روش بیزی برای برآورد و به روز رسانی توابع عملکرد ایمنی تحت شرایط اطلاعات محدود: تجزیه و تحلیل حساسیت

In road safety studies, decision makers must often cope with limited data conditions. In such circumstances, the maximum likelihood estimation (MLE), which relies on asymptotic theory, is unreliable and prone to bias. Moreover, it has been reported in the literature that (a) Bayesian estimates might be significantly biased when using non-informative prior distributions under limited data conditions, and that (b) the calibration of limited data is plausible when existing evidence in the form of proper priors is introduced into analyses. Although the Highway Safety Manual (2010) (HSM) and other research studies provide calibration and updating procedures, the data requirements can be very taxing. This paper presents a practical and sound Bayesian method to estimate and/or update safety performance function (SPF) parameters combining the information available from limited data with the SPF parameters reported in the HSM. The proposed Bayesian updating approach has the advantage of requiring fewer observations to get reliable estimates. This paper documents this procedure. The adopted technique is validated by conducting a sensitivity analysis through an extensive simulation study with 15 different models, which include various prior combinations. This sensitivity analysis contributes to our understanding of the comparative aspects of a large number of prior distributions. Furthermore, the proposed method contributes to unification of the Bayesian updating process for SPFs. The results demonstrate the accuracy of the developed methodology. Therefore, the suggested approach offers considerable promise as a methodological tool to estimate and/or update baseline SPFs and to evaluate the efficacy of road safety countermeasures under limited data conditions.

Safety performance functions (SPFs) often referred to as crash frequency models are an essential component of road safety studies. In practice, roadway or transportation agencies often need to estimate crash frequency models for limited data, that is, data with only a small number of observations and limited number of contributing factors (independent variables). In fact, limited data conditions frequently occur in road safety analyses, mainly due to the lack of funds required to involve a large sample of sites in developing SPFs and/or conducting before–after observational studies (Lord and Bonneson, 2005). Hence, practitioners often need to calibrate statistical models under these restrictions to obtain baseline SPFs. The MLE that relies on asymptotic theory has been shown to be unreliable for limited data conditions (Lord, 2006 and Daziano et al., 2013). However, the full Bayes (FB) paradigm can be employed as a viable alternative to the MLE. Some advantages of the FB context compared to its Frequentist counterpart are, first, that the available information (based on expert criteria, previous studies, etc.) related to the parameters of interest can be incorporated into the analysis by assigning prior distributions to these parameters. This is a vital advantage resulting in unbiased estimates for limited data (Lord and Miranda-Moreno, 2008, Miranda-Moreno et al., 2013 and Heydari et al., 2013). By using suitable priors, thus, the sample size required to conduct a reliable road safety analysis may decrease. Second, Bayesian statistics have a natural characteristic of accommodating hierarchical models. Note that hierarchical models are capable of dealing with complex data structures and their use in the Bayesian methods is common and straightforward (Gelman et al., 2003). Third, solving complex statistical models in the Frequentist framework requires further computational efforts and algorithms such as in a situation in which the MLE cannot solve a problem, and a simulation based solution is necessary. However, the Bayesian perspective, regardless of the model complexity, always applies the Bayes theorem to derive the posterior inference. Thus, the modeler can usually adopt a complicated statistical model, then run Markov chain Monte Carlo (MCMC) simulations to obtain the posteriors without any additional effort to invent or develop a method for this goal. Fourth, the uncertainty in the Frequentist statistics is mainly addressed via confidence intervals that do not imply, as believed by many practitioners, that an estimate occurs in this interval with a certain probability for a given dataset. Nevertheless, this implication can be made under the Bayesian approach. In other words, the Frequentist context cannot conclude, for an observed dataset, the probability for an estimate being in a certain interval. It can only state that a confidence interval contains the estimated value given that a considerable number of trials are repeated. Conversely, Bayesian statistics, for a given dataset, provide directly the probability that an estimate occurs in an interval. Based on this probability, a direct and explicit statement can be made, which is a natural interpretation of results. On the basis of the above-mentioned advantages, this paper adopts a Bayesian updating method. When using FB methods in road safety studies, the majority of research papers employ non-informative priors to analyze accident data. As mentioned above, specifying informative priors can result in more-robust estimates under certain circumstances (Lord and Miranda-Moreno, 2008, Miranda-Moreno et al., 2013 and Heydari et al., 2013). To specify alternative priors, for instance, Miranda-Moreno et al. (2013) built informative priors for the inverse dispersion parameter from the reported values in previous studies. However, doing so, the estimated priors vary from one practitioner to another since different studies are usually taken into account yielding disparate statistical inferences. Heydari et al. (2013), using a large number of quasi-simulated data to draw reliable statistical inferences, investigated the effect of prior specification (based on past evidence) on regression coefficient (e.g., traffic flow) estimates, inverse dispersion parameter estimates, hotspot identification, and goodness-of-fit. The authors concluded that the inverse dispersion parameter is the SPF parameter most sensitive to prior choice, while hotspot identification and deviance information criteria (DIC) are slightly affected by prior specification. Following the aforementioned studies, this paper takes advantage of informative and semi-informative priors to calibrate and update SPFs. Note that great strides have been made in improving updating and re-calibration procedures (Persaud et al., 2002, Hadayeghi et al., 2006, Yu and Abdel-Aty, 2013, Connors et al., 2013 and Wood et al., 2013). However, the data requirements for updating SPFs can be very taxing (for example, Wood et al. (2013) have expressed some concerns in this regard). We propose a practical and efficient Bayesian updating approach that has the advantage of requiring fewer observations to get reliable estimates. This approach combines the information available from limited data with specific SPF parameter values reported in the HSM to estimate and/or update SPFs. Related to the updating problem, Yu and Abdel-Aty (2013) compared different prior selection methods for Bayesian updating of SPFs. The authors grouped their case study into training and test datasets. A two-stage Bayesian updating procedure was then developed and stated to be able to provide the most accurate estimates. In practice, however, finding a training database with characteristics similar to the limited data under investigation may not be always feasible. Connors et al. (2013), based on a case study of rural single carriageway roads, compared the performance of different models (e.g., Poisson-gamma and Poisson-Weibull models) using both the MCMC and the MLE estimation techniques. The authors assumed three informative priors (gamma, log normal, and Weibull) for the inverse dispersion parameter, which have a mean of 1 and cv (coefficient of variation) value of 0.8. Other issues such as goodness-of-fit measures and independency among observations in different years were also investigated in the latter paper. For a detailed discussion on the updating issues, including temporal transferability of SPFs, see Wood et al. (2013). The scope of this research study was to offer a practical and efficient Bayesian method to estimate and update SPF parameters (which, in turn, allows practitioners to evaluate safety treatment effectiveness), especially under limited data conditions. This paper's methodology was tested through an extensive simulation exercise—instead of using a single case study—to draw reliable statistical inferences. For this purpose, we examined the estimation of the SPF parameters and indices of treatment effectiveness in before–after studies, using a large variety of prior assumptions. It was thus possible to compare a large number of prior distributions for model parameters (informative, semi-informative, and non-informative). Despite the complexity of the adopted simulation study, which was inevitable to validate the proposed method, our technique can be easily applied by practitioners. Note that by using this technique, transportation authorities will be able to conduct road safety studies even with a limited number of observations, thereby reducing the cost and time for the decision-making process.

This paper proposes a practical and efficient method that contributes to the unification of the Bayesian updating process. This methodology was developed based on the HSM for multilane undivided roadway segments and is extendable to other road facilities such as intersections. A simulation study was conducted to validate the suggested methodology. This simulation exercise, in turn, adds some knowledge to the literature by comparing a large variety of priors. To this end, we investigated the effect of various prior specifications on the estimation of the SPFs parameters and safety treatment effectiveness in FB before–after studies. This paper examined 15 models that include different assumptions for the prior distribution on the regression coefficients (e.g., traffic flow) and the inverse dispersion parameter when using the Poisson-gamma likelihood. Regardless of the complexity of the simulation exercise carried out in this study, the tested technique can be easily adopted by practitioners. As it was expected, priors with smaller variances provide estimates with narrower credible intervals (i.e., greater precision). Nevertheless, the results indicated that using very small variances penalizes the estimation of the posterior mean (i.e., less accuracy). A 4-step framework is provided to specify informative priors for the inverse dispersion parameter. This paper highlighted that there is no unique prior (or variance value) that can be used for the inverse dispersion parameter. Employing the proposed approach, the inverse dispersion parameter's prior adjusts itself accordingly to every given dataset, resulting in improved estimates. Regarding regression coefficients, a variety of priors were examined. As it was discussed, the various model results can be verified considering the accuracy and the precision of the estimates. Furthermore, this study showed that there is not a single model (prior combination) that leads to better estimates for all SPF parameters. That is, one model, for instance, may provide an accurate estimate of the coefficient associated with traffic flow, but its estimate related to the inverse dispersion parameter may be biased (e.g., model 14). Taking into account the overall performance, the analyses outcomes indicated that Model 10 and 6 with informative priors for the inverse dispersion parameter and semi-informative priors for regression coefficients perform better compared to other models. The provided method to update and estimate SPFs is a great benefit to practitioners in road safety analysis to evaluate and select safety countermeasures. Note that limited data conditions are confronted by road agencies on a regular basis. Therefore, the methodology discussed in this paper offers promise in conducting FB road safety studies; particularly, in estimating and updating SPF parameters under limited data conditions. Employing the steps provided in this research study practitioners will be able to (a) easily define suitable priors for FB road safety studies, (b) analyze limited data characterized by low mean values (e.g., data including fatal-injury accidents) without penalizing the quality of the estimates, and (c) conduct observational before–after studies with a limited number of observations available for treated and/or comparison sites. As a future work, adjustments for temporal trends from the use of national or state-wide statistics will be investigated.