Bayesian model averaging in time-series studies of air pollution and mortality

Extrapolating Metal (Cu, Ni, Zn) Toxicity from Individuals to Populations Across Daphnia Species Using Mechanistic Models: The Roles of Uncertainty Propagation and Combined Physiological Modes of Action

Mechanistic effect modeling is a promising tool to improve the ecological realism of environmental risk assessment. An open question for the mechanistic modeling of metal toxicity is whether the same physiological mode of action (PMoA) could be assumed for closely related species. The implications of various modeling choices, such as the use of parameter point estimates and assumption of simplistic toxicodynamic models, are largely unexplored. We conducted life-table experiments with Daphnia longispina, Daphnia magna, and Daphnia pulex exposed to the single metals Cu, Ni, and Zn, and calibrated toxicokinetic-toxicodynamic (TKTD) models based on dynamic energy budget theory. We developed TKTD models with single and combined PMoAs to compare their goodness-of-fit and predicted population-level sensitivity. We identified the PMoA reproduction efficiency as most probable in all species for Ni and Zn, but not for Cu, and found that combined-PMoA models predicted higher population-level sensitivity than single-PMoA models, which was related to the predicted individual-level sensitivity, rather than to mechanistic differences between models. Using point estimates of parameters, instead of sampling from the probability distributions of parameters, could also lead to differences in the predicted population-level sensitivity. According to model predictions, apical chronic endpoints (cumulative reproduction, survival) are conservative for single-metal population effects across metals and species. We conclude that the assumption of an identical PMoA for different species of Daphnia could be justified for Ni and Zn, but not for Cu. Single-PMoA models are more appropriate than combined-PMoA models from a model selection perspective, but propagation of the associated uncertainty should be considered. More accurate predictions of effects at low concentrations may nevertheless motivate the use of combined-PMoA models. Environ Toxicol Chem 2024;43:338-358. © 2023 The Authors. Environmental Toxicology and Chemistry published by Wiley Periodicals LLC on behalf of SETAC.

Exposure of Akwesasne Mohawk women to polychlorinated biphenyls and hexachlorobenzene is associated with increased serum levels of thyroid peroxidase autoantibodies

Persistent organic pollutants (POPs) including polychlorinated biphenyls (PCBs), hexachlorobenzene (HCB), and dichlorodiphenyltrichloroethane (p,p'-DDT) were reported to influence immunological activity. As endocrine-disrupting chemicals (EDC), these pollutants may disrupt normal thyroid function and act as catalysts for development of autoimmune thyroid disease by directly and indirectly affecting levels of thyroid peroxidase antibodies (TPOAbs). Native American communities are disproportionately exposed to harmful toxicants and are at an increased risk of developing an autoimmune disease. The aim of this study was to determine the association between POPs and TPOAbs in serum obtained from Native American women. This assessment was used to measure whether increased risk of autoimmune thyroid disease occurred as a result of exposure to POPs. Data were collected from 183 Akwesasne Mohawk women, 21-38 years of age, between 2009 and 2013. Multivariate analyses were conducted to determine the association between toxicant exposure and levels of TPOAbs. In multiple logistic regression analyses, exposure to PCB congener 33 was related to elevated risk of individuals possessing above normal levels of TPOAbs. Further, HCB was associated with more than 2-fold higher risk of possessing above normal levels of TPOAbs compared to women with normal levels of TPOAbs. p,p'-DDE was not associated with TPOAb levels within this study. Exposure to PCB congener 33 and HCB was correlated with above normal levels of TPOAbs, a marker of autoimmune thyroid disease. Additional investigations are needed to establish the causes and factors surrounding autoimmune thyroid disease which are multiple and complex.

A Bayesian joint model for compositional mediation effect selection in microbiome data

Analyzing multivariate count data generated by high-throughput sequencing technology in microbiome research studies is challenging due to the high-dimensional and compositional structure of the data and overdispersion. In practice, researchers are often interested in investigating how the microbiome may mediate the relation between an assigned treatment and an observed phenotypic response. Existing approaches designed for compositional mediation analysis are unable to simultaneously determine the presence of direct effects, relative indirect effects, and overall indirect effects, while quantifying their uncertainty. We propose a formulation of a Bayesian joint model for compositional data that allows for the identification, estimation, and uncertainty quantification of various causal estimands in high-dimensional mediation analysis. We conduct simulation studies and compare our method's mediation effects selection performance with existing methods. Finally, we apply our method to a benchmark data set investigating the sub-therapeutic antibiotic treatment effect on body weight in early-life mice.

Policy impact on microplastic reduction in China: Observation and prediction using statistical model in an intensive mariculture bay

Plastic pollution in the environment has spurred debate among scientists, policymakers, and the general public over how industrialization and consumerism are wreaking havoc on our ecosystem, but some policies might assist to ameliorate the problem in the near future. In this study, the decision tree classifier and Bayesian Structural Time Series (BSTS) model was used to anticipate the possible sources of microplastics and their near future state in 26 surface sediment and a sediment core, respectively in Sansha Bay, which has been criticized for its intensive mariculture applications. An inventory of microplastics in the sediment core was estimated, and it was discovered that during the previous six decades, an average of 181.95 tons of microplastics were deposited, with an average deposition (by a layer of sediment) of 179.44 tons/cm. According to the DT classifier, mariculture was the primary source of microplastics, whereas urban and industrial areas were the primary sources of POPs. The Bayesian Structural Time Series (BSTS) model revealed a microplastic downward slope, indicating that regional and national strategies implemented might successfully reduce microplastic pollution regionally.

A review on the application of the exposome paradigm to unveil the environmental determinants of age-related diseases

Age-related diseases account for almost half of all diseases among adults worldwide, and their incidence is substantially affected by the exposome, which is the sum of all exogenous and endogenous environmental exposures and the human body’s response to these exposures throughout the entire lifespan. Herein, we perform a comprehensive review of the epidemiological literature to determine the key elements of the exposome that affect the development of age-related diseases and the roles of aging hallmarks in this process. We find that most exposure assessments in previous aging studies have used a reductionist approach, whereby the effect of only a single environmental factor or a specific class of environmental factors on the development of age-related diseases has been examined. As such, there is a lack of a holistic and unbiased understanding of the effect of multiple environmental factors on the development of age-related diseases. To address this, we propose several research strategies based on an exposomic framework that could advance our understanding—in particular, from a mechanistic perspective—of how environmental factors affect the development of age-related diseases. We discuss the statistical methods and other methods that have been used in exposome-wide association studies, with a particular focus on multiomics technologies. We also address future challenges and opportunities in the realm of multidisciplinary approaches and genome–exposome epidemiology. Furthermore, we provide perspectives on precise public health services for vulnerable populations, public communications, the integration of risk exposure information, and the bench-to-bedside translation of research on age-related diseases.

On inferences about lag effects using lag models in air pollution time-series studies

The choice of lag length is a matter of uncertainty in air pollution time-series studies. Lag models and model selections are widely used for inferences about lag effects, but there is lack of discussion on the integration of the two. We aimed to provide theoretical discussion on the performance of lag models, and the impact of model selections on inferences about lag effects. Bias and model selections based upon information criteria, statistical significance, effect size, and model averaging were discussed in the context of lag analysis. A simulation with eight of PM_2.5-mortality relation scenarios was also conducted in order to explore the performances of lag models and to compare the model selections. The application of lag models with an insufficient lag interval taken into account (i.e. insufficient lag models) provides biased estimates. We provided features of the model selections and showed their pitfalls in lag analysis of air pollution time-series studies. We also discussed limitations of meta-analysis which fails to consider the application of different lag models in individual studies. To foster exploration on air pollution-lag-response relations with relevant tools, we encourage researchers to compare different lag models in terms of effect estimates and variance estimates, and to report their favored models and competing models together based upon scientific knowledge supporting lag-response relations.

Impact of air pollution on cause-specific mortality in Korea: Results from Bayesian Model Averaging and Principle Component Regression approaches

Health effects related to air pollution are a major global concern. Related studies based on reliable exposure assessment methods would potentially enable policy makers to propose appropriate environmental management policies. In this study, integrated Bayesian Model Averaging (BMA) and Principle Component Regression (PCR) were adopted to assess the severity of air pollution impacts on mortality related to circulatory, respiratory and skin diseases in 25 districts of Seoul, South Korea for the years 2005-2015. These methods were consistent in determining the best regression models and most important pollutants related to mortality in those highly susceptible to poor air quality. Specifically, the results demonstrated that pneumonia was highly associated with air pollution, with a large determination coefficient (BMA: 0.46, PCR: 0.51) and high model's posterior probability (0.47). The most reliable prediction model for pneumonia was indicated by the lowest Bayesian Information Criterion. Among the pollutants, particulate matter with an aerodynamic diameter of 10 μm or less (PM₁₀) was associated with serious health risks on evaluation, with the highest posterior inclusion probabilities (range, 80.20 to 100.00%) and significantly positive correlation coefficients (range, 0.14 to 0.34, p < 0.05). In addition, excessive PM₁₀ concentration (approximately 2.54 times the threshold) and a continuous increase in mortality due to respiratory diseases (approximately 1.50-fold in 10 years) were also exhibited. Overall, the results of this study suggest that currently, socio-environmental policies and international collaboration to mitigate health effects of air pollution is necessary in Seoul, Korea. Moreover, consideration of uncertainty of the regression model, which was verified in this research, will facilitate further application of this approach and enable optimal prediction of interactions between human and environmental factors.

Current approaches used in epidemiologic studies to examine short-term multipollutant air pollution exposures

PURPOSE

Air pollution epidemiology traditionally focuses on the relationship between individual air pollutants and health outcomes (e.g., mortality). To account for potential copollutant confounding, individual pollutant associations are often estimated by adjusting or controlling for other pollutants in the mixture. Recently, the need to characterize the relationship between health outcomes and the larger multipollutant mixture has been emphasized in an attempt to better protect public health and inform more sustainable air quality management decisions.

METHODS

New and innovative statistical methods to examine multipollutant exposures were identified through a broad literature search, with a specific focus on those statistical approaches currently used in epidemiologic studies of short-term exposures to criteria air pollutants (i.e., particulate matter, carbon monoxide, sulfur dioxide, nitrogen dioxide, and ozone).

RESULTS

Five broad classes of statistical approaches were identified for examining associations between short-term multipollutant exposures and health outcomes, specifically additive main effects, effect measure modification, unsupervised dimension reduction, supervised dimension reduction, and nonparametric methods. These approaches are characterized including advantages and limitations in different epidemiologic scenarios.

DISCUSSION

By highlighting the characteristics of various studies in which multipollutant statistical methods have been used, this review provides epidemiologists and biostatisticians with a resource to aid in the selection of the most optimal statistical method to use when examining multipollutant exposures.

Considerations for assessing model averaging of regression coefficients

Model choice is usually an inevitable source of uncertainty in model-based statistical analyses. While the focus of model choice was traditionally on methods for choosing a single model, methods to formally account for multiple models within a single analysis are now accessible to many researchers. The specific technique of model averaging was developed to improve predictive ability by combining predictions from a set of models. However, it is now often used to average regression coefficients across multiple models with the ultimate goal of capturing a variable's overall effect. This use of model averaging implicitly assumes the same parameter exists across models so that averaging is sensible. While this assumption may initially seem tenable, regression coefficients associated with particular explanatory variables may not hold equivalent interpretations across all of the models in which they appear, making explanatory inference about covariates challenging. Accessibility to easily implementable software, concerns about being criticized for ignoring model uncertainty, and the chance to avoid having to justify choice of a final model have all led to the increasing popularity of model averaging in practice. We see a gap between the theoretical development of model averaging and its current use in practice, potentially leaving well-intentioned researchers with unclear inferences or difficulties justifying reasons for using (or not using) model averaging. We attempt to narrow this gap by revisiting some relevant foundations of regression modeling, suggesting more explicit notation and graphical tools, and discussing how individual model results are combined to obtain a model averaged result. Our goal is to help researchers make informed decisions about model averaging and to encourage question-focused modeling over method-focused modeling.

Analysis of correlation between pediatric asthma exacerbation and exposure to pollutant mixtures with association rule mining

OBJECTIVES

Traditional studies on effects of outdoor pollution on asthma have been criticized for questionable statistical validity and inefficacy in exploring the effects of multiple air pollutants, alone and in combination. Association rule mining (ARM), a method easily interpretable and suitable for the analysis of the effects of multiple exposures, could be of use, but the traditional interest metrics of support and confidence need to be substituted with metrics that focus on risk variations caused by different exposures.

METHODS

We present an ARM-based methodology that produces rules associated with relevant odds ratios and limits the number of final rules even at very low support levels (0.5%), thanks to post-pruning criteria that limit rule redundancy and control for statistical significance. The methodology has been applied to a case-crossover study to explore the effects of multiple air pollutants on risk of asthma in pediatric subjects.

RESULTS

We identified 27 rules with interesting odds ratio among more than 10,000 having the required support. The only rule including only one chemical is exposure to ozone on the previous day of the reported asthma attack (OR=1.14). 26 combinatory rules highlight the limitations of air quality policies based on single pollutant thresholds and suggest that exposure to mixtures of chemicals is more harmful, with odds ratio as high as 1.54 (associated with the combination day0 SO₂, day0 NO, day0 NO₂, day1 PM).

CONCLUSIONS

The proposed method can be used to analyze risk variations caused by single and multiple exposures. The method is reliable and requires fewer assumptions on the data than parametric approaches. Rules including more than one pollutant highlight interactions that deserve further investigation, while helping to limit the search field.

Spatial variable selection methods for investigating acute health effects of fine particulate matter components

Multi-site time series studies have reported evidence of an association between short term exposure to particulate matter (PM) and adverse health effects, but the effect size varies across the United States. Variability in the effect may partially be due to differing community level exposure and health characteristics, but also due to the chemical composition of PM which is known to vary greatly by location and time. The objective of this article is to identify particularly harmful components of this chemical mixture. Because of the large number of highly-correlated components, we must incorporate some regularization into a statistical model. We assume that, at each spatial location, the regression coefficients come from a mixture model with the flavor of stochastic search variable selection, but utilize a copula to share information about variable inclusion and effect magnitude across locations. The model differs from current spatial variable selection techniques by accommodating both local and global variable selection. The model is used to study the association between fine PM (PM <2.5μm) components, measured at 115 counties nationally over the period 2000-2008, and cardiovascular emergency room admissions among Medicare patients.

Classification and regression trees for epidemiologic research: an air pollution example

BACKGROUND

Identifying and characterizing how mixtures of exposures are associated with health endpoints is challenging. We demonstrate how classification and regression trees can be used to generate hypotheses regarding joint effects from exposure mixtures.

METHODS

We illustrate the approach by investigating the joint effects of CO, NO2, O3, and PM2.5 on emergency department visits for pediatric asthma in Atlanta, Georgia. Pollutant concentrations were categorized as quartiles. Days when all pollutants were in the lowest quartile were held out as the referent group (n = 131) and the remaining 3,879 days were used to estimate the regression tree. Pollutants were parameterized as dichotomous variables representing each ordinal split of the quartiles (e.g. comparing CO quartile 1 vs. CO quartiles 2-4) and considered one at a time in a Poisson case-crossover model with control for confounding. The pollutant-split resulting in the smallest P-value was selected as the first split and the dataset was partitioned accordingly. This process repeated for each subset of the data until the P-values for the remaining splits were not below a given alpha, resulting in the formation of a "terminal node". We used the case-crossover model to estimate the adjusted risk ratio for each terminal node compared to the referent group, as well as the likelihood ratio test for the inclusion of the terminal nodes in the final model.

RESULTS

The largest risk ratio corresponded to days when PM2.5 was in the highest quartile and NO2 was in the lowest two quartiles (RR: 1.10, 95% CI: 1.05, 1.16). A simultaneous Wald test for the inclusion of all terminal nodes in the model was significant, with a chi-square statistic of 34.3 (p = 0.001, with 13 degrees of freedom).

CONCLUSIONS

Regression trees can be used to hypothesize about joint effects of exposure mixtures and may be particularly useful in the field of air pollution epidemiology for gaining a better understanding of complex multipollutant exposures.

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.

Estimating the health effects of exposure to multi-pollutant mixture

PURPOSE

Air pollution constitutes a major public health concern because of its ubiquity and of its potential health impact. Because individuals are exposed to many air pollutants at once that are highly correlated with each other, there is a need to consider the multi-pollutant exposure phenomenon. The characteristics of multiple pollutants that make statistical analysis of health-related effects of air pollution complex include the high correlation between pollutants prevents the use of standard statistical methods, the potential existence of interaction between pollutants, the common measurement errors, the importance of the number of pollutants to consider, and the potential nonlinear relationship between exposure and health.

METHODS

We made a review of statistical methods either used in the literature to study the effect of multiple pollutants or identified as potentially applicable to this problem. We reported the results of investigations that applied such methods.

RESULTS

Eighteen publications have investigated the multi-pollutant effects, 5 on indoor pollution, 10 on outdoor pollution, and 3 on statistical methodology with application on outdoor pollution. Some other publications have only addressed statistical methodology.

CONCLUSIONS

The use of Hierarchical Bayesian approach, dimension reduction methods, clustering, recursive partitioning, and logic regression are some potential methods described. Methods that provide figures for risk assessments should be put forward in public health decisions.

Reassessing the human health benefits from cleaner air

Recent proposals to further reduce permitted levels of air pollution emissions are supported by high projected values of resulting public health benefits. For example, the Environmental Protection Agency recently estimated that the 1990 Clean Air Act Amendment (CAAA) will produce human health benefits in 2020, from reduced mortality rates, valued at nearly $2 trillion per year, compared to compliance costs of $65 billion ($0.065 trillion). However, while compliance costs can be measured, health benefits are unproved: they depend on a series of uncertain assumptions. Among these are that additional life expectancy gained by a beneficiary (with median age of about 80 years) should be valued at about $80,000 per month; that there is a 100% probability that a positive, linear, no-threshold, causal relation exists between PM(2.5) concentration and mortality risk; and that progress in medicine and disease prevention will not greatly diminish this relationship. We present an alternative uncertainty analysis that assigns a positive probability of error to each assumption. This discrete uncertainty analysis suggests (with probability >90% under plausible alternative assumptions) that the costs of CAAA exceed its benefits. Thus, instead of suggesting to policymakers that CAAA benefits are almost certainly far larger than its costs, we believe that accuracy requires acknowledging that the costs purchase a relatively uncertain, possibly much smaller, benefit. The difference between these contrasting conclusions is driven by different approaches to uncertainty analysis, that is, excluding or including discrete uncertainties about the main assumptions required for nonzero health benefits to exist at all.

A Bayesian model averaging approach for estimating the relative risk of mortality associated with heat waves in 105 U.S. cities

Estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change. In this article, we propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct Bayesian model averaging (BMA) to account for the multiplicity of potential models. Applying these methods to data from 105 U.S. cities for the period 1987-2005, we identify those cities having a high posterior probability of increased mortality risk during heat waves, examine the heterogeneity of the posterior distributions of mortality risk across cities, assess sensitivity of the results to the selection of prior distributions, and compare our BMA results to a model selection approach. Our results show that no single model best predicts risk across the majority of cities, and that for some cities heat-wave risk estimation is sensitive to model choice. Although model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection, we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models.

Bootstrap-after-bootstrap model averaging for reducing model uncertainty in model selection for air pollution mortality studies

BACKGROUND

Concerns have been raised about findings of associations between particulate matter (PM) air pollution and mortality that have been based on a single "best" model arising from a model selection procedure, because such a strategy may ignore model uncertainty inherently involved in searching through a set of candidate models to find the best model. Model averaging has been proposed as a method of allowing for model uncertainty in this context.

OBJECTIVES

To propose an extension (double BOOT) to a previously described bootstrap model-averaging procedure (BOOT) for use in time series studies of the association between PM and mortality. We compared double BOOT and BOOT with Bayesian model averaging (BMA) and a standard method of model selection [standard Akaike's information criterion (AIC)].

METHOD

Actual time series data from the United States are used to conduct a simulation study to compare and contrast the performance of double BOOT, BOOT, BMA, and standard AIC.

RESULTS

Double BOOT produced estimates of the effect of PM on mortality that have had smaller root mean squared error than did those produced by BOOT, BMA, and standard AIC. This performance boost resulted from estimates produced by double BOOT having smaller variance than those produced by BOOT and BMA.

CONCLUSIONS

Double BOOT is a viable alternative to BOOT and BMA for producing estimates of the mortality effect of PM.

Model selection and health effect estimation in environmental epidemiology

In air pollution epidemiology, improvements in statistical analysis tools can help improve signal-to-noise ratios, and untangle large correlations between exposures and confounders. For this reason, we welcome a novel model-selection approach that helps to identify the time-windows of exposure to pollutants that produces adverse health effects. However, there are concerns about approaches that select a model based on a given data set, and then estimate health effects in the same data. This can create problems when (1) the sample size is small in relation to the magnitude of the health effects; and (2) candidate predictors are highly correlated and likely to have similar effects. Bayesian Model Averaging has been advocated as a way to estimate health effects that accounts for model uncertainty. However, implementations where posterior model probabilities are approximated using BIC, as well as other default choices, may not reflect the ability of each model to provide an estimate of the health effect that is properly adjusted for confounding. Air pollution studies need to focus on estimating health effects while accounting for the uncertainty in the adjustment for confounding factors. This is true especially when model choice and estimation are performed on the same data. The development of appropriate statistical tools remains an open area of investigation.

Panel discussion review: session three--issues involved in interpretation of epidemiologic analyses--statistical modeling

The Clean Air Act mandates that the US Environmental Protection Agency (EPA) develop National Ambient Air Quality Standards for criteria air pollutants and conduct periodic reviews of the standards based on new scientific evidence. In recent reviews, evidence from epidemiologic studies has played a key role. Epidemiologic studies often provide evidence for effects of several air pollutants. Determining whether there are independent effects of the separate pollutants is a challenge. Among the many issues confronting the interpretation of epidemiologic studies of multi-pollutant exposures and health effects are those specifically related to statistical modeling. The EPA convened a workshop on 13 and 14 December 2006 in Chapel Hill, North Carolina, USA, to discuss these and other issues; Session Three of the workshop was devoted specifically to statistical modeling. Prominent statistical modeling issues in epidemiologic studies of air pollution include (1) measurement error across the co-pollutants; (2) correlation and multi-collinearity among the co-pollutants; (3) the timing of the concentration-response function; (4) confounding; and (5) spatial analyses.

Viewpoint: using gene-environment interactions to dissect the effects of complex mixtures

Teasing out the health effects of constituents of complex mixtures poses formidable statistical challenges owing to the problem of multicollinearity. While statistical devices such as regression on principal components, model selection, and model averaging offer some approaches to this problem, incorporation of external information is likely to be more helpful. I explore a general hierarchical modeling framework that would allow such information as sources, genetic interactions, and toxicology to be included in the higher levels of the model.