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Accident Analysis and Prevention 48 (2012) 464– 471
Contents lists available at SciVerse ScienceDirect
Accident Analysis and Prevention
jo ur n al hom ep a ge: www.elsev ier .com/ locate /aap
he roles of exposure and speed in road safety analysis
in Peia,1, S.C. Wongb,2, N.N. Szeb,∗
Department of Automation, Tsinghua University, Beijing, ChinaDepartment of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong
r t i c l e i n f o
rticle history:eceived 26 March 2011eceived in revised form 28 February 2012ccepted 3 March 2012
eywords:xposure
a b s t r a c t
Speed is a determining factor in road safety analysis. It is generally believed that an increase in speedharms road safety. However, it can also be argued that driving at high speed reduces the length of timeexposure and thus the likelihood of a crash. It is therefore critical to clarify the roles that exposure andspeed play in road safety analysis. This study evaluates the relationship between speed and crash riskwith respect to distance and time exposure, using disaggregated crash and speed data collected from112 road segments in Hong Kong. A joint probability model based on a full Bayesian method is applied
peedrash riskrash severity
oint probability modelull Bayesian method
simultaneously to model crash occurrence and crash severity. In addition, we consider the explanatoryvariables, including road design, weather conditions, and temporal distribution, in the proposed crashprediction model. The results indicate that average speed plays a significant role in crash risk, despiteopposing correlations with respect to distance and time exposure; the correlation between speed andcrash risk is positive when distance exposure is considered, but negative when time exposure is used.However, in both cases, speed is positively associated with the injury severity.
. Introduction
In transportation engineering, speed is one of the basic variableshat index the state and the level of service of roadway facilities.peed plays an important role in traffic management and control. Its also generally believed to be a determining factor in the numbernd severity of crashes (OECD, 1996; National Research Council,010), because driving becomes a more complex and demandingask at high speed and thus the likelihood of driver error is greater.urthermore, the energy released in a high-speed crash is greater,hereby increasing the likelihood of serious injury, death, and sig-ificant damage to property.
Average speed and speed dispersion both influence crash occur-ence and crash severity (Hauer, 2009; National Research Council,010). Yet there is no consensus on the magnitude and direc-ion of the influences of speed (Lave and Elias, 1994; Garber andadirau, 1988; Shinar, 1998; McCarthy, 1998; Taylor et al., 2000,002; Aarts and van Schagen, 2006) and speed dispersion (Lave,
985; Zlatoper, 1991) on the risk of crash. Some studies have sug-ested that the direction of the influences of speed and speedispersion on crash risk change when speed is increased to a∗ Corresponding author. Tel.: +852 2859 2662; fax: +852 2517 0124.E-mail addresses: [email protected] (X. Pei), [email protected]
S.C. Wong), [email protected] (N.N. Sze).1 Tel.: +86 10 6279 5043; fax: +86 10 6277 3835.2 Tel.: +852 2859 1964; fax: +852 2559 5337.
001-4575/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.oi:10.1016/j.aap.2012.03.005
© 2012 Elsevier Ltd. All rights reserved.
certain level (Solomon, 1964). However, Nilsson (2004) used con-trolled before-and-after studies to establish the Power Model todescribe the effect of changes in speed on changes in crash fre-quency and severity. The Power Model was validated by Elviket al. (2004) using the meta-analysis approach. Empirical esti-mates of the exponents of the Power Model have been helpful forextended investigations into the association between speed andcrash risk. A recent study suggested that the exponents of thePower Model might vary with road type. In particular, the expo-nent estimates for the risk of a slight injury crash were smaller thanpredicted by the Power Model on urban arterial roads (Cameronand Elvik, 2010). It seems that the magnitude of the effect ofspeed on road safety in urban road network needs to be furtherexplored.
The availability of disaggregated speed data is essential forthe analysis of the relationship between speed and road safety.With the support of advanced intelligent transport system (ITS)techniques, it is possible to detect traffic crashes based on compre-hensive real-time speed and flow data obtained from a double-loopdetector (Lee et al., 2002; Abdel-Aty et al., 2004, 2008). In this case,speed data are used as a proxy for roadway performance in acutetraffic control and management, but not for inferences on the effectof speed on traffic safety. Disaggregated speed data are also essen-tial for estimating length of time exposure in models of crash risk.
Furthermore, when modeling crash risk, the explanatory factorssuch as road design, weather conditions, and temporal distribu-tion for the association between speed, speed dispersion, and trafficsafety must be considered.
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X. Pei et al. / Accident Analysis
Exposure measures the likelihood of being involved in a dan-erous or hazardous situation. In road safety research, exposures an important factor in the estimation of crash risk (Chapman,973; Wolfe, 1982; Rumar, 2002). The selection of an appropriateroxy for exposure is thus essential to evaluate the influences ofossible risk factors on the crash risk. A wide range of data aresed to proxy exposure, including traffic volume (Miaou, 1994;ountain et al., 1996; Qin et al., 2004, 2006; Wong et al., 2007;
an den Bossche et al., 2005), conflicts (Bie et al., 2005; Wongt al., 2006), travel distance (Li et al., 2003), travel time (Chipmant al., 1993), and population or fuel consumption (Fridstrøm et al.,995).
In prior studies, distance exposure, measured in either vehicleilometers (VKM) or vehicle miles traveled (VMT), was adoptedo model the relationship between speed and crash risk (Abbas,004; Hakim et al., 1991). However, it could be argued that as
ength of time exposure on the road might be reduced whenpeed is increased, then the likelihood of crash involvement mightlso be reduced (Chipman et al., 1992). For this reason, timexposure measured in vehicle hours (VH) could be considered aetter proxy for exposure. In this study, taking advantage of thevailability of disaggregated speed data, it is possible to deter-ine the length of time exposure given that the traffic volume
nd length of road segment are known. Hence, the influence ofoth exposure measures (i.e., distance and time exposure) onhe relationship between speed and road safety can be evalu-ted.
In this study, comprehensive information on crashes, trafficow, and speed were collected from 112 roadway segments inong Kong during the three-month period from July to September009. A joint probability model based on the MCMC approach fullayesian method is established to model crash occurrence andrash severity simultaneously (Pei et al., 2011). The rest of thisaper is organized as follows. In the following section, we firstescribe the study design and the method used for collecting data.e discuss the methods of analysis employed in this study in Sec-
ion 3. The results are presented in Section 4 and their implicationsre discussed in Section 5. Section 6 presents our conclusions andecommendations for future research.
. Data
In preparation for this study, we first established a comprehen-ive crash database containing information on traffic flow, speed,oad design, weather conditions, and temporal distribution foroadway segments in Hong Kong, using geographical informationystem (GIS) techniques.
Detailed traffic flow data were obtained from the annual traf-c census (ATC) system that consists of over 1500 stations andovers 86.8% of all motorways in Hong Kong (Tong et al., 2003;am et al., 2003). The ATC system provides detailed information onverage annual daily traffic (AADT) and its temporal (with respecto month, day of the week, and hour of the day) and directional
ultiplicative factors. In particular, continuous measurements areaken at 112 core stations. These core stations are evenly and widelyistributed across the whole territory and cover a total length of64.6 km (i.e., 8.0% of all motorways in Hong Kong). In the pro-osed model, we derived directional traffic volumes for all roadegments adjacent to the 112 core stations for every 4-h period07:00–11:00 (morning), 11:00–15:00 (noon), 15:00–19:00 (after-oon), 19:00–23:00 (evening), 23:00–03:00 (middle of the night)
nd 03:00–07:00 (dawn)] between July and September 2009. Theample thus comprises a total of 68,990 observations.Speed data measured at the relevant road segments dur-ng the corresponding observation periods were obtained from
revention 48 (2012) 464– 471 465
measurements taken by 480 taxis equipped with global position-ing system (GPS) devices operating on the roadway network inHong Kong. In particular, instantaneous information on the loca-tion, speed, and travel direction of each GPS probe was transmittedto and stored in the control center at 30-s intervals. The averagetravel speed and the standard deviation of travel speed for every 4-h period can be calculated using these raw data. Here, average speedshould account for traffic conditions, while the standard deviationof speed should account for variation in traffic conditions acrossthe spatial–temporal domain. Hence, the speed data collected fromthe taxis can be used as a proxy for actual speed in particular trafficconditions and can facilitate analysis of the role that speed playsin road safety. When a crash occurs, the traffic at the road seg-ment in question will be disrupted, thereby affecting traffic speed.In this case, the estimated speed is estimated on the basis of roaddesign, weather conditions, and temporal distribution when thereis no crash, rather than the speed directly observed during the crashperiod.
Crash data were obtained from the Traffic Information System(TIS) maintained by the Transport Department. This database con-tains precise information on crash, vehicle, and casualty attributesfor all crashes involving personal injury that are reported to thepolice. We applied GIS techniques to map crashes that occurredon the studied road segments. A total of 347 crashes occurred atthe corresponding road segments during the three-mouth periodfrom July to September 2009. In the TIS, crashes are classified intothree categories—fatal, serious, and slight—according to the sever-ity of injuries among the casualties. In this study, we groupedthe fatal and serious crashes together as killed and seriouslyinjured (KSI) crashes. Of the 347 crashes captured, 59 (17.0%) wereKSI.
This study considers both distance and time exposure. Dis-tance exposure was derived by multiplying traffic volume by roadsegment length (in vehicle-kilometers, VKM). Given the aver-age speed and road segment length, it was possible to estimatethe travel time, and time exposure was then derived by multi-plying traffic volume by average travel time (in vehicle-hours,VH).
Information was also collected on the geometric design of theroad segments and traffic control at each segment. The variablesof interest are lane changing opportunity (LCO, which refers tothe total number of possible lane-cuttings based on those set outby different lane markings, as shown in Fig. 1), road curvature(average changing angle), gradient, number of merging ramps,number of diverging ramps, number of intersections (includingsignal junctions, yield junctions and stop sign junctions), the pres-ence of a central divider, the presence of a hard shoulder, thepresence of a bus stop, and the presence of on-street parking. Inaddition, rainfall is also an important environmental factor likelyto have a significant impact on road safety due to its effects ondriver visibility and vehicle braking performance. In this study,detailed rainfall data for each geographical location and periodwere obtained from the Hong Kong Observatory. The influencesof temporal distribution on crash risk, in terms of day of the weekand time of day, are also controlled for. Table 1 summarizes thecharacteristics of the 68,990 observations analyzed in the proposedmodel.
A multicollinearity test for the independent variables was con-ducted prior to the determination of association measures for crashoccurrence and crash severity. In the multicollinearity test, if theVIF value of any independent variable is greater than 10, that vari-able should be removed from the model to avoid biased parameterestimates. The results of the multicollinearity test revealed thatthe VIF values of the independent variables were all less than 10.
Therefore, no evidence could be established for the existence of amulticollinearity problem.
466 X. Pei et al. / Accident Analysis and Prevention 48 (2012) 464– 471
F : A –L s allowl l LCO =
3
3
of
TS
ig. 1. Lane changing opportunities for different road section configurations. NotesCOA = 0; B – continuous line and broken line. Lane changing from Lane 3 to Lane 2 iine. Lane changing between Lane 3 and Lane 4 is allowed: LCOC = 2. Therefore, tota
. Method
.1. Modeling framework
A joint probability model is established to model the crashccurrence and severity simultaneously using the MCMC approachull Bayesian method (Pei et al., 2011).
able 1ummary of the observations incorporated into the proposed model.
Min Max
Number of observations = 68,990Total crashes 0 2
KSI crashes 0 1Average speed (km/h) 0.07 114.56
Standard deviation of speed (km/h) 0.00 62.63
Traffic volume (veh) 42 25,700
Road length (km) 0.17 9.07
ln (VKM) 4.07 11.42
ln (VH) 0.54 9.34
Rainfall (mm) 0.00 103.50
Lane changing opportunity 0.00 7.85
Road curvature 0.00 85.07
Gradient −0.11 0.11
Number of merging ramps 0 4
Number of diverging ramps 0 17
Number of intersections 0 16
Presence of central divider
Presence of hard shoulder
Presence of bus stop
Presence of on-street parking
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday or public holiday
07:00–11:00
11:00–15:00
15:00–19:00
19:00–23:00
23:00–03:00
03:00–07:00
double continuous lines. No lane changing is allowed between Lane 1 and Lane 2:ed, while that from Lane 2 to Lane 3 is not allowed: LCOB = 1; and C – single broken
LCOA + LCOB + LCOC = 3.
For crash occurrence, because the dependent variable—crashfrequency—is discrete, non-negative, and random in nature, countdata models are applied. In particular, to analyze the effects of dis-tance and time exposure, logarithmically transformed VKM and VH
are incorporated into the model to offset these respective effects.As crash occurrence is subject to over-dispersion, a negativebinomial regression model should be applied (Washington
Mean S.D. Count Proportion
0.01 0.070.00 0.03
52.57 23.2913.80 5.27
5007 43731.67 1.598.24 1.274.40 1.111.98 7.292.64 1.68
24.43 16.710.00 0.030.99 1.041.73 2.411.54 2.47
54,359 78.8%11,365 16.5%39,463 57.2%25,067 36.3%
9550 13.84%9542 13.83%9600 13.92%9737 14.11%9893 14.34%
10,100 14.64%10,568 15.32%12,303 17.83%12,743 18.47%12,607 18.27%12,371 17.93%10,512 15.24%
8454 12.25%
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X. Pei et al. / Accident Analysis
t al., 2003), where the probability function is definedy
(yi) = � [(1/˛) + yi]� (1/˛)yi!
(1/˛
(1/˛) + �i
)1/˛(�i
(1/˛) + �i
)yi
(1)
long with the link function, �i = exp(log(exposure) + �X i), wherei is the number of crashes for the ith observation and �i is thexpected number of crashes for the ith observation. Xi denotes theector of explanatory factors and � the vector of correspondingoefficients. ̨ is the over-dispersion parameter.
A hierarchical binomial-logistic model is employed to estimaterash severity. Given that each crash occurrence is considered aernoulli trial, the crash outcome with respect to injury severityhould follow a binomial distribution. The number of KSI crashesKSIi
follows the binomial distribution defined as
KSIi ∼Binomial(�KSI
i , yi), (2)
here yi is the total crash risk for the ith observation and �KSIi
ishe probability of KSI crashes.
Therefore, the probability function of kKSIi
, conditional on yi totalrashes, can be defined as
(kKSIi |yi) =
(yi
kKSIi
)(�KSI
i )kKSI
i (1 − �KSIi )
yi−kKSIi . (3)
ere, we can establish a logit function to model the associationsetween the binomial probability and possible factors defined as
ogit(�KSIi ) = ˇKSIX i, (4)
here ˇKSI is the vector of parameters measuring the correlationsetween explanatory factors and crash severity.
Hence, we have
(kKSIi |yi) =
(yi
kKSIi
) [exp(ˇKSIXi)
1 + exp(ˇKSIXi)
]kKSIi
[1
1 + exp(ˇKSIXi)
]yi−kKSIi
(5
onsequently, the joint probability function for crash occurrencend crash severity can be obtained by combining Eqs. (1) and (5)s
(yi, kKSIi ) = P(yi) × P(kKSI
i |yi)
= � [(1/˛) + yi
� (1/˛)yi!
(1/˛
(1/˛) + exp(log(exposure) + �X i)
)1/˛
×(
exp(log(exposure) + �X i)(1/˛) + exp(log(exposure) + �X i)
)yi
×(
yi
kKSIi
)[exp(kKSI
i ˇKSIX i)][1 + exp(ˇKSIX i)]−yi
. (6)
.2. Model estimation
The full Bayesian method, using a Markov chain Monte CarloMCMC) approach, is applied to the proposed joint probability
odel. The full Bayesian method treats all unknown parameters asandom variables under a prior distribution. Instead of giving pointstimations, as in the classical maximum likelihood approach, theull Bayesian method uses the posterior distribution of the parame-
ers to summarize the results in the proposed model (Gelman et al.,004).According to Bayes’ theorem, by integrating the prior distri-ution of parameters and the likelihood function, the posterior
revention 48 (2012) 464– 471 467
distribution of parameters can be estimated by the following func-tion:
f (�|y) = f (y|�)f (�)f (y)
∝ f (y|�)f (�) (7)
where y is the observed outcome and � is the parameter estimate.Therefore, the likelihood function can be defined as
f (y|�) =n∏
i=1
f (yi|�) (8)
Appropriate specification of the prior distribution f (�) is a criti-cal element of Bayesian inference. For instance, it is essential tohave prior information on and a good understanding of the vari-ables of interest. If prior information for particular parameters isnot available, we can set up a diffuse prior distribution as
�∼Normal(0, �2Im), (9)
where �2 is very large, I is an identity matrix of m dimensions, andm equals the number of parameters.
In a full Bayesian framework, the inferences are determined bythe Markov chain Monte Carlo (MCMC) simulation approach thatdraws the sample from the prior distribution of unknown param-eters in an iterative procedure. Such a process should be repeateduntil the distribution of parameter estimates converges and givesthe posterior distribution. The Metropolis–Hastings algorithm andGibbs sampling are two common sampling approaches that followthe MCMC method. Gibbs sampling can be regarded as a specialcase of the Metropolis–Hastings sampling algorithm, in which acandidate estimate is always accepted rather than being acceptedat a particular probability. In addition, the Gibbs sampling approachis capable of drawing a sample from the distributions of parame-ters other than that of the variable of concern, thus ensuring thatthe estimates will converge more easily. This quality has made thewidely used Gibbs sampling method a ‘golden standard’. Yet Gibbssampling might not be suitable when the form of the parameterdistribution is sophisticated (Ntzoufras, 2009).
Possible ways of evaluating convergence of the estimates inthe iterative procedure include estimating the Markov chain error,plotting the autocorrelation distributions, plotting the generatedsample values, and computing Gelman–Rubin statistics for multi-ple chain simulation (Spiegelhalter et al., 2003).
The validity of the model can be evaluated by comparing theobserved and replicated data. The replicated data can be predictedin the simulation process set out by the posterior predictive dis-tribution. For instance, a chi-square test statistic can be calculatedby
�2(y, �) =n∑
i=1
[yi − E(yi|�)]2
Var(yi|�), (10)
where yi denotes either the observed response or the replicatedresponse. In each iteration t of an MCMC simulation, the differencebetween �2(yrep, �(t)) and �2(y, �(t)) is monitored, as is the cor-responding posterior predictive p-value. The results indicate thatthe proposed model does not fit well if the p-value is close to 0or 1. Gelman et al. (2004) suggests that major failures should beconsidered if the p-value is less than 0.01 or more than 0.99.
4. Results
This study establishes two joint probability models control-
ling for distance and time exposure, respectively, to measurethe relationship between speed, other possible risk factors, andcrash occurrence and severity simultaneously. In the proposed fullBayesian modeling framework with the MCMC approach, three
468 X. Pei et al. / Accident Analysis and Prevention 48 (2012) 464– 471
Table 2Results of the joint probability model for crash occurrence and crash severity.
Distance exposure (VKM) Time exposure (VH)
Crash occurrence Crash severity Crash occurrence Crash severity
Mean (95% CIs) Mean (95% CIs) Mean (95% CIs) Mean (95% CIs)
Constant −13.11* (−14.06, −12.21) −4.24* (−7.48, −0.93) −11.00* (−11.94, −10.15) −3.16* (−6.96, −0.37)Average speed −0.02* (−0.03, −0.01) 0.03* (0.01, 0.06) 0.01* (0.00, 0.01) 0.03* (0.01, 0.06)Standard deviation of speed 0.00 (−0.02, 0.02) −0.01 (−0.12, 0.10) 0.02 (−0.01, 0.04) −0.03 (−0.12, 0.08)Rainfall 0.02* (0.01, 0.03) −0.09* (−0.19, −0.01) 0.02* (0.01, 0.03) −0.09* (−0.18, −0.01)Lane-changing opportunity 0.20* (0.12, 0.28) −0.19 (−0.47, 0.07) 0.18* (0.10, 0.26) −0.18 (−0.44, 0.07)Curvature −0.003 (−0.01, 0.005) 0.01 (−0.01, 0.04) −0.002 (−0.01, 0.01) 0.01 (−0.01, 0.03)Gradient −0.44 (−4.90, 3.86) 2.22 (−10.39, 14.71) 0.04 (−4.48, 4.44) 1.56 (−10.89, 13.94)Number of merging ramps −0.08 (−0.19, 0.02) −0.03 (−0.38, 0.32) −0.10 (−0.22, 0.01) −0.04 (−0.38, 0.32)Number of diverging ramps 0.00 (−0.06, 0.07) −0.22* (−0.43, −0.02) 0.01 (−0.05, 0.08) −0.21* (−0.41, −0.01)Number of intersections 0.03 (−0.04, 0.10) 0.13 (−0.08, 0.33) 0.02 (−0.05, 0.09) 0.10 (−0.10, 0.30)Presence of central divider −0.55* (−1.06, −0.09) −0.64 (−1.98, 0.74) −0.55* (−1.02, −0.06) −0.73 (−2.08, 0.57)Presence of hard shoulder 0.09 (−0.29, 0.47) −0.26 (−1.38, 0.87) −0.20 (−0.58, 0.19) −0.15 (−1.32, 0.99)Presence of bus stop 0.41* (0.13, 0.68) 0.01 (−0.89, 0.94) 0.45* (0.18, 0.73) −0.001 (−0.89, 0.88)Presence of on-street parking 0.35 (−0.07, 0.76) 0.99 (−0.22, 2.18) 0.35 (−0.05, 0.76) 0.89 (−0.34, 2.04)Monday 0.12 (−0.29, 0.53) 0.89 (−0.28, 2.14) 0.12 (−0.29, 0.54) 0.88 (−0.24, 2.10)Tuesday −0.52* (−1.00, −0.06) 1.12 (−0.28, 2.53) −0.52* (−0.99, −0.04) 1.15 (−0.21, 2.53)Wednesday 0.04 (−0.37, 0.45) −1.02 (−2.74, 0.55) 0.03 (−0.38, 0.46) −0.98 (−2.72, 0.56)Thursday 0.16 (−0.24, 0.55) 0.71 (−0.49, 1.98) 0.14 (−0.27, 0.57) 0.74 (−0.49, 2.05)Friday 0.12 (−0.29, 0.53) 0.85 (−0.35, 2.13) 0.12 (−0.28, 0.54) 0.87 (−0.33, 2.18)Saturday −0.11 (−0.55, 0.33) 0.12 (−1.24, 1.50) −0.12 (−0.56, 0.32) 0.11 (−1.26, 1.53)Sunday or public holiday (Control)07:00–11:00 −0.22 (−0.76, 0.41) 1.24 (−0.54, 3.41) −0.13 (−0.67, 0.45) 0.85 (−0.84, 2.82)11:00–15:00 −0.35 (−0.90, 0.27) 0.57 (−1.25, 2.78) −0.25 (−0.82, 0.34) 0.17 (−1.58, 2.15)15:00–19:00 −0.28 (−0.83, 0.33) 1.23 (−0.52, 3.34) −0.19 (−0.74, 0.39) 0.81 (−0.84, 2.78)19:00–23:00 −0.38 (−0.95, 0.24) 1.03 (−0.81, 3.18) −0.29 (−0.83, 0.32) 0.65 (−1.10, 2.61)23:00–03:00 0.41 (−0.14, 1.03) 1.17 (−0.61, 3.39) 0.47 (−0.09, 1.07) 0.78 (−0.99, 2.79)03:00–07:00 (Control)
ccsItacpcep
eci[tI[l
lt[Faba5if−−s
Over-dispersion parameter 0.42* (0.13, 1.33)
* Statistically significant at the 5% level.
hains of 30,000 iterations are adopted in each simulation pro-ess. All posterior estimates are within the range for not producingtrong periodicities and tendencies, as indicated by the trace plots.n addition, Gelman–Rubin convergence diagnostic results indicatehat both the inter-sample and intra-sample variabilities are stablend close to one. In other words, the two proposed models bothonverge. Table 2 reports the results of the two proposed jointrobability models. To assess goodness-of-fit, the p-values of thehi-square statistics are calculated as 0.12 and 0.09 for the mod-ls controlling for distance and time exposure, respectively. Theroposed models generally fit well with the observations.
As shown in Table 2, there are no major differences in the param-ter estimates between the model controlling for distance and thatontrolling for time exposure, other than for average speed. Fornstance, when distance exposure is controlled for, average speedMean = −0.02; 95% CIs = (−0.03, −0.01)] is negatively related tohe likelihood of a crash occurring at the 5% level of significance.n contrast, when controlling for time exposure, average speedMean = 0.01; 95% CIs = (0.00, 0.01)] is positively related to the like-ihood of a crash occurring at the 5% significance level.
For crash occurrence, rainfall is positively related to theikelihood of a crash occurring, controlling for both dis-ance [Mean = 0.02; 95% CIs = (0.01, 0.03)] and time exposureMean = 0.02; 95% CIs = (0.01, 0.03)], at the 5% significance level.urthermore, lane changing opportunities is positively associ-ted with the likelihood of a crash occurring, controlling foroth distance exposure [Mean = 0.20; 95% CIs = (0.12, 0.28)]nd time exposure [Mean = 0.18; 95% CIs = (0.10, 0.26)], at the% significance level. The road segment with central divider
s related to lower likelihood of a crash occurring, controlling
or both distance exposure [Mean = −0.55; 95% CIs = (−1.06,0.09)] and time exposure [Mean = −0.54; 95% CIs = (−1.02,0.06)], at the 5% significance level. In addition, for the roadegment with bus stop, the probability of a crash occurring is
0.23* (0.09, 0.60)
higher, controlling for both distance exposure [Mean = 0.41; 95%CIs = (0.13, 0.68)] and time exposure [Mean = 0.45; 95% CIs = (0.18,0.73)], at the 5% significance level. Moreover, the crash risk onTuesdays is lower than that on Sundays or public holidays, control-ling for both distance exposure [Mean = −0.52; 95% CIs = (−1.00,−0.06)] and time exposure [Mean = −0.52; 95% CIs = (−0.99,−0.04)], at the 5% significance level.
Table 2 also shows that for crash severity, average speed ispositively related to the likelihood of a KSI crash, controlling forboth distance exposure [Mean = 0.03; 95% CIs = (0.01, 0.06)] andtime exposure [Mean = 0.03; 95% CIs = (0.01, 0.06)], at the 5% sig-nificance level. Rainfall is negatively related to the likelihood of aKSI crash, controlling for both distance exposure [Mean = −0.09;95% CIs = (−0.19, −0.01)] and time exposure [Mean = −0.08; 95%CIs = (−0.18, −0.01)], at the 5% significance level. Furthermore,the number of diverging ramps is also negatively related to thelikelihood of a KSI crash, controlling for both distance expo-sure [Mean = −0.22; 95% CIs = (−0.43, −0.02)] and time exposure[Mean = −0.21; 95% CIs = (−0.41, −0.01)], at the 5% significancelevel.
5. Discussion
We establish two joint probability models to reveal the associ-ation between speed, other possible factors, and crash occurrenceand crash severity simultaneously. We also investigate differencesin association measures when controlling for different exposuremeasures.
First, the relationship between speed and the likelihood of acrash occurring differs according to which exposure measure is
adopted. For instance, the results indicate that crash risk is nega-tively associated with average speed when controlling for distanceexposure. Although it is not uncommon for distance exposure tobe controlled for when modeling the association between crash
X. Pei et al. / Accident Analysis and Prevention 48 (2012) 464– 471 469
ntrol
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Fig. 2. Relationship between crash risk and speed, co
isk and speed, results from prior studies were inconclusive on thisoint (Solomon, 1964; Shinar, 1998; McCarthy, 1998; Aarts andan Schagen, 2006; Garber and Gadirau, 1988). Some researchersrgue that roadway segments designed for higher speeds are usu-lly well planned and developed, and should deliver better roadafety performance. Therefore, crash frequency should decreaseith speed. The current study reveals not only a negative rela-
ionship between speed and crash risk (controlling for distancexposure), but also the possible influences of explanatory factorsncluding road design, weather conditions, and temporal distribu-ion on respective relationship. Nevertheless, information on otherossible factors related to crash occurrence, including traffic com-osition and driver behavior, was not available for this study. Theseactors are worthy of exploration in future research.
Notwithstanding the foregoing finding, there is also an argu-ent that this negative relationship between speed and crash
ccurrence, observed when controlling for distance exposure, mayistort the role of speed, because the likelihood of a vehicle travel-
ng at a higher speed being involved in a crash is underestimatedor a given travel distance due to the shorter time of exposureChipman et al., 1992). This issue has received less attentionecause travel time and speed data are generally unavailable for
ling for (a) distance exposure and (b) time exposure.
accident analyses. Given the availability of speed data for this study,we are able to investigate this problem by controlling for time expo-sure on highways. Our analysis finds that crash risk is positivelyassociated with average speed when time exposure is controlledfor. This helps to shed light on the interesting problem of whetherhigher speed is related to a higher crash risk: our results indicatethat given the same time duration, higher vehicle speed would beassociated with higher crash risk when correcting for the reduc-tion in time exposure per unit of distance traveled. This means thathigh-speed roads may not necessarily be safer than lower-speedroads when the time exposure measure is properly accounted for.
Turning to injury severity, it is apparent that both models exam-ined here point to a similar observation whereby crash severity ispositively related to speed, controlling for both distance and timeexposure. This finding is consistent with that of a previous study(Hauer, 2009), which argues that as the amount of energy releasedis greater when the impact speed is higher, the harm done and thelikelihood of mortality and serious injury are greater.
The implications of the relationship between speed, crash occur-rence and crash severity are illustrated in Fig. 2. For instance, therelationships between average speed and (i) total crash risk, (ii)KSI crash risk, and (iii) the risk of a crash causing slight injury are
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valuated. As shown in Fig. 2(a), when controlling for distancexposure, total crash risk decreases with speed, whereas the risk of
KSI crash increases with speed.As shown in Fig. 2(b), when controlling for time exposure, both
otal crash risk and KSI crash risk increase with speed. In partic-lar, the risk of a KSI crash increases at a rate remarkably higherhan that of total crash risk because the likelihood of a KSI crash is
ore sensitive to speed. Moreover, when speed increases beyond5 km/h, the risk of a crash causing slight injury begins to decline.hen speed is around 110 km/h, the risk of a KSI crash is equal
o that of a slight injury crash. Furthermore, when speed increaseseyond 110 km/h, the risk of a KSI crash is much higher than thatf a slight injury crash.
Speed dispersion was also expected to be highly related to crashisk. However, in the proposed model, there is no evidence that thetandard deviation of speed is significantly associated with the like-ihood of crash occurrence or crash severity. However, it is worthoting that the speed dispersion used in this study does not rep-esent the variations in speed seen in a mixed traffic stream withifferent types of vehicles, but is a proxy for the variability of trafficonditions experienced by drivers in a spatio-temporal domain onhe highway.
Since the Power Model has been widely adopted to connectpeed and road safety, we attempt to apply the Power Model to ourodel by incorporating logarithmically transformed speed into the
ink function. For the crash occurrence (i.e., injured crash) model,he exponent estimate of speed is 0.50. Such an estimate is muchower than that of Nilsson’s model (Nilsson, 2004), but is close tohat (i.e., 1.2, 95% CI [0.7, 1.7]) of Cameron and Elvik’s model (2010)or urban/residential roads. For the crash severity (i.e., KSI crash)
odel, the exponent estimate of speed is 1.44. Such an estimate islose to that of Nilsson’s model (i.e., Exponent of KSI crash minusxponent of Casualty Crash = 3 minus 2 = 1). There are two possi-le explanations for why the results of the current model deviaterom those of Nilsson’s model. First, our model measured the risk ofrash and crash severity on urban roads, whereas Nilsson’s modeleasured that of rural highways and freeways. Additionally, ourodel measured the average speed over a very short time window
4 h), whereas Nilsson’s model measured the average speed over aonger period of time. In future, it is worth exploring the differencesn the effects of speed on crash risk with respect to different roadypes.
For weather conditions, the results indicate that rainfall iselated to the likelihood of a crash occurring with a positive sign,ut to the likelihood of a KSI crash with a negative sign. This may beecause rain reduces skidding resistance and driver visibility, andhus increases the overall crash risk. However, the impact forcexerted during collisions may be reduced as driver alertness mayncrease when driving in rainy conditions, and crashes in such
eather conditions are, therefore, possibly less severe (Hermanst al., 2006; Khattak et al., 1998; Edwards, 1998; Fridstrøm et al.,995; Sze and Wong, 2007).
For road design, increased lane-changing opportunities areound to be related to an increase in the likelihood of a crashccurring. This result could be attributable to increased vehiclenteraction due to the increase in the number of permissible lane-utting and overtaking activities, thereby raising the incidence ofraffic conflict and collisions. Similarly, the presence of a bus stops related to a higher crash risk possibly due to frequent pedestrianctivity around the bus stop, leading to a higher risk of vehicle-edestrian conflict. Indeed, of the crashes involving pedestrians
n the observed road segments, 93% occurred on roads with bus
tops. Moreover, there may be potential for more conflict betweenuses entering or leaving bus bays and other vehicles in the mainraffic stream. In addition, our prediction that the presence of a cen-ral divider would be related to a lower risk of a crash occurring isrevention 48 (2012) 464– 471
confirmed, as the opposing traffic flows are separated. Moreover, anincrease in the number of diverging ramps is related to the reduc-tion in the risk of KSI crash. This may be because the increase inroad space for defensive driving behavior at the area near diverg-ing ramps leads to a reduction in the impact of the collision in theevent of a crash.
With respect to temporal distribution, crash risk on Tuesdaysis found to be lower than that on Sundays or public holidays.Although it may not be possible to elaborate explicitly on the vari-ation in crash risk across different time periods, the influencesof unidentified factors that vary over time are controlled for inthe proposed model, including both time of day and day of theweek.
6. Conclusions
Based on comprehensive crash, traffic flow, and speed data forHong Kong’s roadway network, we investigate the effects of differ-ent forms of exposure measures, i.e. time and distance exposure,on the relationship between speed, other possible risk factors andcrash risk. Taking advantage of detailed speed data obtained fromGPS probes, it is possible to reveal the relationship between speedand crash risk when controlling for time exposure. A joint prob-ability model is applied to analyze crash occurrence and crashseverity simultaneously under the MCMC approach full Bayesianframework.
The results indicate the positive relationship between speed andthe risk of a crash occurring when controlling for the effect of timeexposure. This contrasts with the results obtained when controllingfor distance exposure, which indicates the opposite relationship.This finding helps shed light on the interesting argument that con-trolling for distance exposure may distort the overall relationshipbetween crash risk and speed, because of the lower time exposureper unit of distance traveled on high-speed roads. From the obser-vations made in this study, it is apparent that high-speed roadsmay not necessarily be safer than lower-speed roads when a properexposure measure is used. However, the results for crash severityshow that injury severity is generally positively associated withspeed, irrespective of whether distance or time exposure is used.
Our results do not provide evidence suggesting that speed dis-persion, as a proxy for the variability of traffic conditions in thespatio-temporal domain of a highway, is significantly related toeither crash risk or crash severity. However, it would be worth-while exploring variations in the associations between speed, speeddispersion, and crash risk across different crash types in futureresearch.
The joint probability approach adopted in this study integratesboth types of risk assessed in crash models—the risk of an accidentoccurring given a certain level of exposure and the risk of injuryseverity in the event of an accident—into a unified analysis thatgives us a better understanding of the roles played by exposureand speed in the likelihood of traffic accidents occurring and theseverity of injuries suffered by casualties. This approach supportsthe holistic assessment of traffic accidents and helps to identify therelationship between crash risk and possible risk factors. More-over, for the latter case in which a risk factor is related to bothtypes of risk, the analysis helps to give an overall picture of the risktradeoff between accident occurrence and injury severity for thisfactor.
Acknowledgments
The work described in this paper was partially supported bya Research Postgraduate Studentship, an Outstanding ResearcherAward, the Engineering Postdoctoral Fellowship Programme of the
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niversity of Hong Kong, a grant from the Research Grants Councilf the Hong Kong Special Administrative Region, China (Project No.KU7176/07E), and a grant from National Natural Science Founda-
ion of China (Grant No. 61021063). We would also like to thankoncord Pacific Satellite Technologies Limited and Motion Poweredia Limited for providing the GPS taxi data.
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