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Estimating the joint effect of a multivariate, continuous exposure is crucial, particularly in environmental health where interest lies in simultaneously evaluating the impact of multiple environmental pollutants on health. We develop novel methodology that addresses two key issues for estimation of treatment effects of multivariate, continuous exposures. We use nonparametric Bayesian methodology that is flexible to ensure our approach can capture a wide range of data generating processes. Additionally, we allow the effect of the exposures to be heterogeneous with respect to covariates. Treatment effect heterogeneity has not been well explored in the causal inference literature for multivariate, continuous exposures, and therefore we introduce novel estimands that summarize the nature and extent of the heterogeneity, and propose estimation procedures for new estimands related to treatment effect heterogeneity. We provide theoretical support for the proposed models in the form of posterior contraction rates and show that it works well in simulated examples both with and without heterogeneity. We apply our approach to a study of the health effects of simultaneous exposure to the components of PM$_{2.5}$ and find that the negative health effects of exposure to these environmental pollutants is exacerbated by low socioeconomic status and age.

Multi-gene panel testing allows efficient detection of pathogenic variants in cancer susceptibility genes including moderate-risk genes such as ATM and PALB2. A growing number of studies examine the risk of breast cancer (BC) conferred by pathogenic variants of such genes. A meta-analysis combining the reported risk estimates can provide an overall age-specific risk of developing BC, i.e., penetrance for a gene. However, estimates reported by case-control studies often suffer from ascertainment bias. Currently there are no methods available to adjust for such ascertainment bias in this setting. We consider a Bayesian random-effects meta-analysis method that can synthesize different types of risk measures and extend it to incorporate studies with ascertainment bias. This is achieved by introducing a bias term in the model and assigning appropriate priors. We validate the method through a simulation study and apply it to estimate BC penetrance for carriers of pathogenic variants of ATM and PALB2 genes. Our simulations show that the proposed method results in more accurate and precise penetrance estimates compared to when no adjustment is made for ascertainment bias or when such biased studies are discarded from the analysis. The estimated overall BC risk for individuals with pathogenic variants in (1) ATM is 5.77% (3.22%-9.67%) by age 50 and 26.13% (20.31%-32.94%) by age 80; (2) PALB2 is 12.99% (6.48%-22.23%) by age 50 and 44.69% (34.40%-55.80%) by age 80. The proposed method allows for meta-analyses to include studies with ascertainment bias resulting in a larger number of studies included and thereby more robust estimates., Comment: 36 pages. arXiv admin note: text overlap with arXiv:2304.01912

Quantifying the causal effects of continuous exposures on outcomes of interest is critical for social, economic, health, and medical research. However, most existing software packages focus on binary exposures. We develop the CausalGPS R package that implements a collection of algorithms to provide algorithmic solutions for causal inference with continuous exposures. CausalGPS implements a causal inference workflow, with algorithms based on generalized propensity scores (GPS) as the core, extending propensity scores (the probability of a unit being exposed given pre-exposure covariates) from binary to continuous exposures. As the first step, the package implements efficient and flexible estimations of the GPS, allowing multiple user-specified modeling options. As the second step, the package provides two ways to adjust for confounding: weighting and matching, generating weighted and matched data sets, respectively. Lastly, the package provides built-in functions to fit flexible parametric, semi-parametric, or non-parametric regression models on the weighted or matched data to estimate the exposure-response function relating the outcome with the exposures. The computationally intensive tasks are implemented in C++, and efficient shared-memory parallelization is achieved by OpenMP API. This paper outlines the main components of the CausalGPS R package and demonstrates its application to assess the effect of long-term exposure to PM2.5 on educational attainment using zip code-level data from the contiguous United States from 2000-2016., Comment: 22 pages, 8 figures

We develop new methodology to improve our understanding of the causal effects of multivariate air pollution exposures on public health. Typically, exposure to air pollution for an individual is measured at their home geographic region, though people travel to different regions with potentially different levels of air pollution. To account for this, we incorporate estimates of the mobility of individuals from cell phone mobility data to get an improved estimate of their exposure to air pollution. We treat this as an interference problem, where individuals in one geographic region can be affected by exposures in other regions due to mobility into those areas. We propose policy-relevant estimands and derive expressions showing the extent of bias one would obtain by ignoring this mobility. We additionally highlight the benefits of the proposed interference framework relative to a measurement error framework for accounting for mobility. We develop novel estimation strategies to estimate causal effects that account for this spatial spillover utilizing flexible Bayesian methodology. Lastly, we use the proposed methodology to study the health effects of ambient air pollution on mortality among Medicare enrollees in the United States.

Multi-gene panel testing allows many cancer susceptibility genes to be tested quickly at a lower cost making such testing accessible to a broader population. Thus, more patients carrying pathogenic germline mutations in various cancer-susceptibility genes are being identified. This creates a great opportunity, as well as an urgent need, to counsel these patients about appropriate risk reducing management strategies. Counseling hinges on accurate estimates of age-specific risks of developing various cancers associated with mutations in a specific gene, i.e., penetrance estimation. We propose a meta-analysis approach based on a Bayesian hierarchical random-effects model to obtain penetrance estimates by integrating studies reporting different types of risk measures (e.g., penetrance, relative risk, odds ratio) while accounting for the associated uncertainties. After estimating posterior distributions of the parameters via a Markov chain Monte Carlo algorithm, we estimate penetrance and credible intervals. We investigate the proposed method and compare with an existing approach via simulations based on studies reporting risks for two moderate-risk breast cancer susceptibility genes, ATM and PALB2. Our proposed method is far superior in terms of coverage probability of credible intervals and mean square error of estimates. Finally, we apply our method to estimate the penetrance of breast cancer among carriers of pathogenic mutations in the ATM gene., Comment: 37 pages, correction of typo

Investigating the health impacts of wildfire smoke requires data on people's exposure to fine particulate matter (PM$_{2.5}$) across space and time. In recent years, it has become common to use machine learning models to fill gaps in monitoring data. However, it remains unclear how well these models are able to capture spikes in PM$_{2.5}$ during and across wildfire events. Here, we evaluate the accuracy of two sets of high-coverage and high-resolution machine learning-derived PM$_{2.5}$ data sets created by Di et al. (2021) and Reid et al. (2021). In general, the Reid estimates are more accurate than the Di estimates when compared to independent validation data from mobile smoke monitors deployed by the US Forest Service. However, both models tend to severely under-predict PM$_{2.5}$ on high-pollution days. Our findings complement other recent studies calling for increased air pollution monitoring in the western US and support the inclusion of wildfire-specific monitoring observations and predictor variables in model-based estimates of PM$_{2.5}$. Lastly, we call for more rigorous error quantification of machine-learning derived exposure data sets, with special attention to extreme events., Comment: Main text = 26 pages, including 3 figures and 2 tables. References + Supporting Information = 22 pages, including equations, supplemental notes, 2 figures, and 8 tables

Environmental Protection Agency (EPA) air quality (AQ) monitors, the gold standard for measuring air pollutants, are sparsely positioned across the US due to their costliness. Low-cost sensors (LCS) are increasingly being used by the public to fill in the gaps in AQ monitoring; however, LCS are not as accurate as EPA monitors. In this work, we investigate factors impacting the differences between an individual's true (unobserved) exposure to fine particulate matter (PM2.5) and the exposure reported by their nearest AQ instrument, which could be either an EPA monitor or an LCS. Three factors contributing to these differences are (1) distance to the nearest AQ instrument, (2) local variability in AQ, and (3) device measurement error. We examine the contributions of each component to the overall error in reported AQ using simulations based on California data. The simulations explore different combinations of hypothetical LCS placement strategies (at schools, near major roads, and in environmentally and socioeconomically marginalized census tracts) for different numbers of LCS, with varying plausible amounts of LCS device measurement error. For each scenario, we evaluate the accuracy of daily AQ information available from individuals' nearest AQ instrument with respect to absolute errors and misclassifications of the Air Quality Index, stratified by socioeconomic and demographic characteristics. We illustrate how real-time AQ reporting could be improved (or, in some cases, worsened) by using LCS, both for the population overall and for marginalized communities specifically. This work has implications for the integration of LCS into real-time AQ reporting platforms., Comment: Main text = 17 pages, including 3 figures and 1 table. References + Supplemental Information = 33 pages, including supplemental notes, 12 figures and 9 tables

Numerous studies have examined the associations between long-term exposure to fine particulate matter (PM2.5) and adverse health outcomes. Recently, many of these studies have begun to employ high-resolution predicted PM2.5 concentrations, which are subject to measurement error. Previous approaches for exposure measurement error correction have either been applied in non-causal settings or have only considered a categorical exposure. Moreover, most procedures have failed to account for uncertainty induced by error correction when fitting an exposure-response function (ERF). To remedy these deficiencies, we develop a multiple imputation framework that combines regression calibration and Bayesian techniques to estimate a causal ERF. We demonstrate how the output of the measurement error correction steps can be seamlessly integrated into a Bayesian additive regression trees (BART) estimator of the causal ERF. We also demonstrate how locally-weighted smoothing of the posterior samples from BART can be used to create a more accurate ERF estimate. Our proposed approach also properly propagates the exposure measurement error uncertainty to yield accurate standard error estimates. We assess the robustness of our proposed approach in an extensive simulation study. We then apply our methodology to estimate the effects of PM2.5 on all-cause mortality among Medicare enrollees in New England from 2000-2012.

Risk evaluation to identify individuals who are at greater risk of cancer as a result of heritable pathogenic variants is a valuable component of individualized clinical management. Using principles of Mendelian genetics, Bayesian probability theory, and variant-specific knowledge, Mendelian models derive the probability of carrying a pathogenic variant and developing cancer in the future, based on family history. Existing Mendelian models are widely employed, but are generally limited to specific genes and syndromes. However, the upsurge of multi-gene panel germline testing has spurred the discovery of many new gene-cancer associations that are not presently accounted for in these models. We have developed PanelPRO, a flexible, efficient Mendelian risk prediction framework that can incorporate an arbitrary number of genes and cancers, overcoming the computational challenges that arise because of the increased model complexity. We implement an eleven-gene, eleven-cancer model, the largest Mendelian model created thus far, based on this framework. Using simulations and a clinical cohort with germline panel testing data, we evaluate model performance, validate the reverse-compatibility of our approach with existing Mendelian models, and illustrate its usage. Our implementation is freely available for research use in the PanelPRO R package., Comment: 18 pages, 7 figures

Pedigree data contain family history information that is used to analyze hereditary diseases. These clinical data sets may contain duplicate records due to the same family visiting a clinic multiple times or a clinician entering multiple versions of the family for testing purposes. Inferences drawn from the data or using them for training or validation without removing the duplicates could lead to invalid conclusions, and hence identifying the duplicates is essential. Since family structures can be complex, existing deduplication algorithms cannot be applied directly. We first motivate the importance of deduplication by examining the impact of pedigree duplicates on the training and validation of a familial risk prediction model. We then introduce an unsupervised algorithm, which we call SNIP (Sorted NeIghborhood for Pedigrees), that builds on the sorted neighborhood method to efficiently find and classify pairwise comparisons by leveraging the inherent hierarchical nature of the pedigrees. We conduct a simulation study to assess the performance of the algorithm and find parameter configurations where the algorithm is able to accurately detect the duplicates. We then apply the method to data from the Risk Service, which includes over 300,000 pedigrees at high risk of hereditary cancers, and uncover large clusters of potential duplicate families. After removing 104,520 pedigrees (33% of original data), the resulting Risk Service dataset can now be used for future analysis, training, and validation. The algorithm is available as an R package snipR available at https://github.com/bayesmendel/snipR., Comment: 39 pages, 22 figures (including supplementary materials)

Family history is a major risk factor for many types of cancer. Mendelian risk prediction models translate family histories into cancer risk predictions based on knowledge of cancer susceptibility genes. These models are widely used in clinical practice to help identify high-risk individuals. Mendelian models leverage the entire family history, but they rely on many assumptions about cancer susceptibility genes that are either unrealistic or challenging to validate due to low mutation prevalence. Training more flexible models, such as neural networks, on large databases of pedigrees can potentially lead to accuracy gains. In this paper, we develop a framework to apply neural networks to family history data and investigate their ability to learn inherited susceptibility to cancer. While there is an extensive literature on neural networks and their state-of-the-art performance in many tasks, there is little work applying them to family history data. We propose adaptations of fully-connected neural networks and convolutional neural networks to pedigrees. In data simulated under Mendelian inheritance, we demonstrate that our proposed neural network models are able to achieve nearly optimal prediction performance. Moreover, when the observed family history includes misreported cancer diagnoses, neural networks are able to outperform the Mendelian BRCAPRO model embedding the correct inheritance laws. Using a large dataset of over 200,000 family histories, the Risk Service cohort, we train prediction models for future risk of breast cancer. We validate the models using data from the Cancer Genetics Network.

Improving existing widely-adopted prediction models is often a more efficient and robust way towards progress than training new models from scratch. Existing models may (a) incorporate complex mechanistic knowledge, (b) leverage proprietary information and, (c) have surmounted barriers to adoption. Compared to model training, model improvement and modification receive little attention. In this paper we propose a general approach to model improvement: we combine gradient boosting with any previously developed model to improve model performance while retaining important existing characteristics. To exemplify, we consider the context of Mendelian models, which estimate the probability of carrying genetic mutations that confer susceptibility to disease by using family pedigrees and health histories of family members. Via simulations we show that integration of gradient boosting with an existing Mendelian model can produce an improved model that outperforms both that model and the model built using gradient boosting alone. We illustrate the approach on genetic testing data from the USC-Stanford Cancer Genetics Hereditary Cancer Panel (HCP) study., Comment: 46 pages, 4 figures

Motivated by environmental policy questions, we address the challenges of estimation, change point detection, and uncertainty quantification of a causal exposure-response function (CERF). Under a potential outcome framework, the CERF describes the relationship between a continuously varying exposure (or treatment) and its causal effect on an outcome. We propose a new Bayesian approach that relies on a Gaussian process (GP) model to estimate the CERF nonparametrically. To achieve the desired separation of design and analysis phases, we parametrize the covariance (kernel) function of the GP to mimic matching via a Generalized Propensity Score (GPS). The hyper-parameters as well as the form of the kernel function of the GP are chosen to optimize covariate balance. Our approach achieves automatic uncertainty evaluation of the CERF with high computational efficiency, and enables change point detection through inference on derivatives of the CERF. We provide theoretical results showing the correspondence between our Bayesian GP framework and traditional approaches in causal inference for estimating causal effects of a continuous exposure. We apply the methods to 520,711 ZIP-code-level observations to estimate the causal effect of long-term exposures to PM2.5, ozone, and NO2 on all-cause mortality among Medicare enrollees in the US. A computationally efficient implementation of the proposed GP models is provided in the GPCERF R package, which is available on CRAN.

The methodological development of this paper is motivated by the need to address the following scientific question: does the issuance of heat alerts prevent adverse health effects? Our goal is to address this question within a causal inference framework in the context of time series data. A key challenge is that causal inference methods require the overlap assumption to hold: each unit (i.e., a day) must have a positive probability of receiving the treatment (i.e., issuing a heat alert on that day). In our motivating example, the overlap assumption is often violated: the probability of issuing a heat alert on a cooler day is zero. To overcome this challenge, we propose a stochastic intervention for time series data which is implemented via an incremental time-varying propensity score (ItvPS). The ItvPS intervention is executed by multiplying the probability of issuing a heat alert on day $t$ -- conditional on past information up to day $t$ -- by an odds ratio $\delta_t$. First, we introduce a new class of causal estimands that relies on the ItvPS intervention. We provide theoretical results to show that these causal estimands can be identified and estimated under a weaker version of the overlap assumption. Second, we propose nonparametric estimators based on the ItvPS and derive an upper bound for the variances of these estimators. Third, we extend this framework to multi-site time series using a spatial meta-analysis approach. Fourth, we show that the proposed estimators perform well in terms of bias and root mean squared error via simulations. Finally, we apply our proposed approach to estimate the causal effects of increasing the probability of issuing heat alerts on each warm-season day in reducing deaths and hospitalizations among Medicare enrollees in $2,837$ U.S. counties., Comment: 31 pages, 3 figures, 2 tables

Identifying individuals who are at high risk of cancer due to inherited germline mutations is critical for effective implementation of personalized prevention strategies. Most existing models to identify these individuals focus on specific syndromes by including family and personal history for a small number of cancers. Recent evidence from multi-gene panel testing has shown that many syndromes once thought to be distinct are overlapping, motivating the development of models that incorporate family history information on several cancers and predict mutations for more comprehensive panels of genes. Once such class of models are Mendelian risk prediction models, which use family history information and Mendelian laws of inheritance to estimate the probability of carrying genetic mutations, as well as future risk of developing associated cancers. To flexibly model the complexity of many cancer-mutation associations, we present a new software tool called PanelPRO, a R package that extends the previously developed BayesMendel R package to user-selected lists of susceptibility genes and associated cancers. The model identifies individuals at an increased risk of carrying cancer susceptibility gene mutations and predicts future risk of developing hereditary cancers associated with those genes. Additional functionalities adjust for prophylactic interventions, known genetic testing results, and risk modifiers such as race and ancestry. The package comes with a customizable database with default parameter values estimated from published studies. The PanelPRO package is open-source and provides a fast and flexible back-end for multi-gene, multi-cancer risk modeling with pedigree data. The software enables the identification of high-risk individuals, which will have an impact on personalized prevention strategies for cancer and individualized decision making about genetic testing.

Accurate risk stratification is key to reducing cancer morbidity through targeted screening and preventative interventions. Numerous breast cancer risk prediction models have been developed, but they often give predictions with conflicting clinical implications. Integrating information from different models may improve the accuracy of risk predictions, which would be valuable for both clinicians and patients. BRCAPRO and BCRAT are two widely used models based on largely complementary sets of risk factors. BRCAPRO is a Bayesian model that uses detailed family history information to estimate the probability of carrying a BRCA1/2 mutation, as well as future risk of breast and ovarian cancer, based on mutation prevalence and penetrance (age-specific probability of developing cancer given genotype). BCRAT uses a relative hazard model based on first-degree family history and non-genetic risk factors. We consider two approaches for combining BRCAPRO and BCRAT: 1) modifying the penetrance functions in BRCAPRO using relative hazard estimates from BCRAT, and 2) training an ensemble model that takes as input BRCAPRO and BCRAT predictions. We show that the combination models achieve performance gains over BRCAPRO and BCRAT in simulations and data from the Cancer Genetics Network.

PURPOSE: The popularity of germline genetic panel testing has led to a vast accumulation of variant-level data. Variant names are not always consistent across laboratories and not easily mappable to public variant databases such as ClinVar. A tool that can automate the process of variants harmonization and mapping is needed to help clinicians ensure their variant interpretations are accurate. METHODS: We present a Python-based tool, Ask2Me VarHarmonizer, that incorporates data cleaning, name harmonization, and a four-attempt mapping to ClinVar procedure. We applied this tool to map variants from a pilot dataset collected from 11 clinical practices. Mapping results were evaluated with and without the transcript information. RESULTS: Using Ask2Me VarHarmonizer, 4728 out of 6027 variant entries (78%) were successfully mapped to ClinVar, corresponding to 3699 mappable unique variants. With the addition of 1099 unique unmappable variants, a total of 4798 unique variants were eventually identified. 427 (9%) of these had multiple names, of which 343 (7%) had multiple names within-practice. 99% mapping consistency was observed with and without transcript information. CONCLUSION: Ask2Me VarHarmonizer aggregates and structures variant data, harmonizes names, and maps variants to ClinVar. Performing harmonization removes the ambiguity and redundancy of variants from different sources.

PURPOSE: The medical literature relevant to germline genetics is growing exponentially. Clinicians need tools monitoring and prioritizing the literature to understand the clinical implications of the pathogenic genetic variants. We developed and evaluated two machine learning models to classify abstracts as relevant to the penetrance (risk of cancer for germline mutation carriers) or prevalence of germline genetic mutations. METHODS: We conducted literature searches in PubMed and retrieved paper titles and abstracts to create an annotated dataset for training and evaluating the two machine learning classification models. Our first model is a support vector machine (SVM) which learns a linear decision rule based on the bag-of-ngrams representation of each title and abstract. Our second model is a convolutional neural network (CNN) which learns a complex nonlinear decision rule based on the raw title and abstract. We evaluated the performance of the two models on the classification of papers as relevant to penetrance or prevalence. RESULTS: For penetrance classification, we annotated 3740 paper titles and abstracts and used 60% for training the model, 20% for tuning the model, and 20% for evaluating the model. The SVM model achieves 89.53% accuracy (percentage of papers that were correctly classified) while the CNN model achieves 88.95 % accuracy. For prevalence classification, we annotated 3753 paper titles and abstracts. The SVM model achieves 89.14% accuracy while the CNN model achieves 89.13 % accuracy. CONCLUSION: Our models achieve high accuracy in classifying abstracts as relevant to penetrance or prevalence. By facilitating literature review, this tool could help clinicians and researchers keep abreast of the burgeoning knowledge of gene-cancer associations and keep the knowledge bases for clinical decision support tools up to date.

In the context of a binary treatment, matching is a well-established approach in causal inference. However, in the context of a continuous treatment or exposure, matching is still underdeveloped. We propose an innovative matching approach to estimate an average causal exposure-response function under the setting of continuous exposures that relies on the generalized propensity score (GPS). Our approach maintains the following attractive features of matching: a) clear separation between the design and the analysis; b) robustness to model misspecification or to the presence of extreme values of the estimated GPS; c) straightforward assessment of covariate balance. We first introduce an assumption of identifiability, called local weak unconfoundedness. Under this assumption and mild smoothness conditions, we provide theoretical guarantees that our proposed matching estimator attains point-wise consistency and asymptotic normality. In simulations, our proposed matching approach outperforms existing methods under settings of model misspecification or the presence of extreme values of the estimated GPS. We apply our proposed method to estimate the average causal exposure-response function between long-term PM$_{2.5}$ exposure and all-cause mortality among 68.5 million Medicare enrollees, 2000-2016. We found strong evidence of a harmful effect of long-term PM$_{2.5}$ exposure on mortality. Code for the proposed matching approach is provided in the CausalGPS R package, which is available on CRAN and provides a computationally efficient implementation., Comment: Code for the GPS matching approach is provided in the CausalGPS R package, which is available on CRAN. The Github website (https://github.com/wxwx1993/GPSmatching) also contains tutorials and examples to use the R package

Fine particulate matter (PM$_{2.5}$) is one of the criteria air pollutants regulated by the Environmental Protection Agency in the United States. There is strong evidence that ambient exposure to (PM$_{2.5}$) increases risk of mortality and hospitalization. Large scale epidemiological studies on the health effects of PM$_{2.5}$ provide the necessary evidence base for lowering the safety standards and inform regulatory policy. However, ambient monitors of PM$_{2.5}$ (as well as monitors for other pollutants) are sparsely located across the U.S., and therefore studies based only on the levels of PM$_{2.5}$ measured from the monitors would inevitably exclude large amounts of the population. One approach to resolving this issue has been developing models to predict local PM$_{2.5}$, NO$_2$, and ozone based on satellite, meteorological, and land use data. This process typically relies developing a prediction model that relies on large amounts of input data and is highly computationally intensive to predict levels of air pollution in unmonitored areas. We have developed a flexible R package that allows for environmental health researchers to design and train spatio-temporal models capable of predicting multiple pollutants, including PM$_{2.5}$. We utilize H2O, an open source big data platform, to achieve both performance and scalability when used in conjunction with cloud or cluster computing systems.

We propose a new approach for estimating causal effects when the exposure is measured with error and confounding adjustment is performed via a generalized propensity score (GPS). Using validation data, we propose a regression calibration (RC)-based adjustment for a continuous error-prone exposure combined with GPS to adjust for confounding (RC-GPS). The outcome analysis is conducted after transforming the corrected continuous exposure into a categorical exposure. We consider confounding adjustment in the context of GPS subclassification, inverse probability treatment weighting (IPTW) and matching. In simulations with varying degrees of exposure error and confounding bias, RC-GPS eliminates bias from exposure error and confounding compared to standard approaches that rely on the error-prone exposure. We applied RC-GPS to a rich data platform to estimate the causal effect of long-term exposure to fine particles ($PM_{2.5}$) on mortality in New England for the period from 2000 to 2012. The main study consists of $2,202$ zip codes covered by $217,660$ 1km $\times$ 1km grid cells with yearly mortality rates, yearly $PM_{2.5}$ averages estimated from a spatio-temporal model (error-prone exposure) and several potential confounders. The internal validation study includes a subset of 83 1km $\times$ 1km grid cells within 75 zip codes from the main study with error-free yearly $PM_{2.5}$ exposures obtained from monitor stations. Under assumptions of non-interference and weak unconfoundedness, using matching we found that exposure to moderate levels of $PM_{2.5}$ ($8 <$ $PM_{2.5}$ $\leq 10\ {\rm \mu g/m^3}$) causes a $2.8\%$ ($95\%$ CI: $0.6\%, 3.6\%$) increase in all-cause mortality compared to low exposure ($PM_{2.5}$ $\leq 8\ {\rm \mu g/m^3}$).