Full Bayesian Research Articles

Multivariate Full Bayesian Hot Spot Identification and Ranking

The present study introduces a new technique for multivariate identification and ranking of hot spots based on the Mahalanobis distance. This approach aims to extend the univariate potential for safety improvement to cases in which multiple response variables are modeled jointly. Because the literature shows that ranking techniques based on Bayesian methods are superior to those that rely simply on the observed collision count, the proposed method was developed in a full Bayesian (FB) context. The new technique involves the following steps: (a) applying multivariate Poisson–lognormal regression models to the data by means of the FB method, (b) using the estimates of the Poisson posterior means for each site to compute the multivariate (Mahalanobis) distance from what is the normal Poisson mean for similar sites, and (c) preparing an ordered list of potentially hazardous sites. This method was applied to a sample of 173 signalized intersections in the city of Vancouver, British Columbia, Canada, for the years 2008 to 2012. The study also examines the consistency of the technique itself by analyzing the mathematical intersection of ranked sites identified in subsequent time periods. Finally, the consistency of the multivariate FB ranking was assessed against the independent (separate) univariate one that is still dominant in road safety evaluations.

Effects of spatial correlation in random parameters collision count-data models

This study investigated the inclusion of spatial correlation in random parameters collision count-data models. Three different modeling formulations were applied to measure the effects of spatial correlation in random parameters models using three years of collision data collected from two cities, Richmond and Vancouver (British Columbia, Canada). The proposed models were estimated in a Full Bayesian (FB) context using a Markov Chain Monte Carlo (MCMC) simulation. The Deviance Information Criteria (DIC) values and chi-square statistics indicated that all the models were comparable to one another. According to the parameter estimates, a variety of traffic and road geometric covariates were found to significantly influence collision frequencies. For the Richmond dataset, only 38.3% of the total variability was explained by spatial correlation under model with both heterogeneous effects and spatial correlation (Model C), as most of the variations were likely captured by heterogeneous effects and site variation. For the Vancouver dataset, the effects of spatial correlation were much clearer, with a high percentage of the total variability (83.8%) explained by spatial correlation under Model C. Moreover, model estimation results showed that the precision of parameter estimates slightly improved by inclusion of spatial correlation when the sample size was small. However, parameter estimations did not change significantly and goodness of fit did not improve which indicate that it cannot be substantiated with the current datasets that the random parameters model with both heterogeneous effects and spatial correlation is better than other models investigated. Therefore, further studies with different datasets are needed to get more clear understanding of the added benefits of incorporating spatial correlation in random parameters model.

A Full Bayesian multivariate count data model of collision severity with spatial correlation

This study investigated the inclusion of spatial correlation in multivariate count data models of collision severity. The models were developed for severe (injury and fatal) and no-injury collisions using three years of collision data from the city of Richmond and the city of Vancouver. The proposed models were estimated in a Full Bayesian (FB) context via Markov Chain Monte Carlo (MCMC) simulation. The multivariate model with both heterogeneous effects and spatial correlation provided the best fit according to the Deviance Information Criteria (DIC). The results showed significant positive correlation between various road attributes and collision severities. For the Richmond dataset, the proportion of variance for spatial correlation was smaller than the proportion of variance for heterogeneous effects. Conversely, the spatial variance was greater than the heterogeneous variance for the Vancouver dataset. The correlation between severe and no-injury collisions for the total random effects (heterogeneous and spatial) was significant and quite high (0.905 for Richmond and 0.945 for Vancouver), indicating that a higher number of no-injury collisions is associated with a higher number of severe collisions. Furthermore, the multivariate spatial models were compared with two independent univariate Poisson lognormal (PLN) spatial models, with respect to model inference and goodness-of-fit. Multivariate spatial models provide a superior fit over the two univariate PLN spatial models with a significant drop in the DIC value (35.3 for Richmond and 116 for Vancouver). These results advocate the use of multivariate models with both heterogeneous effects and spatial correlation over univariate PLN spatial models.

Collision modification functions: Incorporating changes over time

Collision modification factors (CMFs) are considered the primary tools for estimating the effectiveness of safety treatments at road sites. Three main techniques are commonly used to estimate CMFs: the empirical Bayes (EB) method, the comparison-group (CG) method, and a combination of the EB and CG methods. CMF estimates from these techniques are usually provided with a measure of uncertainty of the estimate, in the form of standard error and confidence interval.However, representing CMFs as point estimates may not be adequate to explain how a safety countermeasure affects collision frequency over time and to evaluate the presence of novelty effects associated with the treatment. Therefore, the main goal of this study is to show how to overcome this drawback through the development of collision modification functions (CMFunctions) which incorporate changes over time for the treatment effectiveness, rather than using a single value.Within a fully Bayesian (FB) context, linear and non-linear intervention models, which acknowledge that the safety treatment effects do not always occur instantaneously but are spread over future time periods, provide a promising methodological framework for estimating CMFunctions with time trend.A case study is presented where the linear and non-linear intervention models were applied to estimate the effectiveness of the “Signal Head Upgrade Program” recently implemented in the city of Surrey (British Columbia, Canada). The results of the case study highlight the advantages of estimating CMFunctions with time trend and impact on the economic evaluation of safety countermeasures. Given the way future benefits are discounted to present values, the results of using a CMFunction can affect the cost effectiveness of safety countermeasures significantly. The results also showed the non-linear intervention models provide a more realistic trend where the treatment effect in the long-run converges to an everlasting treatment impact.

Visualização dos padrões de variação da taxa de mortalidade infantil no Rio Grande do Sul, Brasil: comparação entre as abordagens Bayesiana Empírica e Totalmente Bayesiana

Infant mortality is considered a sensitive health indicator, and knowledge of its geographical profile is essential for formulating appropriate public health policies. Disease mapping aims to describe the geographical distribution of disease incidence and mortality rates. Due to the heavy instability of crude rates in small areas, methods involving Bayesian smoothing of rates are used, drawing on information for the whole area or neighborhood to estimate the event rate. The current study compares empirical Bayesian (EB) and fully Bayesian (FB) methods for infant mortality rates (accumulated data from 2001 to 2004) in Rio Grande do Sul State, Brazil. This study highlights the advantages of Bayesian estimators for viewing and interpreting maps. For the problem at hand, EB and FB methods showed quite similar results and had the great advantage of easy use by health professionals, since they evenly highlight the main spatial patterns in the mortality rate in the State during the target period.

Fully Bayesian Approach to Investigate and Evaluate Ranking Criteria for Black Spot Identification

The fully Bayesian (FB) approach for identification of collision black spots has been available for some time. However, little research has been conducted on the performance of the FB method, especially on criteria for ranking sites. A study was done to fill this void by a thorough evaluation of the FB method for black spot identification. First, an investigation compared the FB approach with the now-traditional empirical Bayesian method. It was confirmed that the FB method was superior for key ranking criteria [the posterior Poisson mean (PM) of crash frequency and potential for safety improvement] based on evaluation criteria, including sensitivity and specificity, and the sum of the PM. Next, eight ranking criteria, which included PM, posterior expected, mode and median ranks, and probability of being the worst, were proposed and evaluated for the best of several FB model variations explored. The mode rank of the posterior distribution of the Poisson mean proved to be the most promising because it tended to provide the best results, especially for top-ranked sites. The sum of the Poisson mean was also found to be a solid evaluation criterion, especially for limited numbers of top-ranked sites.

A fully Bayesian multivariate approach to before–after safety evaluation

This paper presents a fully Bayesian multivariate approach to before–after safety evaluation. Although empirical Bayes (EB) methods have been widely accepted as statistically defensible safety evaluation tools in observational before–after studies for more than a decade, EB has some limitations such that it requires a development and calibration of reliable safety performance functions (SPFs) and the uncertainty in the EB safety effectiveness estimates may be underestimated when a fairly large reference group is not available. This is because uncertainty (standard errors) of the estimated regression coefficients and dispersion parameter in SPFs is not reflected in the final safety effectiveness estimate of EB. Fully Bayesian (FB) methodologies in safety evaluation are emerging as the state-of-the-art methods that have a potential to overcome the limitations of EB in that uncertainty in regression parameters in the FB approach is propagated throughout the model and carries through to the final safety effectiveness estimate. Nonetheless, there have not yet been many applications of fully Bayesian methods in before–after studies. Part of reasons is the lack of documentation for a step-by-step FB implementation procedure for practitioners as well as an increased complexity in computation. As opposed to the EB methods of which steps are well-documented in the literature for practitioners, the steps for implementing before–after FB evaluations have not yet been clearly established, especially in more general settings such as a before–after study with a comparison group/comparison groups. The objectives of this paper are two-fold: (1) to develop a fully Bayesian multivariate approach jointly modeling crash counts of different types or severity levels for a before–after evaluation with a comparison group/comparison groups and (2) to establish a step-by-step procedure for implementing the FB methods for a before–after evaluation with a comparison group/comparison groups. The fully Bayesian multivariate approach introduced in this paper has additional advantages over the corresponding univariate approaches (whether classical or Bayesian) in that the multivariate approach can recover the underlying correlation structure of the multivariate crash counts and can also lead to a more precise safety effectiveness estimate by taking into account correlations among different crash severities or types for estimation of the expected number of crashes. The new method is illustrated with the multivariate crash count data obtained from expressways in Korea for 13 years to assess the safety effectiveness of decreasing the posted speed limit.

Comparison of empirical Bayes and full Bayes approaches for before–after road safety evaluations

The empirical Bayes (EB) approach has now gained wide acceptance among researchers as the much preferred one for the before-after evaluation of road safety treatments. In this approach, the before period crash experience at treated sites is used in conjunction with a crash prediction model for untreated reference sites to estimate the expected number of crashes that would have occurred without treatment. This estimate is compared to the count of crashes observed after treatment to evaluate the effect of the treatment. This procedure accounts for regression-to-the-mean effects that result from the natural tendency to select for treatment those sites with high observed crash frequencies. Of late, a fully Bayesian (FB) approach has been suggested as a useful, though complex alternative to the empirical Bayes approach in that it is believed to require less data for untreated reference sites, it better accounts for uncertainty in data used, and it provides more detailed causal inferences and more flexibility in selecting crash count distributions. This paper adds to the literature on comparing the two Bayesian approaches through empirical applications. The main application is an evaluation of the conversion of road segments from a four-lane to a three-lane cross-section with two-way left-turn lanes (also known as road diets). For completeness, the paper also summarizes the results of an earlier application pertaining to the evaluation of conversion of rural intersections from unsignalized to signalized control. For both applications, the estimated safety effects from the two approaches are comparable.

Missing-Data Methods for Generalized Linear Models

Missing data is a major issue in many applied problems, especially in the biomedical sciences. We review four common approaches for inference in generalized linear models (GLMs) with missing covariate data: maximum likelihood (ML), multiple imputation (MI), fully Bayesian (FB), and weighted estimating equations (WEEs). There is considerable interest in how these four methodologies are related, the properties of each approach, the advantages and disadvantages of each methodology, and computational implementation. We examine data that are missing at random and nonignorable missing. For ML, we focus on techniques using the EM algorithm, and in particular, discuss the EM by the method of weights and related procedures as discussed by Ibrahim. For MI, we examine the techniques developed by Rubin. For FB, we review approaches considered by Ibrahim et al. For WEE, we focus on the techniques developed by Robins et al. We use a real dataset and a detailed simulation study to compare the four methods.

Estimation in Bayesian Disease Mapping

Recent work on Bayesian inference of disease mapping models discusses the advantages of the fully Bayesian (FB) approach over its empirical Bayes (EB) counterpart, suggesting that FB posterior standard deviations of small-area relative risks are more reflective of the uncertainty associated with the relative risk estimation than counterparts based on EB inference, since the latter fail to account for the variability in the estimation of the hyperparameters. In this article, an EB bootstrap methodology for relative risk inference with accurate parametric EB confidence intervals is developed, illustrated, and contrasted with the hyperprior Bayes. We elucidate the close connection between the EB bootstrap methodology and hyperprior Bayes, present a comparison between FB inference via hybrid Markov chain Monte Carlo and EB inference via penalized quasi-likelihood, and illustrate the ability of parametric bootstrap procedures to adjust for the undercoverage in the "naive" EB interval estimates. We discuss the important roles that FB and EB methods play in risk inference, map interpretation, and real-life applications. The work is motivated by a recent analysis of small-area infant mortality rates in the province of British Columbia in Canada.

Small area estimation of incidence of cancer around a known source of exposure with fine resolution data

OBJECTIVESTo describe the small area system developed in Finland. To illustrate the use of the system with analyses of incidence of lung cancer around an asbestos mine. To compare the...

Bayesian estimates of disease maps: how important are priors?

In the fully Bayesian (FB) approach to disease mapping the choice of the hyperprior distribution of the dispersion parameter is a key issue. In this context we investigated the sensitivity of the rate ratio estimates to the choice of the hyperprior via a simulation study. We also compared the performance of the FB approach to mapping disease risk to the conventional approach of mapping maximum likelihood (ML) estimates and p-values. The study was modelled on the incidence data of insulin dependent diabetes mellitus (IDDM) as observed in the communes of Sardinia.