Bayesian distributed lag models: estimating effects of particulate matter air pollution on daily mortality

Summary .  A distributed lag model (DLagM) is a regression model that includes lagged exposure variables as covariates; its corresponding distributed lag (DL) function describes the relationship between the lag and the coefficient of the lagged exposure variable. DLagMs have recently been used in environmental epidemiology for quantifying the cumulative effects of weather and air pollution on mortality and morbidity. Standard methods for formulating DLagMs include unconstrained, polynomial, and penalized spline DLagMs. These methods may fail to take full advantage of prior information about the shape of the DL function for environmental exposures, or for any other exposure with effects that are believed to smoothly approach zero as lag increases, and are therefore at risk of producing suboptimal estimates. In this article, we propose a Bayesian DLagM (BDLagM) that incorporates prior knowledge about the shape of the DL function and also allows the degree of smoothness of the DL function to be estimated from the data. We apply our BDLagM to its motivating data from the National Morbidity, Mortality, and Air Pollution Study to estimate the short-term health effects of particulate matter air pollution on mortality from 1987 to 2000 for Chicago, Illinois. In a simulation study, we compare our Bayesian approach with alternative methods that use unconstrained, polynomial, and penalized spline DLagMs. We also illustrate the connection between BDLagMs and penalized spline DLagMs. Software for fitting BDLagM models and the data used in this article are available online.     

Estimating perinatal critical windows of susceptibility to environmental mixtures via structured Bayesian regression tree pairs

Maternal exposure to environmental chemicals during pregnancy can alter birth and children's health outcomes. Research seeks to identify critical windows, time periods when exposures can change future health outcomes, and estimate the exposure–response relationship. Existing statistical approaches focus on estimation of the association between maternal exposure to a single environmental chemical observed at high temporal resolution (e.g., weekly throughout pregnancy) and children's health outcomes. Extending to multiple chemicals observed at high temporal resolution poses a dimensionality problem and statistical methods are lacking. We propose a regression tree–based model for mixtures of exposures observed at high temporal resolution. The proposed approach uses an additive ensemble of tree pairs that defines structured main effects and interactions between time-resolved predictors and performs variable selection to select out of the model predictors not correlated with the outcome. In simulation, we show that the tree-based approach performs better than existing methods for a single exposure and can accurately estimate critical windows in the exposure–response relation for mixtures. We apply our method to estimate the relationship between five exposures measured weekly throughout pregnancy and birth weight in a Denver, Colorado, birth cohort. We identified critical windows during which fine particulate matter, sulfur dioxide, and temperature are negatively associated with birth weight and an interaction between fine particulate matter and temperature. Software is made available in the R package dlmtree.     

G-estimation of structural nested restricted mean time lost models to estimate effects of time-varying treatments on a failure time outcome

G-estimation of structural nested models (SNMs) plays an important role in estimating the effects of time-varying treatments with appropriate adjustment for time-dependent confounding. As SNMs for a failure time outcome, structural nested accelerated failure time models (SNAFTMs) and structural nested cumulative failure time models have been developed. The latter models are included in the class of structural nested mean models (SNMMs) and are not involved in artificial censoring, which induces several difficulties in g-estimation of SNAFTMs. Recently, restricted mean time lost (RMTL), which corresponds to the area under a distribution function up to a restriction time, is attracting attention in clinical trial communities as an appropriate summary measure of a failure time outcome. In this study, we propose another SNMM for a failure time outcome, which is called structural nested RMTL model (SNRMTLM) and describe randomized and observational g-estimation procedures that use different assumptions for the treatment mechanism in a randomized trial setting. We also provide methods to estimate marginal RMTLs under static treatment regimes using estimated SNRMTLMs. A simulation study evaluates finite-sample performances of the proposed methods compared with the conventional intention-to-treat and per-protocol analyses. We illustrate the proposed methods using data from a randomized controlled trial for cardiovascular disease with treatment changes. G-estimation of SNRMTLMs is a useful tool to estimate the effects of time-varying treatments on a failure time outcome.     

Estimation of causal effects of a time-varying exposure at multiple time points through multivariable mendelian randomization

Mendelian Randomisation (MR) is a powerful tool in epidemiology that can be used to estimate the causal effect of an exposure on an outcome in the presence of unobserved confounding, by utilising genetic variants as instrumental variables (IVs) for the exposure. The effect estimates obtained from MR studies are often interpreted as the lifetime effect of the exposure in question. However, the causal effects of some exposures are thought to vary throughout an individual’s lifetime with periods during which an exposure has a greater effect on a particular outcome. Multivariable MR (MVMR) is an extension of MR that allows for multiple, potentially highly related, exposures to be included in an MR estimation. MVMR estimates the direct effect of each exposure on the outcome conditional on all the other exposures included in the estimation. We explore the use of MVMR to estimate the direct effect of a single exposure at different time points in an individual’s lifetime on an outcome. We use simulations to illustrate the interpretation of the results from such analyses and the key assumptions required. We show that causal effects at different time periods can be estimated through MVMR when the association between the genetic variants used as instruments and the exposure measured at those time periods varies. However, this estimation will not necessarily identify exact time periods over which an exposure has the most effect on the outcome. Prior knowledge regarding the biological basis of exposure trajectories can help interpretation. We illustrate the method through estimation of the causal effects of childhood and adult BMI on C-Reactive protein and smoking behaviour.     

A generalized weighted quantile sum approach for analyzing correlated data in the presence of interactions

A weighted quantile sum (WQS) regression has been used to assess the associations between environmental exposures and health outcomes. However, the currently available WQS approach, which is based on additive effects, does not allow exploring for potential interactions of exposures with other covariates in relation to a health outcome. In addition, the current WQS cannot account for clustering, thus it may not be valid for analysis of clustered data. We propose a generalized WQS approach that can assess interactions by estimating stratum‐specific weights of exposures in a mixture, while accounting for potential clustering effect of matched pairs of cases and controls as well as censored exposure data due to being below the limits of detection. The performance of the proposed method in identifying interactions is evaluated through simulations based on various scenarios of correlation structures among the exposures and with an outcome. We also assess how well the proposed method performs in the presence of the varying levels of censoring in exposures. Our findings from the simulation study show that the proposed method outperforms the traditional WQS, as indicated by higher power of detecting interactions. We also find no strong evidence that the proposed method falsely identifies interactions when there are no true interactive effects. We demonstrate application of the proposed method to real data from the Epidemiological Research on Autism Spectrum Disorder (ASD) in Jamaica (ERAJ) by examining interactions between exposure to manganese and glutathione S‐transferase family gene, GSTP1 in relation to ASD.     

Using bivariate models to understand between- and within-cluster regression coefficients, with application to twin data

In the regression analysis of clustered data it is important to allow for the possibility of distinct between- and within-cluster exposure effects on the outcome measure, represented, respectively, by regression coefficients for the cluster mean and the deviation of the individual-level exposure value from this mean. In twin data, the within-pair regression effect represents association conditional on exposures shared within pairs, including any common genetic or environmental influences on the outcome measure. It has therefore been proposed that a comparison of the within-pair regression effects between monozygous (MZ) and dizygous (DZ) twins can be used to examine whether the association between exposure and outcome has a genetic origin. We address this issue by proposing a bivariate model for exposure and outcome measurements in twin-pair data. The between- and within-pair regression coefficients are shown to be weighted averages of ratios of the exposure and outcome variances and covariances, from which it is straightforward to determine the conditions under which the within-pair regression effect in MZ pairs will be different from that in DZ pairs. In particular, we show that a correlation structure in twin pairs for exposure and outcome that appears to be due to genetic factors will not necessarily be reflected in distinct MZ and DZ values for the within-pair regression coefficients. We illustrate these results in a study of female twin pairs from Australia and North America relating mammographic breast density to weight and body mass index.     

Variable selection for the single-index model

We consider variable selection in the single-index model. Weprove that the popular leave-m-out crossvalidation method hasdifferent behaviour in the single-index model from that in linearregression models or nonparametric regression models. A newconsistent variable selection method, called separated crossvalidation,is proposed. Further analysis suggests that the method has betterfinite-sample performance and is computationally easier thanleave-m-out crossvalidation. Separated crossvalidation, appliedto the Swiss banknotes data and the ozone concentration data,leads to single-index models with selected variables that havebetter prediction capability than models based on all the covariates.     

Multiple low-level exposures: Hg interactions with co-occurring neurotoxic substances in early life

All chemical forms of Hg can affect neurodevelopment; however, low levels of organic Hg (methylmercury-MeHg and ethylmercury-EtHg in Thimerosal-containing vaccines, hereafter ‘TCV’) exposures during early life (pregnancy and lactation) co-occur with other environmental neurotoxic substances. These neurotoxicants may act in parallel, synergistically, or antagonistically to Hg. Nevertheless, the risks of neurotoxicity associated with multiple neuro-toxicants depend on type, time, combinations of exposure, and environmental and/or genetic-associated factors. Neurological developmental disorders, delays in cognition and behavioral outcomes associated with multiple exposures (which include Hg) may show transient or lasting outcomes depending on constitutional and/or environmental factors that can interact to neutralize, aggravate or attenuate these effects; often these studies are challenging to interpret. During pregnancy and lactation, fish-MeHg exposure is frequently confounded with the opposing effects of neuroactive nutrients (in fish) that lead to positive, negative, or no effects on neurobehavioral tests. In infancy, exposures to acute binary mixtures (TCV- EtHg and Al-adjuvants in infant immunizations) are associated with increased risks of tics and other developmental disorders. Despite the certitude that promulgates single environmental neurotoxicants, empirical comparisons of combined exposures indicate that Hg-related outcome is uneven. Hg in combination with other neurotoxic mixtures may elevate risks of neurotoxicity, but these risks arise in circumstances that are not yet predictable. Therefore, to achieve the goals of the Minamata treaty and to safeguard the health of children, low levels of mercury exposure (in any chemical form) needs to be further reduced whether the source is environmental (air- and food-borne) or iatrogenic (pediatric TCVs).     

Efficient measurement error correction with spatially misaligned data

Association studies in environmental statistics often involve exposure and outcome data that are misaligned in space. A common strategy is to employ a spatial model such as universal kriging to predict exposures at locations with outcome data and then estimate a regression parameter of interest using the predicted exposures. This results in measurement error because the predicted exposures do not correspond exactly to the true values. We characterize the measurement error by decomposing it into Berkson-like and classical-like components. One correction approach is the parametric bootstrap, which is effective but computationally intensive since it requires solving a nonlinear optimization problem for the exposure model parameters in each bootstrap sample. We propose a less computationally intensive alternative termed the "parameter bootstrap" that only requires solving one nonlinear optimization problem, and we also compare bootstrap methods to other recently proposed methods. We illustrate our methodology in simulations and with publicly available data from the Environmental Protection Agency.     

A latent variable approach to study gene-environment interactions in the presence of multiple correlated exposures

Many existing cohort studies initially designed to investigate disease risk as a function of environmental exposures have collected genomic data in recent years with the objective of testing for gene-environment interaction (G × E) effects. In environmental epidemiology, interest in G × E arises primarily after a significant effect of the environmental exposure has been documented. Cohort studies often collect rich exposure data; as a result, assessing G × E effects in the presence of multiple exposure markers further increases the burden of multiple testing, an issue already present in both genetic and environment health studies. Latent variable (LV) models have been used in environmental epidemiology to reduce dimensionality of the exposure data, gain power by reducing multiplicity issues via condensing exposure data, and avoid collinearity problems due to presence of multiple correlated exposures. We extend the LV framework to characterize gene-environment interaction in presence of multiple correlated exposures and genotype categories. Further, similar to what has been done in case-control G × E studies, we use the assumption of gene-environment (G-E) independence to boost the power of tests for interaction. The consequences of making this assumption, or the issue of how to explicitly model G-E association has not been previously investigated in LV models. We postulate a hierarchy of assumptions about the LV model regarding the different forms of G-E dependence and show that making such assumptions may influence inferential results on the G, E, and G × E parameters. We implement a class of shrinkage estimators to data adaptively trade-off between the most restrictive to most flexible form of G-E dependence assumption and note that such class of compromise estimators can serve as a benchmark of model adequacy in LV models. We demonstrate the methods with an example from the Early Life Exposures in Mexico City to Neuro-Toxicants Study of lead exposure, iron metabolism genes, and birth weight.     

Toxicity of cadmium and zinc mixtures to the decaudized cercarial life span of Diplostomum spathaceum

The effects of cadmium and zinc mixtures at concentrations ranging from 0.1 to 10,000 microg l(-1) on the life-span of decaudized cercarial bodies (cercariae that have shed their tails) of Diplostomum spathaceum (Trematoda: Diplostomatidae) was investigated. Cercariae were exposed to metal mixtures of equal and unequal concentrations, and a low-dose pre-treatment followed by a high-dose exposure mixtures. Metal mixtures demonstrated variable effects on decaudized cercariae either by increasing or reducing their life-span compared to single metal exposures dependent on concentration and the type of mixed metal treatment. Prolonged exposure to equal metal mixtures at low concentrations (0.1-100 microg l(-1)) resulted in a reduction in the life-span of decaudized cercariae at 0.1 and 100 microg l(-1) in those individuals decaudized during the initial 24 h exposure period compared with those decaudized during the final 24 h period of cercarial survival, whilst in controls there was no significant life-span change between the two time periods. Decaudized cercariae which were exposed to low concentrations (0.1-100 microg l(-1)) of equal metal mixtures were also evaluated for their role as an indicator of larval 'fitness' for migrating through the tissues of their target fish host for those individuals decaudized during the initial 24 h exposure period, and demonstrated only a limited change in their life-span compared to control and single metal exposures. The importance of metal mixtures in parasite establishment in the fish host is discussed.     

The stimulating effect of bright light on physical performance depends on internal time

The human circadian clock regulates the daily timing of sleep, alertness and performance and is synchronized to the 24-h day by the environmental light-dark cycle. Bright light exposure has been shown to positively affect sleepiness and alertness, yet little is known about its effects on physical performance, especially in relation to chronotype. We, therefore, exposed 43 male participants (mean age 24.5 yrs ± SD 2.3 yrs) in a randomized crossover study to 160 minutes of bright (BL: ≈ 4.420 lx) and dim light (DL: ≈ 230 lx). During the last 40 minutes of these exposures, participants performed a bicycle ergometer test. Time-of-day of the exercise sessions did not differ between the BL and DL condition. Chronotype (MSF(sc), mid-sleep time on free days corrected for oversleep due to sleep debt on workdays) was assessed by the Munich ChronoType Questionnaire (MCTQ). Total work was significantly higher in BL (median 548.4 kJ, min 411.82 kJ, max 875.20 kJ) than in DL (median 521.5 kJ, min 384.33 kJ, max 861.23 kJ) (p = 0.004) going along with increased exhaustion levels in BL (blood lactate (+12.7%, p = 0.009), heart rate (+1.8%, p = 0.031), and Borg scale ratings (+2.6%, p = 0.005)) in all participants. The differences between total work levels in BL and DL were significantly higher (p = 0.004) if participants were tested at a respectively later time point after their individual mid-sleep (chronotype). These novel results demonstrate, that timed BL exposure enhances physical performance with concomitant increase in individual strain, and is related not only to local (external) time, but also to an individual's internal time.     

Effects of Artificial Dawn and Morning Blue Light on Daytime Cognitive Performance,Well-being,Cortisol and Melatonin Levels

Light exposure elicits numerous effects on human physiology and behavior, such as better cognitive performance and mood. Here we investigated the role of morning light exposure as a countermeasure for impaired cognitive performance and mood under sleep restriction (SR). Seventeen participants took part of a 48h laboratory protocol, during which three different light settings (separated by 2?wks) were administered each morning after two 6-h sleep restriction nights: a blue monochromatic LED (light-emitting diode) light condition (BL; 100?lux at 470?nm for 20?min) starting 2?h after scheduled wake-up time, a dawn-simulating light (DsL) starting 30?min before and ending 20?min after scheduled wake-up time (polychromatic light gradually increasing from 0 to 250?lux), and a dim light (DL) condition for 2?h beginning upon scheduled wake time (<8?lux). Cognitive tasks were performed every 2?h during scheduled wakefulness, and questionnaires were administered hourly to assess subjective sleepiness, mood, and well-being. Salivary melatonin and cortisol were collected throughout scheduled wakefulness in regular intervals, and the effects on melatonin were measured after only one light pulse. Following the first SR, analysis of the time course of cognitive performance during scheduled wakefulness indicated a decrease following DL, whereas it remained stable following BL and significantly improved after DsL. Cognitive performance levels during the second day after SR were not significantly affected by the different light conditions. However, after both SR nights, mood and well-being were significantly enhanced after exposure to morning DsL compared with DL and BL. Melatonin onset occurred earlier after morning BL exposure, than after morning DsL and DL, whereas salivary cortisol levels were higher at wake-up time after DsL compared with BL and DL. Our data indicate that exposure to an artificial morning dawn simulation light improves subjective well-being, mood, and cognitive performance, as compared with DL and BL, with minimal impact on circadian phase. Thus, DsL may provide an effective strategy for enhancing cognitive performance, well-being, and mood under mild sleep restriction.     

The relationship between co-exposure to multiple heavy metals and liver damage

BackgroundThe impact of heavy metal exposure on human health has attracted widespread attention of researchers, and the impact of heavy metal exposure on liver function has also been confirmed, however, more attention is paid to the impact of single or two heavy metal exposures, and most epidemiological studies focus on heavy metal pollution areas. In this study, rural residents in non-heavy metal-contaminated areas in Northwest China were selected as the research objects to explore the comprehensive effects of co-exposure to multiple heavy metals on the liver, which can provide certain reference and support for related research.ObjectivesThis study used a Bayesian nuclear machine model (BKMR) to evaluate the relationship between exposure to heavy metal mixtures and indicators of liver function in a population in rural Northwest China.ResultsExposure to higher concentrations of metal mixtures was positively correlated with total bilirubin, direct bilirubin, and aspartate aminotransferase, and negatively correlated with alanine aminotransferase, with Pb contributing the most to indicators of liver function. We also observed a possible interaction of Cd with other heavy metals in the effect of heavy metal mixtures on DB levels.ConclusionsConcurrent exposure to higher concentrations of heavy metal mixtures (Cr, Co, Cd, and Pb) in rural China was associated with indicators representing poor liver function, of which the effect of lead on liver function should be focused. More prospective epidemiological studies and animal experiments need to be carried out to determine this relationship and possible mechanism.     

Application of the time-dependent ROC curves for prognostic accuracy with multiple biomarkers

The rapid advancement in molecule technology has led to the discovery of many markers that have potential applications in disease diagnosis and prognosis. In a prospective cohort study, information on a panel of biomarkers as well as the disease status for a patient are routinely collected over time. Such information is useful to predict patients' prognosis and select patients for targeted therapy. In this article, we develop procedures for constructing a composite test with optimal discrimination power when there are multiple markers available to assist in prediction and characterize the accuracy of the resulting test by extending the time-dependent receiver operating characteristic (ROC) curve methodology. We employ a modified logistic regression model to derive optimal linear composite scores such that their corresponding ROC curves are maximized at every false positive rate. We provide theoretical justification for using such a model for prognostic accuracy. The proposed method allows for time-varying marker effects and accommodates censored failure time outcome. When the effects of markers are approximately constant over time, we propose a more efficient estimating procedure under such models. We conduct numerical studies to evaluate the performance of the proposed procedures. Our results indicate the proposed methods are both flexible and efficient. We contrast these methods with an application concerning the prognostic accuracies of expression levels of six genes.     

Using semiparametric‐mixed model and functional linear model to detect vulnerable prenatal window to carcinogenic polycyclic aromatic hydrocarbons on fetal growth

Prenatal exposure to carcinogenic polycyclic aromatic hydrocarbons (c‐PAHs) through maternal inhalation induces higher risk for a wide range of fetotoxic effects. However, the most health‐relevant dose function from chronic gestational exposure remains unclear. Whether there is a gestational window during which the human embryo/fetus is particularly vulnerable to PAHs has not been examined thoroughly. We consider a longitudinal semiparametric‐mixed effect model to characterize the individual prenatal PAH exposure trajectory, where a nonparametric cyclic smooth function plus a linear function are used to model the time effect and random effects are used to account for the within‐subject correlation. We propose a penalized least squares approach to estimate the parametric regression coefficients and the nonparametric function of time. The smoothing parameter and variance components are selected using the generalized cross‐validation (GCV) criteria. The estimated subject‐specific trajectory of prenatal exposure is linked to the birth outcomes through a set of functional linear models, where the coefficient of log PAH exposure is a fully nonparametric function of gestational age. This allows the effect of PAH exposure on each birth outcome to vary at different gestational ages, and the window associated with significant adverse effect is identified as a vulnerable prenatal window to PAHs on fetal growth. We minimize the penalized sum of squared errors using a spline‐based expansion of the nonparametric coefficient function to draw statistical inferences, and the smoothing parameter is chosen through GCV.     

Conditional generalized estimating equations for the analysis of clustered and longitudinal data

Summary .   A common and important problem in clustered sampling designs is that the effect of within-cluster exposures (i.e., exposures that vary within clusters) on outcome may be confounded by both measured and unmeasured cluster-level factors (i.e., measurements that do not vary within clusters). When some of these are ill/not accounted for, estimation of this effect through population-averaged models or random-effects models may introduce bias. We accommodate this by developing a general theory for the analysis of clustered data, which enables consistent and asymptotically normal estimation of the effects of within-cluster exposures in the presence of cluster-level confounders. Semiparametric efficient estimators are obtained by solving so-called conditional generalized estimating equations. We compare this approach with a popular proposal by Neuhaus and Kalbfleisch (1998, Biometrics 54, 638–645) who separate the exposure effect into a within- and a between-cluster component within a random intercept model. We find that the latter approach yields consistent and efficient estimators when the model is linear, but is less flexible in terms of model specification. Under nonlinear models, this approach may yield inconsistent and inefficient estimators, though with little bias in most practical settings.     

Generalized additive distributed lag models: quantifying mortality displacement

There are a number of applied settings where a response is measured repeatedly over time, and the impact of a stimulus at one time is distributed over several subsequent response measures. In the motivating application the stimulus is an air pollutant such as airborne particulate matter and the response is mortality. However, several other variables (e.g. daily temperature) impact the response in a possibly non-linear fashion. To quantify the effect of the stimulus in the presence of covariate data we combine two established regression techniques: generalized additive models and distributed lag models. Generalized additive models extend multiple linear regression by allowing for continuous covariates to be modeled as smooth, but otherwise unspecified, functions. Distributed lag models aim to relate the outcome variable to lagged values of a time-dependent predictor in a parsimonious fashion. The resultant, which we call generalized additive distributed lag models, are seen to effectively quantify the so-called 'mortality displacement effect' in environmental epidemiology, as illustrated through air pollution/mortality data from Milan, Italy.     

Bayesian multiple index models for environmental mixtures

An important goal of environmental health research is to assess the risk posed by mixtures of environmental exposures. Two popular classes of models for mixtures analyses are response-surface methods and exposure-index methods. Response-surface methods estimate high-dimensional surfaces and are thus highly flexible but difficult to interpret. In contrast, exposure-index methods decompose coefficients from a linear model into an overall mixture effect and individual index weights; these models yield easily interpretable effect estimates and efficient inferences when model assumptions hold, but, like most parsimonious models, incur bias when these assumptions do not hold. In this paper, we propose a Bayesian multiple index model framework that combines the strengths of each, allowing for non-linear and non-additive relationships between exposure indices and a health outcome, while reducing the dimensionality of the exposure vector and estimating index weights with variable selection. This framework contains response-surface and exposure-index models as special cases, thereby unifying the two analysis strategies. This unification increases the range of models possible for analysing environmental mixtures and health, allowing one to select an appropriate analysis from a spectrum of models varying in flexibility and interpretability. In an analysis of the association between telomere length and 18 organic pollutants in the National Health and Nutrition Examination Survey (NHANES), the proposed approach fits the data as well as more complex response-surface methods and yields more interpretable results.