Bayesian Distributed Lag Models: Estimating Effects of Particulate Matter Air Pollution on Daily Mortality

Spatial scale effects on associations between built environment and cognitive function: Multi-Ethnic Study of Atherosclerosis

Built environments have the potential to favorably support cognitive function. Despite growing work on this topic, most of the work has ignored variation in the spatial scale of the effect. The issue with spatial scale effects is that the size and shape of the areal unit within which built environment characteristics are measured naturally influence the built environment exposure metric and thus the estimated associations with health. We used spatial distributed lag modeling (DLM) to estimate how associations between built environment exposures (walkable destinations [WD], social destinations [SD]) and change in cognition varied across distance of these destinations from participants' residences. Cognition was assessed as maintained/improved processing speed (PS) and global cognition (GC). Person-level data from Exam 5 (2010-2012) and Exam 6 (2016-2018) of the Multi-Ethnic Study of Atherosclerosis was used (N = 1380, mean age 67). Built environment data were derived from the National Establishment Time Series. Higher availability of walkable and social destinations at closer distance from participants' residence was associated with maintained/improved PS. The adjusted associations between maintained/improved PS and destinations waned with increasing distance from the residence; associations were evident until approximately 1.9-km for WD and 1.5-km for SD. Associations were most apparent for participants living in areas with high population density. We found little evidence for associations between change in GC and built environment at any distance. These results highlight the importance of identifying appropriate spatial scale to understand the mechanisms for built environment-cognition associations.

Semiparametric distributed lag quantile regression for modeling time-dependent exposure mixtures

Studying time-dependent exposure mixtures has gained increasing attentions in environmental health research. When a scalar outcome is of interest, distributed lag (DL) models have been employed to characterize the exposures effects distributed over time on the mean of final outcome. However, there is a methodological gap on investigating time-dependent exposure mixtures with different quantiles of outcome. In this paper, we introduce semiparametric partial-linear single-index (PLSI) DL quantile regression, which can describe the DL effects of time-dependent exposure mixtures on different quantiles of outcome and identify susceptible periods of exposures. We consider two time-dependent exposure settings: discrete and functional, when exposures are measured in a small number of time points and at dense time grids, respectively. Spline techniques are used to approximate the nonparametric DL function and single-index link function, and a profile estimation algorithm is proposed. Through extensive simulations, we demonstrate the performance and value of our proposed models and inference procedures. We further apply the proposed methods to study the effects of maternal exposures to ambient air pollutants of fine particulate and nitrogen dioxide on birth weight in New York University Children's Health and Environment Study (NYU CHES).

Analysis of N-of-1 trials using Bayesian distributed lag model with autocorrelated errors

An N-of-1 trial is a multi-period crossover trial performed in a single individual, with a primary goal to estimate treatment effect on the individual instead of population-level mean responses. As in a conventional crossover trial, it is critical to understand carryover effects of the treatment in an N-of-1 trial, especially when no washout periods between treatment periods are instituted to reduce trial duration. To deal with this issue in situations where a high volume of measurements are made during the study, we introduce a novel Bayesian distributed lag model that facilitates the estimation of carryover effects, while accounting for temporal correlations using an autoregressive model. Specifically, we propose a prior variance-covariance structure on the lag coefficients to address collinearity caused by the fact that treatment exposures are typically identical on successive days. A connection between the proposed Bayesian model and penalized regression is noted. Simulation results demonstrate that the proposed model substantially reduces the root mean squared error in the estimation of carryover effects and immediate effects when compared to other existing methods, while being comparable in the estimation of the total effects. We also apply the proposed method to assess the extent of carryover effects of light therapies in relieving depressive symptoms in cancer survivors.

Assessing the health estimation capacity of air pollution exposure prediction models

BACKGROUND

The era of big data has enabled sophisticated models to predict air pollution concentrations over space and time. Historically these models have been evaluated using overall metrics that measure how close predictions are to monitoring data. However, overall methods are not designed to distinguish error at timescales most relevant for epidemiologic studies, such as day-to-day errors that impact studies of short-term health associations.

METHODS

We introduce frequency band model performance, which quantifies health estimation capacity of air quality prediction models for time series studies of air pollution and health. Frequency band model performance uses a discrete Fourier transform to evaluate prediction models at timescales of interest. We simulated fine particulate matter (PM_2.5), with errors at timescales varying from acute to seasonal, and health time series data. To compare evaluation approaches, we use correlations and root mean squared error (RMSE). Additionally, we assess health estimation capacity through bias and RMSE in estimated health associations. We apply frequency band model performance to PM_2.5 predictions at 17 monitors in 8 US cities.

RESULTS

In simulations, frequency band model performance rates predictions better (lower RMSE, higher correlation) when there is no error at a particular timescale (e.g., acute) and worse when error is added to that timescale, compared to overall approaches. Further, frequency band model performance is more strongly associated (R² = 0.95) with health association bias compared to overall approaches (R² = 0.57). For PM_2.5 predictions in Salt Lake City, UT, frequency band model performance better identifies acute error that may impact estimated short-term health associations.

CONCLUSIONS

For epidemiologic studies, frequency band model performance provides an improvement over existing approaches because it evaluates models at the timescale of interest and is more strongly associated with bias in estimated health associations. Evaluating prediction models at timescales relevant for health studies is critical to determining whether model error will impact estimated health associations.

Exploring the spatial scale effects of built environments on transport walking: Multi-Ethnic Study of Atherosclerosis

We employed a longitudinal distributed lag modeling approach to systematically estimate how associations between built environment features and transport walking decayed with the increase of distance from home to built environment destinations. Data came from a cohort recruited from six U.S. cities (follow-up 2000-2010, N = 3913, baseline mean age 60). Built environment features included all walkable destinations, consisting of common and popular destinations for daily life. We also included two subsets frequent social destinations and food stores to examine if the spatial scale effects differed by varying density for different types of built environment destinations. Adjusted results found that increases in transport walking diminished when built environment destinations were farther, although distance thresholds varied across different types of built environment destinations. Higher availability of walking destinations within 2-km and frequent social destinations within 1.6-km were associated with transport walking. Food stores were not associated with transport walking. This new information will help policymakers and urban designers understand at what distances each type of built environment destinations influences transport walking, in turn informing the development of interventions and/or the placement of amenities within neighborhoods to promote transport walking. The findings that spatial scales depend on specific built environment features also highlight the need for methods that can more flexibly estimate associations between outcomes and different built environment features across varying contexts, in order to improve our understanding of the spatial mechanisms involved in said associations.

Excess of COVID-19 cases and deaths due to fine particulate matter exposure during the 2020 wildfires in the United States

The year 2020 brought unimaginable challenges in public health, with the confluence of the COVID-19 pandemic and wildfires across the western United States. Wildfires produce high levels of fine particulate matter (PM_2.5). Recent studies reported that short-term exposure to PM_2.5 is associated with increased risk of COVID-19 cases and deaths. We acquired and linked publicly available daily data on PM_2.5, the number of COVID-19 cases and deaths, and other confounders for 92 western U.S. counties that were affected by the 2020 wildfires. We estimated the association between short-term exposure to PM_2.5 during the wildfires and the epidemiological dynamics of COVID-19 cases and deaths. We adjusted for several time-varying confounding factors (e.g., weather, seasonality, long-term trends, mobility, and population size). We found strong evidence that wildfires amplified the effect of short-term exposure to PM_2.5 on COVID-19 cases and deaths, although with substantial heterogeneity across counties.

Effect of O3, PM10 and PM2.5 on cardiovascular and respiratory diseases in cities of France, Iran and Italy

At present, both tropospheric ozone (O₃) and particulate matters (PM) are among the most threatening air pollutants for human health in cities. The air pollution effects over public health include increased risk of hospital admissions and mortality for respiratory and cardiovascular diseases even when air pollutant concentrations are below European and international standards. The aim of this study was to (i) estimate the burden of mortality and morbidity for cardiovascular and respiratory diseases attributed to PM_2.5, PM₁₀ and O₃ in nine selected cities in France, Iran and Italy in 2015 and 2016 and to (ii) compare estimated burdens at current O₃ and PM levels with pre-industrial levels. The selected Mediterranean cities are among the most affected by the air pollution in Europe, in particular by rising O₃ while the selected Iranian cities rank as the most polluted by PM in the world. The software AirQ+ was used to estimate the short-term health effects, in terms of mortality and morbidity by using in situ air quality data, city-specific relative risk values and baseline incidence. Compared to pre-industrial levels, long-term exposures to ambient PM_2.5, PM₁₀ and O₃ have substantially contributed to mortality and hospital admissions in selected cities: about 8200 deaths for non-accidental causes, 2400 deaths for cardiovascular diseases, 540 deaths for respiratory diseases, 220 deaths for chronic obstructive pulmonary diseases as well as 18,800 hospital admissions for cardiovascular diseases and 3400 for respiratory diseases were reported in 2015. The study supports the need of city-specific epidemiological data and urgent strategies to mitigate the health burden of air pollution.

Quantifying personal exposure to air pollution from smartphone-based location data

Personal exposure assessment is a challenging task that requires both measurements of the state of the environment as well as the individual's movements. In this paper, we show how location data collected by smartphone applications can be exploited to quantify the personal exposure of a large group of people to air pollution. A Bayesian approach that blends air quality monitoring data with individual location data is proposed to assess the individual exposure over time, under uncertainty of both the pollutant level and the individual location. A comparison with personal exposure obtained assuming fixed locations for the individuals is also provided. Location data collected by the Earthquake Network research project are employed to quantify the dynamic personal exposure to fine particulate matter of around 2500 people living in Santiago (Chile) over a 4-month period. For around 30% of individuals, the personal exposure based on people movements emerges significantly different over the static exposure. On the basis of this result and thanks to a simulation study, we claim that even when the individual location is known with nonnegligible error, this helps to better assess personal exposure to air pollution. The approach is flexible and can be adopted to quantify the personal exposure based on any location-aware smartphone application.

Robust distributed lag models using data adaptive shrinkage

Distributed lag models (DLMs) have been widely used in environmental epidemiology to quantify the lagged effects of air pollution on an outcome of interest such as mortality or cardiovascular events. Generally speaking, DLMs can be applied to time-series data where the current measure of an independent variable and its lagged measures collectively affect the current measure of a dependent variable. The corresponding distributed lag (DL) function represents the relationship between the lags and the coefficients of the lagged exposure variables. Common choices include polynomials and splines. On one hand, such a constrained DLM specifies the coefficients as a function of lags and reduces the number of parameters to be estimated; hence, higher efficiency can be achieved. On the other hand, under violation of the assumption about the DL function, effect estimates can be severely biased. In this article, we propose a general framework for shrinking coefficient estimates from an unconstrained DLM, that are unbiased but potentially inefficient, toward the coefficient estimates from a constrained DLM to achieve a bias-variance trade-off. The amount of shrinkage can be determined in various ways, and we explore several such methods: empirical Bayes-type shrinkage, a hierarchical Bayes approach, and generalized ridge regression. We also consider a two-stage shrinkage approach that enforces the effect estimates to approach zero as lags increase. We contrast the various methods via an extensive simulation study and show that the shrinkage methods have better average performance across different scenarios in terms of mean squared error (MSE).We illustrate the methods by using data from the National Morbidity, Mortality, and Air Pollution Study (NMMAPS) to explore the association between PM$_{10}$, O$_3$, and SO$_2$ on three types of disease event counts in Chicago, IL, from 1987 to 2000.

Statistical Approaches for Investigating Periods of Susceptibility in Children's Environmental Health Research

PURPOSE OF REVIEW

Children's environmental health researchers are increasingly interested in identifying time intervals during which individuals are most susceptible to adverse impacts of environmental exposures. We review recent advances in methods for assessing susceptible periods.

RECENT FINDINGS

We identified three general classes of modeling approaches aimed at identifying susceptible periods in children's environmental health research: multiple informant models, distributed lag models, and Bayesian approaches. Benefits over traditional regression modeling include the ability to formally test period effect differences, to incorporate highly time-resolved exposure data, or to address correlation among exposure periods or exposure mixtures. Several statistical approaches exist for investigating periods of susceptibility. Assessment of susceptible periods would be advanced by additional basic biological research, further development of statistical methods to assess susceptibility to complex exposure mixtures, validation studies evaluating model assumptions, replication studies in different populations, and consideration of susceptible periods from before conception to disease onset.

Critical window variable selection: estimating the impact of air pollution on very preterm birth

Understanding the impact that environmental exposure during different stages of pregnancy has on the risk of adverse birth outcomes is vital for protection of the fetus and for the development of mechanistic explanations of exposure-disease relationships. As a result, statistical models to estimate critical windows of susceptibility have been developed for several different reproductive outcomes and pollutants. However, these current methods fail to adequately address the primary objective of this line of research; how to statistically identify a critical window of susceptibility. In this article, we introduce critical window variable selection (CWVS), a hierarchical Bayesian framework that directly addresses this question while simultaneously providing improved estimation of the risk parameters. Through simulation, we show that CWVS outperforms existing competing techniques in the setting of highly temporally correlated exposures in terms of (i) correctly identifying critical windows and (ii) accurately estimating risk parameters. We apply all competing methods to a case/control analysis of pregnant women in North Carolina, 2005-2008, with respect to the development of very preterm birth and exposure to ambient ozone and particulate matter $<$ 2.5 $\mu$m in aerodynamic diameter, and identify/estimate the critical windows of susceptibility. The newly developed method is implemented in the R package CWVS.

Distributed Lag Interaction Models with Two Pollutants

Distributed lag models (DLMs) have been widely used in environmental epidemiology to quantify the lagged effects of air pollution on a health outcome of interest such as mortality and morbidity. Most previous DLM approaches only consider one pollutant at a time. In this article, we propose distributed lag interaction model (DLIM) to characterize the joint lagged effect of two pollutants. One natural way to model the interaction surface is by assuming that the underlying basis functions are tensor products of the basis functions that generate the main-effect distributed lag functions. We extend Tukey's one-degree-of-freedom interaction structure to the two-dimensional DLM context. We also consider shrinkage versions of the two to allow departure from the specified Tukey's interaction structure and achieve bias-variance tradeoff. We derive the marginal lag effects of one pollutant when the other pollutant is fixed at certain quantiles. In a simulation study, we show that the shrinkage methods have better average performance in terms of mean squared error (MSE) across different scenarios. We illustrate the proposed methods by using the National Morbidity, Mortality, and Air Pollution Study (NMMAPS) data to model the joint effects of PM₁₀ and O₃ on mortality count in Chicago, Illinois, from 1987 to 2000.

Unexpected potential protective associations between outdoor air pollution and cataracts

Air pollution is one of the biggest public health issues, and the eye is continuously exposed to multiple outdoor air pollution. However, to date, no large-scale study has assessed the relationship between air pollutants and cataracts. We investigated associations between outdoor air pollution and cataracts in the Korean population. A population-based cross-sectional study was performed using data from the Korea National Health and Nutrition Examination Survey, including 18,622 adults more than 40 years of age. The presence of cataracts and their subtypes were evaluated by ophthalmologists. Air pollution data (levels of particulate matter, ozone, nitrogen dioxide, and sulfur dioxide) for the 2 years prior to the ocular examinations were collected from national monitoring stations. The associations of multiple air pollutants with cataracts were assessed by multivariate logistic regression analyses. Sociodemographic factors and previously known risk factors for cataracts were controlled as covariates (model 1 included sociodemographic factors, sun exposure, and behavioral factors, while model 2 further included clinical factors). Higher ozone concentrations were protectively associated with overall cataract which included all subtypes [single pollutant model: 0.003 ppm increase-model 1 (OR 0.89, p = 0.014), model 2 (OR 0.87, p = 0.011); multi-pollutant model: 0.003 ppm increase-model 1 (OR 0.80, p = 0.002), model 2 (OR 0.87, p = 0.002)]. Especially, higher ozone concentrations showed deeply protective association with nuclear cataract subtype [0.003 ppm increase-single pollutant model: model 2 (OR 0.84, p = 0.006), multi-pollutant model: model 2 (OR 0.73, p < 0.001)]. Higher tropospheric ozone concentrations showed protective associations with overall cataract and nuclear cataract subtype in the Korean population.

Potential for Bias When Estimating Critical Windows for Air Pollution in Children's Health

Evidence supports an association between maternal exposure to air pollution during pregnancy and children's health outcomes. Recent interest has focused on identifying critical windows of vulnerability. An analysis based on a distributed lag model (DLM) can yield estimates of a critical window that are different from those from an analysis that regresses the outcome on each of the 3 trimester-average exposures (TAEs). Using a simulation study, we assessed bias in estimates of critical windows obtained using 3 regression approaches: 1) 3 separate models to estimate the association with each of the 3 TAEs; 2) a single model to jointly estimate the association between the outcome and all 3 TAEs; and 3) a DLM. We used weekly fine-particulate-matter exposure data for 238 births in a birth cohort in and around Boston, Massachusetts, and a simulated outcome and time-varying exposure effect. Estimates using separate models for each TAE were biased and identified incorrect windows. This bias arose from seasonal trends in particulate matter that induced correlation between TAEs. Including all TAEs in a single model reduced bias. DLM produced unbiased estimates and added flexibility to identify windows. Analysis of body mass index z score and fat mass in the same cohort highlighted inconsistent estimates from the 3 methods.

A New Variance Component Score Test for Testing Distributed Lag Functions with Applications in Time Series Analysis

We propose to test a given constrained distributed lag model (DLM) of the form β = Cθ against an unconstrained alternative using a variance component score test (VCST) and show that VCST is more powerful than the standard likelihood ratio test in a simulation study.

Time-Course of Cause-Specific Hospital Admissions During Snowstorms: An Analysis of Electronic Medical Records From Major Hospitals in Boston, Massachusetts

With global climate change, more frequent severe snowstorms are expected; however, evidence regarding their health effects is very limited. We gathered detailed medical records on hospital admissions (n = 433,037 admissions) from the 4 largest hospitals in Boston, Massachusetts, during the winters of 2010-2015. We estimated the percentage increase in hospitalizations for cardiovascular and cold-related diseases, falls, and injuries on the day of and for 6 days after a day with low (0.05-5.0 inches), moderate (5.1-10.0 inches), or high (>10.0 inches) snowfall using distributed lag regression models. We found that cardiovascular disease admissions decreased by 32% on high snowfall days (relative risk (RR) = 0.68, 95% confidence interval (CI): 0.54, 0.85) but increased by 23% 2 days after (RR = 1.23, 95% CI: 1.01, 1.49); cold-related admissions increased by 3.7% on high snowfall days (RR = 3.7, 95% CI: 1.6, 8.6) and remained high for 5 days after; and admissions for falls increased by 18% on average in the 6 days after a moderate snowfall day (RR = 1.18, 95% CI: 1.09, 1.27). We did not find a higher risk of hospitalizations for injuries. To our knowledge, this is the first study in which the time course of hospitalizations during and immediately after snowfall days has been examined. These findings can be translated into interventions that prevent hospitalizations and protect public health during harsh winter conditions.

A penalized framework for distributed lag non-linear models

Distributed lag non-linear models (DLNMs) are a modelling tool for describing potentially non-linear and delayed dependencies. Here, we illustrate an extension of the DLNM framework through the use of penalized splines within generalized additive models (GAM). This extension offers built-in model selection procedures and the possibility of accommodating assumptions on the shape of the lag structure through specific penalties. In addition, this framework includes, as special cases, simpler models previously proposed for linear relationships (DLMs). Alternative versions of penalized DLNMs are compared with each other and with the standard unpenalized version in a simulation study. Results show that this penalized extension to the DLNM class provides greater flexibility and improved inferential properties. The framework exploits recent theoretical developments of GAMs and is implemented using efficient routines within freely available software. Real-data applications are illustrated through two reproducible examples in time series and survival analysis.

Outdoor Air Pollution and Pterygium in Korea

We investigated relationships between outdoor air pollution and pterygium in Korean adults. This study includes 23,276 adults in population-based cross-sectional data using the Korea National Health and Nutrition Examination Survey 2008-2011. Pterygium was assessed using slit lamp biomicroscopy. Air pollution data (humidity, particulate matter with aerodynamic diameter less than 10 μm [PM₁₀], ozone [O₃], nitrogen dioxide [NO₂], and sulfur dioxide levels [SO₂]) for 2 years preceding the ocular examinations were acquired. Associations of multiple air pollutants with pterygium or pterygium recurrence after surgery were examined using multivariate logistic models, after adjusting for several covariates. Distributed lag models were additionally used for estimating cumulative effects of air pollution on pterygium. None of air pollution factors was significantly associated with pterygium or pterygium recurrence (each P > 0.05). Distributed lag models also showed that air pollution factors were not associated with pterygium or pterygium recurrence in 0-to-2 year lags (each P > 0.05). However, primary pterygium showed a weak association with PM10 after adjusting for covariates (odds ratio [OR] 1.23; [per 5 μg/m³ PM₁₀ increase]; P = 0.023). Aging, male sex, and greater sun exposure were associated with pterygium, while higher education level and myopia were negatively associated with pterygium (each P ≤ 0.001). Male sex and myopia were negatively associated with pterygium recurrence (each P < 0.05). In conclusion, exposure to higher PM10 levels was associated with primary pterygium, although this study observed no significant association between air pollution and overall pterygium or pterygium recurrence in Korean adults.

Risk of fire occurrence in arid and semi-arid ecosystems of Iran: an investigation using Bayesian belief networks

Identifying areas that have a high risk of burning is a main component of fire management planning. Although the available tools can predict the fire risks, these are poor in accommodating uncertainties in their predictions. In this study, we accommodated uncertainty in wildfire prediction using Bayesian belief networks (BBNs). An influence diagram was developed to identify the factors influencing wildfire in arid and semi-arid areas of Iran, and it was populated with probabilities to produce a BBNs model. The behavior of the model was tested using scenario and sensitivity analysis. Land cover/use, mean annual rainfall, mean annual temperature, elevation, and livestock density were recognized as the main variables determining wildfire occurrence. The produced model had good accuracy as its ROC area under the curve was 0.986. The model could be applied in both predictive and diagnostic analysis for answering "what if" and "how" questions. The probabilistic relationships within the model can be updated over time using observation and monitoring data. The wildfire BBN model may be updated as new knowledge emerges; hence, it can be used to support the process of adaptive management.

Incorporating High-Dimensional Exposure Modelling into Studies of Air Pollution and Health

Performing studies on the risks of environmental hazards on human health requires accurate estimates of exposures that might be experienced by the populations at risk. Often there will be missing data and in many epidemiological studies, the locations and times of exposure measurements and health data do not match. To a large extent this will be due to the health and exposure data having arisen from completely different data sources and not as the result of a carefully designed study, leading to problems of both ‘change of support’ and ‘misaligned data’. In such cases, a direct comparison of the exposure and health outcome is often not possible without an underlying model to align the two in the spatial and temporal domains. The Bayesian approach provides the natural framework for such models; however, the large amounts of data that can arise from environmental networks means that inference using Markov Chain Monte Carlo might not be computationally feasible in this setting. Here we adapt the integrated nested Laplace approximation to implement spatio–temporal exposure models. We also propose methods for the integration of large-scale exposure models and health analyses. It is important that any model structure allows the correct propagation of uncertainty from the predictions of the exposure model through to the estimates of risk and associated confidence intervals. The methods are demonstrated using a case study of the levels of black smoke in the UK, measured over several decades, and respiratory mortality.

Environmental effects and individual body condition drive seasonal fecundity of rabbits: identifying acute and lagged processes

The reproduction of many species is determined by seasonally-driven resource supply. But it is difficult to quantify whether the fecundity is sensitive to short- or long-term exposure to environmental conditions such as rainfall that drive resource supply. Using 25 years of data on individual fecundity of European female rabbits, Oryctolagus cuniculus, from semiarid Australia, we investigate the role of individual body condition, rainfall and temperature as drivers of seasonal and long-term and population-level changes in fecundity (breeding probability, ovulation rate, embryo survival). We built distributed lag models in a hierarchical Bayesian framework to account for both immediate and time-lagged effects of climate and other environmental drivers, and possible shifts in reproduction over consecutive seasons. We show that rainfall during summer, when rabbits typically breed only rarely, increased breeding probability immediately and with time lags of up to 10 weeks. However, an earlier onset of the yearly breeding period did not result in more overall reproductive output. Better body condition was associated with an earlier onset of breeding and higher embryo survival. Breeding probability in the main breeding season declined with increased breeding activity in the preceding season and only individuals in good body condition were able to breed late in the season. Higher temperatures reduce breeding success across seasons. We conclude that a better understanding of seasonal dynamics and plasticity (and their interplay) in reproduction will provide crucial insights into how lagomorphs are likely to respond and potentially adapt to the influence of future climate and other environmental change.

Distributed Lag Models: Examining Associations Between the Built Environment and Health

Built environment factors constrain individual level behaviors and choices, and thus are receiving increasing attention to assess their influence on health. Traditional regression methods have been widely used to examine associations between built environment measures and health outcomes, where a fixed, prespecified spatial scale (e.g., 1 mile buffer) is used to construct environment measures. However, the spatial scale for these associations remains largely unknown and misspecifying it introduces bias. We propose the use of distributed lag models (DLMs) to describe the association between built environment features and health as a function of distance from the locations of interest and circumvent a-priori selection of a spatial scale. Based on simulation studies, we demonstrate that traditional regression models produce associations biased away from the null when there is spatial correlation among the built environment features. Inference based on DLMs is robust under a range of scenarios of the built environment. We use this innovative application of DLMs to examine the association between the availability of convenience stores near California public schools, which may affect children's dietary choices both through direct access to junk food and exposure to advertisement, and children's body mass index z scores.

Assessment of critical exposure and outcome windows in time-to-event analysis with application to air pollution and preterm birth study

In reproductive epidemiology, there is a growing interest to examine associations between air pollution exposure during pregnancy and the risk of preterm birth (PTB). One important research objective is to identify critical periods of exposure and estimate the associated effects at different stages of pregnancy. However, population studies have reported inconsistent findings. This may be due to limitations from the standard analytic approach of treating PTB as a binary outcome without considering time-varying exposures together over the course of pregnancy. To address this research gap, we present a Bayesian hierarchical model for conducting a comprehensive examination of gestational air pollution exposure by estimating the joint effects of weekly exposures during different vulnerable periods. Our model also treats PTB as a time-to-event outcome to address the challenge of different exposure lengths among ongoing pregnancies. The proposed model is applied to a dataset of geocoded birth records in the Atlanta metropolitan area between 1999-2005 to examine the risk of PTB associated with gestational exposure to ambient fine particulate matter [Formula: see text]m in aerodynamic diameter (PM[Formula: see text]). We find positive associations between PM[Formula: see text] exposure during early and mid-pregnancy, and evidence that associations are stronger for PTBs occurring around week 30.

Bayesian analysis of time-series data under case-crossover designs: posterior equivalence and inference

Case-crossover designs are widely used to study short-term exposure effects on the risk of acute adverse health events. While the frequentist literature on this topic is vast, there is no Bayesian work in this general area. The contribution of this paper is twofold. First, the paper establishes Bayesian equivalence results that require characterization of the set of priors under which the posterior distributions of the risk ratio parameters based on a case-crossover and time-series analysis are identical. Second, the paper studies inferential issues under case-crossover designs in a Bayesian framework. Traditionally, a conditional logistic regression is used for inference on risk-ratio parameters in case-crossover studies. We consider instead a more general full likelihood-based approach which makes less restrictive assumptions on the risk functions. Formulation of a full likelihood leads to growth in the number of parameters proportional to the sample size. We propose a semi-parametric Bayesian approach using a Dirichlet process prior to handle the random nuisance parameters that appear in a full likelihood formulation. We carry out a simulation study to compare the Bayesian methods based on full and conditional likelihood with the standard frequentist approaches for case-crossover and time-series analysis. The proposed methods are illustrated through the Detroit Asthma Morbidity, Air Quality and Traffic study, which examines the association between acute asthma risk and ambient air pollutant concentrations.

Statistical strategies for constructing health risk models with multiple pollutants and their interactions: possible choices and comparisons

BACKGROUND

As public awareness of consequences of environmental exposures has grown, estimating the adverse health effects due to simultaneous exposure to multiple pollutants is an important topic to explore. The challenges of evaluating the health impacts of environmental factors in a multipollutant model include, but are not limited to: identification of the most critical components of the pollutant mixture, examination of potential interaction effects, and attribution of health effects to individual pollutants in the presence of multicollinearity.

METHODS

In this paper, we reviewed five methods available in the statistical literature that are potentially helpful for constructing multipollutant models. We conducted a simulation study and presented two data examples to assess the performance of these methods on feature selection, effect estimation and interaction identification using both cross-sectional and time-series designs. We also proposed and evaluated a two-step strategy employing an initial screening by a tree-based method followed by further dimension reduction/variable selection by the aforementioned five approaches at the second step.

RESULTS

Among the five methods, least absolute shrinkage and selection operator regression performs well in general for identifying important exposures, but will yield biased estimates and slightly larger model dimension given many correlated candidate exposures and modest sample size. Bayesian model averaging, and supervised principal component analysis are also useful in variable selection when there is a moderately strong exposure-response association. Substantial improvements on reducing model dimension and identifying important variables have been observed for all the five statistical methods using the two-step modeling strategy when the number of candidate variables is large.

CONCLUSIONS

There is no uniform dominance of one method across all simulation scenarios and all criteria. The performances differ according to the nature of the response variable, the sample size, the number of pollutants involved, and the strength of exposure-response association/interaction. However, the two-step modeling strategy proposed here is potentially applicable under a multipollutant framework with many covariates by taking advantage of both the screening feature of an initial tree-based method and dimension reduction/variable selection property of the subsequent method. The choice of the method should also depend on the goal of the study: risk prediction, effect estimation or screening for important predictors and their interactions.

Extending distributed lag models to higher degrees

Distributed lag (DL) models relate lagged covariates to a response and are a popular statistical model used in a wide variety of disciplines to analyze exposure-response data. However, classical DL models do not account for possible interactions between lagged predictors. In the presence of interactions between lagged covariates, the total effect of a change on the response is not merely a sum of lagged effects as is typically assumed. This article proposes a new class of models, called high-degree DL models, that extend basic DL models to incorporate hypothesized interactions between lagged predictors. The modeling strategy utilizes Gaussian processes to counterbalance predictor collinearity and as a dimension reduction tool. To choose the degree and maximum lags used within the models, a computationally manageable model comparison method is proposed based on maximum a posteriori estimators. The models and methods are illustrated via simulation and application to investigating the effect of heat exposure on mortality in Los Angeles and New York.

Distributed lag models for hydrological data

The distributed lag model (DLM), used most prominently in air pollution studies, finds application wherever the effect of a covariate is delayed and distributed through time. We specify modified formulations of DLMs to provide computationally attractive, flexible varying-coefficient models that are applicable in any setting in which lagged covariates are regressed on a time-dependent response. We investigate the application of such models to rainfall and river flow and in particular their role in understanding the impact of hidden variables at work in river systems. We apply two models to data from a Scottish mountain river, and we fit to some simulated data to check the efficacy of our model approach. During heavy rainfall conditions, changes in the influence of rainfall on flow arise through a complex interaction between antecedent ground wetness and a time-delay in rainfall. The models identify subtle changes in responsiveness to rainfall, particularly in the location of peak influence in the lag structure.

Reducing and meta-analysing estimates from distributed lag non-linear models

Background

The two-stage time series design represents a powerful analytical tool in environmental epidemiology. Recently, models for both stages have been extended with the development of distributed lag non-linear models (DLNMs), a methodology for investigating simultaneously non-linear and lagged relationships, and multivariate meta-analysis, a methodology to pool estimates of multi-parameter associations. However, the application of both methods in two-stage analyses is prevented by the high-dimensional definition of DLNMs.

Methods

In this contribution we propose a method to synthesize DLNMs to simpler summaries, expressed by a reduced set of parameters of one-dimensional functions, which are compatible with current multivariate meta-analytical techniques. The methodology and modelling framework are implemented in R through the packages dlnm and mvmeta.

Results

As an illustrative application, the method is adopted for the two-stage time series analysis of temperature-mortality associations using data from 10 regions in England and Wales. R code and data are available as supplementary online material.

Discussion and Conclusions

The methodology proposed here extends the use of DLNMs in two-stage analyses, obtaining meta-analytical estimates of easily interpretable summaries from complex non-linear and delayed associations. The approach relaxes the assumptions and avoids simplifications required by simpler modelling approaches.

Kernel Averaged Predictors for Spatio-Temporal Regression Models

In applications where covariates and responses are observed across space and time, a common goal is to quantify the effect of a change in the covariates on the response while adequately accounting for the spatio-temporal structure of the observations. The most common approach for building such a model is to confine the relationship between a covariate and response variable to a single spatio-temporal location. However, oftentimes the relationship between the response and predictors may extend across space and time. In other words, the response may be affected by levels of predictors in spatio-temporal proximity to the response location. Here, a flexible modeling framework is proposed to capture such spatial and temporal lagged effects between a predictor and a response. Specifically, kernel functions are used to weight a spatio-temporal covariate surface in a regression model for the response. The kernels are assumed to be parametric and non-stationary with the data informing the parameter values of the kernel. The methodology is illustrated on simulated data as well as a physical data set of ozone concentrations to be explained by temperature.

Flexible Distributed Lag Models using Random Functions with Application to Estimating Mortality Displacement from Heat-Related Deaths

As climate continues to change, scientists are left to analyze the effects these changes will have on the public. In this article, a flexible class of distributed lag models are used to analyze the effects of heat on mortality in four major metropolitan areas in the U.S. (Chicago, Dallas, Los Angeles, and New York). Specifically, the proposed methodology uses Gaussian processes as a prior model for the distributed lag function. Gaussian processes are adequately flexible to capture a wide variety of distributed lag functions while ensuring smoothness properties of process realizations. Additionally, the proposed framework allows for probabilistic inference of the maximum lag. Applying the proposed methodology revealed that mortality displacement (or, harvesting) was present for most age groups and cities analyzed suggesting that heat advanced death in some individuals. Additionally, the estimated shape of the DL functions gave evidence that prolonged heat exposure and highly variable temperatures pose a threat to public health.

The temporal lag structure of short-term associations of fine particulate matter chemical constituents and cardiovascular and respiratory hospitalizations

BACKGROUND

In air pollution time-series studies, the temporal pattern of the association of fine particulate matter (PM2.5; particulate matter ≤ 2.5 µm in aerodynamic diameter) and health end points has been observed to vary by disease category. The lag pattern of PM2.5 chemical constituents has not been well investigated, largely because daily data have not been available.

OBJECTIVES

We explored the lag structure for hospital admissions using daily PM2.5 chemical constituent data for 5 years in the Denver Aerosol Sources and Health (DASH) study.

METHODS

We measured PM2.5 constituents, including elemental carbon, organic carbon, sulfate, and nitrate, at a central residential site from 2003 through 2007 and linked these daily pollution data to daily hospital admission counts in the five-county Denver metropolitan area. Total hospital admissions and subcategories of respiratory and cardiovascular admissions were examined. We assessed the lag structure of relative risks (RRs) of hospital admissions for PM2.5 and four constituents on the same day and from 1 to 14 previous days from a constrained distributed lag model; we adjusted for temperature, humidity, longer-term temporal trends, and day of week using a generalized additive model.

RESULTS

RRs were generally larger at shorter lags for total cardiovascular admissions but at longer lags for total respiratory admissions. The delayed lag pattern was particularly prominent for asthma. Elemental and organic carbon generally showed more immediate patterns, whereas sulfate and nitrate showed delayed patterns.

CONCLUSION

In general, PM2.5 chemical constituents were found to have more immediate estimated effects on cardiovascular diseases and more delayed estimated effects on respiratory diseases, depending somewhat on the constituent.

Estimating the effects of environmental exposures using a weighted mean of monitoring stations

The health effects of environmental hazards are often examined using time series of the association between a daily response variable (e.g., death) and a daily level of exposure (e.g., temperature). Exposures are usually the average from a network of stations. This gives each station equal importance, and negates the opportunity for some stations to be better measures of exposure. We used a Bayesian hierarchical model that weighted stations using random variables between zero and one. We compared the weighted estimates to the standard model using data on health outcomes (deaths and hospital admissions) and exposures (air pollution and temperature) in Brisbane, Australia. The improvements in model fit were relatively small, and the estimated health effects of pollution were similar using either the standard or weighted estimates. Spatial weighted exposures would be probably more worthwhile when there is either greater spatial detail in the health outcome, or a greater spatial variation in exposure.

The effects of the 2009 dust storm on emergency admissions to a hospital in Brisbane, Australia

In September 2009 an enormous dust storm swept across eastern Australia. Dust is potentially hazardous to health as it interferes with breathing, and previous dust storms have been linked to increased risks of asthma and even death. We examined whether the 2009 Australian dust storm changed the volume or characteristics of emergency admissions to hospital. We used an observational study design, using time series analyses to examine changes in the number of admissions, and case-only analyses to examine changes in the characteristics of admissions. The admission data were from the Prince Charles Hospital, Brisbane, between 1 January 2009 and 31 October 2009. There was a 39% increase in emergency admissions associated with the storm (95% confidence interval: 5, 81%), which lasted for just 1 day. The health effects of the storm could not be detected using particulate matter levels. We found no significant change in the characteristics of admissions during the storm; specifically, there was no increase in respiratory admissions. The dust storm had a short-lived impact on emergency hospital admissions. This may be because the public took effective avoidance measures, or because the dust was simply not toxic, being composed mainly of soil. Emergency departments should be prepared for a short-term increase in admissions during dust storms.

Does ignoring model selection when assessing the effect of particulate matter air pollution on mortality make us too vigilant?

PURPOSE

To investigate the extent to which standard errors can be underestimated in time-series studies of the association between particulate matter air pollution (PM) and mortality if model selection variation is not accounted for.

METHODS

Actual-time series data from Cook County, Illinois, and Salt Lake County, Utah, for the period 1987 to 2000 were used to generate mortality time series. These series were used to examine the overconfidence resulting from ignoring variability introduced by the model selection process.

RESULTS

When variation associated with a model selection process is not accounted for, we found that the estimated standard errors for the effect of PM on mortality were substantially smaller than the true standard errors that necessarily incorporate model selection variability. Because of this, the true standard errors are approximately 70% larger than the reported standard errors. We also found that not accounting for model selection effects can result in the observed size of tests of no association between PM and mortality being up to about five times the nominal significance level.

CONCLUSIONS

Failing to account properly for the effect of model selection can reduce the accepted burden of proof for concluding a statistically significant association between PM and mortality.

Wavelet-based functional linear mixed models: an application to measurement error-corrected distributed lag models

Frequently, exposure data are measured over time on a grid of discrete values that collectively define a functional observation. In many applications, researchers are interested in using these measurements as covariates to predict a scalar response in a regression setting, with interest focusing on the most biologically relevant time window of exposure. One example is in panel studies of the health effects of particulate matter (PM), where particle levels are measured over time. In such studies, there are many more values of the functional data than observations in the data set so that regularization of the corresponding functional regression coefficient is necessary for estimation. Additional issues in this setting are the possibility of exposure measurement error and the need to incorporate additional potential confounders, such as meteorological or co-pollutant measures, that themselves may have effects that vary over time. To accommodate all these features, we develop wavelet-based linear mixed distributed lag models that incorporate repeated measures of functional data as covariates into a linear mixed model. A Bayesian approach to model fitting uses wavelet shrinkage to regularize functional coefficients. We show that, as long as the exposure error induces fine-scale variability in the functional exposure profile and the distributed lag function representing the exposure effect varies smoothly in time, the model corrects for the exposure measurement error without further adjustment. Both these conditions are likely to hold in the environmental applications we consider. We examine properties of the method using simulations and apply the method to data from a study examining the association between PM, measured as hourly averages for 1-7 days, and markers of acute systemic inflammation. We use the method to fully control for the effects of confounding by other time-varying predictors, such as temperature and co-pollutants.

Revisiting a population-dynamic model of air pollution and daily mortality of the elderly in Philadelphia

Epidemiological studies find that elderly, susceptible, and previously impaired individuals are more sensitive to transient air pollution exposures than healthy persons. However, any associated changes in life expectancy remain largely unresolved. Murray and Nelson published a model of daily mortality and air pollution that addresses mortality displacement or harvesting by directly considering population dynamics on the basis of the assumption that a period of illness or frailty precedes most elderly deaths. The underlying concept is that a person's response to an environmental exposure also depends on his/her physiological ability to withstand stress at that time. They used Kalman filtering to estimate an unobservable quantity--the size of the frail subpopulation from which elderly (ages > or = 65 yr) nontraumatic deaths are assumed to derive. They found a small subpopulation, relatively robust to environmental variations over 14 yr, with remaining life expectancies of 8-31 days in this frail status. Here, this model and dataset are expanded to examine the ramifications in more detail (including seasonality), to consider peak ozone as an additional pollutant, and to consider remaining life expectancies of the this frail subpopulation on a daily basis. Previous studies of mortality displacement and of Philadelphia mortality-air-pollution associations are also summarized in general, and agreement with the Murray-Nelson model was found, thus supporting its validity. The estimated additional mortality associated with a given environmental exposure persists for a few days at most but is not always compensated by subsequent mortality deficits. It is concluded that the pollution-associated mortality increases of a few percent in this dataset are consistent with losses of remaining life expectancy of up to a few days. It is also recommended that a more complex population-dynamic model be implemented to examine the extent to which previous short-term environmental exposures and seasonal trends may also influence morbidity and thus entry into the frail at-risk subpopulation.

Integrated exposure modeling: a model using GIS and GLM

Traffic exhaust is a source of air contaminants that have adverse health effects. Quantification of traffic as an exposure variable is complicated by aerosol dispersion related to variation in layout of roads, traffic density, meteorology, and topography. A statistical model is presented that uses Geographic Information Systems (GIS) technology to incorporate variables into a generalized linear model that estimates distribution of traffic-related pollution. Exposure from a source is expressed as an integral of a function proportional to average daily traffic and a nonparametric dispersion function, which takes the form of a step, polynomial, or spline model. The method may be applied using standard regression techniques for fitting generalized linear models. Modifiers of pollutant dispersion such as wind direction, meteorology, and landscape features can also be included. Two examples are given to illustrate the method. The first employs data from a study in which NO(2) (a known pollutant from automobile exhaust) was monitored outside of 138 Connecticut homes, providing a model for estimating NO(2) exposure. In the second example, estimated levels of nitrogen dioxide (NO(2)) from the model, as well as a separate spatial model, were used to analyze traffic-related health effects in a study of 761 infants.

Local and global effects of climate on dengue transmission in Puerto Rico

The four dengue viruses, the agents of dengue fever and dengue hemorrhagic fever in humans, are transmitted predominantly by the mosquito Aedes aegypti. The abundance and the transmission potential of Ae. aegypti are influenced by temperature and precipitation. While there is strong biological evidence for these effects, empirical studies of the relationship between climate and dengue incidence in human populations are potentially confounded by seasonal covariation and spatial heterogeneity. Using 20 years of data and a statistical approach to control for seasonality, we show a positive and statistically significant association between monthly changes in temperature and precipitation and monthly changes in dengue transmission in Puerto Rico. We also found that the strength of this association varies spatially, that this variation is associated with differences in local climate, and that this relationship is consistent with laboratory studies of the impacts of these factors on vector survival and viral replication. These results suggest the importance of temperature and precipitation in the transmission of dengue viruses and suggest a reason for their spatial heterogeneity. Thus, while dengue transmission may have a general system, its manifestation on a local scale may differ from global expectations.

Dengue viruses are a major health problem throughout the tropical and subtropical regions of the world. Because they are transmitted by mosquitoes that are sensitive to changes in rainfall and temperature, transmission intensity may be regulated by weather and climate. Laboratory studies have shown this to be biologically plausible, but studies of transmission in real-life situations have been inconclusive. Here we demonstrate that increased temperature and rainfall are associated with increased dengue transmission in subsequent months across Puerto Rico. We also show that differences in local climate within Puerto Rico can explain local differences observed in the relationship between weather and dengue transmission. This finding is important because it suggests that the determinants of transmission occur on a local level such that although dengue viruses have a basically universal transmission cycle, changes in temperature or rainfall may have diverse local effects.