective of this project is to explore the impact of weekly air pollution on respiratory and cardiovascular death counts in Los Angeles County, CA, while also considering meteorological factors. The document is organized in the following manner: The third section will provide a comprehensive description of the dataset, including a detailed descriptive data analysis. The methodology section will focus on model selection, primarily utilizing Generalized Linear Models (GLMs) and Stan model, supplemented by simulation examples to demonstrate model performance. The results section will detail how air pollution and meteorological factors affect health outcomes. The discussion will interpret these findings, compare them with existing research, address limitations, and offer recommendations for future research and policy.

Table 1 presents the distribution of deaths, meteorological factors, and air pollutants for Los Angeles County, California, from January 1, 2018, to December 17, 2022 (259 weeks in total).

Over the five-year study period, the mean weekly concentrations of various air pollutants were observed to be 11.73 $\mu$g/m³ for PM2.5, 26.49 $\mu$g/m³ for PM10, 0.35 ppm for CO, 0.36 ppb for SO2, 12.80 ppb for NO2, 0.03 ppm for O3, respectively. Despite Los Angeles’s reputation as a highly polluted region in the United States, our analysis revealed that all measured air pollutants from 2018 to 2022 remained below their respective standard criteria levels.
A closer examination of the data through timeseries plots revealed interesting seasonal patterns. Specifically, the concentrations of CO and NO2 tended to be lower during hotter periods, while the concentrations of other pollutants increased. This inverse relationship underscores the complex interactions between temperature and pollutant levels.

Analyzing trends over time, we observed that SO2 concentrations decreased from 2018 to 2019, then plateaued, before rising again after 2019. O3 levels were stable from 2018 until 2020, after which they experienced a sudden increase, maintaining this higher level thereafter. PM2.5 levels were relatively steady but showed peaks during the winters of 2020 and 2021. Other pollutants exhibited steady levels throughout the study period. Notably, all pollutants displayed clear seasonal patterns, further emphasizing the influence of temporal factors on air quality.
In terms of meteorological factors, the mean daily temperature, pressure, and humidity during the study period were 17.27 ^∘C, 987.75 hPa, and 56.71$\%$, respectively. These environmental variables play a crucial role in understanding the broader context of air pollution and its potential health impacts.
Table 1 also illustrates the weekly distribution of deaths from COPD, Pneumonia, Neoplasms, and Heart Failure. Between January 1, 2018, and December 17, 2022, a total of 40,982 deaths were recorded, with 10,287 attributed to COPD, 8,038 attributed to Pneumonia, 11,541 attributed to Neoplasms, and 11,116 attributed to Heart Failure. On average, the study area experienced about 158 deaths per week. Meanwhile, seasonal analysis revealed that the number of deaths was higher during the colder months compared to the warmer months, highlighting a potential seasonal influence on mortality rates.

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  $Y_{t}$: Death counts in week $t$.

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  $P_{t}$: Observed PM2.5 level in week $t$.

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  $X_{t}$: True (unobserved) PM2.5 level in week $t$.

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  $C_{t}$: Vector of covariates in week $t$.

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  $T_{t}$: Binary categorical variable for temperature conditions (1 = Hot, 0 = Cold).

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  $\text{offset}_{t}$: Known offset term (e.g., population size).

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  $t=1,\dots,n$: Time index.

Averaging is utilized to derive time series values from raw data, especially when the data may contain errors. This section explains the process of calculating both daily and weekly averages, not only for PM2.5 but also for all other air pollutants and meteorological factors.

Figure 1 shows the monitoring station locations (marked by black circles), while the white area represents Los Angeles County. Below, a detailed illustration of this conversion is provided specifically for PM2.5.

Continuous temperature data is converted to a binary variable for analysis. In this conversion, a temperature is classified as ”hot” (1) if it is above the mean temperature and ”cold” (0) if it is below or equal to the mean temperature. This transformation facilitates a simplified analysis of temperature’s impact on mortality rates, specifically allowing us to investigate whether low temperatures are associated with an increase in death rates.

In a Bayesian framework, we place prior distributions on all the unknown parameters in the model:

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  Mean of Log-Pollution Process:

  $$\mu\sim\mathcal{N}(0,\sigma^{2}_{\mu}),$$

  where $\sigma^{2}_{\mu}$ is a large variance reflecting prior uncertainty.

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  Autoregressive Coefficient:

  $$\phi\sim\mathcal{N}(0,\sigma^{2}_{\phi}),$$

  where $\sigma^{2}_{\phi}$ reflects prior uncertainty about the AR coefficient.

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  Effect of Pollution on Mortality:

  $$\gamma\sim\mathcal{N}(0,\sigma^{2}_{\gamma}),$$

  where $\sigma^{2}_{\gamma}$ reflects prior uncertainty about the effect of pollution on death counts.

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  Covariate Effects:

  $$\delta_{temperature}\sim\mathcal{N}(0,\sigma^{2}_{\delta_{temperature}}),$$

  where $\sigma^{2}_{temperature}$ reflects prior uncertainty about the effect of binary temperature on death counts.

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  Seasonal Effects:

  $$\delta_{1}\sim\mathcal{N}(0,\sigma^{2}_{\delta_{1}}),$$

  $$\delta_{2}\sim\mathcal{N}(0,\sigma^{2}_{\delta_{2}}),$$

  where $\sigma^{2}_{\delta}$ reflects prior uncertainty about the seasonal effects from sine and cosine terms.

Insufficient built-in results can lead to convergence issues in Bayesian modeling for several reasons. First, if the priors for the model parameters are too vague or poorly specified, the model may struggle to find reasonable estimates, which is particularly critical in complex models with multiple parameters. High-dimensional models can complicate the joint posterior distribution, making it challenging for the sampling algorithm to effectively explore the parameter space. Additionally, poor initialization, where the initial values of the parameters are far from high posterior density regions, can result in slow mixing or chains getting stuck in local optima. Model mis-specification also plays a role; if the model does not accurately represent the underlying data-generating process, the sampling may diverge or yield implausible parameter estimates, especially in intricate models like ARIMA for time series.

Furthermore, limitations in the sampling algorithm, such as Hamiltonian Monte Carlo (HMC) struggling in poorly conditioned models with highly correlated parameters, can lead to low acceptance rates or divergent transitions. Lack of reparameterization may hinder efficiency, while insufficient warmup periods can bias results and lead to non-convergence. Lastly, data-related issues, including missing values or outliers, can affect convergence by complicating the model’s ability to accommodate the data structure.

To address these convergence challenges, it is essential to reassess priors for informativeness, consider reparameterization to improve conditioning, increase warmup and iterations for more thorough exploration, and check for divergence to identify problematic regions in the parameter space. Experimenting with different initial values can also enhance convergence by providing the sampler with better starting points.

Overall, ensuring adequate built-in results requires careful attention to model specification, priors, and sampling strategies to facilitate effective exploration of the parameter space.

In our analysis, the simulation results demonstrated a commendable performance of the Stan model, effectively addressing many of the convergence challenges outlined above. The use of carefully chosen informative priors significantly improved the model’s ability to converge to reasonable parameter estimates, even in the presence of complex, high-dimensional data structures.

The Hamiltonian Monte Carlo algorithm displayed improved efficiency, achieving higher acceptance rates and reducing the incidence of divergent transitions. This success can be attributed to the thoughtful reparameterization of the model, which enhanced the condition of the posterior distribution and allowed for more effective exploration of the parameter space.

Additionally, the model exhibited robust mixing properties, as evidenced by trace plots that showed well-behaved chains converging to stable distributions across multiple iterations. The diagnostics revealed that the warmup period was sufficient, allowing the sampler to reach high posterior density regions before collecting samples for inference.

We also performed sensitivity analyses by varying initial values, which confirmed that the model was robust to different starting points, further enhancing our confidence in the convergence and reliability of the estimates.

Overall, the simulation results underscore the effectiveness of the Stan model in overcoming convergence issues through meticulous model specification, strategic use of priors, and a robust sampling framework, ultimately leading to credible and actionable insights from the data.

In this study, we developed a sophisticated Bayesian hierarchical model to investigate the impact of air pollution on mortality rates, addressing the complexities introduced by measurement error in pollution data on the log scale. Our model integrates autoregressive (AR) and moving average (MA) components to capture temporal dependencies in true pollution levels and includes harmonic terms to account for seasonal variations. This approach provided a robust framework for understanding the association between air pollution and health outcomes.

As shown in Table 2, the primary finding of our analysis is a significant positive association between true pollution levels and mortality rates. The posterior mean of the coefficient for the true pollution levels, denoted as $\hat{\gamma}$, was estimated at $11.75$ with a 95% credible interval of $[10.64,12.93]$. This result suggests that increases in pollution levels are associated with higher mortality rates. The credible interval indicates the uncertainty around this estimate, reflecting the model’s robustness in capturing the true effect of pollution.

The model’s performance was notably enhanced by incorporating ARMA terms, which successfully accounted for temporal dependencies in the pollution data. The inclusion of these terms resulted in a significant reduction in the residual sum of squares (RSS), improving the goodness-of-fit as reflected in the increased $R^{2}$ values. This improvement demonstrates the model’s ability to capture the underlying temporal patterns in the pollution data, thus providing a more accurate representation of the pollution-mortality relationship.

Seasonal variations were effectively modeled using harmonic terms. The estimated coefficients for these seasonal components showed significant seasonal patterns, which aligned with known cycles in pollution levels. This aspect of the model helped in accounting for periodic fluctuations in pollution, thus refining the estimates of the true pollution effects on mortality.

The Hamiltonian Monte Carlo (HMC) method, particularly the No-U-Turn Sampler (NUTS), employed for posterior inference showed successful convergence through Rhat, as confirmed by diagnostic checks. This ensured that the parameter estimates derived from the posterior distributions were reliable. The HMC method iteratively updated estimates of the parameters, including the coefficients for ARMA components, seasonal effects, and measurement error variances, leading to robust and precise estimates.

Measurement error was modeled explicitly by assuming $\log(P_{t})=\log(X_{t})+\epsilon_{t}$, where $\epsilon_{t}\sim\mathcal{N}(0,\sigma^{2}_{\epsilon})$ represents the measurement error on the log scale. This approach allowed us to account for the discrepancy between observed pollution levels $P_{t}$ and the true levels $X_{t}$. The measurement error model significantly improved the precision of the estimated pollution effects, as evidenced by a decrease in the mean squared error (MSE) between the true and estimated pollution levels.

Our results highlight the critical role of accounting for measurement error and temporal dependencies in environmental health studies. By incorporating these elements into our model, we were able to obtain more accurate estimates of the health impacts of air pollution. This model provides valuable insights into the relationship between air pollution and mortality, which is crucial for developing effective public health policies and air quality regulations.

In conclusion, this study underscores the importance of using advanced statistical methods to address measurement error and temporal dynamics in environmental health research. The current version of our Bayesian hierarchical model specifically focuses on chronic obstructive pulmonary disease (COPD). Future work will expand this model to include additional elements and explore the effects of air pollutants on pneumonia, neoplasm, and heart failure. The results from these analyses will be included in forthcoming versions of this research, providing a more comprehensive understanding of the health impacts of air pollution.

The methodology outlined in this section addresses the critical issue of measurement error in air pollution studies, particularly when using Poisson log-linear generalized linear models (GLMs) to assess the impact of pollution on mortality. By incorporating a measurement error model on the log scale, we account for the inherent inaccuracies in observed pollution data, leading to more reliable estimates of the true association between pollution and health outcomes. The integration of an autoregressive moving average (ARMA) process with harmonic seasonal components provides a robust framework for modeling the temporal dependencies and seasonal variations in pollution levels, which are often observed in environmental time series data.

One key advantage of this approach is its ability to separate the true pollution signal from the noise introduced by measurement error, which is crucial in epidemiological studies where the effects of pollution are often subtle and can be confounded by inaccuracies in the data. The use of Bayesian inference, particularly the Hamiltonian Monte Carlo (HMC) method with the No-U-Turn Sampler (NUTS), allows for the incorporation of prior knowledge and the quantification of uncertainty in the model parameters, including the pollution effect on mortality. This is particularly important in public health studies, where policy decisions are often based on estimates that must account for both measurement error and uncertainty.

The model’s flexibility is further enhanced by the inclusion of covariates, which allows for the adjustment of potential confounders and the exploration of interactions between pollution levels and other risk factors. The Bayesian framework also facilitates the extension of the model to include other sources of uncertainty, such as spatial variability in pollution levels or the use of alternative distributions for the death counts, which may be overdispersed relative to the Poisson distribution.

Several extensions to the model are possible, offering opportunities for further research and refinement. One potential extension is the incorporation of spatial components to model the variation in pollution levels and mortality across different geographic regions. This would allow for a more detailed analysis of the spatial heterogeneity in pollution effects and could be achieved by extending the current ARMA model to a spatiotemporal framework, potentially using Gaussian processes or spatial autoregressive models.

Furthermore, the Poisson log-linear model could be replaced with a more flexible model that allows for overdispersion, such as a negative binomial model or a quasi-Poisson model. This would be particularly useful in cases where the observed variance in death counts exceeds the mean, a common scenario in epidemiological data. Such an extension would provide more accurate estimates of the effect of pollution on mortality, particularly in the presence of overdispersed count data.

Lastly, incorporating non-linear effects of pollution on mortality could be another valuable extension. This could be achieved by replacing the linear term $\gamma\log(X_{t})$ with a non-linear function (e.g., splines or polynomial terms) that allows for more complex relationships between pollution levels and health outcomes. This extension could capture potential threshold effects or saturation points, where the health impacts of pollution may change at different levels