bayesian road safety analysis: incorporation of past ......email: [email protected]...

Bayesian road safety analysis: incorporation of past experiences and effect of hyper-prior choice

Luis F. Miranda-Moreno1

Assistant Professor Department of Civil Engineering and Applied Mechanics

McGill University Macdonald Engineering Building

817 Sherbrooke St. W. Montreal, Quebec H3A 2K6

Phone: 514-398-6589 Email: [email protected]

Dominique Lord Assistant Professor

Zachry Department of Civil Engineering Texas A&M University, College Station

3136 TAMU, TX, US Tel. (979) 458-3949

E-mail: [email protected]

Liping Fu

Associate Professor Department of Civil Engineering University of Waterloo, Waterloo

200 University Avenue West N2L 3G1, ON, Canada

Tel: (519) 888-4567 ext 3984 Email: [email protected]

Submitted to AA&P (Revised Version) April 27, 2009

This paper was presented at the 87th Annual Meeting of the Transportation Research Board (paper no. 08-1788)

1 Corresponding author. Fax: (418) 656-7412

Miranda-Moreno, Lord and Fu

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Abstract

This paper addresses two key issues when applying hierarchical Bayes methods for traffic

safety analysis, namely, (a) how to incorporate available information from previous

studies (past experiences) for the specification of informative hyper-priors in accident

data modeling; and, (b) what are the practical implications of the choice of hyper-priors

on the results of a road safety analysis. In this paper, we first illustrate how to incorporate

an integrated summary of the available evidences from previous studies for the

development of informative hyper-priors in the dispersion parameter. The performance of

different hyper-priors was then investigated, including Gamma and Uniform probability

distributions. For this purpose, different simulation scenarios were defined and tested

using crash data collected at 3-legged rural intersections in California and crash data

collected for rural 4-lane highway segments in Texas. By doing so, we show how the

accuracy of parameter estimates are considerably improved, in particular when working

with limited accident data, i.e., crash datasets with low mean and small sample size. The

results however show that the past knowledge incorporated in the hyper-priors can only

slightly improve the accuracy of the hotspot identification methods. In addition, when the

number of roadway elements or the time period (e.g., number of years of crash data) is

relatively large, the choice of the hyper-prior distribution does not have a significant

impact on the final results of a traffic safety analysis.

Keywords: Bayesian approach, hyper-prior choice, past experiences


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1. INTRODUCTION

In traffic safety studies, Bayesian methods have been widely applied for both identifying

hotspot locations and evaluating the effectiveness of countermeasures (e.g., Higle and

Witkowski, 1988; Hauer, 1997; Schluter, et al. 1997; Persaud, et al. 1999; Heydecker and

Wu, 2001; Miaou and Song, 2005; Miranda-Moreno, 2005; Nathan and Gary, 2006;

Song, et al. 2006; Persaud and Lyon, 2007). Within the class of Bayesian methods, we

can distinguish two main approaches commonly used in traffic safety studies: the

empirical Bayes (EB) approach and the hierarchical Bayes approach (also referred as full

Bayesian analysis). One important difference between these two approaches is in the way

the prior parameters are determined. In the EB approach, the prior parameters are

estimated using a maximum likelihood technique or any other techniques (e.g., method of

moments) involving the use of the accident data, e.g. see Hauer (1997); Miranda-Moreno

et al. (2005); Lord (2006), Lord and Park (2008). Alternatively, in a hierarchical Bayesian

analysis, the parameters of the prior distributions depend in turn on additional parameters

with their own priors- e.g., see Berger, J.O. (1985), Carlin and Louis (2000) and Rao

(2003). In traffic safety, the hierarchical Bayes approach is usually utilized when working

with hierarchical Poisson models (Miaou and Song, 2005). In these types of models, the

model parameters at the second level (e.g., regression or dispersion parameters in the

hierarchical Poisson/Gamma model) are also supposed to follow some underlying

distributions (called also hyper-priors), adding another level of randomness.

Despite its modeling flexibility, the HB approach requires the specification of priors for

the parameters included in the last level of the model hierarchy, which are referred also


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as “hyper-priors” (Carlin and Louis, 2000; Gelman, et al. 2003; Rao, 2003; Lord and

Miranda-Moreno, 2007). In practice, the so-called “vague” or “non-informative” hyper-

priors are commonly used with the idea to let the data “speak for itself”. In spite of this

common practice (e.g., see Nathan and Gary, 2006; Song, et al. 2006; Lord and Miranda-

Moreno, 2007), the specification of hyper-priors may not be trivial when modeling

accident data characterized by a low sample mean (LSM) and small sample size (SSS).

Under these conditions, the use of vague hyper-priors can be problematic leading to

inaccurate posterior estimates – e.g., see Lamber et al. (2005) and Lord and Miranda-

Moreno (2007). In addition, the results may be sensitive to the distribution choice.

In the traffic safety literature, much of the attention has focused on the development of

alternative model settings and safety measures, the selection of functional forms, and the

estimation of accident modification factors among others - see e.g., Persaud, 1986; Miaou

and Lord, 2003; Cheng and Washington, 2005; Miaou and Song, 2005; Miranda-Moreno,

2006; Persaud and Lyon, 2007. In traffic safety research, little work has so far

investigated the effect of the hyper-prior choice adopted on model parameters. Moreover,

there is a lack of practical guidance for building informative hyper-priors using available

evidences from past traffic studies and/ expert’s opinions, as a potential solution to the

LSM and SSS problem. In other fields such as epidemiology, biology and reliability

engineering, both the effect of distribution choice and the incorporation of prior

knowledge have attracted a lot of attention (Ashby, et al. 2002; Lambert, et al., 2005; Van

Dongen, 2006; McMahon, et al. 2006; Guikema, 2007). Even so, in traffic safety, very

little research has been done in this respect. The work of Oh and Washington (2006) is


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one of the few attempts to incorporate experts’ opinions (expert knowledge elicitation)

for safety analyses.

According to these observations, the aim of this paper is two-fold:

• To investigate the performance of alternative hyper-prior distributions for

modeling the dispersion parameter in a hierarchical Poisson model under LSM

and SSS.

• To propose a simple framework to incorporate available evidences from past

similar studies to formulate informative hyper-priors for the dispersion parameter.

To achieve our objectives, a simulation study is carried out to compare alternative

distribution choices (e.g. gamma versus uniform) or hyper-prior specifications (e.g.,

vague versus informative priors) for the dispersion parameter in a hierarchical

Poisson/Gamma model. The performance of alternative hyper-prior specifications is

evaluated according to the model capacity to detect the “true” hotspots. A comparative

performance is also made in terms of parameter estimation accuracy.

2. HIERARCHICAL POISSON MODELS FOR ACCIDENT DATA

In the traffic safety literature, different modeling settings have been proposed for

analyzing crash data, ranging from the standard negative binomial to more complex

models such as the zero-inflated Poisson, hierarchical mixed Poisson, latent-class

Poisson, Poisson mixture and Conway-Maxwell-Poisson models (e.g., see Hauer, 1997;

Miranda-Moreno, 2006; Lord, et al. 2007; Song, et al. 2006; Park and Lord, 2009).


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Among them, the hierarchical Poisson/Gamma is perhaps the most popular. A simple

version of this model is defined above.

Considering that the number of accidents at site i (Yi ) over a given time period Ti, is

Poisson distributed, a hierarchical Poisson/Gamma model may be defined as follows

(Rao, 2002; Miranda-Moreno, 2006):

i. )T(Poisson~,T|Y iiiii θ⋅θ

)eT(Poisson~ iiiεμ⋅

ii. iεe ~ Gamma(φ, φ)

iii. (.)~ πφ and 1)(f ∝β (1)

where, iμ = f ( ,F,F 2i1i );βix and β = '

k0 ),...,( ββ is a vector of regression coefficients to

be estimated from the data. 1iF and 2iF are entering traffic flows in intersecting directions

at intersections (Miaou and Lord, 2003). In addition, ix is a vector of covariates

representing site-specific attributes. Ti is the period of observation at site i, which is

usually assumed to be the same for all locations in the hotspot detection activity (but does

not have to be). In this case, the model error, ieε , is assumed to follow a Gamma(.)

distribution with both shape and scale parameters to be equal leading to 0.1]e[E i =ε and

φ=ε /1]e[Var i (Winkelmann, 2003). This error assumption is defined in order to obtain a

hierarchical version of the traditional negative binomial model, which has been widely

used in traffic safety studies. Note again that in this hierarchical model, regression (β)

and dispersion (φ) parameters are assumed to be random. That is, a hyper-prior


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distribution, denoted by (.)π , is assumed on the dispersion parameter φ and, f(β) denotes

the hyper-prior on the regression coefficients β, which is commonly assumed to be flat or

non-informative, e.g., βj ~N(0,1000).

Note that with the introduction of random variations in the mean, the hierarchical

modeling framework has the potential to address over-dispersion caused by unobserved

or unmeasured heterogeneity. In the road safety literature, more complex hierarchical

Poisson models with additive random effects have been implemented to account for

spatial correlation (Miaou and Song, 2005; Song, et al. 2006). To account for spatial

dependency, Gaussian conditional autoregressive (CAR) models are very popular.

However, for illustrative purpose, this research is based on the simple hierarchical model

defined in (1) and more complex models are out of the scope of our study. To obtain

parameter estimates of model (1), posterior inference are carried on by Markov Chain

Monte Carlo (MCMC) simulation methods such as Gibbs sampling and Metropolis–

Hastings algorithm (Rao, 2003; Carlin and Louis, 2000; Gelman, 2003), which are

already implemented in the software package WinBUGS.

Note that in the hierarchical Bayes setting defined by equation 1, the dispersion

parameter φ is assumed to be random, (.)~ πφ . In this study, we investigate what is the

impact of the use of alternative distributions on φ including the following:

(i). φ−−φΓ

=φ 000

b1a

0

a

00 e)a(b)b,a;(f , φ > 0, a0 > 0, b > 0 (Gamma distribution)


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(ii). 20

00 )a(

a)a;(fφ+

=φ , φ > 0, a0 > 0, (Christiansen’ distribution)

(iii). 0

0 ab1)a;(f−

=φ , a0 < φ < b (Uniform distribution) (2)

where, a0 and b are the parameters of these alternative distributions. Remark that the

distribution 2-(ii) was first suggested by Christensen and Morris (1997), where a0 is the

hyper-prior guess for the median of φ. For this particular distribution, small values of a0

are less informative. A conservative choice letting the data speak by itself would be to

choose a0 small enough so that a0 is less than the median of φ (Nandram, et al. 2005).

Later in section 2.5, we see how this prior (ii) can be defined on the basis of a Pareto

distribution.

Note that these alternative distributions on φ have been used in previous research (e.g.,

Miaou and Song, 2005; Nathan and Gary, 2006; Song, et al. 2006). However, their

impact on the final outcome of a safety analysis has not been investigated. As shown in

previous studies (Lambert, et al. 2005; Lord and Miranda-Moreno, 2007), hyper-prior

assumptions can play an important role in the analysis. In particular when working with

small amount of data, the use of vague hyper-priors can lead to very inaccurate posterior

estimates. This paper investigates: (i) the practical implication of using alternative

distributions on φ and (ii) how to incorporate past evidences as prior information to

improve the accuracy of posterior estimates when modeling crash data with LSM and

SSS.


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3. INCORPORATION OF PRIOR KNOWLEDGE ON φ

There are two main steps in the incorporation of past evidences and expert’s opinions for

the construction of informative hyper-priors (Schluter, et al. 1997; Nandram, et al. 2005):

1. Adoption of distributions that adequately describe the parameter of interest, in this

case φ; and,

2. Incorporation of prior information (from past studies and/or elicitation of expert

opinions) about the parameter of interest.

For step 1, we have already seen some possible distributions for φ suggested in previous

research. For step 2, we need to address the following questions (Schluter, et al. 1997):

i. What should be the average value of φ? The answer to this question requires

collecting the past knowledge from previous studies, experts’ belief or opinions

about the location of the mean of φ.

ii. How variable is φ ? For that, we have to establish a range where the underlying

values of φ are expected to fall. This question centers on the past experiences (when

existing) about the bounds for φ.

To illustrate how to incorporate information from past experiences into prior knowledge,

we first start by considering a Gamma distribution for φ, with mean and variance given

by

i. b/a)(E 0=φ=η

ii. 202 b/a)(Var =φ=σ and sr 2 ≤σ≤ (3)

where, as before, a0 and b are the Gamma parameters assumed to be fixed. Then,

combining these two equations, we can write:


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ba0 ⋅η= and 2/b ση= (4)

Considering then that various past studies have reported some estimates of φ, which are

determined using accident data from similar roadway elements and jurisdictions. For

instance, a number of previous studies have developed Negative Binomial (NB) models

for three-legged rural intersections (Berger, 1985; Vogt and Bared, 1998; Vogt, 1999;

Lyon, et al. 2003; Kim, et al. 2006). Some of these studies are listed in Table 1. Note that

in this table, the estimates of the dispersion parameter (φ) vary approximately between

0.5 and 4.3 (minimum and maximum, respectively). From this set of values, it can be

asserted that the average of the reported evidences in the literature is approximately 1.9,

with a variance of 2σ =1.5. Hence, by setting η = 1.9 and 2σ =1.5, we obtain a0=2.5 and

b=1.3. Note that the other hyper-prior distributions in (2) are developed the same way,

but are not described in this paper. In addition, hyper-priors incorporating a synthesis of

evidences from past studies, as shown here, are referred as “informative”.

4. A SIMULATION METHODOLOGY

As discussed previously, some of the main objectives of traffic safety Bayesian analyses

include estimating the safety performance for the locations of interest and calibrating

model parameters to either develop accident modification factors (AMFs) or establish a

relationship between crashes and site-specific covariates (safety performance functions,

SPF) (Hauer, 1997; Nathan and Gary, 2007; Persaud and Lyon, 2007; Lord, et al. 2007).

These outcomes are the basis for a traffic safety analysis such as the evaluation of

countermeasures and hotspot identification. In this section, a simulation framework is

introduced to evaluate the impact of hyper-prior specification on the dispersion


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parameter. This impact is investigated in terms of parameter estimate accuracy and

hotspot identification.

4.1. Accident risk estimator

Various ranking criteria or safety measures can be used for hotspot identification. Among

the most popular safety measures, we find the posterior expected number of accidents, the

posterior probability of excess, the posterior expectation of ranks and the posterior

probability that a site is the worst, see e.g., (Schluter, et al. 1997; Heydecker and Wu,

2001; Miranda-Moreno, et al. 2005; Miaou and Song, 2005). The posterior expectation of

θi, denoted here by i~θ , is perhaps the most popular in the safety literature and is defined

as,

i~θ = ∫

∞θθθ=θ

0 iiiiid)y|(p]y|[E (5)

Once a safety measure has been defined, two strategies can be used for selecting sites

indentified as hotspots (Higle and Witkowski, 1998; Schluter, et al. 1997; Cheng and

Washington, 2005; Miranda-Moreno, 2006): 1) selecting sites on the basis of the budget

available to conduct safety inspections and implement countermeasures; and 2) selecting

a list of sites based on some specified cutoff or threshold value. In the former, hotspots

are selected sequentially from the ranked list until all budgeted resources are exhausted.

The latter ensures the selection of a list of locations that are deemed dangerous at some

critical level “c”, leaving variable the number of locations to be selected (Higle and

Witkowski, 1988; Schluter, et al. 1997; Heydecker and Wu, 2001). This second strategy

is often the result of local safety policies that stipulate tolerance levels of accident risks. It


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is the most appropriate when one wants to identify a list of sites exceeding a certain

critical value, denoted here by “c”. This strategy can be represented by the following rule:

If c~i >θ , site i belongs to the hotspot list

For this work, our simulation framework considers the threshold-based strategy. Note that

the advantage of working with simulated data is that the “true” safety state of each

location is known. In consequence, one knows if each location is truly a hotspot (i.e.,

dangerous) or not (Cheng and Washington, 2005; Miranda-Moreno, 2006). Thus, the

“true” hotspots are first identified from the true accident frequency distribution, which are

then compared with the “detected” hotspots determined with alternative hyper-priors. The

performance of alternative distributions on φ is evaluated by comparing the “detected”

versus “true” hotspots. Finally, the hotspot differences are examined to assess the

performance of each approach according to some measures such as false-discovery and

false-negative error rates, and others, as defined above.

4.2 Simulation design

The proposed simulation-based methodology considers a selected set of sites with known

safety status. That is, the model parameters and probability distribution of mean accident

frequency at each site are presupposed and then used to establish the “true” hotspots and

generate accident “history”. The latter is then used for identifying the “detected” hotspots

after model calibration. The methodology includes the following steps:

Step 1. Select a random set of n sites (e.g., n=50, 100, 200…) from an existing

inventory database with known site-specific attributes ( 'ix ) for each site i.


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Step 2. Assume that accidents at the given locations follow a Poisson/Gamma

model with known fixed parameters φ̂ and β̂ (these parameters are defined

previous to the simulation). Based on these parameters and the site-specific

attributes ( 'ix ), generate the “true” mean accident frequency for each site i as

follows2:

)ˆ,ˆ(Gamma~ˆ|e i φφφε

ieitruei

εμ=θ for iμ = f('ix ; β̂ )

Step 3. Specify a global critical c-value depending of the dataset under analysis.

This can simply be specified as the observed accident average in the dataset, e.g.,

c = 0.5, 1.0, 1.5. Note that “c” applies to the crash mean of site i and not to yi, the

observed number of accidents. That is, once c has been pre-specified, we apply

the following selection rule:

If trueiθ > c set hi = 1 and site i is defined as “true” hotspot

Else set hi = 0, and site i is defined as “non-true” hotspot

Then, we calculate the “true” total number of hotspots (n1) as:

∑=

=n

1ii1 hn

Step 4. For each site, simulate accident “history” based on the following:

)(Poisson~|Y trueitruei

simi θθ

2 To simulate data according to a different model, accident data was also generated from a Poisson/Lognormal model with multiplicative error, exp(εi) ~LogNormal (-0.5τ2, τ2). This is illustrated in Miranda-Moreno (2006). However, the results generated according to this lognormal model seems to be very consistent with the gamma model. This means that results are not sensitive to the data generation model. Therefore, the discuss presented in this paper focuses only in a Gamma error term.


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Step 5. According to the simulated data - simiY - and the site-specific attributes

( 'ix ), the model parameters (β and φ) along with i~θ are estimated by drawing

samples directly from the posterior distributions using MCMC algorithms and

taking after the expected values of the posterior draws.

Once i~θ is obtained for each site and based on a given model (with particular

hyper-prior specifications), we identify the “detected” hotspots as follows:

If i~θ > c set di = 1, the site i is a “detected” hotspot

Else set di = 0, the site i is a “non-detected” hotspot

Then, we calculate the total number of “detected” hotspots (D) as,

∑=

=n

1iidD

Step 6. For each simulation replication, compute and record the set of error

measures: False Discovery Rate (FDR), False Negative Rate (FNR), sensitivity

(SENS) and specificity (SPEC), which are defined in Equation 6.

In order to obtain statistically reliable estimates, each scenario is replicated 100 times,

repeating steps 1 to 6. The average of the 100 replications is reported in the section of

results presented later.

4.3 Performance evaluation criteria

Table 2 summarizes the possible outcomes over the n sites for each simulation.

According to these outcomes, the following four measures are defined to evaluate the

comparative performance of the alternative hyper-priors in terms of the power to detect


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the “true” hotspots: FDR, FNR, SENS and SPEC, for details we can refer to Miranda-

Moreno (2006). These accuracy measures are defined using the notations provided in

Table 2 as follows:

i. DVFDR = ,

ii. )Dn(

RFNR−

= ,

iii. 1n

SSENS = ,

iv. 0n

USPEC = , (6)

where,

V - # of false positives or Type I errors

R - # of false negatives or Type II errors

S - # of sites correctly classified as hotspots

U - # of sites correctly classified as non-hotspots

D - # of sites detected hotspots as hotspots

n, n0, and n1 - Defined in Table 2.

FDR - Proportion of Type I errors among the “detected” hotspots.

FNR - Proportion of Type II errors among the “detected” non-hotspots. Note

that relative small values of FDR and FNR are expected when a

method or a model performs well.


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SENS - Sensitivity or proportion of sites that have been correctly detected as

hotspots, which can also be interpreted as the method capacity to

detect a “true” hotspot in a group under analysis. This criterion is

difficult to control in practice and values greater than 0.80 are difficult

to obtain.

SPEC

- In a similar manner as the SENS, the specificity (SPEC) represents the

proportion of non-hotspots that have been correctly classified as “non-

hotspots. Since the number of locations detected as hotspots is

typically very small, it is expected to obtain SPEC values close to 1.

4.4 Computational details

The simulated data are generated using the software R - a language specialized in

statistical computing available in http://www.r-project.org. In the present work, once the

data are generated, the posterior quantities of interest are estimated using WinBUGS 1.4

(Spiegelhalter, et al. 2003) 3. This software is specialized for the Bayesian analysis of

complex statistical models using Markov chain Monte Carlo (MCMC) methods.

Moreover, to make a connection between R and WinBUGS, the R2WinBUGS package is

employed. This package provides convenient functions to call WinBUGS from R (Sturtz,

2005).

For each scenario involved in the analysis, the simulation was replicated 100 times,

according to the procedure defined above. At the end of the 100 replications, the average

and variance are computed in order to assess the precision of the parameter estimates. For 3 R and Winbugs codes are available upon request.


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instance, for each regression parameter k, we compute the mean ∑ ==100

1j kj100/)β̂( , where,

j stands for each of the 100 simulations. Furthermore, for the MCMC simulations, we

specify:

• number of Markov chains = 3

• number of iterations per chain = 8,000

• length of burn-in = 4,000 (i.e., number of iterations to discard at the beginning)

• number of iterations between saving of results = 2000 for each chain (given that

the number of thins equals 2, which represents the thinning rate). Thus, 6000

iterations were used for parameter estimations.

4.5 Dataset

To accomplish our objectives, a rich set of simulation scenarios (with different samples

sizes and year of information) were established using two datasets. The first dataset

contains crash data collected at three-legged rural intersections in California. The second

dataset includes data collected for rural 4-lane divided and undivided highways in Texas.

• California data

A sample of three-legged rural intersections located in California is the focus of this

simulation study. Accident history, traffic flow and geometric design data were obtained

from the Federal Highway Administration’s (FHWA) Highway Safety Information

System (HSIS) managed by the University of North Carolina in Chapel Hill, NC. For a

five-year study period (1997-2001), a total of 5,752 three-legged rural intersections were


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included in the database. Intersections that contained missing values were discarded from

the dataset, resulting in a sample of 5,642 three-legged rural intersections for the same

five-year period with a total of 6,185 reported crashes (all crash severities). More details

about this dataset can be found in Lord and Park (2007).

For the purpose of our study, we obtain the “true” model parameters using accident data

of different period of time, T = 2, 3, 4 and 5 years. In addition, we use the best functional

form in agreement with the result presented by Miaou and Lord (2003) and Lord and Park

(2007), where μi is given by:

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i1

i2i2i10i F

F)FF(β

β ⎟⎠⎞⎜

⎝⎛+β=μ (7)

where, i1F and i2F are the average minor and major entering traffic daily volumes, and

β0, β1 and β2 are regression parameters. The “true” regression estimates (β̂ ) used in our

simulations are presented in Table 3. These parameters were calibrated by maximum

likelihood using the standard Negative Binomial model. Furthermore, to adequately

assess the impact of hyper-prior choice on posterior inference, different sample sizes and

time period are taken into account including, n = 50, 100, 200, etc., and T = 2, 3, 4 and 5.

For each simulation, 100 samples are selected randomly from the whole dataset.

In our simulation study, the advantages of informative hyper-priors for φ are explored. As

previously discussed, an informative hyper-prior can be built based on past evidences -

reported in the traffic safety literature. As seen previously (Table 1), various studies have

been carried on using data from rural intersections from different jurisdictions or US

states. From these documented evidences, section 3 shows that the estimated values of


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φ=1/α for this particular type of intersection fall in the interval [0.5,4.3]. Based on that,

the following informative and non-informative hyper-priors for φ are established:

- Informative Uniform with a0 =0.1 and b=5.0,

- Informative Christiansen’s distribution with a0 = 2.0,

- Informative Gamma with a0=2.5 and b=1.3.

- Non-informative Gamma with a0=0.001 and b=0.001.

The a0 and b parameters of the uniform distribution are chosen in order that the

distribution covers the interval of the reported φ estimates. The choice of a0 and b for the

informative Gamma has been explained in section 3. For the Christiansen’s distribution,

Equation 2-ii, a value of a0 =2.0 is chosen, which is equal to the median of the φ values

reported on the literature. Note that this hyper-prior can be derived from a Pareto

distribution as follows. Define Z as a random variable following a Pareto distribution,

1z/k)z(f +δδδ= , with k=2 and δ=1, i.e., )1,2(Pareto~Z . From which one can show

that φ = Z - 2. Then, if )1,2(Pareto~Z , the density of φ = Z - 2 is 2)2/(2)(f φ+=φ .

Finally, for the non-informative Gamma, a0 and b are chosen so that the variance of φ is

very large.

• Texas data

Our second sample of roadway elements contains accident data collected at four-lane

rural undivided and divided segments in Texas. These data have been provided by the

Texas Department of Public Safety (DPS) and the Texas Department of Transportation

(TxDOT). It has been used for the project NCHRP, 17-29 (Dominique et al. 2008). The

final database included 3,220 segments (≥ 0.1 mile) and 15,753 crashes during five years


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of accident data (which results in an average of 4.89 crashes per segment). The only two

variables reported in these data are traffic flow (average annual daily traffic, AADT) and

length in miles for segment. More details about this dataset can be found in Lord, et al.

2008.

The functional form used for modeling crash risk for this dataset is the same used in some

previous studies (Persaud, et al. 2004; Lord, et al. 2008), which is the following:

1)F(L ii0iββ=μ , (8)

where, Fi is the flow traveling on segment i, Li is the length in miles for segment i, and β0

and β1 are regression coefficients. The “true” regression estimates used in our simulations

are ln(β0) = -6.56 and β1 = 0.926 and φ = 3.231. These quantities are estimated using the

full accident data, negative binomial model and maximum likelihood (ML) technique.

As for the California intersections, available information from previous studies is

incorporated to formulate informative hyper-priors for φ. Note that different published

studies have employed accident data from two-line or four-lane rural segments to develop

Negative Binomial (NB) models (Persaud, et al. 2004; Caliendo, et al. 2007). Among of

the previous work, we can mention the study by Persaud et al. (2004), which calibrated

different NB models based on crash data collected in different US states. From the

calibrated models, these authors reported φ -values (1/α) equal to 2.7, 1.69, 2.08 and 2.63

for California, Colorado, Minnesota and Washington state, respectively. Another recent

study published by Caliendo et al. (2007) reported crash-prediction models for four-lane


21

rural segments in Italy. In this case, models were calibrated based on accident data

observed during a 5-year period. The reported value for the NB-φ parameter was 3.62.

From these five estimates, a mean and variance equal to 2.5 and 0.5 is estimated,

respectively. According to these values, the parameters a0 and b are fixed in the same way

as for the California case study. Hence, the informative and non-informative hyper-priors

for φ in the Texas dataset are:

- Informative Uniform with a0 = 1.0 and b=5.0,

- A special case of Christiansen’s distribution (a0 =1),

- Informative Gamma with a0=4.8 and b=12.2,

- Non-informative Gamma with a0=0.001 and b=0.001.

5. RESULTS

The simulation results for different scenarios are summarized in Tables 4 to 9. These

scenarios consider different sample sizes (n=50, 100 and 200 locations) and time periods.

Based on these results, the major findings are summarized as follows.

• Hotspot identification

Table 4 shows the simulation outcomes for the California dataset when working with

small number of intersections (n=50) and different years of data T=2, 3, 4 and 5. This

table presents the error measures defined above, i.e., FDR, FNR, SENS and SPEC, which

range between 22-35%, 7-10%, 64-75% and 89-93%, respectively. Note that the lower

the FDR and FNR are, the better the results are; conversely, the higher the SENS and

SPEC are, the better the outcome is. From these results, the error measures under the


22

informative Uniform and Gamma are slightly better than the Christiansen (Pareto) hyper-

prior. However, the worst results are obtained when assuming non-informative Gamma

with very small values of a0 and b. For instance, when model calibration is done based on

a small sample of n=50 intersections and 2 or 3 years of data, the informative Gamma

produces smaller False Discovery Rates (FDR=0.318 and 0.248, respectively) than the

non-informative Gamma (FDR=0.358 and 0.312, respectively). However, when

increasing the number of years of data and keeping n =50, all error measures are very

similar (for both informative and non-informative hyper-priors). This particular result

illustrates that: 1) more years of data are included in the analysis, more precise the

identification of hotspots will be, especially when working with small samples, 2) The

use of informative hyper-prior improves the detection of hotspots when working with

small samples and few years of data, 3) The effect of the distribution choice (Gamma,

Uniform or Christiansen) for φ seems to have a marginal impact (e.g., under informative

hyper-priors very small differences among the alternative distributions).

Table 5 presents the simulation scenarios for larger sample sizes (n=100 and 200

intersections). From these results, we can see that the performance of the alternative

informative or non-informative hyper-priors is very similar. This is an illustrative

example of how the role of the hyper-prior assumptions becomes less important as the

sample size increases. This agrees with a previous work (Lord, 2006), which states that

the problem of low mean can be also solved by increasing the number of roadway

elements involved in the analysis, i.e., the sample size.


23

Table 6 presents some of the results obtained for the Texas dataset. Similar to the

California dataset, different sample sizes where considered, including n=30, 50 and 100

segments. In this case, the time period was always the same, 5 years. It is important to

note that the mean accident frequency for this dataset is relatively high and equal to 4.89

crashes per segment. Hence, this dataset is not characterized by a low mean value. From

this table, we can observe that the different error measures for the informative Gamma,

Christiansen and Uniform distributions provide very similar results. Here, the

performance of the non-informative Gamma is also very close to the informative hyper-

priors. Again, this shows the advantages of using informative hyper-priors, in particular is

situations where data availability is limited (small number of observations combined with

low mean sample).

• Parameter estimates

Table 7 shows the parameter estimates results for the California dataset. When

considering an informative Gamma or Uniform distributions, posterior estimates of φ are

closer to the true values than those obtained with the alternative hyper-priors, especially

for cases with limited data availability. For instance, for n =50 and T=2 years, the average

estimate of φ under the informative Gamma is 1.48. This value is much closer to the true

parameter (φtrue = 0.717) than the average values of φ obtained with the Christiansen or

non-informative Gamma, which are of 6.0 and 44.3, respectively. This illustrates how the

use of informative hyper-priors can lead to much more accurate parameter estimates. As

in the previous case, the impact of the hyper-priors specification becomes less important

as the sample size increase - see results presented in Table 8 for the California data for


24

n=100 and 200 intersections. The results of the California dataset show again how the use

of informative hyper-priors leads to more accurate posterior estimates of φ - see Table 9.

This simulation study shows how the inclusion of past experiences can improve

considerably the accuracy of dispersion parameter estimates, in particular, when working

with small sample sizes, combined with low sample mean values. However, the results

suggest that incorporating hyper-prior information improves only slightly the detection of

hotspots.

6. FINAL DISCUSSION AND CONCLUSIONS

In traffic safety research, a fair amount of work has been published recently using the

hierarchicalBayes (HB) approach as an alternative approach to the traditional empirical

Bayes (EB). Under the HB setting, rather vague or non-informative hyper-priors are

usually assumed for model parameters in order to let the data “speak for itself.” This

research has however illustrated that this rule of thumb may not work well with crash

data characterized by low sample mean values and small sample size, which is common

in many traffic safety studies. When working with little data availability, vague hyper-

priors on the dispersion parameter can lead to high error rates in the hotspot identification

(e.g., FDR and FNR) and very imprecise parameter estimates. This supports the findings

reported recently by (Lord and Miranda-Moreno, 2007) which found that crash data

characterized by a LSM combined with a SSS can seriously affect the estimation of the

posterior mean of the dispersion parameter of hierarchical Poisson models.


25

As a potential solution to the problem of low sample mean and small sample size, this

paper suggests some practical guidance on how to formulate informative hyper-priors

based on past evidences (parameter estimates reported in the traffic safety literature)

obtained from previous studies. By doing so, we illustrate how informative hyper-priors

for the dispersion parameter can lead to more precise parameter estimates. However, the

improvement in term of hotspot identification seems to be very modest. These results are

achieved by simulating data according to different sample sizes and years of accident data

and using three-legged rural intersections in California and four-lane rural segments in

Texas. For traffic safety analysts, our approach suggests a way to improve the accuracy

of parameter estimates when working with limited accident data. Our approach can also

be useful to reduce the accident data needs (number of roadway elements and years of

data required in the analysis).

A simulation framework using different error measures was developed to determine the

performance of the alternative hyper-priors in terms of their capacity to detect hotspots

and bias with respect to true model parameters. Among the alternative informative hyper-

priors involved in our simulation study, it was found that informative Gamma and

Uniform performed slightly better than the Christiansen hyper-prior when working with

limited data (e.g., 50 sites and 2 years of accident data). Results obtained with non-

informative hyper-priors were the worst. However, we illustrated how, when the dataset

is sufficiently large (e.g. over 200 sites), the choice of hyper-prior and the incorporation

of prior information becomes less important, showing a minimum impact on the outcome

of the analysis.


26

For future work, a more sophisticated approach would be to directly integrate different

accident raw datasets accounting for heterogeneity between studies. The performance of

other hotspot identification criteria, distributions and more complex models will also be

investigated (see Park and Lord, 2009). Although we consider one of the simplest

hierarchical models, the simulation framework proposed in this work can be employed

for this purpose.


27

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Table 1: Dispersion parameters reported in previous works using different datasets from three-legged rural intersections Study α φ=1/α Vogt and Bored (1998) 0.48 2.08

0.53 1.90

Vogt (1999) 0.39 2.57 0.36 2.81

0.23 4.29

Lyon et al. (2003) 0.26 3.80 0.42 2.36 Kim et al. (2006) 0.41 2.42

Lord and Park (2007)

1.40 2.01 0.94 1.31 1.46 1.30

0.71 0.50 1.06 0.76 0.68 0.77


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Table 2: Outcomes when n sites are classified according to a given

hotspot identification method # of sites

“detected” as non-hotspots

# of sites “detected” as

hotspots

# of “true” non-hotspots U V 0

n

# of “true” hotspots

R S 1n

n – D D n

where: n - Total number of sites in the set under analysis

0n - Number of “true” non-hotspots

1n - Number of “true” hotspots


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Table 3: “True” model coefficients for the California intersections

Parameters

Number of years of accident data

2-year period

3-year period

4-year period

5-year period

ln(β0) -9.154 -9.003 -9.052 -8.759

β1 1.112 1.130 1.169 1.162

β2 0.415 0.421 0.416 0.418

α 1.394 1.351 1.306 1.287

φ=1/α 0.717 0.740 0.766 0.777


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Table 4: California data - hotspot identification errors, for n=50 sites and T=2, 3 , 4 and 5 years

Hyper-prior

Time period = 2 years

FDR FNR SENS SPEC

Inf. Uniform (a0 =0.1, b=5) 0.334 0.094 0.677 0.877

Christiansen (a0 =2.0) 0.331 0.097 0.668 0.892

Inf. Gamma (a0=2.5 and b=1.3) 0.318 0.094 0.680 0.892

Non-inf. Gamma (a0 =0.001, b=0.001) 0.358 0.108 0.625 0.871


FDR FNR SENS SPEC

Inf. Uniform (a0 =0.1, b=5) 0.265 0.095 0.691 0.910

Christiansen (a0=2.0) 0.257 0.094 0.698 0.916

Inf. Gamma (a0=2.5 and b=1.3) 0.248 0.095 0.692 0.918

Non-inf. Gamma (a0 =0.001, b=0.001) 0.312 0.098 0.691 0.903


FDR FNR SENS SPEC

Inf. Uniform (a0 =0.1, b=5) 0.255 0.090 0.693 0.924

Christiansen (a0 =2.0) 0.245 0.088 0.701 0.927

Inf. Gamma (a0=2.5 and b=1.3) 0.241 0.088 0.699 0.930

Non-inf. Gamma (a0 =0.001, b=0.001) 0.282 0.091 0.713 0.914


FDR FNR SENS SPEC

Inf. Uniform (a0 =0.1, b=5) 0.227 0.079 0.740 0.928

Christiansen (a0 =2.0) 0.224 0.077 0.748 0.929

Inf. Gamma (a0 =2.5 and b=1.3) 0.223 0.080 0.739 0.930

Non-inf. Gamma (a0 =0.001, b=0.001) 0.230 0.079 0.744 0.923 Note: all results are mean values obtained from 100 simulations


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Table 5: California data - hotspot identification errors for n=100 and 200 sites, and 5 years

Hyper-prior

n = 100 sites, and T = 5 years

FDR FNR SENS SPEC

Inf. Uniform (a0 =0.1, b=5) 0.220 0.073 0.717 0.946

Christiansen (a0 =2.0) 0.221 0.072 0.720 0.946

Inf. Gamma (a0=2.5 and b=1.3) 0.216 0.073 0.716 0.947


FDR FNR SENS SPEC

Inf. Uniform (a0 =0.1, b=5) 0.209 0.084 0.752 0.930

Christiansen (a0 =2.0) 0.209 0.084 0.754 0.930

Inf. Gamma (a0=2.5 and b=1.3) 0.210 0.085 0.750 0.930 Note: all results are mean values obtained from 100 simulations


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Table 6: Texas data - hotspot identification errors for

n=30, 50 and 100 sites

Hyper-prior


FDR FNR SENS SPEC

Inf. Uniform (a0 =1.0, b=5) 0.110 0.043 0.889 0.957

Christiansen (a0 =2.6) 0.114 0.044 0.887 0.955

Inf. Gamma (a0=4.8, b=12.2) 0.129 0.048 0.880 0.948

Non-inf. Gamma (a0=0.001, b=0.001) 0.131 0.051 0.861 0.940


FDR FNR SENS SPEC

Inf. Uniform (a0 =1.0, b=5) 0.125 0.043 0.881 0.959

Christiansen (a0 =2.6) 0.125 0.044 0.878 0.957

Inf. Gamma (a0 =4.8, b=12.2) 0.127 0.047 0.871 0.952

Non-inf. Gamma (a0=0.001, b=0.001) 0.137 0.049 0.864 0.945


SPEC SPEC SENS SPEC

Inf. Uniform (a0 =1.0, b=5) 0.112 0.041 0.884 0.954

Christiansen (a0 =2.6) 0.118 0.041 0.883 0.955

Inf. Gamma (a0 =4.8, b=12.2) 0.112 0.042 0.879 0.954

Non-inf. Gamma (a0=0.001, b=0.001) 0.114 0.040 0.883 0.950 Note: all results are mean values obtained from 100 simulations


37

Table 7: California data - true versus posterior estimated values of model parameters

(n=50 sites and T=2, 3 , 4 and 5 years)

Hyper-prior n =50, time period = 2 years

Real values: Ln(β0)=- -9.154 β1=1.112 β2=0.415 φ=0.717

Average estimates:

Inf. Uniform (a0 =0.1, b=5) -9.076 1.102 0.433 1.841

Christiansen (a0 =2.0) -9.028 1.102 0.441 6.004

Inf. Gamma (a0 =2.5 and b=1.3) -8.949 1.086 0.428 1.478

Non-inf. Gamma (a0 =0.001, b=0.001) -8.324 1.104 0.497 44.350

n =50, time period = 3 years

Real values Ln(β0)=- -9.003 β1=1.130 β2=0.421 φ=0.740

Average estimates:

Inf. Uniform (a0 =0.1, b=5) -8.987 1.125 0.424 1.808

Christiansen (a0 =2.0) -8.945 1.124 0.429 4.693

Inf. Gamma (a0=2.5 and b=1.3) -8.907 1.117 0.424 1.479

Non-inf. Gamma (a0 =0.001, b=0.001) -8.192 1.060 0.465 31.674


Real values Ln(β0)= -9.052 β1=1.169 β2=0.416 φ=0.766

Inf. Uniform (a0 =0.1, b=5) -8.836 1.136 0.429 1.749

Christiansen (a0 =2.0) -8.971 1.157 0.439 3.737

Inf. Gamma (a0=2.5 and b=1.3) -8.937 1.147 0.428 1.432

Non-inf. Gamma (a0 =0.001, b=0.001) -8.982 1.153 0.417 26.407



Inf. Uniform (a0 =0.1, b=5) -8.483 1.128 0.432 1.352

Christiansen (a0 =2.0) -8.591 1.145 0.440 1.987

Inf. Gamma (a0=2.5 and b=1.3) -8.549 1.138 0.436 1.203

Non-inf. Gamma (a0=0.001, b=0.001) -8.411 1.138 0.453 13.321 Note: all results are mean values obtained from 100 simulations


38

Table 8: California data - true versus posterior estimated values of model parameters (n=100 and 200 sites, and 5 years)

Hyper-prior n = 100 sites, and T = 5 years

Real values: Ln(β0)= -8.759 β1=1.162 β2=0.418 φ=0.777

Average estimates:

Inf. Uniform (a0 =0.1, b=5) -8.781 1.158 0.413 1.214

Christiansen (a0 =2.0) -8.856 1.168 0.414 1.108Inf. Gamma (a0 =2.5 and b=1.3) -8.825 1.163 0.413 1.118



Average estimates:

Inf. Uniform (a0 =0.1, b=5) -8.680 1.149 0.405 0.756

Christiansen (a0 =2.0) -8.776 1.119 0.406 0.717Inf. Gamma (a0 =2.5 and b=1.3) -8.722 1.116 0.406 0.771 Note: all results are mean values obtained from 100 simulations


39

Table 9: Texas data - true versus posterior estimated values of model parameters (n=30, 50 and 100 sites, and 5 years)

Hyper-prior n = 30 sites, and T = 5 years

Real values: ln(β0) = -6.56 β1 = 0.926 φ = 3.231

Average estimates:

Inf. Uniform (a0 =1.0, b=5) -6.093 0.874 3.234

Christiansen (a0 =2.6) -5.903 0.853 7.535

Inf. Gamma (a0=4.8, b=12.2) -6.138 0.825 1.856

Non-inf. Gamma (a0=0.001, b=0.001) -6.245 0.910 49.043


Real values ln(β0) = -6.56 β1 = 0.926 φ = 3.231

Average estimates:

Inf. Uniform (a0 =1.0, b=5) -6.763 0.947 3.303

Christiansen (a0 =2.6) -6.780 0.949 4.754

Inf. Gamma (a0=4.8, b=12.2) -6.459 0.914 2.048

Non-inf. Gamma (a0=0.001, b=0.001) -6.849 0.957 10.373


Real values ln(β0) = -6.56 β1 = 0.926 φ = 3.231

Average estimates:

Inf. Uniform (a0 =1.0, b=5) -6.409 0.909 3.430

Christiansen (a0 =2.6) -6.325 0.900 3.757

Inf. Gamma (a0=4.8, b=12.2) -6.314 0.898 2.380

Non-inf. Gamma (a0=0.001, b=0.001) -6.637 0.935 3.979 Note: all results are mean values obtained from 100 simulations

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