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Background: The postnatal period is a critical period for both mothers and their newborns for their health. Lack of early postnatal care (PNC) services during a 2-day period is a life-threatening situation for both the mother and the babies. However, no data have been examined for PNCs in East Africa. Hence, using the more flexible Bayesian multilevel modeling approach, this study aims to investigate the pooled prevalence and potential factors for PNC utilization among women after delivery in East African countries., Methods: We retrieved secondary data from the Kids Record (KR) demographic and health surveys (DHS) data from 2015 to 2022 from 10 East African countries. A total of 77,052 weighted women were included in the study. We used R 4.3.2 software for analysis. We fitted Bayesian multilevel logistic regression models. Techniques such as Rhat, effective sample size, density, time series, autocorrelation plots, widely applicable information criterion (WAIC), deviance information criterion (DIC), and Markov Chain Monte-Carlo (MCMC) simulation were used to estimate the model parameters using Hamiltonian Monte-Carlo (HMC) and its extensions, No-U-Turn Sampler (NUTS) techniques. An adjusted odds ratio (AOR) with a 95% credible interval (CrI) in the multivariable model to select variables that have a significant association with PNC was used., Results: The overall pooled prevalence of PNC within 48 hrs. of delivery was about 52% (95% CrI: 39, 66). A higher rate of PNC usage was observed among women aged 25-34 years (AOR = 1.21; 95% CrI: 1.15, 1.27) and 35-49-years (AOR = 1.61; 95% CrI: 1.5, 1.72) as compared to women aged 15-24 years; similarly, women who had achieved primary education (AOR = 1.96; 95% CrI: 1.88, 2.05) and secondary/higher education (AOR = 3.19; 95% CrI: 3.03, 3.36) as compared to uneducated women; divorced or widowed women (AOR = 0.83; 95% CrI: 0.77, 0.89); women who had currently working status (AOR = 0.9; 95% CrI: 0.87, 0.93); poorer women (AOR = 0.88; 95% CrI: 0.84, 0.92), middle-class women (AOR = 0.83; 95% CrI: 0.79, 0.87), richer women (AOR = 0.77; 95% CrI: 0.73, 0.81), and richest women (AOR = 0.59; 95% CrI: 0.55, 0.63) as compared to the poorest women; women who had media exposure (AOR = 1.32; 95% CrI: 1.27, 1.36), were having 3-5 children (AOR = 0.89; 95% CrI: 0.84, 0.94), had >5 children (AOR = 0.69; 95% CrI: 0.64, 0.75), had first birth at age < 20 years (AOR = 0.82; 95% CrI: 0.79, 0.84), had at least one ANC visit (AOR = 1.93; 95% CrI: 1.8, 2.08), delivered at health facilities (AOR = 2.57; 95% CrI: 2.46, 2.68), had average birth size (AOR = 0.94; 95% CrI: 0.91, 0.98) and small birth size child (AOR = 0.88; 95% CrI: 0.84, 0.92), had twin newborns (AOR = 1.15; 95% CrI: 1.02, 1.3), and fourth and above birth order (AOR = 0.88; 95% CrI: 0.82, 0.95) were individual-driven women who have been independently associated with PNC, respectively. Regarding community-level variables, rural women (AOR = 0.76; 95% CrI: 0.72, 0.79), high media exposure communities (AOR = 1.1; 95% CrI: 1.04, 1.18), communities with high wealth levels (AOR = 0.88 95% CrI: 0.83, 0.94), communities with high antenatal care (ANC) utilization (AOR = 1.13, 95% CrI: 1.07, 1.19), and long distance to health facilities (AOR = 1.5; 95% CrI: 1.38, 1.63) were among the community factors associated with PNC, respectively., Conclusion: One of the significant public health priorities in East Africa continues to be the underutilization of immediate PNC. The government ought to prioritize improving maternity and child health services, collaborating with interested parties in the area, reducing health disparities, educating mothers about child health, and other connected issues that are very beneficial., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2024 Terefe, Asgedom, Arage, Aychiluhm and Ayele.)

In conventional subgroup analyses, subgroup treatment effects are estimated using data from each subgroup separately without considering data from other subgroups in the same study. The subgroup treatment effects estimated this way may be heterogenous with high variability due to small sample sizes in some subgroups and much different from the treatment effect in the overall population. A Bayesian hierarchical model (BHM) can be used to derive more precise, and less heterogenous estimates of subgroup treatment effects that are closer to the treatment effect in the overall population. BHM assumes exchangeability in treatment effect across subgroups after adjusting for effect modifiers and other relevant covariates. In this article, we will discuss the technical details for applying one-way and multi-way BHM using summary-level statistics, and patient-level data for subgroup analysis. Four case studies based on four new drug applications are used to illustrate the application of these models in subgroup analyses for continuous, dichotomous, time-to-event, and count endpoints., (© 2024 John Wiley & Sons Ltd.)

Optimizing doses for multiple indications is challenging. The pooled approach of finding a single optimal biological dose (OBD) for all indications ignores that dose-response or dose-toxicity curves may differ between indications, resulting in varying OBDs. Conversely, indication-specific dose optimization often requires a large sample size. To address this challenge, we propose a Randomized two-stage basket trial design that Optimizes doses in Multiple Indications (ROMI). In stage 1, for each indication, response and toxicity are evaluated for a high dose, which may be a previously obtained maximum tolerated dose, with a rule that stops accrual to indications where the high dose is unsafe or ineffective. Indications not terminated proceed to stage 2, where patients are randomized between the high dose and a specified lower dose. A latent-cluster Bayesian hierarchical model is employed to borrow information between indications, while considering the potential heterogeneity of OBD across indications. Indication-specific utilities are used to quantify response-toxicity trade-offs. At the end of stage 2, for each indication with at least one acceptable dose, the dose with highest posterior mean utility is selected as optimal. Two versions of ROMI are presented, one using only stage 2 data for dose optimization and the other optimizing doses using data from both stages. Simulations show that both versions have desirable operating characteristics compared to designs that either ignore indications or optimize dose independently for each indication., (© The Author(s) 2024. Published by Oxford University Press on behalf of The International Biometric Society.)

Although transcriptomics data is typically used to analyze mature spliced mRNA, recent attention has focused on jointly investigating spliced and unspliced (or precursor-) mRNA, which can be used to study gene regulation and changes in gene expression production. Nonetheless, most methods for spliced/unspliced inference (such as RNA velocity tools) focus on individual samples, and rarely allow comparisons between groups of samples (e.g. healthy vs. diseased). Furthermore, this kind of inference is challenging, because spliced and unspliced mRNA abundance is characterized by a high degree of quantification uncertainty, due to the prevalence of multi-mapping reads, ie reads compatible with multiple transcripts (or genes), and/or with both their spliced and unspliced versions. Here, we present DifferentialRegulation, a Bayesian hierarchical method to discover changes between experimental conditions with respect to the relative abundance of unspliced mRNA (over the total mRNA). We model the quantification uncertainty via a latent variable approach, where reads are allocated to their gene/transcript of origin, and to the respective splice version. We designed several benchmarks where our approach shows good performance, in terms of sensitivity and error control, vs. state-of-the-art competitors. Importantly, our tool is flexible, and works with both bulk and single-cell RNA-sequencing data. DifferentialRegulation is distributed as a Bioconductor R package., (© The Author(s) 2024. Published by Oxford University Press.)

Factors contributing to social inequalities are associated with negative mental health outcomes and disparities in mental well-being. We propose a Bayesian hierarchical controlled interrupted time series to evaluate the impact of policies on population well-being whilst accounting for spatial and temporal patterns. Using data from the UKs Household Longitudinal Study, we apply this framework to evaluate the impact of the UKs welfare reform implemented in the 2010s on the mental health of the participants, measured using the GHQ-12 index. Our findings indicate that the reform led to a 2.36% (95% CrI: 0.57%-4.37%) increase in the national GHQ-12 index in the exposed group, after adjustment for the control group. Moreover, the geographical areas that experienced the largest increase in the GHQ-12 index are from more disadvantage backgrounds than affluent backgrounds., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 The Author(s). Published by Elsevier Ltd.. All rights reserved.)

Multi-gene panel testing allows many cancer susceptibility genes to be tested quickly at a lower cost making such testing accessible to a broader population. Thus, more patients carrying pathogenic germline mutations in various cancer-susceptibility genes are being identified. This creates a great opportunity, as well as an urgent need, to counsel these patients about appropriate risk-reducing management strategies. Counseling hinges on accurate estimates of age-specific risks of developing various cancers associated with mutations in a specific gene, ie, penetrance estimation. We propose a meta-analysis approach based on a Bayesian hierarchical random-effects model to obtain penetrance estimates by integrating studies reporting different types of risk measures (eg, penetrance, relative risk, odds ratio) while accounting for the associated uncertainties. After estimating posterior distributions of the parameters via a Markov chain Monte Carlo algorithm, we estimate penetrance and credible intervals. We investigate the proposed method and compare with an existing approach via simulations based on studies reporting risks for two moderate-risk breast cancer susceptibility genes, ATM and PALB2. Our proposed method is far superior in terms of coverage probability of credible intervals and mean square error of estimates. Finally, we apply our method to estimate the penetrance of breast cancer among carriers of pathogenic mutations in the ATM gene., (© The Author(s) 2024. Published by Oxford University Press on behalf of The International Biometric Society.)

Probabilistic approaches to hazard assessment use species sensitivity distributions (SSDs) to characterize hazard for toxicants exposure for different species within a community. Many of the assumptions at the core of SSDs are unrealistic, among them the assumption that the tolerance levels of all species in a specific ecological community are a priori exchangeable for each new toxic substance. Here we propose the use of a particular test to detect situations where such an assumption is violated. Then, a new method based on non-nested random effects model is required to identify novel SSDs capable of taking into account species non-exchangeability. Credible intervals, representing SSD uncertainty, could be determined based on our procedure. This leads to new and reliable estimates of the environmental hazard. We present a Bayesian modeling approach to address model inference issues, using Markov chain Monte Carlo sampling., (© 2021 Wiley-VCH GmbH.)

Severe vehicle crashes have resulted in a large-scale social and economic loss. As a result, the reduction of those crashes has become one of the key objectives of policy makers. Although traffic barriers have been utilized to reduce the run-of-road crash severity, still those crashes account for a high number of severe crashes. Previous studies of traffic barrier crashes often either ignore the heterogeneity across different traffic barrier types, or just focus on specific types of traffic barriers. Thus, this study developed a Bayesian Hierarchical model (BHM) to identify the contributory factors impacting the severity of traffic barrier crashes while accounting for that heterogeneity. The assessment of model fit, inter-class correlation (ICC) coefficient, and deviance information criterion (DIC) all favoured the use of the BHM. Besides accounting for the heterogeneity between barrier types, the interaction across variables shoulder width and traffic barrier height were incorporated into the analysis. Due to the differences in traffic barrier design and vehicle performance across different roadway classifications, only Wyoming interstate traffic barrier were considered. Results indicated that there is an important interaction term between traffic barrier height and shoulder width so that the impact of these two predictors should not be separated. In addition, having a citation record, negotiating a curve, being a female driver, non-speed compliance, alcohol involvement, and showing emotional signs at the time of crash were factors increasing the severity of traffic barrier crashes. On the other hand, having a turn before hitting a traffic barrier, being a younger driver, and driving in adverse weather conditions were factors that significantly decrease the severity of traffic barrier crashes.

Objectives: Existing Canadian social determinants of health (SDOH) indicators do not quantify uncertainty to identify priority areas. The objectives of this methodologic study were: (1) to estimate and map small area (dissemination area) shared and variable-specific SDOH indicators with measures of uncertainty using a Bayesian model that accounts for spatial dependence; (2) to quantify geographic variation in the SDOH indicators and their contribution to a shared indicator; and (3) to assess the SDOH indicators' associations with behavioural risk factors and their consistency with the Ontario Marginalization Index (ON-Marg)., Methods: Lower education-, income-, unemployment-, living alone- and visible minority-related variables used in existing Canadian SDOH indices were fit as dependent variables to a Bayesian model to produce area-based SDOH indicators that were mapped with measures of uncertainty in two study areas. The fractions of spatial variation explained by the model components were computed. Bayesian analysis of variance was used to examine the SDOH indicator associations with behavioural risk factors and their consistency with ON-Marg examined using Pearson's correlation coefficient., Results: The shared component was strongly associated with material deprivation (i.e., income) in each study area; however, variable-specific SDOH indicators were important too. The SDOH indicators were associated with behavioural risk factors for chronic disease, particularly alcohol consumption and smoking, and the shared component estimates were consistent with the ON-Marg material deprivation., Conclusions: The Bayesian approach to produce SDOH indicators met the three study objectives and as such provides a new approach to prioritize areas that may experience health inequalities.

Spatio-temporal data in sport is increasing rapidly, however suitable statistical methods for analysing this data are underdeveloped. The current study establishes the need for spatial statistical methods, propose a Bayesian hierarchical model as an appropriate method for comparing spatial variables, and test this model across three spatial scales. The need for spatial statistical methods was established through the identification of spatial autocorrelation. This necessitated the use of a Bayesian hierarchical model to test for an association between spatial ball movement entropy and spatial effectiveness. Posterior distribution results showed a generally positive association such that increases in entropy were associated with increases in effectiveness. The strength and confidence of the associations were impacted by the spatial scale, with the 6 × 6 grid showing the most conclusive evidence of a positive relationship; the 4 × 4 grid was mostly positive, however with a large variation; and finally, the basket-centric scale results were less conclusive. The results of the current study demonstrate the suitability of a Bayesian hierarchical model for testing for associations or differences between spatial variables. With the increase in spatial analyses in sport, this study presents an appropriate statistical method for dealing with complex problems associated with spatial analyses.

Before a first-in-man trial is conducted, preclinical studies are performed in animals to help characterise the safety profile of the new medicine. We propose a robust Bayesian hierarchical model to synthesise animal and human toxicity data, using scaling factors to translate doses administered to different animal species onto an equivalent human scale. After scaling doses, the parameters of dose-toxicity models intrinsic to different animal species can be interpreted on a common scale. A prior distribution is specified for each translation factor to capture uncertainty about differences between toxicity of the drug in animals and humans. Information from animals can then be leveraged to learn about the relationship between dose and risk of toxicity in a new phase I trial in humans. The model allows human dose-toxicity parameters to be exchangeable with the study-specific parameters of animal species studied so far or non-exchangeable with any of them. This leads to robust inferences, enabling the model to give greatest weight to the animal data with parameters most consistent with human parameters or discount all animal data in the case of non-exchangeability. The proposed model is illustrated using a case study and simulations. Numerical results suggest that our proposal improves the precision of estimates of the toxicity rates when animal and human data are consistent, while it discounts animal data in cases of inconsistency.

One difficulty for real-time tracking of epidemics is related to reporting delay. The reporting delay may be due to laboratory confirmation, logistical problems, infrastructure difficulties, and so on. The ability to correct the available information as quickly as possible is crucial, in terms of decision making such as issuing warnings to the public and local authorities. A Bayesian hierarchical modelling approach is proposed as a flexible way of correcting the reporting delays and to quantify the associated uncertainty. Implementation of the model is fast due to the use of the integrated nested Laplace approximation. The approach is illustrated on dengue fever incidence data in Rio de Janeiro, and severe acute respiratory infection data in the state of Paraná, Brazil., (© 2019 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.)

We analyze time-course protein activation data to track the changes in protein expression over time after exposure to drugs such as protein inhibitors. Protein expression is expected to change over time in response to the intervention in different ways due to biological pathways. We therefore allow for clusters of proteins with different treatment effects, and allow these clusters to change over time. As the effect of the drug wears off, protein expression may revert back to the level before treatment. In addition, different drugs, doses, and cell lines may have different effects in altering the protein expression. To model and understand this process we develop random partitions that define a refinement and coagulation of protein clusters over time. We demonstrate the approach using a time-course reverse phase protein array (RPPA) dataset consisting of protein expression measurements under different drugs, dose levels, and cell lines. The proposed model can be applied in general to time-course data where clustering of the experimental units is expected to change over time in a sequence of refinement and coagulation., (© 2019 International Biometric Society.)

Noncompliance to assigned treatment is a common challenge in analysis and interpretation of randomized clinical trials. The complier average causal effect (CACE) approach provides a useful tool for addressing noncompliance, where CACE is defined as the average difference in potential outcomes for the response in the subpopulation of subjects who comply with their assigned treatments. In this article, we present a Bayesian hierarchical model to estimate the CACE in a meta-analysis of randomized clinical trials where compliance may be heterogeneous between studies. Between-study heterogeneity is taken into account with study-specific random effects. The results are illustrated by a re-analysis of a meta-analysis comparing the effect of epidural analgesia in labor versus no or other analgesia in labor on the outcome cesarean section, where noncompliance varied between studies. Finally, we present simulations evaluating the performance of the proposed approach and illustrate the importance of including appropriate random effects and the impact of over- and under-fitting., (© 2019 International Biometric Society.)

Combining different treatment regimens provides an effective approach to induce a synergistic treatment effect and overcome resistance to monotherapy. The challenge is that, given the large number of existing monotherapies, the number of possible combinations is huge and new potentially more efficacious compounds may become available any time during drug development. To address this challenge, we propose a flexible Bayesian drug combination platform design with adaptive shrinkage (ComPAS), which allows for dropping futile combinations, graduating effective combinations, and adding new combinations during the course of the trial. A new adaptive shrinkage method is developed to adaptively borrow information across combinations and efficiently identify the efficacious combinations based on Bayesian model selection and hierarchical models. Simulation studies show that ComPAS identifies the effective combinations with higher probability than some existing designs. ComPAS provides an efficient and flexible platform to accelerate drug development in a seamless and timely fashion., (© 2018 John Wiley & Sons, Ltd.)

Background: Global monitoring efforts have relied on national estimates of modern contraceptive prevalence rate (mCPR) for many low-income countries. However, most contraceptive delivery programs are implemented by health departments at lower administrative levels, reflecting a persisting gap between the availability of and need for subnational mCPR estimates., Methods: Using woman-level data from multiple semi-annual national survey rounds conducted between 2013 and 2016 in five sub-Saharan African countries (Burkina Faso, Ethiopia, Ghana, Kenya, and Uganda) by the Performance, Monitoring and Accountability 2020 project, we propose a Bayesian Hierarchical Model with a standard set of covariates and temporally correlated random effects to estimate the level and trend of mCPR for first level administrative divisions in each country., Results: There is considerable narrowing of the uncertainty interval (UI) around the model-based estimates, compared to the estimates directly based on the survey data. We find substantial variations in the estimated subnational mCPRs. Uganda, for example, shows a gain in mCPR of 6.4% (95% UI: 4.5-8.3) based on model estimates of 20.9% (19.6-22.2) in mid-2014 and 27.3% (26.0-28.8) in mid-2016, with change across 10 regions ranging from - 0.6 points in Karamoja to 9.4 points in Central 2 region. The lower bound of the UIs of the change over four rounds was above 0 in 6 regions. Similar upward trends are observed for most regions in the other four countries, and there is noticeable within-country geographic variation., Conclusions: Reliable subnational estimates of mCPR empower health departments in evidence-based policy making. Despite nationally increasing mCPRs, regional disparities exist within countries suggesting uneven contraceptive access. Raising investments in disadvantaged areas may be warranted to increase equity in access to modern contraceptive methods.

Precision medicine endeavors to conform therapeutic interventions to the individuals being treated. Implicit to the concept of precision medicine is heterogeneity of treatment benefit among patients and patient subpopulations. Thus, precision medicine challenges conventional paradigms of clinical translational which have relied on estimates of population-averaged effects to guide clinical practice. Basket trials comprise a class of experimental designs used to study solid malignancies that are devised to evaluate the effectiveness of a therapeutic strategy among patients defined by the presence of a particular drug target (often a genetic mutation) rather than a particular tumor histology. Acknowledging the potential for differential effectiveness on the basis of traditional criteria for cancer subtyping, evaluations of treatment effectiveness are conducted with respect to the "baskets" which collectively represent a partition of the targeted patient population consisting of discrete subtypes. Yet, designs of early basket trials have been criticized for their reliance on basketwise analysis strategies that suffered from limited power in the presence of imbalanced enrollment as well as failed to convey to the clinical community evidentiary measures for consistent effectiveness among the studied clinical subtypes. This article presents novel methodology for sequential basket trial design formulated with Bayesian monitoring rules. Interim analyses are based a novel hierarchical modeling strategy for sharing information among a collection of discrete potentially nonexchangeable subtypes. The methodology is demonstrated by analysis as well as permutation and simulation studies based on a recent basket trial designed to estimate the effectiveness of vemurafenib in BRAF V600 mutant non-melanoma among six primary disease sites and histologies., (© 2018 John Wiley & Sons, Ltd.)

Background: Although human leukocyte antigen (HLA) genotyping based on amplicon, whole exome sequence (WES), and RNA sequence data has been achieved in recent years, accurate genotyping from whole genome sequence (WGS) data remains a challenge due to the low depth. Furthermore, there is no method to identify the sequences of unknown HLA types not registered in HLA databases., Results: We developed a Bayesian model, called ALPHLARD, that collects reads potentially generated from HLA genes and accurately determines a pair of HLA types for each of HLA-A, -B, -C, -DPA1, -DPB1, -DQA1, -DQB1, and -DRB1 genes at 3rd field resolution. Furthermore, ALPHLARD can detect rare germline variants not stored in HLA databases and call somatic mutations from paired normal and tumor sequence data. We illustrate the capability of ALPHLARD using 253 WES data and 25 WGS data from Illumina platforms. By comparing the results of HLA genotyping from SBT and amplicon sequencing methods, ALPHLARD achieved 98.8% for WES data and 98.5% for WGS data at 2nd field resolution. We also detected three somatic point mutations and one case of loss of heterozygosity in the HLA genes from the WGS data., Conclusions: ALPHLARD showed good performance for HLA genotyping even from low-coverage data. It also has a potential to detect rare germline variants and somatic mutations in HLA genes. It would help to fill in the current gaps in HLA reference databases and unveil the immunological significance of somatic mutations identified in HLA genes.

Drug self-administration experiments are a frequently used approach to assessing the abuse liability and reinforcing property of a compound. It has been used to assess the abuse liabilities of various substances such as psychomotor stimulants and hallucinogens, food, nicotine, and alcohol. The demand curve generated from a self-administration study describes how demand of a drug or non-drug reinforcer varies as a function of price. With the approval of the 2009 Family Smoking Prevention and Tobacco Control Act, demand curve analysis provides crucial evidence to inform the US Food and Drug Administration's policy on tobacco regulation, because it produces several important quantitative measurements to assess the reinforcing strength of nicotine. The conventional approach popularly used to analyze the demand curve data is individual-specific non-linear least square regression. The non-linear least square approach sets out to minimize the residual sum of squares for each subject in the dataset; however, this one-subject-at-a-time approach does not allow for the estimation of between- and within-subject variability in a unified model framework. In this paper, we review the existing approaches to analyze the demand curve data, non-linear least square regression, and the mixed effects regression and propose a new Bayesian hierarchical model. We conduct simulation analyses to compare the performance of these three approaches and illustrate the proposed approaches in a case study of nicotine self-administration in rats. We present simulation results and discuss the benefits of using the proposed approaches.

Background: The basket trial evaluates the treatment effect of a targeted therapy in patients with the same genetic or molecular aberration, regardless of their cancer types. Bayesian hierarchical modeling has been proposed to adaptively borrow information across cancer types to improve the statistical power of basket trials. Although conceptually attractive, research has shown that Bayesian hierarchical models cannot appropriately determine the degree of information borrowing and may lead to substantially inflated type I error rates., Methods: We propose a novel calibrated Bayesian hierarchical model approach to evaluate the treatment effect in basket trials. In our approach, the shrinkage parameter that controls information borrowing is not regarded as an unknown parameter. Instead, it is defined as a function of a similarity measure of the treatment effect across tumor subgroups. The key is that the function is calibrated using simulation such that information is strongly borrowed across subgroups if their treatment effects are similar and barely borrowed if the treatment effects are heterogeneous., Results: The simulation study shows that our method has substantially better controlled type I error rates than the Bayesian hierarchical model. In some scenarios, for example, when the true response rate is between the null and alternative, the type I error rate of the proposed method can be inflated from 10% up to 20%, but is still better than that of the Bayesian hierarchical model., Limitation: The proposed design assumes a binary endpoint. Extension of the proposed design to ordinal and time-to-event endpoints is worthy of further investigation., Conclusion: The calibrated Bayesian hierarchical model provides a practical approach to design basket trials with more flexibility and better controlled type I error rates than the Bayesian hierarchical model. The software for implementing the proposed design is available at http://odin.mdacc.tmc.edu/~yyuan/index&#95;code.html.

Propensity score methods are an important tool to help reduce confounding in non-experimental studies and produce more accurate causal effect estimates. Most propensity score methods assume that covariates are measured without error. However, covariates are often measured with error. Recent work has shown that ignoring such error could lead to bias in treatment effect estimates. In this paper, we consider an additional complication: that of differential measurement error across treatment groups, such as can occur if a covariate is measured differently in the treatment and control groups. We propose two flexible Bayesian approaches for handling differential measurement error when estimating average causal effects using propensity score methods. We consider three scenarios: systematic (i.e., a location shift), heteroscedastic (i.e., different variances), and mixed (both systematic and heteroscedastic) measurement errors. We also explore various prior choices (i.e., weakly informative or point mass) on the sensitivity parameters related to the differential measurement error. We present results from simulation studies evaluating the performance of the proposed methods and apply these approaches to an example estimating the effect of neighborhood disadvantage on adolescent drug use disorders.

Reliable subnational mortality estimates are essential in the study of health inequalities within a country. One of the difficulties in producing such estimates is the presence of small populations among which the stochastic variation in death counts is relatively high, and thus the underlying mortality levels are unclear. We present a Bayesian hierarchical model to estimate mortality at the subnational level. The model builds on characteristic age patterns in mortality curves, which are constructed using principal components from a set of reference mortality curves. Information on mortality rates are pooled across geographic space and are smoothed over time. Testing of the model shows reasonable estimates and uncertainty levels when it is applied both to simulated data that mimic U.S. counties and to real data for French départements. The model estimates have direct applications to the study of subregional health patterns and disparities.

Functional data are defined as realizations of random functions (mostly smooth functions) varying over a continuum, which are usually collected on discretized grids with measurement errors. In order to accurately smooth noisy functional observations and deal with the issue of high-dimensional observation grids, we propose a novel Bayesian method based on the Bayesian hierarchical model with a Gaussian-Wishart process prior and basis function representations. We first derive an induced model for the basis-function coefficients of the functional data, and then use this model to conduct posterior inference through Markov chain Monte Carlo methods. Compared to the standard Bayesian inference that suffers serious computational burden and instability in analyzing high-dimensional functional data, our method greatly improves the computational scalability and stability, while inheriting the advantage of simultaneously smoothing raw observations and estimating the mean-covariance functions in a nonparametric way. In addition, our method can naturally handle functional data observed on random or uncommon grids. Simulation and real studies demonstrate that our method produces similar results to those obtainable by the standard Bayesian inference with low-dimensional common grids, while efficiently smoothing and estimating functional data with random and high-dimensional observation grids when the standard Bayesian inference fails. In conclusion, our method can efficiently smooth and estimate high-dimensional functional data, providing one way to resolve the curse of dimensionality for Bayesian functional data analysis with Gaussian-Wishart processes., (© 2017, The International Biometric Society.)

In systems biology, it is of great interest to identify new genes that were not previously reported to be associated with biological pathways related to various functions and diseases. Identification of these new pathway-modulating genes does not only promote understanding of pathway regulation mechanisms but also allow identification of novel targets for therapeutics. Recently, biomedical literature has been considered as a valuable resource to investigate pathway-modulating genes. While the majority of currently available approaches are based on the co-occurrence of genes within an abstract, it has been reported that these approaches show only sub-optimal performances because 70% of abstracts contain information only for a single gene. To overcome such limitation, we propose a novel statistical framework based on the concept of ontology fingerprint that uses gene ontology to extract information from large biomedical literature data. The proposed framework simultaneously identifies pathway-modulating genes and facilitates interpreting functions of these new genes. We also propose a computationally efficient posterior inference procedure based on Metropolis-Hastings within Gibbs sampler for parameter updates and the poor man's reversible jump Markov chain Monte Carlo approach for model selection. We evaluate the proposed statistical framework with simulation studies, experimental validation, and an application to studies of pathway-modulating genes in yeast. The R implementation of the proposed model is currently available at https://dongjunchung.github.io/bayesGO/. Copyright © 2017 John Wiley & Sons, Ltd., (Copyright © 2017 John Wiley & Sons, Ltd.)

As the sequencing cost continued to drop in the past decade, RNA sequencing (RNA-seq) has replaced microarray to become the standard high-throughput experimental tool to analyze transcriptomic profile. As more and more datasets are generated and accumulated in the public domain, meta-analysis to combine multiple transcriptomic studies to increase statistical power has received increasing popularity. In this article, we propose a Bayesian hierarchical model to jointly integrate microarray and RNA-seq studies. Since systematic fold change differences across RNA-seq and microarray for detecting differentially expressed genes have been previously reported, we replicated this finding in several real datasets and showed that incorporation of a normalization procedure to account for the bias improves the detection accuracy and power. We compared our method with the popular two-stage Fisher's method using simulations and two real applications in a histological subtype (invasive lobular carcinoma) of breast cancer comparing PR+ versus PR- and early-stage versus late-stage patients. The result showed improved detection power and more significant and interpretable pathways enriched in the detected biomarkers from the proposed Bayesian model.

Background/aim: Meta-analyses are frequently performed on adverse event data and are primarily used for improving statistical power to detect safety signals. However, in the evaluation of drug safety for New Drug Applications, simple pooling of adverse event data from multiple clinical trials is still commonly used. We sought to propose a new Bayesian hierarchical meta-analytic approach based on consideration of a hierarchical structure of reported individual adverse event data from multiple randomized clinical trials., Methods: To develop our meta-analysis model, we extended an existing three-stage Bayesian hierarchical model by including an additional stage of the clinical trial level in the hierarchical model; this generated a four-stage Bayesian hierarchical model. We applied the proposed Bayesian meta-analysis models to published adverse event data from three premarketing randomized clinical trials of tadalafil and to a simulation study motivated by the case example to evaluate the characteristics of three alternative models., Results: Comparison of the results from the Bayesian meta-analysis model with those from Fisher's exact test after simple pooling showed that 6 out of 10 adverse events were the same within a top 10 ranking of individual adverse events with regard to association with treatment. However, more individual adverse events were detected in the Bayesian meta-analysis model than in Fisher's exact test under the body system "Musculoskeletal and connective tissue disorders." Moreover, comparison of the overall trend of estimates between the Bayesian model and the standard approach (odds ratios after simple pooling methods) revealed that the posterior median odds ratios for the Bayesian model for most adverse events shrank toward values for no association. Based on the simulation results, the Bayesian meta-analysis model could balance the false detection rate and power to a better extent than Fisher's exact test. For example, when the threshold value of the posterior probability for signal detection was set to 0.8, the false detection rate was 41% and power was 88% in the Bayesian meta-analysis model, whereas the false detection rate was 56% and power was 86% in Fisher's exact test., Limitations: Adverse events under the same body system were not necessarily positively related when we used "system organ class" and "preferred term" in the Medical Dictionary for Regulatory Activities as a hierarchical structure of adverse events. For the Bayesian meta-analysis models to be effective, the validity of the hierarchical structure of adverse events and the grouping of adverse events are critical., Conclusion: Our proposed meta-analysis models considered trial effects to avoid confounding by trial and borrowed strength from both within and across body systems to obtain reasonable and stable estimates of an effect measure by considering a hierarchical structure of adverse events.

In this article a multi-subject vector autoregressive (VAR) modeling approach was proposed for inference on effective connectivity based on resting-state functional MRI data. Their framework uses a Bayesian variable selection approach to allow for simultaneous inference on effective connectivity at both the subject- and group-level. Furthermore, it accounts for multi-modal data by integrating structural imaging information into the prior model, encouraging effective connectivity between structurally connected regions. They demonstrated through simulation studies that their approach resulted in improved inference on effective connectivity at both the subject- and group-level, compared with currently used methods. It was concluded by illustrating the method on temporal lobe epilepsy data, where resting-state functional MRI and structural MRI were used. Hum Brain Mapp 38:1311-1332, 2017. © 2016 Wiley Periodicals, Inc., (© 2016 Wiley Periodicals, Inc.)

The prevalence of anthelmintic resistance has increased in recent years, as a result of the extensive use of anthelmintic drugs to reduce the infection of parasitic worms in livestock. In order to detect the resistance, the number of parasite eggs in animal faeces is counted. Typically a subsample of the diluted faeces is examined, and the mean egg counts from both untreated and treated animals are compared. However, the conventional method ignores the variabilities introduced by the counting process and by different infection levels across animals. In addition, there can be extra zero counts, which arise as a result of the unexposed animals in an infected population or animals. In this paper, we propose the zero-inflated Bayesian hierarchical models to estimate the reduction in faecal egg counts. The simulation study compares the Bayesian models with the conventional faecal egg count reduction test and other methods such as bootstrap and quasi-Poisson regression. The results show the Bayesian models are more robust and they perform well in terms of both the bias and the coverage. We further illustrate the advantages of our proposed model using a case study about the anthelmintic resistance in Swedish sheep flocks., (Copyright © 2017 The Authors. Published by Elsevier B.V. All rights reserved.)

In recent years, disease mapping studies have become a routine application within geographical epidemiology and are typically analysed within a Bayesian hierarchical model formulation. A variety of model formulations for the latent level have been proposed but all come with inherent issues. In the classical BYM (Besag, York and Mollié) model, the spatially structured component cannot be seen independently from the unstructured component. This makes prior definitions for the hyperparameters of the two random effects challenging. There are alternative model formulations that address this confounding; however, the issue on how to choose interpretable hyperpriors is still unsolved. Here, we discuss a recently proposed parameterisation of the BYM model that leads to improved parameter control as the hyperparameters can be seen independently from each other. Furthermore, the need for a scaled spatial component is addressed, which facilitates assignment of interpretable hyperpriors and make these transferable between spatial applications with different graph structures. The hyperparameters themselves are used to define flexible extensions of simple base models. Consequently, penalised complexity priors for these parameters can be derived based on the information-theoretic distance from the flexible model to the base model, giving priors with clear interpretation. We provide implementation details for the new model formulation which preserve sparsity properties, and we investigate systematically the model performance and compare it to existing parameterisations. Through a simulation study, we show that the new model performs well, both showing good learning abilities and good shrinkage behaviour. In terms of model choice criteria, the proposed model performs at least equally well as existing parameterisations, but only the new formulation offers parameters that are interpretable and hyperpriors that have a clear meaning., (© The Author(s) 2016.)

We propose a novel Bayesian hierarchical model for brain imaging data that unifies voxel-level (the most localized unit of measure) and region-level brain connectivity analyses, and yields population-level inferences. Functional connectivity generally refers to associations in brain activity between distinct locations. The first level of our model summarizes brain connectivity for cross-region voxel pairs using a two-component mixture model consisting of connected and nonconnected voxels. We use the proportion of connected voxel pairs to define a new measure of connectivity strength, which reflects the breadth of between-region connectivity. Furthermore, we evaluate the impact of clinical covariates on connectivity between region-pairs at a population level. We perform parameter estimation using Markov chain Monte Carlo (MCMC) techniques, which can be executed quickly relative to the number of model parameters. We apply our method to resting-state functional magnetic resonance imaging (fMRI) data from 32 subjects with major depression and simulated data to demonstrate the properties of our method., (© 2015, The International Biometric Society.)

Human immunodeficiency virus (HIV) infection is a severe infectious disease actively spreading globally, and acquired immunodeficiency syndrome (AIDS) is an advanced stage of HIV infection. The HIV testing rate, that is, the probability that an AIDS-free HIV infected person seeks a test for HIV during a particular time interval, given no previous positive test has been obtained prior to the start of the time, is an important parameter for public health. In this paper, we propose a Bayesian hierarchical model with two levels of hierarchy to estimate the HIV testing rate using annual AIDS and AIDS-free HIV diagnoses data. At level one, we model the latent number of HIV infections for each year using a Poisson distribution with the intensity parameter representing the HIV incidence rate. At level two, the annual numbers of AIDS and AIDS-free HIV diagnosed cases and all undiagnosed cases stratified by the HIV infections at different years are modeled using a multinomial distribution with parameters including the HIV testing rate. We propose a new class of priors for the HIV incidence rate and HIV testing rate taking into account the temporal dependence of these parameters to improve the estimation accuracy. We develop an efficient posterior computation algorithm based on the adaptive rejection metropolis sampling technique. We demonstrate our model using simulation studies and the analysis of the national HIV surveillance data in the USA., (Copyright © 2015 John Wiley & Sons, Ltd.)

Bayesian statistical approaches to mixed treatment comparisons (MTCs) are becoming more popular because of their flexibility and interpretability. Many randomized clinical trials report multiple outcomes with possible inherent correlations. Moreover, MTC data are typically sparse (although richer than standard meta-analysis, comparing only two treatments), and researchers often choose study arms based upon which treatments emerge as superior in previous trials. In this paper, we summarize existing hierarchical Bayesian methods for MTCs with a single outcome and introduce novel Bayesian approaches for multiple outcomes simultaneously, rather than in separate MTC analyses. We do this by incorporating partially observed data and its correlation structure between outcomes through contrast-based and arm-based parameterizations that consider any unobserved treatment arms as missing data to be imputed. We also extend the model to apply to all types of generalized linear model outcomes, such as count or continuous responses. We offer a simulation study under various missingness mechanisms (e.g., missing completely at random, missing at random, and missing not at random) providing evidence that our models outperform existing models in terms of bias, mean squared error, and coverage probability then illustrate our methods with a real MTC dataset. We close with a discussion of our results, several contentious issues in MTC analysis, and a few avenues for future methodological development., (Copyright © 2015 John Wiley & Sons, Ltd.)

We show that Bayesian population reconstruction, a recent method for estimating past populations by age, works for data of widely varying quality. Bayesian reconstruction simultaneously estimates age-specific population counts, fertility rates, mortality rates, and net international migration flows from fragmentary data, while formally accounting for measurement error. As inputs, Bayesian reconstruction uses initial bias-reduced estimates of standard demographic variables. We reconstruct the female populations of three countries: Laos, a country with little vital registration data where population estimation depends largely on surveys; Sri Lanka, a country with some vital registration data; and New Zealand, a country with a highly developed statistical system and good quality vital registration data. In addition, we extend the method to countries without censuses at regular intervals. We also use it to assess the consistency of results between model life tables and available census data, and hence to compare different model life table systems.

We propose a method for obtaining joint probabilistic projections of migration for all countries, broken down by age and sex. Joint trajectories for all countries are constrained to satisfy the requirement of zero global net migration. We evaluate our model using out-of-sample validation and compare point projections to the projected migration rates from a persistence model similar to the method used in the United Nations' World Population Prospects, and also to a state-of-the-art gravity model.

Data used for modelling the household transmission of infectious diseases, such as influenza, have inherent multilevel structures and correlated property, which make the widely used conventional infectious disease transmission models (including the Greenwood model and the Reed-Frost model) not directly applicable within the context of a household (due to the crowded domestic condition or socioeconomic status of the household). Thus, at the household level, the effects resulting from individual-level factors, such as vaccination, may be confounded or modified in some way. We proposed the Bayesian hierarchical random-effects (random intercepts and random slopes) model under the context of generalised linear model to capture heterogeneity and variation on the individual, generation, and household levels. It was applied to empirical surveillance data on the influenza epidemic in Taiwan. The parameters of interest were estimated by using the Markov chain Monte Carlo method in conjunction with the Bayesian directed acyclic graphical models. Comparisons between models were made using the deviance information criterion. Based on the result of the random-slope Bayesian hierarchical method under the context of the Reed-Frost transmission model, the regression coefficient regarding the protective effect of vaccination varied statistically significantly from household to household. The result of such a heterogeneity was robust to the use of different prior distributions (including non-informative, sceptical, and enthusiastic ones). By integrating out the uncertainty of the parameters of the posterior distribution, the predictive distribution was computed to forecast the number of influenza cases allowing for random-household effect., (Copyright © 2014 Elsevier Inc. All rights reserved.)

We propose a Bayesian hierarchical model for the calculation of incidence counts from mortality data by a convolution equation that expresses mortality through its relationship with incidence and the survival probability density. The basic idea is to use mortality data together with an estimate of the survival distribution from cancer incidence to cancer mortality to reconstruct the numbers of individuals who constitute previously incident cases that give rise to the observed pattern of cancer mortality. This model is novel because it takes into account the uncertainty from the survival distribution; thus, a Bayesian-mixture cure model for survival is introduced. Furthermore, projections are obtained starting from a Bayesian age-period-cohort model. The main advantage of the proposed approach is its consideration of the three components of the model: the convolution equation, the survival mixture cure model and the age-period-cohort projection within a directed acyclic graph model. Furthermore, the estimation are obtained through the Gibbs sampler. We applied the model to cases of women with stomach cancer using six age classes [15-45], [45-55], [55-65], [65-75], [75-85] and [85-95] and validated it by using data from the Tuscany Cancer Registry. The model proposed and the program implemented are convenient because they allow different cancer disease to be analysed because the survival time is modelled by flexible distributions that are able to describe different trends., (Copyright © 2014 John Wiley & Sons, Ltd.)

The spatial organisation of the chromosomes in the nucleus is influenced by chromatin regions binding to the nucleic lamina, i.e., the inner part of the nucleic envelope. To investigate the architecture of chromosomes in the interphase nucleus, it is thus of high interest to detect such chromatin segments. This goal can be achieved by considering the fibrous protein Lamin B as a surrogate, since regions of high abundance of Lamin B can indicate chromatin segments attached to the nucleic lamina. We analyse ChIP-Seq (Chromatin-Immunoprecipitation Sequencing) data from an experiment that is designed to record Lamin B abundance. We introduce a Bayesian segmentation procedure in which a Markov Chain Monte Carlo (MCMC) algorithm is used for inference about the desired segmentation. The procedure is based on a Bayesian hierarchical model. Inference allows the distinction between regions of high versus low levels of Lamin B, and therefore, gives an insight into the binding of the chromatin to the nucleic envelope. An implementation of this approach is available in the statistical software environment R. This article is part of a special issue entitled: Computational proteomics in the post-identification era. Guest Editors: Martin Eisenacher and Christian Stephan., (© 2013.)

Population cycles have long interested biologists. The ruffed grouse, Bonasa umbellus, is one such species whose populations cycle over most of their range. Thus, much effort has been expended to understand the mechanisms that might control cycles in this and other species. Corticosterone metabolites are widely used in studies of animals to measure physiological stress. We evaluated corticosterone metabolites in feces of territorial male grouse as a potential tool to study mechanisms governing grouse cycles. However, like most studies of corticosterone in wild animals, we did not know the identity of all individuals for which we had fecal samples. This presented an analytical problem that resulted in either pseudoreplication or confounding. Therefore, we derived an analytical approach that accommodated for uncertainty in individual identification. Because we had relatively low success capturing birds, we estimated turnover probabilities of birds on territorial display sites based on capture histories of a limited number of birds we captured. Hence, we developed a study design and modeling approach to quantify variation in corticosterone levels among individuals and through time that would be applicable to any field study of corticosterone in wild animals. Specifically, we wanted a sampling design and model that was flexible enough to partition variation among individuals, spatial units, and years, while incorporating environmental covariates that would represent potential mechanisms. We conducted our study during the decline phase of the grouse cycle and found high variation among corticosterone samples (11.33-443.92 ng/g [x=113.99 ng/g, SD=69.08, median=99.03 ng/g]). However, there were relatively small differences in corticosterone levels among years, but levels declined throughout each breeding season, which was opposite our predictions for stress hormones correlating with a declining population. We partitioned the residual variation into site, bird, and repetition (i.e., multiple samples collected from the same bird on the same day). After accounting for years and three general periods within breeding seasons, 42% of the residual variation among observations was attributable to differences among individual birds. Thus, we attribute little influence of site on stress level of birds in our study, but disentangling individual from site effects is difficult because site and bird are confounded. Our model structures provided analytical approaches for studying species having different ecologies. Our approach also demonstrates that even incomplete information on individual identity of birds within samples is useful for analyzing these types of data., (Copyright © 2013 Elsevier Inc. All rights reserved.)

We build a Bayesian hierarchical model for relating disease to a potentially harmful exposure, by using data from studies in occupational epidemiology, and compare our method with the traditional group-based exposure assessment method through simulation studies, a real data application, and theoretical calculation. We focus on cohort studies where a logistic disease model is appropriate and where group means can be treated as fixed effects. The results show a variety of advantages of the fully Bayesian approach and provide recommendations on situations where the traditional group-based exposure assessment method may not be suitable to use., (Copyright © 2013 John Wiley & Sons, Ltd.)

BACKGROUND:It is widely recognized that there are multiple risk factors for early-life mortality. In practice most interventions to curb early-life mortality target births based on a single risk factor, such as poverty. However, most premature deaths are not from the targeted group. Thus interventions target many births that are at not at high risk and miss many births at high risk. METHODS:Using data from the second wave of Demographic and Health Surveys from India and a hierarchical Bayesian model, we estimate infant mortality risk for 73.320 infants in India as a function of 4 risk factors. We show how this information can be used to improve program targeting. We compare our novel approach against common programs that target groups based on a single risk factor. RESULTS:A conventional approach that targets mothers in the lowest quintile of income correctly identifies only 30% of infant deaths. By contrast, using four risk factors simultaneously we identify a group of births of the same size that includes 57% of all deaths. Using the 2012 census to translate these percentages into numbers, there were 25.642.200 births in 2012 and 4.4% died before the age of one. Our approach correctly identifies 643.106 of 1.128.257 infant deaths while poverty only identifies 338.477 infant deaths. CONCLUSION:Our approach considerably improves program targeting by identifying more infant deaths than the usual approach that targets births based on a single risk factor. This leads to more efficient program targeting. This is particularly useful in developing countries, where resources are lacking and needs are high.

BackgroundThe Human Microbiome has been variously associated with the immune-regulatory mechanisms involved in the prevention or development of many non-infectious human diseases such as autoimmunity, allergy and cancer. Integrative approaches which aim at associating the composition of the human microbiome with other available information, such as clinical covariates and environmental predictors, are paramount to develop a more complete understanding of the role of microbiome in disease development.ResultsIn this manuscript, we propose a Bayesian Dirichlet-Multinomial regression model which uses spike-and-slab priors for the selection of significant associations between a set of available covariates and taxa from a microbiome abundance table. The approach allows straightforward incorporation of the covariates through a log-linear regression parametrization of the parameters of the Dirichlet-Multinomial likelihood. Inference is conducted through a Markov Chain Monte Carlo algorithm, and selection of the significant covariates is based upon the assessment of posterior probabilities of inclusions and the thresholding of the Bayesian false discovery rate. We design a simulation study to evaluate the performance of the proposed method, and then apply our model on a publicly available dataset obtained from the Human Microbiome Project which associates taxa abundances with KEGG orthology pathways. The method is implemented in specifically developed R code, which has been made publicly available.ConclusionsOur method compares favorably in simulations to several recently proposed approaches for similarly structured data, in terms of increased accuracy and reduced false positive as well as false negative rates. In the application to the data from the Human Microbiome Project, a close evaluation of the biological significance of our findings confirms existing associations in the literature.

BackgroundEpidemiological studies suggest that air pollution is adversely associated with pregnancy outcomes. Such associations may be modified by spatially-varying factors including socio-demographic characteristics, land-use patterns and unaccounted exposures. Yet, few studies have systematically investigated the impact of these factors on spatial variability of the air pollution's effects. This study aimed to examine spatial variability of the effects of air pollution on term birth weight across Census tracts and the influence of tract-level factors on such variability.MethodsWe obtained over 900,000 birth records from 2001 to 2008 in Los Angeles County, California, USA. Air pollution exposure was modeled at individual level for nitrogen dioxide (NO2) and nitrogen oxides (NOx) using spatiotemporal models. Two-stage Bayesian hierarchical non-linear models were developed to (1) quantify the associations between air pollution exposure and term birth weight within each tract; and (2) examine the socio-demographic, land-use, and exposure-related factors contributing to the between-tract variability of the associations between air pollution and term birth weight.ResultsHigher air pollution exposure was associated with lower term birth weight (average posterior effects: -14.7 (95 % CI: -19.8, -9.7) g per 10 ppb increment in NO2 and -6.9 (95 % CI: -12.9, -0.9) g per 10 ppb increment in NOx). The variation of the association across Census tracts was significantly influenced by the tract-level socio-demographic, exposure-related and land-use factors. Our models captured the complex non-linear relationship between these factors and the associations between air pollution and term birth weight: we observed the thresholds from which the influence of the tract-level factors was markedly exacerbated or attenuated. Exacerbating factors might reflect additional exposure to environmental insults or lower socio-economic status with higher vulnerability, whereas attenuating factors might indicate reduced exposure or higher socioeconomic status with lower vulnerability.ConclusionsOur Bayesian models effectively combined a priori knowledge with training data to infer the posterior association of air pollution with term birth weight and to evaluate the influence of the tract-level factors on spatial variability of such association. This study contributes new findings about non-linear influences of socio-demographic factors, land-use patterns, and unaccounted exposures on spatial variability of the effects of air pollution.