Reduced Bayesian Hierarchical Models: Estimating Health Effects of Simultaneous Exposure to Multiple Pollutants

Quantifying the health effects associated with simultaneous exposure to many air pollutants is now a research priority of the US EPA. Bayesian hierarchical models (BHM) have been extensively used in multisite time series studies of air pollution and health to estimate health effects of a single pollutant adjusted for potential confounding of other pollutants and other time-varying factors. However, when the scientific goal is to estimate the impacts of many pollutants jointly, a straightforward application of BHM is challenged by the need to specify a random-effect distribution on a high-dimensional vector of nuisance parameters, which often do not have an easy interpretation. In this paper we introduce a new BHM formulation, which we call "reduced BHM", aimed at analyzing clustered data sets in the presence of a large number of random effects that are not of primary scientific interest. At the first stage of the reduced BHM, we calculate the integrated likelihood of the parameter of interest (e.g. excess number of deaths attributed to simultaneous exposure to high levels of many pollutants). At the second stage, we specify a flexible random-effect distribution directly on the parameter of interest. The reduced BHM overcomes many of the challenges in the specification and implementation of full BHM in the context of a large number of nuisance parameters. In simulation studies we show that the reduced BHM performs comparably to the full BHM in many scenarios, and even performs better in some cases. Methods are applied to estimate location-specific and overall relative risks of cardiovascular hospital admissions associated with simultaneous exposure to elevated levels of particulate matter and ozone in 51 US counties during the period 1999-2005.

Computational Techniques for Spatial Logistic Regression with Large Datasets

In epidemiological work, outcomes are frequently non-normal, sample sizes may be large, and effects are often small. To relate health outcomes to geographic risk factors, fast and powerful methods for fitting spatial models, particularly for non-normal data, are required. We focus on binary outcomes, with the risk surface a smooth function of space. We compare penalized likelihood models, including the penalized quasi-likelihood (PQL) approach, and Bayesian models based on fit, speed, and ease of implementation. A Bayesian model using a spectral basis representation of the spatial surface provides the best tradeoff of sensitivity and specificity in simulations, detecting real spatial features while limiting overfitting and being more efficient computationally than other Bayesian approaches. One of the contributions of this work is further development of this underused representation. The spectral basis model outperforms the penalized likelihood methods, which are prone to overfitting, but is slower to fit and not as easily implemented. Conclusions based on a real dataset of cancer cases in Taiwan are similar albeit less conclusive with respect to comparing the approaches. The success of the spectral basis with binary data and similar results with count data suggest that it may be generally useful in spatial models and more complicated hierarchical models.

Underestimation of Standard Errors in Multi-Site Time Series Studies

Multi-site time series studies of air pollution and mortality and morbidity have figured prominently in the literature as comprehensive approaches for estimating acute effects of air pollution on health. Hierarchical models are generally used to combine site-specific information and estimate pooled air pollution effects taking into account both within-site statistical uncertainty, and across-site heterogeneity. Within a site, characteristics of time series data of air pollution and health (small pollution effects, missing data, highly correlated predictors, non linear confounding etc.) make modelling all sources of uncertainty challenging. One potential consequence is underestimation of the statistical variance of the site-specific effects to be combined. In this paper we investigate the impact of variance underestimation on the pooled relative rate estimate. We focus on two-stage normal-normal hierarchical models and on under- estimation of the statistical variance at the first stage. By mathematical considerations and simulation studies, we found that variance underestimation does not affect the pooled estimate substantially. However, some sensitivity of the pooled estimate to variance underestimation is observed when the number of sites is small and underestimation is severe. These simulation results are applicable to any two-stage normal-normal hierarchical model for combining information of site-specific results, and they can be easily extended to more general hierarchical formulations. We also examined the impact of variance underestimation on the national average relative rate estimate from the National Morbidity Mortality Air Pollution Study and we found that variance underestimation as much as 40% has little effect on the national average.

LEARNING FROM NEAR MISSES IN MEDICATION ERRORS: A BAYESIAN APPROACH

Medical errors originating in health care facilities are a significant source of preventable morbidity, mortality, and healthcare costs. Voluntary error report systems that collect information on the causes and contributing factors of medi- cal errors regardless of the resulting harm may be useful for developing effective harm prevention strategies. Some patient safety experts question the utility of data from errors that did not lead to harm to the patient, also called near misses. A near miss (a.k.a. close call) is an unplanned event that did not result in injury to the patient. Only a fortunate break in the chain of events prevented injury. We use data from a large voluntary reporting system of 836,174 medication errors from 1999 to 2005 to provide evidence that the causes and contributing factors of errors that result in harm are similar to the causes and contributing factors of near misses. We develop Bayesian hierarchical models for estimating the log odds of selecting a given cause (or contributing factor) of error given harm has occurred and the log odds of selecting the same cause given that harm did not occur. The posterior distribution of the correlation between these two vectors of log-odds is used as a measure of the evidence supporting the use of data from near misses and their causes and contributing factors to prevent medical errors. In addition, we identify the causes and contributing factors that have the highest or lowest log-odds ratio of harm versus no harm. These causes and contributing factors should also be a focus in the design of prevention strategies. This paper provides important evidence on the utility of data from near misses, which constitute the vast majority of errors in our data.

Seasonal Analyses of Air Pollution and Mortality in 100 U.S. Cities

Time series models relating short-term changes in air pollution levels to daily mortality counts typically assume that the effects of air pollution on the log relative rate of mortality do not vary with time. However, these short-term effects might plausibly vary by season. Changes in the sources of air pollution and meteorology can result in changes in characteristics of the air pollution mixture across seasons. The authors develop Bayesian semi-parametric hierarchical models for estimating time-varying effects of pollution on mortality in multi-site time series studies. The methods are applied to the updated National Morbidity and Mortality Air Pollution Study database for the period 1987--2000, which includes data for 100 U.S. cities. At the national level, a 10 micro-gram/m3 increase in PM(10) at lag 1 is associated with a 0.15 (95% posterior interval: -0.08, 0.39),0.14 (-0.14, 0.42), 0.36 (0.11, 0.61), and 0.14 (-0.06, 0.34) percent increase in mortality for winter, spring, summer, and fall, respectively. An analysis by geographical regions finds a strong seasonal pattern in the northeast (with a peak in summer) and little seasonal variation in the southern regions of the country. These results provide useful information for understanding particle toxicity and guiding future analyses of particle constituent data.

Modeling multilevel sleep transitional data via Poisson log-linear multilevel models

This paper proposes Poisson log-linear multilevel models to investigate population variability in sleep state transition rates. We specifically propose a Bayesian Poisson regression model that is more flexible, scalable to larger studies, and easily fit than other attempts in the literature. We further use hierarchical random effects to account for pairings of individuals and repeated measures within those individuals, as comparing diseased to non-diseased subjects while minimizing bias is of epidemiologic importance. We estimate essentially non-parametric piecewise constant hazards and smooth them, and allow for time varying covariates and segment of the night comparisons. The Bayesian Poisson regression is justified through a re-derivation of a classical algebraic likelihood equivalence of Poisson regression with a log(time) offset and survival regression assuming piecewise constant hazards. This relationship allows us to synthesize two methods currently used to analyze sleep transition phenomena: stratified multi-state proportional hazards models and log-linear models with GEE for transition counts. An example data set from the Sleep Heart Health Study is analyzed.

Bayesian Hierarchical Distributed Lag Models for Summer Ozone Exposure and Cardio-Respiratory Mortality

In this paper, we develop Bayesian hierarchical distributed lag models for estimating associations between daily variations in summer ozone levels and daily variations in cardiovascular and respiratory (CVDRESP) mortality counts for 19 U.S. large cities included in the National Morbidity Mortality Air Pollution Study (NMMAPS) for the period 1987 - 1994. At the first stage, we define a semi-parametric distributed lag Poisson regression model to estimate city-specific relative rates of CVDRESP associated with short-term exposure to summer ozone. At the second stage, we specify a class of distributions for the true city-specific relative rates to estimate an overall effect by taking into account the variability within and across cities. We perform the calculations with respect to several random effects distributions (normal, t-student, and mixture of normal), thus relaxing the common assumption of a two-stage normal-normal hierarchical model. We assess the sensitivity of the results to: 1) lag structure for ozone exposure; 2) degree of adjustment for long-term trends; 3) inclusion of other pollutants in the model;4) heat waves; 5) random effects distributions; and 6) prior hyperparameters. On average across cities, we found that a 10ppb increase in summer ozone level for every day in the previous week is associated with 1.25 percent increase in CVDRESP mortality (95% posterior regions: 0.47, 2.03). The relative rate estimates are also positive and statistically significant at lags 0, 1, and 2. We found that associations between summer ozone and CVDRESP mortality are sensitive to the confounding adjustment for PM_10, but are robust to: 1) the adjustment for long-term trends, other gaseous pollutants (NO_2, SO_2, and CO); 2) the distributional assumptions at the second stage of the hierarchical model; and 3) the prior distributions on all unknown parameters. Bayesian hierarchical distributed lag models and their application to the NMMAPS data allow us estimation of an acute health effect associated with exposure to ambient air pollution in the last few days on average across several locations. The application of these methods and the systematic assessment of the sensitivity of findings to model assumptions provide important epidemiological evidence for future air quality regulations.

ON MARGINALIZED MULTILEVEL MODELS AND THEIR COMPUTATION

Clustered data analysis is characterized by the need to describe both systematic variation in a mean model and cluster-dependent random variation in an association model. Marginalized multilevel models embrace the robustness and interpretations of a marginal mean model, while retaining the likelihood inference capabilities and flexible dependence structures of a conditional association model. Although there has been increasing recognition of the attractiveness of marginalized multilevel models, there has been a gap in their practical application arising from a lack of readily available estimation procedures. We extend the marginalized multilevel model to allow for nonlinear functions in both the mean and association aspects. We then formulate marginal models through conditional specifications to facilitate estimation with mixed model computational solutions already in place. We illustrate this approach on a cerebrovascular deficiency crossover trial.

Multilevel Latent Class Models with Dirichlet Mixing Distribution

Latent class analysis (LCA) and latent class regression (LCR) are widely used for modeling multivariate categorical outcomes in social sciences and biomedical studies. Standard analyses assume data of different respondents to be mutually independent, excluding application of the methods to familial and other designs in which participants are clustered. In this paper, we develop multilevel latent class model, in which subpopulation mixing probabilities are treated as random effects that vary among clusters according to a common Dirichlet distribution. We apply the Expectation-Maximization (EM) algorithm for model fitting by maximum likelihood (ML). This approach works well, but is computationally intensive when either the number of classes or the cluster size is large. We propose a maximum pairwise likelihood (MPL) approach via a modified EM algorithm for this case. We also show that a simple latent class analysis, combined with robust standard errors, provides another consistent, robust, but less efficient inferential procedure. Simulation studies suggest that the three methods work well in finite samples, and that the MPL estimates often enjoy comparable precision as the ML estimates. We apply our methods to the analysis of comorbid symptoms in the Obsessive Compulsive Disorder study. Our models' random effects structure has more straightforward interpretation than those of competing methods, thus should usefully augment tools available for latent class analysis of multilevel data.

LIKELIHOOD ESTIMATION OF CONJUGACY RELATIONSHIPS IN LINEAR MODELS WITH APPLICATIONS TO HIGH-THROUGHPUT GENOMICS

In the simultaneous estimation of a large number of related quantities, multilevel models provide a formal mechanism for efficiently making use of the ensemble of information for deriving individual estimates. In this article we investigate the ability of the likelihood to identify the relationship between signal and noise in multilevel linear mixed models. Specifically, we consider the ability of the likelihood to diagnose conjugacy or independence between the signals and noises. Our work was motivated by the analysis of data from high-throughput experiments in genomics. The proposed model leads to a more flexible family. However, we further demonstrate that adequately capitalizing on the benefits of a well fitting fully-specified likelihood in the terms of gene ranking is difficult.

Studying Effects of Primary Care Physicians and Patients on the Trade-Off Between Charges for Primary Care and Specialty Care Using a Hierarchical Multivariate Two-Part Model

Objective. To examine effects of primary care physicians (PCPs) and patients on the association between charges for primary care and specialty care in a point-of-service (POS) health plan. Data Source. Claims from 1996 for 3,308 adult male POS plan members, each of whom was assigned to one of the 50 family practitioner-PCPs with the largest POS plan member-loads. Study Design. A hierarchical multivariate two-part model was fitted using a Gibbs sampler to estimate PCPs' effects on patients' annual charges for two types of services, primary care and specialty care, the associations among PCPs' effects, and within-patient associations between charges for the two services. Adjusted Clinical Groups (ACGs) were used to adjust for case-mix. Principal Findings. PCPs with higher case-mix adjusted rates of specialist use were less likely to see their patients at least once during the year (estimated correlation: –.40; 95% CI: –.71, –.008) and provided fewer services to patients that they saw (estimated correlation: –.53; 95% CI: –.77, –.21). Ten of 11 PCPs whose case-mix adjusted effects on primary care charges were significantly less than or greater than zero (p < .05) had estimated, case-mix adjusted effects on specialty care charges that were of opposite sign (but not significantly different than zero). After adjustment for ACG and PCP effects, the within-patient, estimated odds ratio for any use of primary care given any use of specialty care was .57 (95% CI: .45, .73). Conclusions. PCPs and patients contributed independently to a trade-off between utilization of primary care and specialty care. The trade-off appeared to partially offset significant differences in the amount of care provided by PCPs. These findings were possible because we employed a hierarchical multivariate model rather than separate univariate models.

A BAYESIAN HIERARCHICAL MODEL FOR CONSTRAINED DISTRIBUTED LAG FUNCTIONS: ESTIMATING THE TIME COURSE OF HOSPITALIZATION ASSOCIATED WITH AIR POLLUTION EXPOSURE

Numerous time series studies have provided strong evidence of an association between increased levels of ambient air pollution and increased levels of hospital admissions, typically at 0, 1, or 2 days after an air pollution episode. An important research aim is to extend existing statistical models so that a more detailed understanding of the time course of hospitalization after exposure to air pollution can be obtained. Information about this time course, combined with prior knowledge about biological mechanisms, could provide the basis for hypotheses concerning the mechanism by which air pollution causes disease. Previous studies have identified two important methodological questions: (1) How can we estimate the shape of the distributed lag between increased air pollution exposure and increased mortality or morbidity? and (2) How should we estimate the cumulative population health risk from short-term exposure to air pollution? Distributed lag models are appropriate tools for estimating air pollution health effects that may be spread over several days. However, estimation for distributed lag models in air pollution and health applications is hampered by the substantial noise in the data and the inherently weak signal that is the target of investigation. We introduce an hierarchical Bayesian distributed lag model that incorporates prior information about the time course of pollution effects and combines information across multiple locations. The model has a connection to penalized spline smoothing using a special type of penalty matrix. We apply the model to estimating the distributed lag between exposure to particulate matter air pollution and hospitalization for cardiovascular and respiratory disease using data from a large United States air pollution and hospitalization database of Medicare enrollees in 94 counties covering the years 1999-2002.

RANDOM EFFECTS MODELS IN A META-ANALYSIS OF THE ACCURACY OF DIAGNOSTIC TESTS WITHIN A GOLD STANDARD IN THE PRESENCE OF MISSING DATA

In evaluating the accuracy of diagnosis tests, it is common to apply two imperfect tests jointly or sequentially to a study population. In a recent meta-analysis of the accuracy of microsatellite instability testing (MSI) and traditional mutation analysis (MUT) in predicting germline mutations of the mismatch repair (MMR) genes, a Bayesian approach (Chen, Watson, and Parmigiani 2005) was proposed to handle missing data resulting from partial testing and the lack of a gold standard. In this paper, we demonstrate an improved estimation of the sensitivities and specificities of MSI and MUT by using a nonlinear mixed model and a Bayesian hierarchical model, both of which account for the heterogeneity across studies through study-specific random effects. The methods can be used to estimate the accuracy of two imperfect diagnostic tests in other meta-analyses when the prevalence of disease, the sensitivities and/or the specificities of diagnostic tests are heterogeneous among studies. Furthermore, simulation studies have demonstrated the importance of carefully selecting appropriate random effects on the estimation of diagnostic accuracy measurements in this scenario.