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category: literaturenote citekey: heydaribayesianmethodologyestimateupdate2014 title: "Bayesian methodology to estimate and update safety performance functions under limited data conditions: A sensitivity analysis" authors: "Heydari, Shahram; Miranda-Moreno, Luis F.; Lord, Dominique; Fu, Liping" year: 2014 date: 2014-00-00 2014 doi: 10.1016/j.aap.2013.11.001 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2013.11.001" zotero_key: VTD436ET zotero_storage: TV25HGV7 collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Heydari, Shahram"

status: converted

Contents lists available at ScienceDirect

Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

Bayesian methodology to estimate and update safety performance functions under limited data conditions: A sensitivity analysis-

Shahram Heydari a,∗, Luis F. Miranda-Moreno b,1, Dominique Lordc,2, Liping Fua,3

  • a Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue W., Waterloo, ON N2L 3G1, Canada
  • b Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke St. W., Montreal, QC H3A 2K6, Canada
  • c Zachary Department of Civil Engineering, Texas A&M University, College Station, TX, USA

a r t i c l e i n f o

Article history: Received 21 May 2013 Received in revised form 27 October 2013 Accepted 1 November 2013

Keywords: Full Bayes road safety analysis Prior distribution SPF parameter estimation and update Index of treatment effectiveness

a b s t r a c t

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.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

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

luis.miranda-moreno@mcgill.ca (L.F. Miranda-Moreno), d-lord@tamu.edu (D. Lord), lfu@uwaterloo.ca (L. Fu).

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; 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; 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

A preliminary version of this paper has been presented at the International Road Safety and Simulation Conference in Rome, Italy (2013).

∗ Corresponding author. Tel.: +1 519 888 4567.

E-mail addresses: shahram.heydari@uwaterloo.ca (S. Heydari),

1 Tel.: +1 514 398 6589.

2 Tel.: +1 979 458 3949.

3 Tel.: +1 519 888 4567.

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 effortto 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; 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; 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 andAbdel-Aty (2013) compared different prior selection methods for Bayesian updating of SPFs. The authors grouped their case study into training and test datasets.Atwo-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.

2. Methodology

When dealing with limited data, one may use the HSM parameters (or any other similar guideline) to obtain a baseline model. Doing so, however, one may introduce bias in SPF parameter estimates since every dataset has its own traits. For example, a dataset from a particular jurisdiction in one part of a country may be characterized differently (e.g., different reporting criteria, etc.) than in another jurisdiction. As a result, the regression coefficients will never be exactly equal to those reported in the HSM. Therefore, a better practice would be to calibrate an SPF for each dataset separately. Now, if the number of observations is limited (i.e., a critical condition in which data is characterized by low mean values and/or small sample size), we suggest the use of HSM parameters as prior information (to estimate proper prior distributions) to calibrate an SPF for each specific database. To this end, a methodology is provided to successfully estimate and update SPF parameters from the HSM under limited data conditions. To include the low mean value problem in the analysis, fatal-injury accidents were used. Nevertheless,the proposed methodology can also be utilized for property damage only or total accidents.

In this paper, the authors examined the proposed methodology using the most critical condition;i.e., data including a small number of observations having a lowmean value. Fromthe HSM, we utilized the reported parameters associated with the contributing factors (e.g., traffic flow) and the function provided for the estimation of the dispersion parameter. Upon doing so, the expected magnitude of a parameter of interest can be introduced into the analysis. Since HSM parameters (see Table 1) are estimated through a comprehensive scientific study, these parameters can be used as a valid reference (prior belief)to build informative priors for unknown SPF parameters. The assumption, in this paper, was thatthe base conditions indicated in the HSM were satisfied (i.e., all crash modification

Table 1 Parameter values reported in the HSM for undivided 4-lane roadway segments.

Constant (a0) Traffic flow (a1) C
Fatal-injury accidents −9.410 1.094 1.796

factors (CMFs) and the calibration factor were assumed to be equal to 1.0). Therefore, it was possible to use a baseline SPF (Eq. (1)), which also helped to decrease the number of model parameters in the simulation study. Because ofthe very large number of scenarios to be analyzed via MCMC simulations (21,000 models were fitted), reducing the number of variables was almost inevitable. Note that in many situations, practitioners only have access to a few independent variables and might still need to build baseline statistical models to explain the occurrence of accidents. Obviously, once suitable baseline SPFs are built, CMFs can be applied to make necessary adjustments. In addition, one should take into account that all the quasi-simulated datasets (in the simulation study) were produced using Eq. (1) and generated accidents were only affected by variables presented in Eq. (1); hence, the omitted variables problem (Lord and Mannering, 2010; Mitra and Washington, 2012) could not emerge in this research.

$$\mu_i = \exp(a_0 + a_1 \ln(AADT_i) + \ln(L_i)) \tag{1}$$

where i = expected fatal-injury crash frequency for road segment i; a0 and a1 = regression coefficients to be estimated; Li = segment length (mile) for road segment i; and AADTi = annual average daily traffic (vehicles per day) for road segment i.

Weunderline thatit was out ofthe scope ofthis study to consider SPFs with multiple variables, and that the main objective was to present a method to estimate SPF parameters for limited data. This method is illustrated for multilane highway segments; however, it can easily be extended to other road facilities (e.g., intersections) and regression coefficients such as lane width as well. Heydari et al. (2013) demonstrated that the effect of the inverse dispersion parameter on improving the accuracy of estimates is much larger than the effect of regression coefficients. In other words, using only informative priors for the inverse dispersion parameter, it is possible to significantly improve the SPF estimates. From this study, therefore, one may only take advantage of the suggested technique to build priors for the inverse dispersion parameter to obtain more reliable estimates (see Section 2.2). Here, the SPF reported in Eq. (1) was calibrated employing the hierarchical Poisson-gamma regression. This regression model under a hierarchical Bayesian approach is described as follows:

  • yi|i ∼ Poisson (i);
  • i = iri;
  • ri ∼ Gamma (ϕ, ϕ);
  • ϕ ∼ Gamma (˛, ˇ) or ϕ ∼ exponential ().

where yi and i are the observed and expected crash frequency, respectively. i = f(xi, a) where xi and a are the vectors of site characteristics and regression coefficients, respectively. The multiplicative random effect ri is typically assumed to follow a gamma distribution with identical shape and scale parameters so that its mean and variance are, respectively, 1 and 1/ϕ where ϕ is the inverse dispersion parameter (Mitra and Washington, 2007; Anastasopoulos and Mannering, 2008). When adopting a hierarchical model, the inverse dispersion parameter, ϕ, is also assumed to follow a hyper-prior, usually a gamma distribution with the hyper-parameters ˛ and ˇ or an exponential distribution with the hyper-parameter (Lord and Miranda-Moreno, 2008).

2.1. A brief introduction to Bayesian posterior inference

The statistical inference under the Bayesian inference is obtained by combining the data likelihood with the prior distribution of the model parameters. The joint posterior distribution—for the parameter vector a and hyper-parameter vector given a vector of observed data y—can be written as:

$$f(\boldsymbol{a}, | \boldsymbol{y}) \propto f(\boldsymbol{y} | \boldsymbol{a}) \pi(\boldsymbol{a} |) \pi()$$ (2)

where f(a,-|y) is the posterior inference; f(y|a) is the data likelihood; (a|-) is the prior assumption on the vector of parameters a; (-) is the prior assumption on the vector of hyper-parameters -. Since the estimation of the posterior requires high dimensional integrations that cannot be obtained mathematically, MCMC methods such as Gibbs sampling or metropolis-Hastings algorithm are used (Gelman et al., 2003; Carlin and Louis, 2009).

2.2. Prior distribution for the inverse dispersion parameter (ϕ)

In this section, it is demonstrated how to construct alternative hyper-priors for the inverse dispersion parameter. As explained earlier, the gamma distribution on ϕ is defined as ϕ ∼ Gamma (˛, ˇ) where˛andˇare the shape andscaleparameters, respectively.And the exponential hyper-prior on ϕ is described as ϕ ∼ Exponential () where is the inverse of the expectation. Eq. (3), which is provided in the HSM, estimates the inverse dispersion parameter as a function of segment length for multilane undivided roadway segments.

$$\varphi = \exp(C + \ln(L)) \tag{3}$$

where C is a regression coefficient—provided in the HSM (Table 1)—to define the overdispersion parameter; and L is the segment length.

The following steps were adopted to construct priors for ϕ:

  • 1. calculate the inverse dispersion parameter for each site based on Eq. (3);
  • 2. compute the mean and variance of the values obtained in step 1 (for entire group of sites);
  • 3. choose a hyper-prior, which is gamma or exponentially distributed;
  • 4. estimate the hyper-prior parameters based on the mean (m) and variance () values obtained in step 2 as follows:
  • a. a gamma hyper-prior in which the shape parameter ˛= m2/ and the scale parameter ˇ = m/;
  • b. an exponential hyper-prior in which the parameter = 1/m.

Note that based on the above-mentioned steps, different priors are obtained for each dataset. In other words, the proposed prior for the inverse dispersion parameter adjusts itself to the data.

2.3. Prior distribution for regression coefficients (a0 and a1)

In the proposed methodology, we consider the regression coefficients—such as traffic flow—reported in the HSM (Table 1) for a specific type of facility (e.g., undivided 4-lane highway segments) and accidents (e.g., fatal-injury crashes) as the mean value for priors associated with these parameters. In road safety studies, a normal distribution with a mean of zero is usually assumed for regression coefficients (Mitra and Washington, 2007; El-Basyouny and Sayed, 2009). In this study, normal distributions with mean equal to parameter values reported in the HSM were also used. The variance was then defined arbitrarily, but reasonably on the basis of a priori belief of the magnitude of SPF parameters. For instance, the HSM suggests a traffic flow parameter of 1.094 for undivided 4-lane highway segments to modelfatal-injury crashes. In our technique, this value was set as the mean of the prior distribution for traffic follow parameter. Consequently, different variances of 1000, 100, 10, 5, and 2.5 were employed to determine non-informative, semi-informative, and informative priors for this parameter.

2.4. Models with different prior distributions

Various models (with different priors) that were examined in this paper are described below. Initially, 20 models were studied from which 15 models were found to be the most relevantto report.

Model 1

This model is the most common prior choice utilized in the FB road safety analysis. It assumes a non-informative prior for all model parameters: ϕ ∼ Gamma (0.001, 0.001), a0 ∼ normal (0, 1000), and a1 ∼ normal (0, 1000).

Model 2

This model has a non-informative prior for regression coefficients and an informative gamma prior for the inverse dispersion parameter: ϕ ∼ Gamma (˛, ˇ), a0 ∼ normal (0, 1000), and a1 ∼ normal (0, 1000). The estimation of gamma hyper-parameters ˛ and ˇ is described in Section 2.2.

Model 3

This model is similar to Model 2 except for the fact that the variance of ϕ is increased by two folds to cover a larger area around themean:ϕ ∼ Gamma (˛ ,ˇ ), a0 ∼ normal(0, 1000), anda1 ∼ normal (0, 1000). Gamma hyper-parameters ˛ and ˇ are equal to /2 and ˇ/2, respectively.

Model 4

This model is similar to Model 2, but it assumes an informative exponential prior for ϕ: ϕ ∼ Exponential (), a0 ∼ normal (0, 1000), anda1 ∼ normal(0, 1000). The estimationofisdescribedinSection 2.2.

Model 5

In this model, ϕ has an informative prior similar to Model 2. Regression coefficients have a mean defined based on Section 2.3 and a variance of 100, say, a semi-informative prior: ϕ ∼ Gamma (˛, ˇ), a0 ∼ normal (−9.410, 100), and a1 ∼ normal (1.094, 100).

Model 6

This model is similar to Model 5; however, the variance of regression coefficients is 10, say, an informative prior: ϕ ∼ Gamma (˛, ˇ), a0 ∼ normal (−9.410, 10), and a1 ∼ normal (1.094, 10).

Model 7 This model has an informative prior for all parameters: ϕ ∼ Gamma (˛, ˇ), a0 ∼ normal (−9.410, 5), and a1 ∼ normal (1.094, 5).

Model 8

This model has an informative prior for all parameters: ϕ ∼ Gamma (˛, ˇ), a0 ∼ normal(−9.410, 2.5), and a1 ∼ normal(1.094, 2.5).

Model 9

This model assumes an informative exponential distribution on ϕ and a semi-informative prior on regression coefficients: ϕ ∼ Exponential (), a0 ∼ normal (−9.410, 100), and a1 ∼ normal (1.094, 100).

Model 10

This model assumes an informative exponential distribution on ϕ and an informative prior on regression coefficients: ϕ ∼ Exponential (), a0 ∼ normal (−9.410, 10), and a1 ∼ normal (1.094, 10).

Model 11

This model is similar to Model 10, but the variance of regression coefficients is smaller: ϕ ∼ Exponential (), a0 ∼ normal (−9.410, 5), and a1 ∼ normal (1.094, 5).

Model 12

This model is similar to Model 10, but the variance of regression coefficients is smaller: ϕ ∼ Exponential (), a0 ∼ normal (−9.410, 2.5), and a1 ∼ normal (1.094, 2.5).

Table 2 Summary statistics of New York data.

Mean Std. dev. Min. Max.
Traffic flow (vehicles/day) 8502.2 6057.9 267 49,670
Segment length (mile) 0.48 0.69 0.01 4.96
Fatal-injury crashes 1.81 2.89 0.00 28.00

Model 13

This model assumes a semi-informative prior for ϕ and noninformative priors for regression coefficients: ϕ ∼ Gamma (0.1, 0.1), a0 ∼ normal (0, 1000), and a1 ∼ normal (0, 1000).

Model 14

This model has a semi-informative prior for all parameters: ϕ ∼ Gamma (0.1, 0.1), a0 ∼ normal (−9.410, 100), and a1 ∼ normal (1.094, 100).

Model 15

This model is similar to Model 14, but regression coefficients have informative priors: ϕ ∼ Gamma (0.1, 0.1), a0 ∼ normal (−9.410, 10), and a1 ∼ normal (1.094, 10).

The above-mentioned priors were introduced in the analysis employing the statistical software WinBUGS (Spiegelhalter et al., 2003) for performing Bayesian inference using Gibbs sampling. Fig. 1 shows the kernel density for some of the prior distributions used in this study. The type of distribution and its parameterizations (mean and variance) lead to different shapes (Fig. 1). Therefore, these priors cover the space of the unknown parameters differently resulting in dissimilar posteriors. A total number of 30,000 iterations were produced from which the first 10,000 were discarded as burn-in iterations. Two chains with different initial values were used; hence, the posterior inferences were obtained based on 40,000 samples, for each parameter.

2.5. Data and demonstration of the problem

A case study of 183 undivided 4-lane highway segments from the State of New York Department of Transportation Database was selected for this research study. Based on this case study, a simulation exercise was carried out to examine the proposed updating technique. The summary statistics ofthe data is reported in Table 2. The accident database comprised fatal-injury crashes (low mean value problem) for a period of three years, 1999–2001.

To clearly demonstrate the nature of the problem discussed in this paper regarding limited data, a sample of 20 observations was selected randomly from the case study. Fig. 2 illustrates the analysis results, in the form of caterpillar plots, for this sample using various prior specifications. Itis shown, in Fig. 2,that different posterior inferences (in terms of the posterior mean and uncertainty around the mean) are drawn from different prior assumptions. Note that the differences in the estimated mean values and more importantly the credible intervals are explicit, especially for the inverse dispersion parameter ϕ (which is shown in a logarithmic form). On the basis of the variation in the results shown in Fig. 2, the objective of this paper is to specify proper priors for SPF parameters to obtain more reliable estimates under limited data conditions.

2.6. Simulation study to show the robustness of the methodology

A data simulation study was undertaken using statistical software R (R Core Team, 2013) and WinBUGS to examine models 1–15. This simulation study consisted in creating a large number of datasets (to be able to make reliable statistical inferences) with characteristics similar to undivided 4-lane roadway segments to investigate the validity of the proposed methodology. Based on a case study of 4-lane undivided roadway segments from New York, fatal-injury accidents representing low mean values were used in

Fig. 1. Kernel density for various prior distributions: beta1 ∼ Normal (0, 1000), beta2 ∼ Normal (1.094, 100), beta3 ∼ Normal (1.094, 10), phi1 ∼ Gamma (0.001, 0.001); phi2 ∼ Gamma (0.1, 0.1); and phi3 ∼ Gamma (4.723, 2.986). Note: beta and phi represent regression coefficients and the inverse dispersion parameter, respectively.

the analyses. Take into account that site characteristics (segment length and traffic flow) were the same as those existing in the case study (real data). In each run of the simulation, site characteristics were selected randomly from the case study according to a predefined sample size (e.g., 20 observations). The accident frequencies were then generated using (a) the SPF reported in Eq. (1), (b) site characteristics of the sample, and (c) true estimates of parameters. The true estimates were obtained from the analysis of the entire case study using the MLE and the Poisson-gamma regression. Consequently, based on the true estimates of the parameters, accident data were generated. Thus, the mean of the generated accidents is very close to the mean of accidents in real data (case study). Note that for various sample sizes consisting of 20, 50, and 100 observations, 200 datasets were replicated using the Poisson-gamma model.

Following the aforementioned steps, regression and the inverse dispersion parameters were estimated for each replicated dataset, in a fully Bayesian framework, using the hierarchical Poissongamma approach considering the various models (prior specifications) discussed in Section 2.4. Because of the large number of quasi-simulated datasets to be analyzed, R2WinBUGS (Sturtz et al., 2005) was used to connect R and WinBUGS automatically. The term "quasi-simulated data" is used since site characteristics are based on real observed data (from case study), and only accidents are

Fig. 2. Mean and 95% credible intervals for SPF parameters – a sample of 20 observations from New York data.

generated randomly. The analyses outcomes were then compared with the true estimates (used as reference values)to recognize how different priors perform. The aim was also to identify, for each sample size, the prior specification approach (models 1–15) that provides better estimates, being the one that is closer to the true estimates. Besides monitoring the posterior mean of the parameters, their 95% credible intervals were also obtained and compared to investigate how different priors capture the uncertainty around the posterior mean values. To summarize, various steps to conduct the simulation study for this section were as follows:

  • 1. estimationof parameters for the entire case study,using theMLE, to obtain the true estimates of parameters;
    1. application of the true estimates in Eq. (1) to obtain i;
    1. generation of the multiplicative random effect ri based on the true estimates of ϕ, obtained from step 1, using the gamma distribution, ri ∼ Gamma(ϕ,ϕ);
    1. calculation of the expected accident frequency, i = iri;
    1. generation of synthetic accident frequencies yi based on the expected accident frequency i using the Poisson distribution, yi|i ∼ Poisson (i);
    1. generation of the quasi-simulated datasets by combining and matching synthetic accident frequencies (generated from the previous step) with site attributes obtained from the case study.

  • 2.7. Testing the proposed methodology in FB before–after studies

In this paper, we also investigated how the proposed method performs in the estimation of safety countermeasures

Fig. 3. Mean and 95% credible intervals of regression parameter estimates (quasi-simulated data) for: (a) the sample of 20 observations; (b) the sample of 50 observations; and (c) the sample of 100 observations. Dashed line represents the true estimate obtained from calibrating the entire case study using the MLE.

effectiveness. It was assumed that the quasi-simulated data (including 20, 50, and 100 observations for each of which 200 datasets were replicated) were subject to a hypothetical safety treatment with a known index of effectiveness * = 0.7. For both treated and comparison sites, two assumptions were, first, that there was not any variation in the site characteristics (e.g., traffic flow) in the after period; thus, the entire change in the crash frequency was due to the hypothetical treatment with a known index of effectiveness. Second, that the before and after periods were of the same duration. On the basis of these assumptions, the adjustment factor (Eq. (5)) was 1. Therefore, comparison sites were canceled out from the simulation exercise for FB before–after studies. In before–after studies, the index of effectiveness is mainly used to determine the effect of safety treatment on crash frequencies. Under the FB context, credible intervals of this index can be directly obtained from the analysis. Quantifying the uncertainty around the estimated index is an important component in assessing safety countermeasures, which helps evaluate the likely size of a safety effect. The effectiveness index is estimated as follows.

Let be the average expected crash frequency of treated sites in the after period if treatment has not been implemented (Hauer, 1997). Parameter can be estimated from Eq. (4).

$$\pi = \gamma_{tb} \cdot AF \tag{4}$$

where tb is the average expected crash frequencies of the treated sites in the before period, and AF is the adjustment factor defined in Eq. (5):

$$AF = \gamma{ca}/\gamma{cb} \tag{5}$$

where cb and ca represent the average estimated (posterior) crash frequencies for the comparison sites in the before and after periods, respectively. t and c denote treated and comparison (control group) sites, respectively.

The index of effectiveness in crash frequency can be estimated from Eq. (6).

$$\theta^* = \gamma_{ta}/\pi \tag{6}$$

where ta is the average expected accidentfrequency ofthe treated sites in the after period.

Using the expected values instead of the observed ones helps overcome the regression-to-the-mean problem (Hauer, 1997; Park et al., 2010), as long as no entry criteria are used (Lord and Kuo, 2012). For more discussions on Fully Bayesian before–after studies see, for example, Pawlovich et al. (2006); Aul and Davis (2006); Li et al. (2008); Lan et al. (2009); Persaud et al. (2010); Park et al. (2010); El-Basyouny and Sayed (2012). The simulation experiment, related to before–after studies, involved 7 main steps, as described below.

  • 1. Randomly select a sample (including 20, 50, and 100 observations) from the case study assuming that these are treated sites (note that, at this point, site characteristics for the selected sample are known);
  • 2. Repeat steps 2–6 of the previous simulation exercise discussed in Section 2.6. Upon doing so, the accident frequency for each site in the selected sample is generated (after this step, the selected sample, which includes site characteristics and accident frequencies, is ready to be analyzed);
  • 3. Based on the models reported in Section 2.4, draw the posterior expected accident frequencies for treated sites in the before period (analyze data obtained from steps 1 and 2);
    1. Apply a hypothetical index of effectiveness (e.g., 0.7) to treated sites in the before period to obtain the accident data in the after period, for treated sites;
    1. Using the models presented in Section 2.4, estimate the posterior expected accident frequencies for treated sites in the after period;
    1. From the results of steps 3 and 5, calculate the index of effectiveness and its credible intervals (the adjustment factor is 1 as discussed earlier);
    1. Compare the estimated values with the true value of *. Recall that, here, the true value is the hypothetical index of

Fig. 4. Mean and 95% credible intervals of the inverse dispersion parameter estimates (quasi-simulated data) for: (a) the sample of 20 observations; (b) the sample of 50 observations; and (c) the sample of 100 observations. Note: dashed line represents the true estimate obtained from calibrating the entire case study using the MLE.

effectiveness. Moreover, compare credible intervals obtained from different models.

3. Results and discussions

This paper examined the performance ofthe suggested prior distributions, which are used for the updating process, in terms of SPF parameter estimates and safety treatment assessment; therefore, the analyses outcomes are discussed in two sections.

3.1. SPF parameters estimates

Figs. 3 and 4 show the posterior mean and 95% credible intervals of the SPF parameters obtained from the simulation study. The quasi-simulated data contained various samples of 20, 50, and 100 observations. For each scenario of sample size, the values reported in the above-mentioned figures are the average of the 200 quasisimulated data. It can be seen in Fig. 3 that regression coefficients have almost similar mean values when various models (priors) are employed. However, there are significant disagreements between the estimated credible intervals. Note that as sample size increases, this disagreement decreases. Fig. 3 illustrates that Models 8 and 12 (models with informative priors having small variances) provide the posterior mean values with the narrowest credible intervals; thus,the most precise estimates. One should take into accountthat, although it might be obvious that priors with smaller variances provide estimates with narrower intervals, these priors may introduce bias in estimating the posterior mean. Therefore, they might be as

Fig. 5. Scatter plot matrix of model parameters and the expected crash frequency (200 quasi-simulated data for: (a) the sample of 20 observations; and (c) the sample of 100 observations.

inaccurate as non-informative priors depending on data properties. In this paper,the simulation study was, in effect, employed to verify both the accuracy and precision of the specified priors, which are then used for the updating purpose. Section 3.3 provides a guideline/summary to select proper priors (including priors' variances) on the basis of the simulation study carried out in this study.

Fig. 4 shows the estimated posterior mean and credible intervals for the inverse dispersion parameter considering different scenarios of sample size. From Fig. 4, it can be inferred that the inverse dispersion parameter is very sensitive to prior specifications. This finding is in accordance with previous studies (Lord and Miranda-Moreno, 2008; Miranda-Moreno et al., 2013; Heydari et al., 2013). In particular, Model 1, which assumes non-informative prior distributions for the SPF parameters, appears discordant with the other models (Fig. 4). Note that there is less disagreement in the dispersion parameter estimates—mean and credible intervals—as the number of observations in the data increases. A logarithmic scale is adopted in Fig. 4 because of the considerable differences in ϕ estimates obtained from deferent models. To provide further insights to readers, Fig. 5 illustrates scatter plots of the agreement between the posterior mean estimates where one can notice three important comparative aspects:(1) how different model assumptions are in accordance, (2) how the sample size affects this accordance, and (3) the sensitivity of various SPF parameters to model assumptions. For instance, it can be seen in Fig. 5 that the agreement between Models 1 and 8 for traffic flow increases as the number of observations varies from 20 to 100. It can be also observed that, as stated earlier, the inverse dispersion parameter is much more affected by prior assumption compared to regression coefficients such as traffic flow. Due to the type of the plots (i.e., matrix plot) and three parameters for which these plots are shown, reporting all 15 models in the graph was not technically possible because of the size limitations of the paper. For illustrative purposes, thus, two sample sizes (20 and 100 observations) and a few models were selected, which cover main priors used in this paper (i.e., non-informative and informative ones).

In addition to the graphical demonstration ofthe results, Table 3 highlights the analysis outcomes (in the form of estimated values) for the sample of 20 observations, being the most critical condition (i.e., small number of observations with low mean values). In particular, Table 3 reports posterior mean values, standard deviations, and 95% credible intervals estimated from various models. Table 3 also indicates the true values of the aforementioned statistics calculated on the basis of the adopted case study, using the MLE technique. One can notice/verify the deviation of the posterior mean estimates from the true mean values. For example, Models 1 and 8 estimate the inverse dispersion parameter to be equal to 119.574 and 3.385, respectively, while the true value of the posterior mean for this parameter is 3.138. Furthermore, it can be seen in Table 3 that models with very similar posterior mean values may capture the uncertainty around the mean differently (see, e.g., traffic flow estimates from Models 1 and 6).

3.2. Index of treatment effectiveness estimates

The posterior mean and 95% credible intervals of the index of effectiveness estimated based on various prior specifications (Models 1 to 15) are shown in Fig. 6. It should be noticed that credible intervals containing the value 1 imply the treatment does not have any significant effect on safety of an entity. On the other hand, those intervals that are below value 1 indicate a significant improvement in safety (i.e., positive effect on safety or reduction in the crash frequency). Alternatively, the credible intervals that are completely above value 1 imply an inverse effect on safety—an increase in crash frequency. As it can be seen in Fig. 6, all the credible intervals estimated by different models are

Fig. 6. Posterior mean and 95% credible intervals ofthe estimated index of effectiveness for quasi-simulated data:(a)the sample of 20 observations;(b)the sample of 50 observations; and (c) the sample of 100 observations. Note: dashed line represents the true estimate (i.e., 0.7).

below the line indicating the value 1. Hence, the estimated indices are as expected statistically significant (the hypothetical index of effectiveness was set to 0.7). Note that the estimated posterior mean values are very close to the true value of 0.7 and different models provide almost similar credible intervals, indicating that the specified priors perform well in before–after analyses. From Fig. 6, it can be inferred that as sample size increases the index of effectiveness is estimated more precisely—the uncertainty (length of credible intervals) around the posterior mean decreases—and there is less difference between various estimates.

3.3. Selecting variances for priors; a guideline for practitioners

In this paper, specifying priors for SPF parameters were separately discussed for the inverse dispersion parameter and

Table 3 SPF parameter estimates, quasi-simulated data with 20 observations.

Constant (a0) Traffic flow (a1) Inverse dispersion parameter (ϕ)
Model Mean Sd. dev. 2.5% 97.5% Mean Sd. dev. 2.5% 97.5% Mean Sd. dev. 2.5% 97.5%
1 −9.006 3.854 −17.090 −1.886 1.180 0.426 0.391 2.071 119.574 273.608 1.224 875.149
2 −9.168 4.212 −17.930 −1.358 1.199 0.466 0.332 2.167 3.398 2.244 0.831 9.263
3 −9.148 4.194 −17.868 −1.392 1.196 0.464 0.336 2.160 4.262 3.400 0.823 13.416
4 −9.160 4.265 −18.053 −1.244 1.198 0.473 0.319 2.183 3.337 2.147 0.814 8.910
5 −9.124 3.820 −16.980 −1.965 1.195 0.423 0.399 2.064 3.413 2.245 0.839 9.250
6 −8.995 2.452 −13.867 −4.258 1.182 0.273 0.655 1.724 3.448 2.245 0.859 9.301
7 −9.109 1.936 −12.940 −5.358 1.196 0.216 0.777 1.624 3.428 2.243 0.848 9.274
8 −9.214 1.462 −12.094 −6.361 1.208 0.164 0.887 1.531 3.385 2.202 0.853 9.112
9 −9.111 3.809 −16.936 −1.967 1.193 0.422 0.399 2.059 3.382 2.157 0.829 8.963
10 −8.999 2.462 −13.904 −4.237 1.183 0.274 0.653 1.728 3.395 2.163 0.834 9.008
11 −9.109 1.933 −12.931 −5.345 1.196 0.216 0.776 1.622 3.373 2.154 0.833 8.963
12 −9.218 1.460 −12.089 −6.371 1.208 0.164 0.888 1.531 3.343 2.140 0.820 8.911
13 −9.103 4.306 −18.051 −0.915 1.191 0.477 0.284 2.181 5.114 4.936 0.766 18.628
14 −8.983 2.439 −13.851 −4.286 1.181 0.271 0.659 1.722 5.298 5.064 0.803 19.174
15 −9.053 3.778 −16.826 −1.923 1.187 0.418 0.395 2.046 5.207 4.976 0.796 18.840
T.E. −8.358 1.184 −10.678 −6.038 1.113 0.131 0.857 1.369 3.138 1.30 1.755 6.689

Note: all values are the average obtained from 200 replicated datasets. T.E. represent the true estimates obtained from analyzing the entire dataset (New York data).

regression coefficients such as traffic flow. The proposed methodology provides a straightforward technique (the 4-step procedure explained in Section 2.2) to estimate suitable priors for the inverse dispersion parameter. When a gamma prior is assumed for this parameter, the 4-step procedure estimates also the variance that is needed in the specification of the gamma prior. This study highlighted thatthere is not a specific prior (or single value for variance) that can be assumed for the inverse dispersion parameter prior. This finding is in accordance with the HSM, which estimates the inverse dispersion parameter based on a mathematical relationship for each dataset (Eq. (3)). The 4-step procedure developed in this paper is an interesting way to define this particular prior, which suitably adjusts itself to a given dataset. For a dataset in hand and for the inverse dispersion parameter, therefore, practitioners could estimate the priormeanand variance based onthe suggested 4-step framework.

We discuss the issue of selecting variances for regression coefficients' priors on two fronts: precision and accuracy. See, for example,the estimates related to traffic volume for the sample of 20 observations in Table 3. Model 8, which has informative priors with small variances for all SPF parameters, estimates a narrower credible interval (a more precise estimate with a smaller uncertainty around the mean) for the traffic flow coefficient (i.e., C.I. [0.887, 1.531]) compared to Model 6 (i.e., C.I. [0.655, 1.724]). Whereas, Model 8 provides a less accurate estimate of the posterior mean (i.e., 1.208) compared to Model 6 (i.e., 1.182) given that the true estimate is 1.113. One can also verify the distance between the point indicting the posterior mean and the dashed line indicating the true value in Fig. 3. Hence, although informative priors with very small variances (e.g., Model 8) lead to more precise estimates, they may penalize the estimation of the posteriors. Therefore, one can choose the best model—selecting or finding a balance between precision and accuracy—depending on the focus of an investigation or study. Another concern is the fact that a selected model may not provide an accurate estimate for all SPF parameters. For example, verify Model 13's results in Table 3 where the estimated inverse dispersion parameter of 5.114 is biased. From this study, it can be clearly implied that using a non-informative prior for the inverse dispersion parameter is not recommended.

Following the aforementioned discussion, we recommend the use of models 10 and 6 given their overall precision and accuracy in estimating different SPF parameters (Table 3), for limited data conditions. As stated in Section 2.4, these models have an informative prior for the inverse dispersion parameter and a semi-informative prior (with a variance of 10) for regression coefficients such as traffic flow.

4. Conclusions

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 Poissongamma 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 thatthere 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.

References