Empirical Bayesian Research Articles

Measuring Nursing Home Performance Using Administrative Data.

To improve the quality of nursing home care, reliable estimates of outcomes are essential. Obtaining such estimates requires optimal use of limited data, especially for small homes. We analyze the variation in mortality and hospital admissions across nursing homes in the Netherlands during the years 2010-2013. We use administrative data on all nursing home clients. We apply mixed-effects survival models, empirical Bayes estimation, and machine-learning techniques to optimally use the available longitudinal data. We find large differences in both outcomes across nursing homes, yet the estimates are surrounded by substantial uncertainty. We find no correlation between performance on mortality and avoidable hospital admissions, suggesting that these are related to different aspects of quality. Hence, caution is needed when evaluating the performance of individual nursing homes, especially when the number of outcome indicators is limited.

Hospital-level variation in mesh use for ventral and incisional hernia repair.

Placement of prosthetic mesh during ventral and incisional hernia repair has been shown to reduce the incidence of postoperative hernia recurrence. Consequently, multiple consensus guidelines recommend the use of mesh for ventral hernias of any size. However, the extent to which real-world practice patterns reflect these recommendations is unclear. We performed a retrospective review of the Michigan Surgical Quality Collaborative Hernia Registry (MSQC-HR) to identify patients undergoing clean ventral or incisional hernia repair between January 1, 2020 and December 31, 2021. The primary outcome was mesh use. We used two-step hierarchical logistic regression modeling with empirical Bayes estimates to evaluate the association of hospital-level mesh use with patient, operative, and hernia characteristics. A total of 5262 patients underwent ventral and incisional hernia repair at 65 hospitals with a mean age of 53.8 (14.5) years, 2292 (43.6%) females, and a mean hernia width of 3.2 (3.4) cm. Mean hospital volume was 81 (49) cases. Mesh was used in 4098 (77.9%) patients. At the patient level, hernia width and surgical approach were significantly associated with mesh use. Specifically, mesh use was 6.2% (95% CI 4.8-7.5%) more likely with each additional centimeter of hernia width and 28.0% (95% CI 26.1-29.8%) more likely for minimally invasive repair compared to open repair. At the hospital level, there was wide variation in mesh use, ranging from 38.0% (95% CI 31.5-44.9%) to 96.4% (95% CI 95.3-97.2%). Hospital-level mesh use was not associated with differences in hernia size (β = - 0.003, P = 0.978), surgical approach (β = - 1.109, P = 0.414), or any other patient factors. Despite strong evidence supporting the use of mesh in ventral and incisional hernia repair, there is substantial variation in mesh use between hospitals that is not explained by differences in patient characteristics or operative approach. This suggests that opportunities exist to standardize surgical practice to better align with evidence supporting the use of mesh in the management of these hernias.

An empirical Bayes approach to stochastic blockmodels and graphons: shrinkage estimation and model selection.

The graphon (W-graph), including the stochastic block model as a special case, has been widely used in modeling and analyzing network data. Estimation of the graphon function has gained a lot of recent research interests. Most existing works focus on inference in the latent space of the model, while adopting simple maximum likelihood or Bayesian estimates for the graphon or connectivity parameters given the identified latent variables. In this work, we propose a hierarchical model and develop a novel empirical Bayes estimate of the connectivity matrix of a stochastic block model to approximate the graphon function. Based on our hierarchical model, we further introduce a new model selection criterion for choosing the number of communities. Numerical results on extensive simulations and two well-annotated social networks demonstrate the superiority of our approach in terms of parameter estimation and model selection.

The empirical Bayes estimators of the parameter of the uniform distribution with an inverse gamma prior under Stein’s loss function

For the hierarchical uniform and inverse gamma model, we calculate the Bayes posterior estimator of the parameter of the uniform distribution under Stein’s loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein’s Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain empirical Bayes estimators of the parameter of the uniform distribution by two methods. Note that the estimators of the hyperparameters of the model by the Maximum Likelihood Estimation (MLE) method are summarized in a theorem, whose proof involves the upper incomplete gamma function and a special case of the Meijer G-function. In numerical simulations, we address from four perspectives. First, we exemplify the two inequalities of the Bayes posterior estimators and the PESLs. Second, we illustrate that the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters. Third, we calculate the goodness-of-fit of the model for the simulated data. Fourth, we plot the marginal densities of the model for various hyperparameters. Finally, we utilize the current prices of the 300 component stocks of Shenzhen 300 Index to illustrate our theoretical studies.

Investigation of Inlaid Pavement Marker Performance and Safety Effectiveness

The objectives of this study were to assess inlaid pavement marker (IPM) performance and estimate their safety effectiveness in St. Louis, Missouri. IPM performance was evaluated through a count of marker presence and a feedback survey from participants who viewed dry and wet night videos of IPM sections in the St Louis area. The safety effectiveness of IPMs was evaluated using a rigorous state-of-the-art empirical Bayes (EB) before–after crash analysis. The marker presence assessment, unfortunately, did not provide any conclusive trends in IPM performance. Newer IPM sections had higher percentages of missing markers than older sections, and there were too many unknown variables to determine the source of the unexpected differences. The unanimous conclusion of the nighttime video visibility survey was that drivers and passengers traveling on a wet night feel that IPMs are very important to the visibility of the roadway’s lane lines. For the EB analysis, installation and reference site data were used to examine the effects for specific crash types, including total, fatal and injury, wet pavement, nighttime, nighttime wet pavement, lane departure, wet pavement lane departure, and nighttime lane departure. Based on the aggregate results, IPMs, when installed with pavement resurfacing, significantly reduce all crash types examined.

An empirical Bayes approach to quantifying the impact of transportation network companies (TNCs) operations on travel demand

Assessing the impacts of new and disruptive technologies on the transportation system is crucial for planners and policymakers. This study offers an innovative method for estimating the impact of transportation network company (TNC, e.g., Uber and Lyft) operations on travel demand. Among various measures of travel demand, vehicle‐miles traveled (VMT) is tested due to its broad applications in evaluating changes in transportation policy. The evaluation is based on counterfactual theory, which compares VMT estimates after the TNCs introduction to a region to what the VMT would have been without the TNCs. The latter of the two is a counterfactual, and this study develops and demonstrates the empirical Bayes (EB) approach for obtaining counterfactual VMT estimates. The EB approach is widely used to estimate crash frequency after a particular traffic safety treatment is applied to a roadway location. We reinterpret the traffic safety treatment as being akin to the introduction of TNCs in the Atlanta region and the estimation of crash frequency as analogous to the resulting change in VMT. Since the crash experience of a roadway site may be affected by several factors, like the VMT of a region, a robust counterfactual estimate is necessary for conducting a before-after study. A counterfactual VMT estimate is obtained by combining two VMT estimates from 1) the cross‐sectional analysis for Atlanta and its regional peers and 2) the time-series analysis based on the longitudinal trend from the Atlanta region. We measure the difference between the counterfactual VMT estimate and the reported VMT estimate as an indicator of TNC impact. We find that the VMT estimates in a counterfactual scenario without TNCs are lower than the actual VMT estimates between 2012 and 2018. We estimate that the TNCs may be accounting for extra 0.6 percent average annual growth in VMT. The findings may support future TNC-related planning and policymaking. We expect the approach to be useful in estimating the effects of other disruptions, such as connected and autonomous vehicles (CAVs) introduction and lasting impact of the pandemic, on VMT.

Important lack of difference in tacrolimus and mycophenolic acid pharmacokinetics between Aboriginal and Caucasian kidney transplant recipients.

To examine whether differences in tacrolimus and mycophenolic acid (MPA) pharmacokinetics contribute to the poorer kidney transplant outcomes experienced by Aboriginal Australians. Concentration-time profiles for tacrolimus and MPA were prospectively collected from 43 kidney transplant recipients: 27 Aboriginal and 16 Caucasian. Apparent clearance (CL/F) and distribution volume (V/F) for each individual were derived from concentration-time profiles combined with population pharmacokinetic priors, with subsequent assessment for between-group difference in pharmacokinetics. In addition, population pharmacokinetic models were developed using the prospective dataset supplemented by previously developed structural models for tacrolimus and MPA. The change in NONMEM objective function was used to assess improvement in goodness of model fit. No differences were found between Aboriginal and Caucasian groups or empirical Bayes estimates, for CL/F or V/F of MPA or tacrolimus. However, a higher prevalence of CYP3A5 expressers (26% compared with 0%) and wider between-subject variability in tacrolimus CL/F (SD=5.00 compared with 3.25 L/h/70 kg) were observed in the Aboriginal group, though these differences failed to reach statistical significance (p= .07 and p= .08). There were no differences in typical tacrolimus or MPA pharmacokinetics between Aboriginal and Caucasian kidney transplant recipients. This means that Bayesian dosing tools developed to optimise tacrolimus and MPA dosing in Caucasian recipients may be applied to Aboriginal recipients. In turn, this may improve drug exposure and thereby transplant outcomes in this group. Aboriginal recipients appeared to have greater between-subject variability in tacrolimus CL/F and a higher prevalence of CYP3A5 expressers, attributes that have been linked with inferior outcomes.

Assessing the safety effect of red-light camera deactivation: a geographically weighted negative binomial regression approach

Municipalities across the country have debated the safety effect of automatic red-light cameras (RLC) and their political and financial implications. Most empirical studies have used the Empirical Bayesian (EB) approach to assess the safe effects to facilitate policy debates. While popular, the EB method has several limitations in data requirement, reference site selection, and control of confounding factors. Moreover, empirical studies of the RLC deactivation effects are limited. This study fills these gaps using the Moran’s I statistic and the Geographically Weighted Negative Binomial Regression (GWNBR) approach for data in the City of Arlington, Texas. The results indicate that the total, injury, and angle crashes in Arlington are on the rise over the study period and that crashes are higher at RLC deactivation intersections than those at other intersections. The direct safety effect of removing RLCs is statistically significant. The spillover effect is observed but statistically insignificant. Speed limit plays an important role in road safety. The findings have significant implications for safety research and practices.

PhyloMDA: an R package for phylogeny-aware microbiome data analysis

BackgroundModern sequencing technologies have generated low-cost microbiome survey datasets, across sample sites, conditions, and treatments, on an unprecedented scale and throughput. These datasets often come with a phylogenetic tree that provides a unique opportunity to examine how shared evolutionary history affects the different patterns in host-associated microbial communities.ResultsIn this paper, we describe an R package, phyloMDA, for phylogeny-aware microbiome data analysis. It includes the Dirichlet-tree multinomial model for multivariate abundance data, tree-guided empirical Bayes estimation of microbial compositions, and tree-based multiscale regression methods with relative abundances as predictors.ConclusionphyloMDA is a versatile and user-friendly tool to analyze microbiome data while incorporating the phylogenetic information and addressing some of the challenges posed by the data.

Bias correction for multiple covariate analysis using empirical bayesian estimation in mixed-effects models for longitudinal data

The naïve empirical Bayes method has been widely used as an ad hoc tool in fitting linear mixed-effect models, which is much computationally efficient than the maximum likelihood estimation method. However, the shrinkage effect of the empirical Bayes method causes bias in the estimates of the fixed effects. Bias-correction has been proposed for the mixed-effects model when only one covariate is present. In this paper, we derive the shrinkage factor of the empirical Bayes predictors of the random effects and the variance-covariance matrix of the corrected estimates when the model has more than one covariate. The empirical Bayes estimates and test statistics are then corrected using the derived factor. Theoretical derivations, simulation studies and a real data application demonstrate the validity of the proposed method in that the corrected estimates are unbiased and the corrected tests have correct p-values.

A kernel-based identification approach for a class of nonlinear systems with quantized output data

In this paper, we consider the problem of identification for a class of nonlinear systems with Hammerstein structure, which is a combination of some basis functions for the static nonlinearity and finite impulse response models for the linear block under quantized output data. Following the empirical Bayes method, we model the impulse response of the system using a Gaussian process whose covariance matrix depends on some suitable kernels. Then, we compute the minimum mean-square error estimation of the impulse response via a special case of Markov Chain Monte Carlo sampling approach, namely, Gibbs sampler. The estimation of the impulse response depends on the kernel hyperparameters, the noise variance and the coefficients of the static nonlinearity. These unknowns are estimated by the marginal likelihood maximization, which is solved via the expectation-conditional maximization algorithm combining the Gibbs sampler. The expectation-conditional maximization method is a variation of the standard expectation-maximization method for solving maximum-likelihood problems. Numerical simulations show the effectiveness of the proposed scheme.

The empirical Bayes estimators of the variance parameter of the normal distribution with a conjugate inverse gamma prior under Stein’s loss function

For the hierarchical normal and inverse gamma model, we calculate the Bayes posterior estimator of the variance parameter of the normal distribution under Stein’s loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein’s Loss (PESL). We also obtain the Bayes posterior estimator of the variance parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the variance parameter of the normal distribution with a conjugate inverse gamma prior by two methods. In numerical simulations, we have illustrated five aspects: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data; the numerical comparisons of the Bayes posterior estimators and the PESLs of the oracle, moment, and MLE methods; and the plots of the marginal densities for various hyperparameters. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we utilize the %bodyfat data of 250 men of various ages to illustrate our theoretical studies.

Efficient testing and effect size estimation for set-based genetic association inference via semiparametric multilevel mixture modeling.

In genetic association studies, rare variants with extremely low allele frequencies play a crucial role in complex traits. Therefore, set-based testing methods that jointly assess the effects of groups of single nucleotide polymorphisms (SNPs) were developed to increase the powers of the association tests. However, these powers are still insufficient, and precise estimations of the effect sizes of individual SNPs are largely impossible. In this article, we provide an efficient set-based statistical inference framework that addresses both of these important issues simultaneously using an empirical Bayes method with semiparametric multilevel mixture modeling. We propose to utilize the hierarchical model that incorporates variations in set-specific effects and to apply the optimal discovery procedure (ODP) that achieves the largest overall power in multiple significance testing. In addition, we provide an optimal "set-based" estimator of the empirical distribution of effect sizes. The efficiency of the proposed methods is demonstrated through application to a genome-wide association study of coronary artery disease and through simulation studies. The results demonstrated numerous rare variants with large effect sizes for coronary artery disease, and the number of significant sets detected by the ODP was much greater than those identified by existingmethods.

Queue Length Estimation for Signalized Intersections under Partially Connected Vehicle Environment

Queue length is a crucial measurement of traffic signal control at urban intersections. Conventional queue length estimation methods mostly still rely on fixed detectors. The development of connected vehicles (CV) provides massive amounts of vehicle trajectory data, and the queue length estimation based on CV data has received considerable attention in recent years. However, most existing CV-based methods require the prior knowledge of the penetration rate of CV and vehicle arrivals, but the estimation of these prior distributions has not been well studied. To address this issue, this paper proposes a cycle-based queue length estimation method under partially connected vehicle (CV) environment, with the prior vehicle arrivals being unknown. The empirical Bayes method is adopted to estimate the arrival rate by leveraging the observed queued CV information such as the number and positions. The hyperparameter estimation problem of the prior distribution is solved by the maximum likelihood estimation (MLE) method. To validate the proposed queue length estimation method, a simulation environment with partially connected vehicles is established using VISSIM and Python for data generating. The results in terms of normalized mean absolute errors (NMAE) and normalized root mean square errors (NRMSE) show that the proposed method could produce accurate and reliable estimated queue length under various CV penetration rates.

Ranking and Selection from Pairwise Comparisons: Empirical Bayes Methods for Citation Analysis

In this paper, Stigler's (1994) model of citation flows among journals adapted to the pairwise comparison model of Bradley and Terry (1952) is employed to do ranking and selection of journal influence based on nonparametric empirical Bayes methods. Comparisons with several other ranking are also made.

Neurocognitive impairment associated with traditional and novel androgen receptor signaling inhibitors ± androgen deprivation therapy: a pharmacovigilance study.

Conflicting evidence exists regarding whether hormone therapy for prostate cancer is associated with neurotoxicity. Thus, we aim to characterize the association between different types of hormone therapy and neurocognitive impairment in a real-world pharmacovigilance database. We queried VigiBase, the World Health Organization's international pharmacovigilance database, for reports of neurocognitive impairment among men who took hormone therapy from 1968 to 2021. We performed disproportionality analysis comparing rates of neurocognitive impairment with different types of hormone therapy versus other VigiBase drugs. Traditional hormonal therapy was defined as androgen deprivation therapy (ADT: gonadotropin-releasing-hormone agonists or antagonists) or first-generation androgen receptor (AR) antagonists. Novel AR signaling inhibitors (ARSIs) were defined as ARSIs with or without ADT. Differences were assessed using reporting odds ratio (ROR) with 95% confidence intervals (CI) and Empirical Bayes Estimator (EBE) values ≥1.0 signifying statistical significance. Odds of neurocognitive impairment were significantly elevated with traditional hormone therapy (ROR 1.47, 95% CI 1.34-1.62, EBE = 1.35) and novel ARSIs (ROR 2.40, 95% CI 2.28-2.54, EBE = 2.26). Odds of neurocognitive impairment were significantly elevated with enzalutamide (ROR 2.89, 95% CI 2.73-3.05, EBE = 2.70) and numerically increased with apalutamide (ROR 3.31, 95% CI 1.57-7.00, EBE = 0.98), but were decreased with abiraterone (ROR 0.68, 95% CI 0.55-0.84, EBE = 0.57). This study demonstrates elevated odds of neurocognitive impairment with hormone therapy in a real-world data set. Neurotoxicity risk was higher with novel ARSIs than traditional agents, and higher with enzalutamide than abiraterone. Due to limitations inherent to disproportionality analysis (measuring associations, not risk) and incomplete data prohibiting the ability to control for factors such as age or use of secondary drugs (e.g., concurrent use of novel ARSIs with ADT), results are exploratory in nature. The amalgamation of these and other conflicting data may contribute to clinical decision-making for men with prostate cancer eligible for treatment with these therapies, especially those with significant neurologic comorbidities.

Derivation of the Empirical Bayesian method for the Negative Binomial-Lindley generalized linear model with application in traffic safety

The expected crash frequency is the long-term average crash count for a specific site. It is extensively used to systematically evaluate the crash risk associated with roadway elements. To estimate the expected crashes, the Empirical Bayesian (EB) approach is typically employed. The EB method is a computationally convenient approximation to the Full Bayesian (FB) method, which gained popularity due to its simple interpretation, computational efficiency, and the ability to account for the regression to the mean bias. However, the common EB method used in traffic safety analysis is only applicable when the traditional Negative Binomial (NB) model is used. The NB model, however, is not a suitable choice when data is highly dispersed, skewed, or has a large number of zero observations. The Negative Binomial-Lindley (NB-L) model is a mixture of the NB and Lindley distributions and has shown superior fit compared to the NB model, especially when the dataset is characterized by excess zero observations. Even though several studies have used the NB-L in developing crash prediction models, the application of the NB-L in other safety-related tasks (e.g., hot spot identification) is largely neglected. This study proposed a framework to develop the EB method for the NB-L model and subsequently estimate the expected crash values. A comparison between the EB and FB estimates was performed to validate the approximation framework in general. The results indicated that the proposed EB framework is able to estimate expected crashes with comparable precision to the FB estimate, but with much less computational cost. In addition, a site ranking analysis using the EB estimates was conducted to validate the proposed approximation method in safety studies. However, it should be noted that any other type of safety analysis that requires access to the expected crashes can benefit from the proposed EB method. This study concluded that the proposed EB framework can properly approximate the underlying FB approach and can reasonably be considered as an alternative to the traditional EB formula derived from the NB model. The results of this study can help to extend the application of the advanced predictive models beyond predicting crashes to other safety-related tasks, with no additional computational efforts.

OnF-modelling-based empirical Bayes estimation of variances

SummaryWe consider the problem of empirical Bayes estimation of multiple variances when provided with sample variances. Assuming an arbitrary prior on the variances, we derive different versions of the Bayes estimators using different loss functions. For one particular loss function, the resulting Bayes estimator relies on the marginal cumulative distribution function of the sample variances only. When replacing it with the empirical distribution function, we obtain an empirical Bayes version called the $F$-modelling-based empirical Bayes estimator of variances. We provide theoretical properties of this estimator, and further demonstrate its advantages through extensive simulations and real data analysis.

Hotspot identification on urban arterials at the meso level

Urban arterials form the main structure of city street networks, and typically have high traffic volume and high crash frequency. To reduce the number of crashes, hotspot identification (HSID) is the first step in the traffic safety management process, and often utilizes crash prediction models. Classical crash prediction models investigate the relationship between roadway characteristics and traffic safety at the micro level, that is, they treat road segments and intersections as isolated units. Micro-level analysis has limitations, however, when examining urban arterial crashes: 1) signal spacing is typically short for urban arterials in dense street networks, and there are interactions between intersections and road segments that classical models do not accommodate; and 2) for practical engineering, a hotspot to which countermeasures are applied generally consists of several adjacent intersections and road segments instead of a single intersection or road segment. To address these concerns, signalized intersections and their adjacent road segments were combined into meso-level units, which were adopted to investigate traffic safety data from 21 urban arterials in Shanghai, China. To determine if the meso-level unit is the most suitable research unit for identifying hotspots on urban arterials, and if so, which HSID method can most consistently identify them, this study identified micro-level (separate intersection and road segment) and meso-level (combined intersection and road segment) hotspots using crash frequency, empirical Bayesian (EB), potential for safety improvement (PSI), and full Bayesian (FB) methods. To evaluate the performance of the HSID methods, hotspot consistency over two years was tested. The results showed that 1) EB and PSI performed better than the other methods no matter which research unit was used; 2) substantial inconsistency between the identified micro- and meso-level hotspots. To identify both hazardous corridors and individual intersections and road segments on urban arterials, a combination of micro- and meso-level hotspots should be recommended to local transportation authorities.

Development of Road Diet Segment and Intersection Crash Modification Factors

Road diets can offer potential safety improvements for both pedestrians and vehicles. The additional space provided by reducing the number of vehicular through lanes can be reallocated to other uses such as bicycle lanes, parking, sidewalks, transit use, turn lanes, curb extensions, parklets, or pedestrian refuge islands. This study evaluated the safety effectiveness of road diets in Virginia using the empirical Bayes method, focusing on the common road diet conversion from a four-lane roadway to a three-lane roadway with added bike lanes. A total of 36 segment sites and 39 intersections were identified in Virginia where road diet installations were implemented between the years 2009 to 2018. The analysis showed segment crash modification factors (CMFs) of 0.62 for total crashes and 0.36 for fatal and injury (FI) crashes. Across all intersection types, the CMFs were 0.65 for total crashes and 0.54 for FI crashes. All CMFs were found to be statistically significant at a 95% confidence level. When intersections were separated into signalized and unsignalized intersections, no significant safety benefit was found for unsignalized intersections. Based on the results, it was concluded that road diets can potentially reduce crashes, and public agencies should consider the safety benefits of road diets when justifying roadway improvements.