Full Bayesian Research Articles

Cost free hyper-parameter selection/averaging for Bayesian inverse problems with vanilla and Rao-Blackwellized SMC samplers

In Bayesian inverse problems, one aims at characterizing the posterior distribution of a set of unknowns, given indirect measurements. For non-linear/non-Gaussian problems, analytic solutions are seldom available: Sequential Monte Carlo samplers offer a powerful tool for approximating complex posteriors, by constructing an auxiliary sequence of densities that smoothly reaches the posterior. Often the posterior depends on a scalar hyper-parameter, for which limited prior information is available. In this work, we show that properly designed Sequential Monte Carlo (SMC) samplers naturally provide an approximation of the marginal likelihood associated with this hyper-parameter for free, i.e. at a negligible additional computational cost. The proposed method proceeds by constructing the auxiliary sequence of distributions in such a way that each of them can be interpreted as a posterior distribution corresponding to a different value of the hyper-parameter. This can be exploited to perform selection of the hyper-parameter in Empirical Bayes (EB) approaches, as well as averaging across values of the hyper-parameter according to some hyper-prior distribution in Fully Bayesian (FB) approaches. For FB approaches, the proposed method has the further benefit of allowing prior sensitivity analysis at a negligible computational cost. In addition, the proposed method exploits particles at all the (relevant) iterations, thus alleviating one of the known limitations of SMC samplers, i.e. the fact that all samples at intermediate iterations are typically discarded. We show numerical results for two distinct cases where the hyper-parameter affects only the likelihood: a toy example, where an SMC sampler is used to approximate the full posterior distribution; and a brain imaging example, where a Rao-Blackwellized SMC sampler is used to approximate the posterior distribution of a subset of parameters in a conditionally linear Gaussian model.

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.

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.

Relative Maximum Likelihood updating of ambiguous beliefs

This paper proposes and axiomatizes a new updating rule: Relative Maximum Likelihood (RML) updating for ambiguous beliefs represented by a set of priors (C). This rule takes the form of applying Bayes’ rule to a subset of C. This subset is a linear contraction of C towards its subset ascribing the maximal probability to the observed event. The degree of contraction captures the extent of willingness to discard priors based on likelihood when updating. Two well-known updating rules of multiple priors, full Bayesian (FB) and Maximum Likelihood (ML), are included as special cases of RML. An axiomatic characterization of conditional preferences generated by RML updating is provided when the preferences admit Maxmin Expected Utility representations. The axiomatization relies on weakening the axioms characterizing FB and ML. The axiom characterizing ML is identified for the first time in this paper, addressing a long-standing open question in the literature.

The Explanatory Generalized Graded Unfolding Model: Incorporating Collateral Information to Improve the Latent Trait Estimation Accuracy.

Collateral information has been used to address subpopulation heterogeneity and increase estimation accuracy in some large-scale cognitive assessments. The methodology that takes collateral information into account has not been developed and explored in published research with models designed specifically for noncognitive measurement. Because the accurate noncognitive measurement is becoming increasingly important, we sought to examine the benefits of using collateral information in latent trait estimation with an item response theory model that has proven valuable for noncognitive testing, namely, the generalized graded unfolding model (GGUM). Our presentation introduces an extension of the GGUM that incorporates collateral information, henceforth called Explanatory GGUM. We then present a simulation study that examined Explanatory GGUM latent trait estimation as a function of sample size, test length, number of background covariates, and correlation between the covariates and the latent trait. Results indicated the Explanatory GGUM approach provides scoring accuracy and precision superior to traditional expected a posteriori (EAP) and full Bayesian (FB) methods. Implications and recommendations are discussed.

Meso-level hotspot identification for suburban arterials

Accurate identification of crash hotspots forms the foundation of roadway safety improvement. The Highway Safety Manual micro-level approach uses individual intersections and road segments as analysis units, and correspondingly identifies some isolated road entities as hotspots. However, because traffic police and administrative agencies routinely conduct safety improvement based on multiple continuous segments and intersections, the identification of hotspots at the micro-level is inefficient for field application. To better meet this need, this study proposes a new meso-level approach to identify hotspots, specifically on suburban arterials. Meso-level analysis units of three different configurations (201, 150, and 100 units) were obtained by combining a set number of intersections and their adjacent segments according to crash distribution and homogeneity. Their influence areas were determined according to the proportion of urbanized land in areas perpendicularly adjacent to the arterials. Three Bayesian Poisson-lognormal conditional autoregressive models (PLN-CAR) considering spatial correlations were developed for each unit configuration, using the full Bayesian (FB) method to ameliorate random fluctuation in crash counts. Potential for safety improvement (PSI) values were calculated based on the modeling results and were used to identify hotspots. Two measures, i.e., the concentrated degree of hotspots (CDH) and the hotspot identification accuracy (HIA), were proposed to make a quantitative and comparative evaluation. Results showed that 1) arterials with more parallel roads suffer lower crash risk, and 2) considering both the hotspot distribution and the identification accuracy, the 150 meso-level unit configuration was the best. The proposed meso-level hotspot identification method promises to be adaptive to safety improvement practices on suburban arterials.

Investigating the impacts of crash prediction models on quantifying safety effectiveness of Adaptive Signal Control Systems

Introduction: Adaptive Signal Control System (ASCS) can improve both operational and safety benefits at signalized corridors. Methods: This paper develops a series of models accounting for model forms and possible predictors and implements these models in Empirical Bayes (EB) and Fully Bayesian (FB) frameworks for ASCS safety evaluation studies. Different models are validated in terms of the ability to reduce the potential bias and variance of prediction and improve the safety effectiveness estimation accuracy using real-world crash data from non-ASCS sites. This paper then develops the safety effectiveness of ASCS at six different corridors with a total of 65 signalized intersections with the same type of ASCS, in South Carolina. Results: Validation results show that the FB model that accounts for traffic volume, roadway geometric features, year factor, and spatial effects shows the best performance among all models. The study findings reveal that ASCS reduces crash frequencies in the total crash, fatal and injury crash, and angle crash for most of the intersections. The safety effectiveness of ASCS varies with different intersection features (i.e., AADT at major streets, number of legs at an intersection, the number of through lanes on major streets, the number of access points on minor streets, and the speed limit at major streets). Conclusions: ASCS is associated with crash reductions, and its safety effects vary with different intersection features. Practical Applications: The findings of this research encourage more ASCS deployments and provide insights into selecting ASCS deployment sites for reducing crashes considering the variation of the safety effectiveness of ASCS.

Before–After Evaluation of Left-Turn Lane Extension Considering Injury Severity and Collision Type

Left-turn lanes are commonly used to provide space to accommodate vehicle deceleration and provide adequate storage of turning vehicles. The objective of this study is to evaluate the safety effectiveness of extending the length of left-turn lanes at signalized intersection approaches. Five years of collision data including injury severity and collision type from three treatment sites and 31 comparison sites in the City of Surrey, Canada were used in the study. The analysis focused on target crashes including left-turn-related rear-end and left-turn-related sideswipe collisions. A full Bayesian (FB) before–after analysis was conducted for all collisions, severity levels, and collision types. Multivariate Poisson–lognormal linear intervention models were used for the analysis. The treatment effectiveness index was calculated to quantitatively measure the effectiveness of the safety treatment. The FB before–after results showed that the treatment-related collisions were reduced by 57.4% following the implementation of extended left-turn lane. The reduction in injuries and fatalities collisions (63.8%) was greater than that in property damage only collisions (55.7%). The decrease in rear-end collisions (62.8%) was greater than that in sideswipe collisions (58.1%). The findings indicate a remarkable improvement in safety after the length extension of the left-turn lane.

A systematic review of Bayesian spatial-temporal models on cancer incidence and mortality.

This study aimed to review the types and applications of fully Bayesian (FB) spatial-temporal models and covariates used to study cancer incidence and mortality. This systematic review searched articles published within Medline, Embase, Web-of-Science and Google Scholar between 2014 and 2018. A total of 38 studies were included in our study. All studies applied Bayesian spatial-temporal models to explore spatial patterns over time, and over half assessed the association with risk factors. Studies used different modelling approaches and prior distributions for spatial, temporal and spatial-temporal interaction effects depending on the nature of data, outcomes and applications. The most common Bayesian spatial-temporal model was a generalized linear mixed model. These models adjusted for covariates at the patient, area or temporal level, and through standardization. Few studies (4) modelled patient-level clinical characteristics (11%), and the applications of an FB approach in the forecasting of spatial-temporally aligned cancer data were limited. This review highlighted the need for Bayesian spatial-temporal models to incorporate patient-level prognostic characteristics through the multi-level framework and forecast future cancer incidence and outcomes for cancer prevention and control strategies.

Full Bayesian Before-After Analysis of Safety Effects of Variable Speed Limit System

Variable speed limits (VSL) is a major Intelligent Transportation System (ITS) technology for controlling freeway mainline traffic, which have been increasingly used to improve traffic safety and operations efficiency of freeway traffic management. The primary objective of this study is to evaluate the safety impacts of the VSL system implemented on Interstate 5 in Seattle, United States since 2010. A Full Bayesian (FB) before-after analysis was conducted based on 9,787 crashes that occurred in a 72-month study period. The analysis was conducted for all crashes, crash severity levels, crash types, and crash causes. The FB before-after results implied that the total crash count was reduced by 32.23% with a standard deviation of 3.58% after the VSL system was applied on the freeway. The count of crashes with no injury decreased more than crashes with severe injury and possible injury. The effect of rear-end crash reduction was the most beneficial among all crash types, while the effect on sideswipe crash reduction was the least. The study also compared the traffic speed features in the before and after periods to fully evaluate the impacts of the VSL system on traffic operations. The result indicated that, with VSL control, the difference in speed was reduced by the VSL system. The results of this study are particularly valuable for policy and control strategy development, and cost-benefit evaluation associated with VSL system implementations.

Modeling correlation and heterogeneity in crash rates by collision types using full bayesian random parameters multivariate Tobit model

Crashes present different collision types. There usually exist unobserved risk factors which could jointly affect crash rates of different types, resulting in correlation and heterogeneity issues across observations. The primary objective of the study is to propose a novel random parameters multivariate Tobit (RPMV-Tobit) model for evaluating risk factors on crash rates of different collision types. Crash data from 367 freeway diverge areas in a three-year period were obtained for modeling. Three major types of collisions including rear-end, sideswipe, and angle collisions were considered. The RPMV-Tobit model was structured to simultaneously accommodate correlations between crash rates across collision types and unobserved heterogeneity across observations. The RPMV-Tobit model was compared with a multivariate Tobit (MV-Tobit) model, a random effect multivariate Tobit (REMV-Tobit) model, and independent univariate Tobit (IU-Tobit) models under the Bayesian framework. The results showed that MV-Tobit model outperforms the IU-Tobit models on fitting crash rates, indicating that accounting for the correlation between crash types can improve model fit. The RPMV-Tobit model and REMV-Tobit model perform better than the MV-Tobit model, suggesting that accounting for the unobserved heterogeneous can further improve model fit. The improvement of model performance with the RPMV-Tobit model is higher than that with the REMV-Tobit model. The impacts of each risk factor on crash rates were estimated and some differences were found across different collision types. The lane-balanced design, number of lanes on mainline, speed limit, and speed difference present significant heterogeneous effects on crash rates. Findings suggest that the RPMV-Tobit model is a superior approach for comprehensive crash rates modeling and traffic safety evaluation purposes.

Comparison of Bayesian techniques for the before–after evaluation of the safety effectiveness of short 2+1 road sections

In evaluating the effectiveness of a road safety treatment, the regression to the mean phenomenon is a cause for concern because of a possible overestimation of benefits. Therefore, Bayesian approaches are usually suggested as the most appropriate methodologies for before—after studies as they account for regression to the mean effects. The empirical Bayes (EB) methodology examines the estimation of the expected number of crashes that would have occurred without treatment and compares them with the crashes observed at the treated sites. Even if there is no significant regression to the mean bias, the EB technique requires a reliable and large dataset with sufficient years of observation and number of treated sites, adequate for estimating the safety effects of a treatment with acceptable standard errors. In this framework, a full Bayesian (FB) approach can mitigate the problem of using small datasets by providing more detailed causal inferences and more flexibility in selecting crash count distributions, acknowledging that a more complex methodology must be applied. With the aim of estimating the safety improvements of new, short 2 + 1 road sections in Poland limited by the existing road network, EB and FB estimations are compared and different safety performance function (SPF) model forms are used in order to evaluate the performance of the two methodologies. Results indicated that, even if crash modification factors (CMFs) resulted in similar average values, the EB trend is to underestimate CMFs compared with the more complex methodology, while overall the FB approach provided a lower standard deviation. The differences are more pronounced between the EB and FB approaches when a simple SPF model form is used for the analysed dataset. Moreover, for this specific dataset, the difference between the FB method and the EB method using a refined regression model with more variables was negligible.

An empirical evaluation of multivariate spatial crash frequency models

Many studies have employed spatial, temporal, or a combination of both specifications for analysis of roadway crashes at different spatial levels. However, there is lack of a comprehensive study which compares the crash estimation performance of different spatial weight matrices and their combination with various temporal treatments. The current study fills the research gap by comparing different Full Bayesian (FB) multivariate spatiotemporal crash models. The pedestrian and bicyclist crash data across an eight-year period for 58 counties in California were used as a case study. Three groups of models were developed based on temporal treatment, where each group comprised of 17 models differing on the basis of different adjacency- and distance-based spatial weight matrices. The first group of multivariate models incorporated only unstructured random error term and spatially structured conditional autoregressive (CAR) term. The second group built upon the former and introduced a linear time trend to develop a spatiotemporal model, while the third group allowed the interaction of space and time. The predictive performance of the alternate models across and within groups was assessed by employing several evaluation criteria.The modeling results demonstrated the robustness of models based on the similar signs and closeness of coefficients for the posterior estimates of parameters. For overall model comparison, the pure-distance model D0.5 demonstrated the best performance for different evaluation criteria based on training and test errors across three groups. The variability in performance of other distance models suggested that caution must be exercised for the choice of exponents. The correlation analysis revealed the presence of positive correlations among the criteria based on training errors, as well as with cross-validation. However, a very strong positive correlation was observed between the criteria based on effective number of parameters and posterior deviance, indicating that an increased number of parameters may not lead to improved model fit. This finding reinforced the importance of selecting the optimum weight matrix for spatial correlation as a more complex structure may not lead to expected advantages at model performance. For comparison among three groups of different temporal treatments, the third group demonstrated the best performance and conveyed the benefits of incorporating the spatial and temporal interaction. The results from ANOVA (analysis of variance) and HSD (Honest Significant Differences) tests also established the existence of statistical differences for the superiority of space-time interactions models. However, the box and whisker plots demonstrated high variability among the models of the third group, suggesting that some models may not benefit from interaction term. For comparison among adjacency- and distance-based models, the distance-based models were mostly observed to be superior. However, the greater variability of model performance associated with distance-based models suggested for careful consideration during their selection. Additionally, it is important to note that the results observed in this study are specific to the county-level crash data of California. As such, the study does not recommend generalization of the results for extension to other spatial levels of roadway network, and readers and future research studies are advised to exercise caution before implementing the models.

Ranking cities for safety investigation by potential for safety improvement

ABSTRACTThe authors performed a city-level hot-spot identification by using the 4-year data of 265 cities in California. It is intended to equip road safety professionals with more useful information to compare the safety performance of city as a whole. Potential for safety improvement (PSI) was adopted as a measure of crash risk to compare alternate identification of hot spot (HSID) methods, including the Empirical Bayes (EB) and three full Bayesian (FB) alternatives, negative binomial, Poisson log-normal, and the Poisson temporal random effect, for ranking the safety performance of cities. Five evaluation tests which contain the Site Consistency Test, the Method Consistency Test, the Total Rank Difference Test, the Total Performance Difference Test, and the Total Score Test were applied to evaluate the performance of the four HSID methods. Moreover, two cutoff levels, top 5% and 10% cities, were employed for more reliable results. Overall, the study results are consistent with the results of previous quantitative evaluations focused on micro-level HSID. The three FB approaches significantly outperform the EB counterpart. The method accounting for temporal random effect produces more reliable HSID results than those without considering the serial correlations in collision counts.

Generalized linear models with covariate measurement error and unknown link function

ABSTRACTGeneralized linear models (GLMs) with error-in-covariates are useful in epidemiological research due to the ubiquity of non-normal response variables and inaccurate measurements. The link function in GLMs is chosen by the user depending on the type of response variable, frequently the canonical link function. When covariates are measured with error, incorrect inference can be made, compounded by incorrect choice of link function. In this article we propose three flexible approaches for handling error-in-covariates and estimating an unknown link simultaneously. The first approach uses a fully Bayesian (FB) hierarchical framework, treating the unobserved covariate as a latent variable to be integrated over. The second and third are approximate Bayesian approach which use a Laplace approximation to marginalize the variables measured with error out of the likelihood. Our simulation results show support that the FB approach is often a better choice than the approximate Bayesian approaches for adjusting for measurement error, particularly when the measurement error distribution is misspecified. These approaches are demonstrated on an application with binary response.

Macroscopic hotspots identification: A Bayesian spatio-temporal interaction approach

This study proposes a Bayesian spatio-temporal interaction approach for hotspot identification by applying the full Bayesian (FB) technique in the context of macroscopic safety analysis. Compared with the emerging Bayesian spatial and temporal approach, the Bayesian spatio-temporal interaction model contributes to a detailed understanding of differential trends through analyzing and mapping probabilities of area-specific crash trends as differing from the mean trend and highlights specific locations where crash occurrence is deteriorating or improving over time. With traffic analysis zones (TAZs) crash data collected in Florida, an empirical analysis was conducted to evaluate the following three approaches for hotspot identification: FB ranking using a Poisson-lognormal (PLN) model, FB ranking using a Bayesian spatial and temporal (B-ST) model and FB ranking using a Bayesian spatio-temporal interaction (B-ST-I) model. The results show that (a) the models accounting for space-time effects perform better in safety ranking than does the PLN model, and (b) the FB approach using the B-ST-I model significantly outperforms the B-ST approach in correctly identifying hotspots by explicitly accounting for the space-time variation in addition to the stable spatial/temporal patterns of crash occurrence. In practice, the B-ST-I approach plays key roles in addressing two issues: (a) how the identified hotspots have evolved over time and (b) the identification of areas that, whilst not yet hotspots, show a tendency to become hotspots. Finally, it can provide guidance to policy decision makers to efficiently improve zonal-level safety.

Before–After Safety Evaluation Using Full Bayesian Macroscopic Multivariate and Spatial Models

Many studies have addressed spatial correlation in traffic collision modeling. It has been generally concluded that the inclusion of spatial correlation improves model goodness of fit and the precision of parameter estimates. However, the application in before–after safety evaluation has rarely been documented in the traffic safety literature. The objectives of the presented study were to ( a) apply both the univariate and multivariate full Bayesian (FB) spatial models in before–after safety evaluation and ( b) compare the results with those of nonspatial FB models. A reduction of the posted speed limit in urban residential neighborhoods in Edmonton, Alberta, Canada was used as a case study for the before–after safety evaluation. Yearly collision data and other neighborhood characteristics data were collected for a group of treated and reference neighborhoods to develop macroscopic models. The four models considered in this study were ( a) Poisson–lognormal, ( b) Poisson–lognormal with conditional autoregressive (CAR) distribution, ( c) multivariate Poisson–lognormal, and ( d) multivariate Poisson–lognormal with CAR distribution. The results showed that the multivariate Poisson–lognormal with CAR distribution model for collision severities outperformed the other three models according to the deviance information criteria. Parameter estimates showed slight differences across the models. However, for the current data set, the results of the before–after safety evaluation showed similar findings across the models. Estimated collision reductions were 13%, 24%, and 12% for total, severe, and property-damage-only collisions, respectively.

Multivariate random parameters collision count data models with spatial heterogeneity

This study investigated the effects of including spatial heterogeneity in multivariate random parameters models and their influence on different collision severity levels. The models were developed for severe (injury and fatal) and no-injury collisions using three years of collision data from the city of Vancouver. Three different modeling formulations were applied to measure the effects of spatial heterogeneity in a multivariate random parameters model. The proposed models were estimated in a Full Bayesian (FB) context using Markov Chain Monte Carlo (MCMC) simulation. The Deviance Information Criteria (DIC) values indicated that all the models were comparable to one another. Therefore, no particular model can be distinctly preferred over others. According to parameter estimates, a variety of traffic and road geometric covariates were found to significantly influence collision severities. The variance for spatial heterogeneity was higher than the variance for heterogeneous effects. The correlation between severe and no-injury collisions for the total random effects (heterogeneous and spatial) was significant and quite high, indicating that higher no-injury collisions are associated with higher severe collisions. These results support the incorporation of spatial heterogeneity in multivariate random parameters models. Furthermore, the multivariate random parameters spatial models were compared with two independent univariate random parameters spatial models with respect to model inference and goodness of fit. The multivariate spatial models outperformed the two univariate spatial models with a very significant drop in the DIC value.

Efficient estimation and prediction for the Bayesian binary spatial model with flexible link functions.

Spatial generalized linear mixed models (SGLMMs) are popular models for spatial data with a non-Gaussian response. Binomial SGLMMs with logit or probit link functions are often used to model spatially dependent binomial random variables. It is known that for independent binomial data, the robit regression model provides a more robust (against extreme observations) alternative to the more popular logistic and probit models. In this article, we introduce a Bayesian spatial robit model for spatially dependent binomial data. Since constructing a meaningful prior on the link function parameter as well as the spatial correlation parameters in SGLMMs is difficult, we propose an empirical Bayes (EB) approach for the estimation of these parameters as well as for the prediction of the random effects. The EB methodology is implemented by efficient importance sampling methods based on Markov chain Monte Carlo (MCMC) algorithms. Our simulation study shows that the robit model is robust against model misspecification, and our EB method results in estimates with less bias than full Bayesian (FB) analysis. The methodology is applied to a Celastrus Orbiculatus data, and a Rhizoctonia root data. For the former, which is known to contain outlying observations, the robit model is shown to do better for predicting the spatial distribution of an invasive species. For the latter, our approach is doing as well as the classical models for predicting the disease severity for a root disease, as the probit link is shown to be appropriate. Though this article is written for Binomial SGLMMs for brevity, the EB methodology is more general and can be applied to other types of SGLMMs. In the accompanying R package geoBayes, implementations for other SGLMMs such as Poisson and Gamma SGLMMs are provided.

Full Bayesian evaluation of the safety effects of reducing the posted speed limit in urban residential area

Full Bayesian (FB) before–after evaluation is a newer approach than the empirical Bayesian (EB) evaluation in traffic safety research. While a number of earlier studies have conducted univariate and multivariate FB before–after safety evaluations and compared the results with the EB method, often contradictory conclusions have been drawn. To this end, the objectives of the current study were to (i) perform a before–after safety evaluation using both the univariate and multivariate FB methods in order to enhance our understanding of these methodologies, (ii) perform the EB evaluation and compare the results with those of the FB methods and (iii) apply the FB and EB methods to evaluate the safety effects of reducing the urban residential posted speed limit (PSL) for policy recommendation. In addition to three years of crash data for both the before and after periods, traffic volume, road geometry and other relevant data for both the treated and reference sites were collected and used. According to the model goodness-of-fit criteria, the current study found that the multivariate FB model for crash severities outperformed the univariate FB models. Moreover, in terms of statistical significance of the safety effects, the EB and FB methods led to opposite conclusions when the safety effects were relatively small with high standard deviation. Therefore, caution should be taken in drawing conclusions from the EB method. Based on the FB method, the PSL reduction was found effective in reducing crashes of all severities and thus is recommended for improving safety on urban residential collector roads.