Estimating Safety by the Empirical Bayes Method: A Tutorial

In order to systematically analyze the trends in crash occurrence, Utah Department of Transportation (UDOT) has developed over the years a state-wide crash database and a GIS-based Data Almanac to search through the crash database as well as other databases including annual average daily traffic (AADT). The Data Almanac is a powerful search engine for extracting necessary crash data. Using the crash data provides one way to describe what has happened to the sections of highways that the analysts are evaluating. However, evaluating what might happen when improvements are made requires going one step further. The Interactive Highway Safety Design Model (IHSDM) developed by the Federal Highway Administration (FHWA), may be used to help the engineers conduct crash prediction and design consistency evaluation and intersection safety review as part of highway safety audits. This software provides an opportunity to design, operation, and safety audits, This software provides an opportunity to design, operation, and safety engineers for evaluating sections of two-lane highways from various aspects and identifying most cost effective improvement alternatives to such audited sections.

This study tries to develop a Korea highway safety evaluation proto type model for quantified safety evaluation. We develop a base model and crash modification factors (CMFs) through reviewing the procedures and methodologies in Highway Safety Manual. At the same time, we try to carefully consider collectable data and roadway environment of Korea that are different from the USA. CMFs are developed through models and expert's judgement. Developed CMFs are for driveway density, pedestrian crosswalk, existence of median, terrain types, and land-use types. Also, verification of the developed base model and CMFs were performed using two routes of Korea's rural two-lane highways while comparing with roadway safety design synthesis and Crash Prediction Module (CPM) of Interactive Highway Safety Design Model (IHSDM). Applicability analysis was then performed in two aspects of predictability and adaptability of the proto type model. As a result, CMFs and predicted accident estimates of CPM are similar with the developed Korea proto type model. However, the base model and CMFs of the developed proto type for Korea are more precise than the others when evaluating sections where the number of actual accidents is higher. It is found that considering CMF for land-use type is justifiable when developing a Korea highway safety evaluation proto type model. This study validates the possibility of developing a full scale of Korea highway safety evaluation model.

A national-level safety analysis tool is needed to complement existing analytical tools for assessment of the safety impacts of roadway design alternatives. FHWA has sponsored the development of the Interactive Highway Safety Design Model (IHSDM), which is roadway design and redesign software that estimates the safety effects of alternative designs. Considering the importance of IHSDM in shaping the future of safety-related transportation investment decisions, FHWA justifiably sponsored research with the sole intent of independently validating some of the statistical models and algorithms in IHSDM. Statistical model validation aims to accomplish many important tasks, including ( a) assessment of the logical defensibility of proposed models, ( b) assessment of the transferability of models over future time periods and across different geographic locations, and ( c) identification of areas in which future model improvements should be made. These three activities are reported for five proposed types of rural intersection crash prediction models. The internal validation of the model revealed that the crash models potentially suffer from omitted variables that affect safety, site selection and countermeasure selection bias, poorly measured and surrogate variables, and misspecification of model functional forms. The external validation indicated the inability of models to perform on par with model estimation performance. Recommendations for improving the state of the practice from this research include the systematic conduct of carefully designed before-and-after studies, improvements in data standardization and collection practices, and the development of analytical methods to combine the results of before-and-after studies with cross-sectional studies in a meaningful and useful way.

The Interactive Highway Safety Design Model (IHSDM) is a suite of software developed by FHWA for monitoring and analyzing two-lane rural highways in the United States. IHSDM consists of six modules: Policy Review, Crash Prediction (CPM), Design Consistency, Traffic Analysis, Intersection Review, and Driver/Vehicle. Of the six modules, CPM was selected for possible use in safety audits of two-lane rural highways. Many departments of transportation (DOTs) often lack the resources to calibrate the model's factors; hence, the module was tested to see if it could be used as is, without model calibration. Three two-lane rural highway sections in Utah were selected for a sample application of CPM to safety audits. The results of this evaluation show that the CPM can produce reasonably reliable crash predictions if appropriate input data, especially alignment data, reflect existing conditions with reasonable accuracy. Using crash records available from the multiyear crash database maintained by the Utah Department of Transportation (UDOT) and CPM's crash prediction capability, UDOT's traffic and safety engineers can locate hot spots for detailed safety audit and thus improve the safety audit task's focus and effectiveness. The study concludes that CPM could be useful for engineering decision making during safety audits of two-lane rural highways. Interpreting the outputs from CPM, however, requires knowledge and experience in highway design.

Hotspot identification (HSID) is a critical part of network-wide safety evaluations. Typical methods for ranking sites are often rooted in using the Empirical Bayes (EB) method to estimate safety from both observed crash records and predicted crash frequency based on similar sites. The performance of the EB method is highly related to the selection of a reference group of sites (i.e., roadway segments or intersections) similar to the target site from which safety performance functions (SPF) used to predict crash frequency will be developed. As crash data often contain underlying heterogeneity that, in essence, can make them appear to be generated from distinct subpopulations, methods are needed to select similar sites in a principled manner. To overcome this possible heterogeneity problem, EB-based HSID methods that use common clustering methodologies (e.g., mixture models, K-means, and hierarchical clustering) to select “similar” sites for building SPFs are developed. Performance of the clustering-based EB methods is then compared using real crash data. Here, HSID results, when computed on Texas undivided rural highway cash data, suggest that all three clustering-based EB analysis methods are preferred over the conventional statistical methods. Thus, properly classifying the road segments for heterogeneous crash data can further improve HSID accuracy.

Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates

Assessments of the potential health impacts of contaminants and other environmental risk factors are often based on comparisons of disease rates among collections of spatially aligned areas. These comparisons are valid only if the observed rates adequately reflect the true underlying area‐specific risk. In areas with small populations, observed incidence values can be highly unstable and true risk differences among areas can be masked by spurious fluctuations in the observed rates. We examine the use of Bayes and empirical Bayes methods for stabilizing incidence rates observed in geographically aligned areas. While these methods improve stability, both the Bayes and empirical Bayes approaches produce a histogram of the estimates that is too narrow when compared to the true distribution of risk. Constrained empirical Bayes estimators have been developed that provide improved estimation of the variance of the true rates. We use simulations to compare the performance of Bayes, empirical Bayes, and constrained empirical Bayes approaches for estimating incidence rates in a variety of multivariate Gaussian scenarios with differing levels of spatial dependence. The mean squared error of estimation associated with the simulated observed rates was, on average, five times greater than that of the Bayes and empirical Bayes estimates. The sample variance of the standard Bayes and empirical Bayes estimates was consistently smaller than the variance of the simulated rates. The constrained estimators produced collections of rate estimates that dramatically improved estimation of the true dispersion of risk. In addition, the mean square error of the constrained empirical Bayes estimates was only slightly greater than that of the unconstrained rate estimates. We illustrate the use of empirical and constrained empirical Bayes estimators in an analysis of lung cancer mortality rates in Ohio.

[EN] The quality of traffic on two-lane roads is closely linked to the possibility of making the overtaking maneuver that is especially critical in addition to the level of risk posed to do it. Options to facilitate overtaking are providing a path with geometric conditions for overtaking can be performed on the road with enough opportunities or provide a cross section that facilitates this advancement by auxiliary lanes. In this research, the second option with passing and climbing lanes was discussed.\nTo characterize the passing maneuver on steep grades an experimental study that allowed access to data and important information for calibrating TWOPAS, a microscopic traffic simulation model of two-lane roads, which is main element of the Traffic Analysis Module (TAM) that integrates the Interactive Highway Safety Design Model (IHSDM) was developed. TWOPAS was validated in other two-lane road with very different characteristics to the road considered during calibration thereby evaluating the usefulness of this simulation model was completed.\nConsidering a target of this research the TWOPAS was applied in the analysis of auxiliary lanes in a multitude of scenarios, around 98,000, where traffic volumes, longitudinal gradients, initiation and length of the auxiliary lane and percentage of trucks were varied; the influence of random variation of traffic generated wealth of information for the scope of this paper and considered for further analysis. A computational tool, the PEMAT, which allows automatic execution of TWOPAS (TAM), capturing the results of each simulation and generating outputs in terms of operational indicators was created.\nAmong the results of the analysis of auxiliary lanes are factors to consider the effect of passing and climbing lanes in the Percentage Time Spent Following (PTSF) and the Average Travel Speed (ATS) to adapt the procedure of Highway Capacity Manual (HCM) in 2010 and to estimate the level of service on two-lane road sections with individual auxiliary lanes.\nThe manuals for the calculation of the capacity and level of service on conventional roads does not cover the diversity of geometric and traffic conditions so that a procedure for analyzing the transit operation in any conventional road was proposed to facilitate overtaking by consideration of a number of auxiliary lanes in both directions of traffic simulation model with TWOPAS; thus overtaking can be solved for any configuration of road and transit improvements in methods in manuals or technical guidance.

A fully Bayesian multivariate approach to before–after safety evaluation

Identification of crash hot spots is the critical component of the highway safety management process. Errors in hot spot identification (HSID) may result in the inefficient use of resources for safety improvements. One HSID method that is based on the empirical Bayesian (EB) method has been widely used as an effective approach for identifying crash-prone sites. For the EB method, the negative binomial (NB) model is usually needed to obtain the EB estimates. Recently, some studies have shown that the Sichel (SI) model can be easily used in the EB modeling framework and potentially yield better EB estimates. The objective of this study was to compare the performance of the two crash prediction (SI and NB) models in identifying hot spots with the EB method. To accomplish the objective of this study, empirical crash data collected at highway segments in Texas were used to generate simulated crash counts. Three commonly used HSID methods (simple ranking, confidence interval, and EB) were applied with the use of simulated data. False positives, false negatives, and false identifications were calculated and compared across the methods. The simulation results in this study suggested that the SI-based EB method could consistently provide a better HSID result than the NB-based EB method. Moreover, EB methods yielded the lowest error percentage of the three HSID methods. This study confirmed that the EB technique was an effective method for identifying hazardous sites. On the basis of the findings in this study, it is recommended that transportation safety researchers consider the SI model as an alternative crash prediction model when the EB approach is used.

Important ideas are liable to be overlooked unless they are named. The Empirical Bayes (EB) method, in its original form, was well named and well developed by Robbins and was given a further boost by the approval of Neyman (1962). The name was all the more appealing for seeming at first like an oxymoron. After 1962, there was a burgeoning of literature on the topic, helped along by Robbins, his graduate students, their graduate students, and so on, in a scholastic genealogical tree. There were 308 papers with “empirical Bayes” or “empirical Bayesian” in the title or as a keyword listed in Current Index to Statistics for the years 1975-1988, but as explained below, many of these papers do not deal with empirical Bayes as originally defined by Robbins. The literature is too vast to be comprehensively surveyed even in a short book, and I shall concentrate mainly on one example, the “Poisson case” defined below. This was the first case considered by Robbins and the main theme of his paper becomes clear when this case is well understood.

Purpose – The focus of this chapter is on state-of-the-art methodology for observational before–after evaluations. The purpose of such evaluations is to acquire information on the safety effects of site-specific treatments that could be used by road agencies for future interventions that may have an impact on safety. Methodology – Information, illustrated by a detailed example, is presented mainly on what is still regarded as the gold standard – the Empirical Bayes (EB) method. The methodology accounts for observed changes in crash frequencies that may be due to regression-to-the-mean, changes in traffic volume and time trends. Two related approaches – a comparison group method and the Full Bayes approach, which have been legitimately used for evaluations under special circumstances – are discussed in brief. Findings – The EB before–after methodology remains the gold standard but must be applied properly. Research implications – Some issues, for example, the selection of reference groups, still require further research. Practical implications – Key issues in the application of the before–after evaluation methods in general are also discussed.

Traffic Modeling and Safety Analysis on Motorway Ramps

The performance of nonparametric estimators is heavily dependent on a bandwidth parameter. In nonparametric Bayesian methods this parameter can be specified as a hyperparameter of the nonparametric prior. The value of this hyperparameter may be made dependent on the data. The empirical Bayes method is to set its value by maximizing the marginal likelihood of the data in the Bayesian framework. In this paper we analyze a particular version of this method, common in practice, that the hyperparameter scales the prior variance. We characterize the behavior of the random hyperparameter, and show that a nonparametric Bayes method using it gives optimal recovery over a scale of regularity classes. This scale is limited, however, by the regularity of the unscaled prior. While a prior can be scaled up to make it appropriate for arbitrarily rough truths, scaling cannot increase the nominal smoothness by much. Surprisingy the standard empirical Bayes method is even more limited in this respect than an oracle, deterministic scaling method. The same can be said for the hierarchical Bayes method.

We present a method for combining unbiased sample data with possibly biased auxiliary information. The estimator we derive is similar in spirit to the James–Stein estimator. We prove that the estimator dominates the sample mean under quadratic loss. When the auxiliary information is unbiased, our estimator has risk slightly greater than the usual combined estimator. As the bias increases, however, the risk of the usual estimator is unbounded, while the risk of our estimator is bounded by the risk of the sample mean. We show how our estimator can be considered an approximation to the best linear combination of the sample data and the auxiliary information, allude to how it can be derived as an empirical Bayes estimator, and suggest a method for constructing confidence sets. Finally, the performance of our estimator is compared to that of the sample mean and the usual combined estimator using real forestry data.