If

CLP

a

ARR2A

KCRFC

1

apsair

tr

abFpflchiid

0h

Accident Analysis and Prevention 63 (2014) 121– 132

Contents lists available at ScienceDirect

Accident Analysis and Prevention

journa l h om epage: www.elsev ier .com/ locate /aap

nvestigating the relationship between run-off-the-road crashrequency and traffic flow through different functional forms

arlos Roque ∗, João Lourenc o Cardosoaboratório Nacional de Engenharia Civil, Departamento de Transportes, Núcleo de Planeamento, Tráfego e Seguranc a, Av do Brasil 101, 1700-066 Lisboa,ortugal

r t i c l e i n f o

rticle history:eceived 5 July 2013eceived in revised form4 September 2013

a b s t r a c t

Crash prediction models play a major role in highway safety analysis. These models can be used forvarious purposes, such as predicting the number of road crashes or establishing relationships betweenthese crashes and different covariates. However, the appropriate choice for the functional form of thesemodels is generally not discussed in research literature on road safety. In case of run-off-the-road crashes,

ccepted 31 October 2013

eywords:rash prediction modelsun-off-the-road crashunctional formonfidence intervals

empirical evidence and logical considerations lead to conclusion that the relationship between expectedfrequency and traffic flow is not monotonously increasing.

© 2013 Elsevier Ltd. All rights reserved.

. Introduction

Crash prediction models play a major role in highway safetynalysis. These models can be used for various purposes, such asredicting the number of road crashes or establishing relation-hips between these crashes and different covariates. However, theppropriate choice for the functional form of these models, relat-ng crash frequency and traffic flow, is generally not present fromesearch literature on road safety.

As suggested by Lord et al. (2005a), it may be preferable to begino develop models that consider the fundamental crash processather than making efforts for the most-fitted model.

In case of run-off-the-road (ROR) crashes, empirical evidencend logical considerations lead to conclusion that the relationshipetween expected frequency and traffic flow is not a linear one.or low traffic flows one may expect the number of ROR crasheser unit of time to be proportional to traffic flow. But, as trafficows increase it becomes more and more difficult not to hit anotherar. Hence, for ROR crashes, proportionality cannot be expected toold at high flows. In traffic congestion, ROR crashes are, in fact,

mpossible, except for very low skid resistance conditions, such as

ce and snow covered pavements. This is a reflex of the fact thatrivers behave differently in sparse, heavy or congested traffic and

∗ Corresponding author. Tel.: +351 218 443 970; fax: +351 218 443 029.E-mail addresses: croque@lnec.pt, carlos.alm.roque@gmail.com (C. Roque).

001-4575/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.aap.2013.10.034

that ROR crash frequency depends on the actual state of the system(flow, speed and density) in time and space (Hauer, 1997).

Two main approaches have been followed to model the relation-ship between roadway and roadside characteristics and ROR crashrisk (Pardillo-Mayora et al., 2010).

One method, usually referred to as encroachment-based, uses aset of conditional probabilities of the sequence of events that lead toa ROR crash following the encroachment of an errant vehicle on theroadside (Mak, 1995; Mak et al., 1998; Ray et al., 2012). The mainobstacle in the development of this type of models is the short-age of encroachment data. Data collected in the 1960s and 70s inNorth America (Hutchinson and Kennedy, 1966; Cooper, 1980) arestill the main source of information on these manoeuvres (Pardillo-Mayora et al., 2010). In the development of the recently updatedversion of the Roadside Safety Analysis Programme (RSAP) – a com-puter programme for performing cost benefit analyses on roadsidedesign developed under NCHRP Project 22–27 – the Cooper datawas re-analyzed to attempt to resolve some of its longstandingproblems (Ray et al., 2012).

A second approach is the development of generalized linearregression models fitted to cross-sectional data, to estimate RORcrash frequencies using exposure and relevant highway and road-side variables as covariates. The frequency of crashes in a givenhighway segment is treated as a random variable which takes dis-

crete integer non-negative values distributed following a Poissondistribution. A generalization of the Poisson form that allows thevariance of the model to be over-dispersed results in the NegativeBinomial (NB) model. Lee and Mannering (2002) and Geedipally

1 alysis a

aRu

�

((eTcfsxsma�co1aa

mtmicO(vtraTtmbswatgtsto

sfbtRtDrddActowNt

22 C. Roque, J.L. Cardoso / Accident An

nd Lord (2010) used Poisson and NB regression models to developOR crash prediction models. In both cases, crash prediction modelsse the following functional form:

i = ˇ0Q ˇ1i e

∑i

�ixi (1)

The mean number of the crashes per unit of time for segment i�i) is a function of traffic flow, Qi, and a set of risk factors, xi. In Eq.1) the effect of traffic flow on crashes is modelled in terms of anlasticity, which is a power, ˇ1, to which traffic volume is raised.he effects of various risk factors that influence the probability ofrashes, given exposure, is generally modelled as an exponentialunction, that is as e (the base of natural logarithms) raised to aum of the product of coefficients, � i, and values of the variables,i, denoting risk factors. In its most basic form, traffic flow is theole explanatory variable in crash prediction models. For road seg-ents, several crash prediction models also include segment length

s covariate which in some cases is assumed as an offset, withi = 1. Depending on the road network characteristics, there areases in which this simplification is not appropriate, both from theperational and purely statistical points of view (Mountain et al.,996). According to Reurings et al. (2005), the segment length, theccess density, the carriageway width and the shoulder width are

desirable minimum list of risk factors for road segments.Eq. (1) is the conventional functional form of crash prediction

odels (Reurings et al., 2005). However, as noted by Hauer (2004),he crash phenomenon does not necessarily have to follow a simple

onotonous, mathematical function. If the model functional forms not appropriate, the regression coefficients obtained are inac-urate, and in several cases the estimated values are ambiguous.ne way to mitigate this problem is to set the traffic flow interval

lower and upper bounds) for which the modelled equations arealid; an alternative way is to fit equations with a different func-ional form, with overall shapes that are in better agreement withoad operation characteristics. However, ADT is the average of thennual distribution of traffic volumes on selected road segments.herefore, it is just a moment statistic of a highly non-uniform dis-ribution, where seasonal, monthly, daily, hourly periodic trends

ay be detected. From traffic census, it is known that the ratiosetween daylight and nighttimes traffic or between winter andummertime traffic are not the same for all segments on a road net-ork. For that reason, it is still open to debate whether ADT is an

ppropriate macroscopic variable to solely represent exposure andhe underlying crash mechanisms directly related to traffic volume,iven the increasing availability of automatically collected datahat may be used to calculate complimentary traffic distributiontatistics. Nevertheless, one must acknowledge that the mentionedraffic time trends show, at least partially, a scale factor dependingn the ADT value.

Following a comprehensive and systematic bibliographicearch, few studies addressing the issue of crash prediction modelunctional form selection were identified. So far, little interest haseen dedicated to study alternative functional forms to expresshe relationship between specific crash types and traffic flow.eurings and Janssen (2007) compared models using the func-ional form expressed in Eq. (1) with models where Annual Averageaily Traffic (AADT) could be considered as a property of the car-

iageway under consideration and hence as a sort of continuousummy-variable. They concluded that not only these last modelsid not have the desired structure but also that adding the variableADT/1000 was indeed an improvement of the models for urbanarriageways. Kononov et al. (2011) related traffic flow parame-ers, such as speed and density, to the choice of the functional form

f crash prediction models. It compared models for urban free-ays developed with sigmoid and exponential functional forms.eural networks (NN) were used to explore the underlying rela-

ionship between accidents and other variables for urban freeway

nd Prevention 63 (2014) 121– 132

segments. The results were then compared with models calibratedby using these same data with generalized linear modelling andan NB error structure. The functional form generated through thetraining of NNs suggests that a sigmoid may be a reasonable approx-imation of the operational characteristics of crash occurrence onurban freeways.

The objective of this paper is to document the application of NBgeneralized linear models with different functional forms in theanalysis of ROR crash data, exploring the underlying relationshipsbetween ROR crashes and traffic flow for interurban road segments.

The study objective was accomplished using observed andsimulated datasets. The models were applied to single and dual car-riageway road datasets. Subsequently, the models were applied tosimulated datasets to show their general performance (Geedipallyet al., 2012).

2. Methodology

This section describes the probabilistic structure of the negativebinomial models, the functional form used for linking ROR crashesto covariates, the procedure employed for estimating the confi-dence intervals, and characteristics of the Monte Carlo simulationstudy.

2.1. Negative binomial models

In applying Poisson regression to crash frequency analysis, let yijbe the number of ROR crashes on highway element i during periodj. The Poisson model is (Washington et al., 2011):

p(yij) =exp(−�ij)�

yij

ij

yij!, (2)

where P(yij) is the probability of y crashes occurring on highwayelement i during time period j and �ij is the expected value of yij:

E(yij) = �ij = exp(ˇXij), (3)

for a roadway section i in time period j, is the vector of param-eters to be estimated and Xij is a vector of explanatory variablesdescribing roadway section geometric and environmental charac-teristics, as well as other relevant features such as traffic, that mayaffect crash frequency.

A feature of the Poisson distribution refers to the equalitybetween the counts expected value and its variance. However, itis not always possible to assume that �ij is constant. On the onehand, the decreasing trend in time of accident risk, as observedin many countries, weakens the validity of the hypothesis of con-stancy in time of probability of occurrence. On the other hand, thereare unknown factors that may contribute to crash occurrence aswell as factors which, although known, are quantified with mea-surement errors, in both cases justifying that the individual riskson each entity in a homogeneous group of entities are not identi-cal. Thus, the ratio of the variance to the expected value differs fromone, i.e., overdispersion or subdispersion are observed (Roque andCardoso, 2013).

The negative binomial model is an extension of the Poissonregression model that accommodates data overdispersion. Thenegative binomial model is derived by rewriting Poisson param-eter for each observation i at a given time interval j as (Washington

et al., 2011):

�ij = exp(ˇXij + εij) (4)

C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132 123

Nu

mb

er

of

RO

R c

ra

sh

es

Nu

mb

er

of

RO

R c

ra

sh

es

wat

V

2

acwrr

c(Amrcpc

m

�

wocL

arf

bm

rtdi

�

p

ADT

Nu

mb

er

of

RO

R c

rash

es

ADT

Nu

mb

er

of

RO

R c

rash

es

ADTADT

Fig. 1. The curve of the function �i = ˇ0ADTˇ1i

Lˇ2i

when ˇ1 is below 1.

here exp (εij) is a random error term that follows a Gamma prob-bility distribution with mean 1 and variance ˛. This term allowshe variance to differ from the mean as below:

AR[yij

]= E

[yij

] [1 + ˛E

[yij

]]= E

[yij

]+ ˛E

[yij

]2(5)

.2. Functional forms

Two functional forms were considered. In both cases, the Aver-ge Daily Traffic (ADT) and the segment length (L) are the covariatesonsidered. Based on crash data considered, the segment lengthas assumed to have a non-linear relation with the crash occur-

ence. Thus, the segment length is considered to be a covariateather than considered as an offset.

The first statistical model considered in this study, hereinafteralled Functional Form 1 (FF1), is similar to the one referred in Eq.1), except that there are only two independent variables, namelyverage Daily Traffic (ADT) and segment length (L). Although suchodel suffers from omitted variables bias (because many non-flow

elated factors are known to affect crash frequency, which are notonsidered), the empirical assessment carried out in this study stillrovides useful information for the development of improved RORrash prediction models.

The mean of the ROR crashes per reference time period for seg-ent i (�i) can be calculated by:

i = ˇ0ADTˇ1i Lˇ2

i (6)

here, ADTi = Average Daily Traffic of segment i, Li = length (in km)f segment i, ˇ0 = intercept (to be estimated), ˇ1 = coefficient asso-iated with ADT (to be estimated), ˇ2 = coefficient associated with

(to be estimated).It is usually assumed that the total number of crashes increases

t a decreasing rate as the traffic flow increases (see Fig. 1). Thiselationship is characterized in predictive models by a coefficientor the traffic flow parameter (ˇ1) below 1.

In the second statistical model considered, the relationshipetween the number of ROR crashes, �i, and ADT, is described by aodification of the Ricker (1954) model.The Ricker (1954) model was initially used in ecological

esearch, where it is important to estimate the survival probabili-ies of plants and animals and where population densities usuallyo not show an indefinitely increasing trend but show local maxima

n the explanatory variable space.Applied to crash prediction, the Ricker model is expressed by:

i = ˇ0ADTie−ˇ1ADTi (7)

Graphically this is a curve that passes through the origin thatresents a maximum and decreases asymptotically to zero. The

Fig. 2. The curve of the function �i = ˇ0ADTˇ1i

eˇ2ADT

10,000 Lˇ3i

when ˇ1 is larger than 1and ˇ2 is negative.

Ricker model is not monotonously increasing. Consequently thereare two values of traffic flow corresponding to each value of thenumber of crashes between zero and the maximum.

The modified Ricker model in the second functional form – here-inafter referred to as Functional Form 2 (FF2) – is calculated byadding to the referred model a parameter, ˇ1, and considering thesegment length (L) as a covariate:

�i = ˇ0ADTˇ1i eˇ2

ADT10,000 Lˇ3

i (8)

where, �i = Mean number of ROR crashes in segment i,ADTi = Average Daily Traffic of segment i, Li = length (in km) ofsegment i, ˇ0 = intercept (to be estimated), ˇ1 = coefficient asso-ciated with ADT (to be estimated), ˇ2 = coefficient also associatedwith ADT (to be estimated), ˇ3 = coefficient associated with L (tobe estimated).

In this case ADT/10,000 is used to avoid possible overflow duringthe estimation process. In accordance with empirical evidence andlogical considerations described above, ˇ1 should be larger than 1and ˇ2 should be negative (see Fig. 2).

With this functional form, plotting the number of ROR crashesagainst ADT, for a given road segment, gives an inverted “U-Shaped”curve. The ROR crash frequency increases with ADT up to a maxi-mum; at ADT levels above the one corresponding to this maximumnumber of ROR crashes, the number of ROR crashes will decreasewith increasing ADT.

2.3. Goodness-of-fit statistics

As pointed out by Mitra and Washington (2007) and Lord andPark (2008) it is recommended to use more than one goodness-of-fit measure for evaluating models. In this analysis, differentmethods were used for evaluating the goodness-of-fit and predic-tive performance of the fitted models, as described bellow:

(1) Akaike Information Criterion (AIC), which measures the good-ness of fit of an estimated statistical model and is defined as(Akaike, 1974):

AIC = −2 log L + 2p (9)

where L is the maximized value of the likelihood function for theestimated model, and p is the number of parameters in the sta-tistical model. The AIC methodology attempts to find the modelthat best explains the data with a minimum of free parametersand thus it penalizes models with a large number of parame-

ters. The model with the lowest AIC is considered to be the bestmodel among all available models.

(2) Sum of model deviances (G2), being the model with the lowestG2, the one with the best fit. If G2 is equal to zero, the model has

1 alysis a

(

(

fpodcc

dncifeArrowwotbb

24 C. Roque, J.L. Cardoso / Accident An

a perfect fit (Washington et al., 2011). This statistic is given as:

G2 = 2n∑

i=1

yi ln

(yi

�i

)(10)

where yi is the observed number and �i is the expected numberof crashes.

3) Mean Absolute Deviation (MAD), which gives a measure of theaverage magnitude of variability of prediction (Washingtonet al., 2011). Smaller values are preferred to larger values. Itis computed using the following equation:

MAD =

n∑i=1

∣∣εi

∣∣n

(11)

where εi corresponds to the difference between the expectedand the observed number of crashes.

4) The proportion of systematic variation explained by the modelmay be quantified using the Elvik’s index. There is system-atic variation in the number of crashes whenever the varianceexceeds the mean, meaning that overdispersion exists. Theamount of overdispersion can be described by:

Var(x) = � × (1 + � �) (12)

where x is the observed number of accidents; � is the averagenumber of crashes; and is � the overdispersion parameter. Eq.(11) can also be written as:

� = Var(x)/y − 1�

(13)

Elvik’s index is calculated by comparing the over-dispersionparameter in the original dataset (�0) with the over-dispersionparameter of a fitted model (�m) (Elvik, 2007). The proportionof systematic variation explained by the model is calculated bythe ratio between the two over-dispersion parameters previ-ously determined, being Elvik’s indexes closer to one associatedwith better-fitted models.

The methods described above evaluate the overall fit and per-ormance of the models. However, it is important that the modelredicts well fitted values for all practical values of a variable, notnly a global good fit. The CURE (CUmulative REsiduals) method, asescribed in Hauer and Bamfo (1997), is particularly useful in thisircumstance. CURE is a graphical method that was also used foromparing the two models performance.

This method consists of plotting the CURE for each indepen-ent variable of interest (usually ADT). The difference between theumber of observed and predicted crashes for an entity (in thisase, road section) and time period is called a ‘residual’. By exam-ning residuals one should be able to learn how well the functionalorm has been chosen and how well the model fits the data. To gen-rate a CURE plot, road sections are sorted in ascending order byDT. Because of the random nature of crash counts, the CURE lineepresents a so-called random walk. For a model that fits well in allanges of the variable, the CURE plot should move up and down andscillates around zero. If the CURE value progressively increasesithin a range of values of the independent variable, it means that,ithin that range, the model predicts fewer crashes than have been

bserved. In opposition, a decreasing CURE line indicates that, inhat range, fewer crashes have been observed than are predictedy the model. If the model fit is poor, it can possibly be improvedy a suitable modification of the functional form (Hauer, 2004)

nd Prevention 63 (2014) 121– 132

2.4. Estimation of confidence intervals

Once goodness of fit is established for each model, it is of inter-est to provide confidence intervals and prediction intervals formodels parameters and covariates, respectively. Wood (2005) hasproposed a method for estimating the confidence intervals for themean response (�), for the gamma mean (m), and the predictedresponse (y) at a new segment having similar characteristics as thesegments used in the dataset from which the model was developed.

The following table gives the equations for calculating the 95%confidence and prediction intervals for NB models. In this table, �iis the logarithm of the estimated mean response �i for segment i,while � is the inverse dispersion parameter estimated during thefitting process.

The primary purpose of this paper is to evaluate whether or notthere will be any significant difference in modelling ROR crasheswith different functional forms. To examine this difference, con-fidence intervals for the Poisson mean, the gamma mean and thepredicted response were calculated for FF1 and FF2, in addition tothe methods used for evaluating the goodness-of-fit and predictiveperformance of the models.

2.5. Modelling

The modelling procedure was accomplished using a 3-stage pro-cess:

(1) In the initial step, the two models were estimated using R(version 2.15.0) (R Development Core Team, 2011). “MASS”(Venables and Ripley, 2002) R package was used.

(2) The confidence interval for Poisson mean �i, gamma meanm, and the prediction interval y at a new road section withthe same traffic flow characteristics was then calculated usingTable 1 equations for FF1 and FF2 models. Var(�i) is calculatedusing XI−1XT where I−1 represents the variance–covariancematrices and X is the matrix containing the observed val-ues. The variance–covariance matrices were calculated by R (RDevelopment Core Team, 2011).

(3) The differences between confidence interval widths calculatedby the two models were then estimated.

2.6. Simulated crash data

A Monte Carlo simulation study was used to verify the resultsproduced using the empirical data. Using simulation, we can assessthe general performance of the proposed models (Geedipally et al.,2012).

The simulation design was accomplished using the simulationprotocol used by Geedipally and Lord (2010):

(1) The traffic flow and segment length from the dual carriagewaydataset are used as the independent variables.

(2) Generate the “true” mean for ROR injury crashes at each seg-ment using the two functional forms considered (FF1 and FF2).The parameters are defined prior to the simulation. Theseparameters are directly taken from the dual carriageway NBmodels in Tables 4 and 5 for ROR injury crashes on two-lane uni-directional sections of the Portuguese motorway net. Similarly,using the defined parameters, generate the “true” confidenceintervals with the dual carriageway NB models.

(3) Simulate ROR injury crash counts for each site and for eachfunctional form. Following the simulation, the parameters and

confidence intervals are re-estimated by the two models (asdescribed above in the modelling procedure).

(4) Steps (2) and (3) are repeated for 30 times (as in Lord, 2006) toobtain statistically reliable estimates. The average values for the

C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132 125

Table 195% confidence and prediction intervals (Wood, 2005).

Parameter Intervals

�

[�i

e1.96√

Var(�i ), �ie1.96

√Var(�i)

]

m

[max

{0, �i − 1.96

√�2

iVar(�i) + �2

iVar(�i )+�2

i�

}, �i + 1.96

√�2

iVar(�i) + �2

iVar(�i )+�2

i�

][ [ √

+ �i

]]

3

cmsipbifiss

crsPn(btscsva

tfcotlAck

4

b

TDr

Table 3Descriptive statistics of explanatory variables and crash data for unidirectional dualcarriageway road segments.

Variable Minimum Maximum Mean (Std. dev.) Total

Segment length (km) 0.32 33.28 6.56 (5.88) 3996.00ADT (vehicles/day) 1056 78824 10689.61 (9901.45) –ROR injury crashes 0 45 5.00 (5.79) 3047

Table 4Estimates of FF1 model coefficients.

Type of carriageway ln(ˇ0) ˇ1 ˇ2 ˛

Single −4.5555 0.5413 0.9261 2.6060Dual −6.4932 0.7246 0.8745 2.2791

Table 5Estimates of FF2 model coefficients.

Type of carriageway ln(ˇ0) ˇ1 ˇ2 ˇ3 ˛

model. This table shows that models with FF2 are found to provide

y 0, �i +√

19 �2iVar(�i) + �2

iVar(�i )+�2

i�

parameters and confidence intervals are finally used. For eachfunctional form, the average confidence intervals obtained arethen compared with the true intervals.

. Data description

This section describes the characteristics of the single and dualarriageway datasets. In both cases, data characteristics and sum-ary statistics for the ROR injury crash data are presented. Only

ingle-vehicle ROR crashes involving roadside features were usedn this study. Roadside features include impact attenuators, bridgearapet ends, bridge rails, guardrail fences, guardrail ends, medianarriers, highway traffic sign posts, overhead sign supports, util-

ty poles, culverts, curbs, ditches, embankments, trees, and otherxed objects. Multi-vehicle crashes were excluded: head-on andideswipe opposite crashes are not included in ROR crashes oningle carriageway roads.

The first dataset contained ROR injury crash and traffic dataollected in a four-year period (2002–2005) at 608 rural single car-iageway segments in Portugal. This dataset comprises mostly ruralingle carriageway segments situated in various regions acrossortugal. The general speed limit on these segments is 90 km/h;evertheless, on some short sections lower speed limits are posteddown to 60 km/in some cases). Access to trunk roads is only possi-le through grade separated interchanges; on other road categorieshere are signalized or stop-controlled at grade intersections. Crossection characteristics are not uniform, depending on the roadategory: carriageway widths vary between 6 and 9 metres; andhoulders of up to 2.5 m may be paved or non-paved. Table 2 pro-ides relevant descriptive statistics for key explanatory variablesnd the crash data for single carriageway data.

The second dataset contained run-off-the-road injury crasheshat occurred on two-lane unidirectional sections of the Portuguesereeway network in a four-year period (2007–2010). This datasetomprises dual carriageway motorway segments situated in vari-us regions across Portugal. All segments have full access control,wo lanes per carriageway and paved shoulders (with widths ofess than 2.5 m and 4.0 m for left and right shoulders, respectively).ccess to and from the motorway is only possible through inter-hange ramps. Table 3 provides relevant descriptive statistics forey explanatory variables and the crash data for dual carriageways.

. Modelling results

This section describes the results of the comparison analysisetween the two functional forms crash models using empirical

able 2escriptive statistics of independent variables and crash data for single carriageway

oad segments.

Variable Minimum Maximum Mean (Std. dev.) Total

Segment length (km) 0.17 62.67 13.47 (8.70) 8189.21ADT (vehicles/day) 283 47900 6927.82 (6700.03) –ROR injury crashes 0 129 11.07(11.54) 6733

Single −6.4685 0.8112 −0.4537 0.9046 2.7362Dual −9.9360 1.1612 −0.4096 0.8472 2.4351

data as well as the output of the Monte Carlo simulation study.The confidence intervals were calculated for injury crashes for thesingle and dual carriageway empirical data and for the simulationstudy.

4.1. Empirical data

Table 4 contains the parameter estimates of FF1 models withtheir associated standard errors for single and dual carriagewayroad sections. Likewise, Table 5 provides the same parameter esti-mates of FF2 models.

With FF1, the coefficient (ˇ1) shows that the ROR injury crashesincrease at a decreasing rate as traffic flow increases, whereas withFF2, at first ROR injury crashes increase with ADT, until reach-ing a maximum and thereafter decreases with a diminishing rate.The dispersion parameter (˛) reveals that the ROR injury crasheswhen modelled with FF2 are slightly more dispersed than the samecrashes modelled with FF1.

Table 6 gives the goodness-of-fit statistics for each type of

a better statistical fit than models with FF1.Plots of the models predictions and CURE plots are provided

in Figs. 3 and 4 for single and dual carriageways. In this example,

Table 6Goodness-of-fit statistics.

Type ofcarriageway

Functionalform

AIC G2 MAD Elvik’s index

Single FF1 3799.4 3028.415 5.498 0.681FF2 3776.9 2923.452 5.406 0.696

Dual FF1 2943.0 1732.323 2.661 0.572FF2 2923.4 1674.070 2.625 0.600

126 C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132

ry cra

amget

Fetwmd

tftdca

Tmgrb

Fig. 3. Comparison of crash prediction models for ROR inju

1 km section length (between interchanges) was considered inodel prediction plots; overall, different values for segment length

enerate shapes similar to the ones presented in this document. Inach figure two graphs (on the left sides) correspond to the FF1; andhe two graphs on the right sides correspond to the FF2 models.

The CURE plots for the crash prediction models, generated withF2, consistently showed a model fit superior to that of the FF1 mod-ls. The model with FF2 was significantly less biased throughouthe entire range of ADT; it showed a moderately amplified randomalk without excursions outside of the 2� limits. In contrast to theodel with FF2, FF1 models showed much more bias and frequently

eparted outside of the 2� boundaries.For both the single and dual carriageway road segments data,

he models with FF2 provided better estimates of expected crashrequencies at different levels of ADT in both data sets. It is wortho note that the influence of access control in crash frequency pre-iction models for single carriageways was also analyzed. In bothases the modified Ricker model shows better quality fit, similarlys with the aggregate models presented above.

Fig. 5 shows the confidence intervals for the Poisson mean ‘�’.his figure illustrates that the width of the intervals for Poisson

ean ‘�’ for ROR injury crashes on 1 kilometre segments of sin-

le and dual carriageways (i.e., the distance between the meanesponse and the upper confidence interval boundary) predictedy the FF1 models is slightly wider than the width predicted by

shes on single carriageways with (left) FF1 and (right) FF2.

FF2 models, for flows more than 20,000 vehicles/day and 40,000vehicles/day, for single and dual carriageways, respectively.

Fig. 6 shows the confidence intervals for the gamma mean ‘m’and for the predictive response ‘y’.

The figure illustrates that the confidence interval for the gammamean ‘m’ for ROR injury crashes in single carriageway road seg-ments predicted by FF2 model is clearly narrower than the FF1model for flows greater than 15,000 vehicles/day. For dual carriage-way road segments, the interval predicted with the FF1 model ismuch wider than the one predicted with the FF2 model for flowsabove 30,000 vehicles/day.

Fig. 6 also shows the confidence intervals for the predictiveresponse ‘y’ for FF1 and FF2 models. The figure illustrates that pre-dictions with FF2 models have smaller confidence intervals thanthose models with FF1 for more than half of the time for single car-riageways and, at least, two thirds of the time for dual carriageways.According to Lord (2008), the predicted response and its associatedvariance are particularly important values to be computed, sincein most cases, analysts will apply the model to an observation (orsite) that was not used for developing the statistical models.

In sum, the comparison analysis showed that using two FF1

models to predict ROR injury crashes, for single and dual carriage-way road segments, most often provides larger confidence intervalsthan using models with FF2, especially for the gamma mean valuesand predictive response.

C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132 127

ury cra

4

twt

m(

idFrvv

ddmeA(upAc

Fig. 4. Comparison of crash prediction models for ROR inj

.2. Simulation

Fig. 7 shows the comparison between the true crash means andhe crash means predicted by FF1 and FF2 models. Values predictedith FF2 models are slightly closer to the true crash mean than

hose with FF1 models.Fig. 8 shows that 95% confidence intervals predicted with FF2

odel predicts are slightly closer to true intervals for Poisson mean�), for ADT higher than 40,000 vehicles/day.

Fig. 9 illustrates the comparative analysis of the 95% confidencentervals for upper boundaries for the Gamma mean (m) and pre-ictive response (y). Intervals predicted with the Gamma meanF2 model are closer to true intervals. Nevertheless, for predictiveesponses, FF1 model predicted intervals are closer to true inter-als than those from the FF2 model, for flows in excess of 40,000ehicles/day.

The results of the analysis with the empirical and simulatedata show that FF2 model has higher variance of mean and pre-icted response. However, the CURE plots for the crash predictionodels consistently showed the models with FF2 provided better

stimates of expected crash frequencies at different levels of ADT.ccordingly, the FF2 models confidence intervals will be narrower

with a very significant difference in case of the gamma mean val-

es and predictive response). Although FF1 confidence intervals forredictive response (y) are closer to the “true intervals” for higherDT values, the crash means predicted by a model with FF2 areloser to the true values and confidence intervals for Gamma mean

shes on dual carriageways with (left) FF1 and (right) FF2.

(m) with this functional form are also closer to the “true intervals”.Thus, this study supports the idea that ROR injury crashes shouldbe modelled with a functional form similar to FF2.

5. Discussion

The results of the analysis presented above raise two importantissues that deserve further discussion.

First, crash prediction models traditionally relate crash occur-rence to exposure measures such as AADT, or ADT. Exposure maybe defined as the number of traffic events in which there is areasonable chain of event that could lead to a collision betweenroad users or a collision with dangerous obstacles. Currently used,volume-based exposure measures, suffer many limitations. In caseof aggregate measurements of traffic volume, such as ADT orAADT, temporal variations in demand (seasonal, monthly, daily andhourly) are not considered. Actually, a road with intense traffic flowduring peak periods might have a different ROR crash potential thanother with the same ADT but with traffic flow more evenly spreadout during daytime. Also, the distribution of weekday to weekendtraffic volume may vary from one location to another or from day-time to night-time. To the extent that the actual hourly volume isan important factor in explaining the number of crashes, the hourly

volume can accurately account for this effect in a way that ADT orother aggregate exposure measures cannot (Qin et al., 2006)

Furthermore, the fact that not all vehicles are interactingunsafely is not explicitly taken into account (El-Basyouny and

128 C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132

mean

Sboe

addaltAaoc

wdioukttTawa

Fig. 5. 95-Percentile confidence intervals for the Poisson

ayed, 2013). In fact, if there are varying degrees of interactionetween vehicles within a traffic stream, then, for example, theccurrence of crashes involving a single vehicle may have a differ-nt likelihood from those involving multiple vehicles.

These model limitations, due primarily to the cost of collectingnd combining more detailed traffic and crash data using tra-itional systems, have been an obstacle to the development ofetailed statistical models of the relationship between traffic flownd ROR crash frequency. Recent developments in traffic data col-ection and management systems allow for the development andest of other traffic distribution statistics that may complementDT. Resorting to very small time periods, however, may gener-te new data consistency problems, as high accuracy in accidentccurrence time reporting and full synchronization of traffic dataollection stations and accident reporting teams are required.

Mensah and Hauer (1998) investigated what problems arisehen data is in a form such that the causal link between the depen-ent variable (the effect) and the independent variables (causes) is

ndirect in the sense that the effect is observed over a “long” periodf time during which the cause has assumed widely different val-es, but for which period only the average value of the cause isnown. They concluded that by using additional information abouthe distribution of the hourly flows on such road segments overhe year, the bias due to traffic flow averaging may be reduced.

hey also analyzed the so called function averaging, a problem thatrises when a single crash prediction model is estimated for data tohich two or more distinct models apply. They recommended to

void this problem by, if possible, attempting to fit separate models

(�) for a 1 km segment of single and dual carriageways.

to say, daytime and nighttime conditions, instead of a single model.Also Garbarino et al. (2001) highlighted the importance of daytimeand nighttime conditions by demonstrating that the distribution ofroad crashes may be correlated with the sleep propensity curve.

As referred by Kononov et al. (2012) it would also be extremelyvaluable to know how safety varies with the volume-to-capacity(V/C) ratio and what V/C ratio provide the lower accident rates,in particular to more thoroughly assess the safety implications ofhighway improvements designed to increase capacity. In a studyon freeways by Lord et al. (2005b) it was found that as den-sity and V/C ratio increase, the number of single-vehicle crashesdecreases.

Garber and Subramanyan (2001) related crashes to lane occu-pancy and concluded that peak crash rates do not occur duringpeak flows. In the particular case of ROR crashes this conclusion isintuitive. Kononov et al. (2011) observed that on uncongested free-ways the number of crashes increases moderately with an increasein traffic; though, once some critical traffic density is reached, thenumber of crashes begins to increase at a much higher rate with anincrease in traffic.

The second issue is related with the development of crashprediction models with a functional form that consider the funda-mental relation between the ROR crash process and traffic flow.Miaou and Lord (2003) pointed out the need to develop new

approaches for modelling of vehicle crashes on roadway net-works that are based on the logic (e.g., reason, consistency, andcoherency), flexibility, extensibility, and interpretability of thefunctional form.

C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132 129

the p

voabrma

Fig. 6. 95-Percentile confidence intervals for the Gamma mean (m) and for

It is very important to look at the relationship between theariables and whether the distribution or model is logicallyr theoretically sound, the so-called “goodness-of logic” (Miaound Lord, 2003). FF2 follows the “goodness-of logic”, starting

y the logic of “no traffic flows, no crashes” and allowing theelationship between ROR crashes and traffic flows to be non-onotonic. In fact, the number of ROR crashes increases, peaks

nd decreases as ADT increases, which implies that, on one

Fig. 7. Comparison of crash means for a 1 km segment o

redictive response (y) for a 1 km segment of single and dual carriageways.

hand, similar ROR crash frequencies may occur under very dif-ferent traffic flow conditions and, on the other, that in somecases the number of ROR crashes may increase when ADT isreduced.

As shown, there is merit in improving the functional form of ADTbased crash frequency prediction models, even though ADT may beregarded as too simple an explanatory variable to represent trafficdistribution time series.

f dual carriageways with (left) FF1 and (right) FF2.

130 C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132

F kilomP

6

ocpdm

wPfgw

(

(

ig. 8. Comparison of 95-percentile confidence intervals for dual carriageway oneoisson mean (�).

. Summary and conclusions

This paper presents some evidence that the functional formf a ROR injury crash prediction models influences the quality ofrash predictions as a function of traffic flow. For that purpose, thisaper documented a research study that examined the potentialifferences between two different functional forms for the referredodels.To accomplish the comparison analysis, ROR injury crash data

ere collected on single and dual carriageway road sections inortugal. Then, models were estimated for the two functionalorms along with the confidence intervals for the Poisson mean (�),amma mean (m), and predictive response (y). A simulation studyas also conducted to complement the comparative analysis.

The following results were obtained from the analysis:

1) CURE plots of models with FF2 consistently showed better qual-ity model fit when compared with models with FF1.

2) There is a clear difference in the prediction of confidence inter-

vals for the gamma mean, and predictive response betweenmodels with FF2 and models with FF1. Globally, a model withFF1 provides wider confidence intervals than a model withFF2.

etre segments. (a) Lower boundary for Poisson mean (�). (b) Upper boundary for

(3) Besides narrower confidence intervals were observed withmodels with FF2, the crash means and the confidenceintervals for Gamma mean (m) predicted with FF2 are closerto the “true crash means” and “true intervals”, respectively.

(4) This paper supports the idea that modelling run-off-the-roadcrashes with a modified Ricker model functional form thatshapes the curve between crash frequency and traffic flow ina concave way, shows better quality model fit than a func-tional form with the assumption that model component fortraffic flow continuously increases with traffic flow. This func-tional form offers a reasonably good estimate of the relationshipbetween safety and exposure in a ROR crash.

As future work, there are some interesting topics to focus onand explore, namely the collection of hourly traffic volumes andROR crash data in combination, allowing for the development ofcrash prediction models to assess how ROR crashes varies with thevolume-to-capacity ratio. Also, there is merit in carrying out simi-lar analyses using data collected on other types of crashes, such asmulti-vehicle crashes, and other functional forms. Finally, to iden-

tify and test readily available AADT parameters (supplementary tothe mean ADT) that may be used to represent the operational mech-anisms that explain improved functional forms for the relationshipbetween traffic and crash frequencies.

C. Roque, J.L. Cardoso / Accident Analysis and Prevention 63 (2014) 121– 132 131

F l carrp

A

SFTfo

A

tTp

R

A

C

E

ig. 9. Comparison of 95-percentile confidence intervals for a 1 km section of duaredictive response (y).

cknowledgements

The authors gratefully acknowledge the scholarshipFRH/BD/82228/2011 by the Portuguese Science and Technologyoundation Agency (FCT-Fundac ão para a Ciência e a Tecnologia).he authors express their gratitude to two anonymous reviewersor their valuable remarks that helped to improve the final versionf this paper.

ppendix A. Supplementary data

Supplementary data associated with this article can be found, inhe online version, at http://dx.doi.org/10.1016/j.aap.2013.10.034.hese data include GoogleMapsTM pinpoints from segment exam-les of the datasets described in this article.

eferences

kaike, 1974. A new look at the statistical model identification. IEEE Transactionson Automatic Control 19, 716–723.

ooper, P., 1980. Analysis of Roadside Encroachments — Single Vehicle Run-offAccident Data Analysis for Five Provinces. British Columbia Research Council,Vancouver, British Columbia, Canada.

l-Basyouny, K., Sayed, T., 2013. Safety performance functions using traffic conflicts.Safety Science 51, 160–164.

iageways. (a) Upper boundary for the Gamma mean (m). (b) Upper boundary for

Elvik, R., 2007. State-of-the-art Approaches to Road Accident Black spot Manage-ment and Safety Analysis of Road Networks. Institute of Transport Economics,Norway, TØI report 883/2007.

Garbarino, S., Nobili, L., Beelke, M., Carli, F.D., Ferrillo, F., 2001. The contributing roleof sleepiness in highway vehicle accidents. Sleep 24 (2), 203–206.

Garber, N.J., Subramanyan, S., 2001. Incorporating crash risk in selecting congestion-mitigation strategies: Hampton roads area (Virginia) case study. TransportationResearch Record 1746, 1–5.

Geedipally, S.R., Lord, D., 2010. Investigating the effect of modeling single-vehicleand multi-vehicle crashes separately on confidence intervals of Poisson-gammamodels. Accident Analysis & Prevention 42, 1273–1282.

Geedipally, S.R., Lord, D., Dhavala, S.S., 2012. The negative binomial-Lindley gener-alized linear model: characteristics and application using crash data. AccidentAnalysis & Prevention 45, 258–265.

Hauer, E., 1997. Observational Before – After Studies in Road Safety. Pergamon,Oxford, UK, ISBN 0-08-0430538.

Hauer, E., Bamfo, J., 1997. Two tools for finding what function links the dependentvariable to the explanatory variables. In: Proceedings of International Coopera-tion on Theories and Concepts in Traffic Conference, Lund, Sweden.

Hauer, E., 2004. Statistical Road Safety Modelling. Transportation Research Board ofthe National Academies, Washington, D.C, pp. 81–87, Transportation ResearchBoard No. 1897.

Hutchinson, J.W., Kennedy, T.W., 1966. Medians of Divided Highways – Frequencyand Nature of Vehicle Encroachments. University of Illinois, Engineering Exper-iment Station Bulletin 487.

Kononov, J., Lyon, C., Allery, B.K., 2011. Relation of flow, speed, and density of urban

freeways to functional form of a safety performance function. TransportationResearch Record 2236, 11–19.

Kononov, J., Hersey, S., Reeves, D., Allery, B.K., 2012. Relationship between free-way flow parameters and safety and its implications for hard shoulder running.Transportation Research Record 2280, 10–17.

1 alysis a

L

L

L

L

L

L

M

M

M

M

M

32 C. Roque, J.L. Cardoso / Accident An

ee, J., Mannering, F., 2002. Impact of roadside features on the frequency and sever-ity of run-off-roadway accidents: an empirical analysis. Accident Analysis &Prevention 34, 149–161.

ord, D., 2006. Modeling motor vehicle crashes using Poisson-gamma models: exam-ining the effects of low sample mean values and small sample size on theestimation of the fixed dispersion parameter. Accident Analysis & Prevention38 (4), 751–766.

ord, D., 2008. Methodology for estimating the variance and confidence intervals ofthe estimate of the product of baseline models and AMFs. Accident Analysis &Prevention 40 (3), 1013–1017.

ord, D., Washington, S., Ivan, J., 2005a. Poisson, poisson-gamma and zero-inflatedregression models of motor vehicle crashes: balancing statistical fit and theory.Accident Analysis & Prevention 37 (1), 35–46.

ord, D., Manar, A., Vizioli, A., 2005b. Modeling crash-flow-density and crash-flow-v/c ratio for rural and urban freeway segments. Accident Analysis & Prevention37 (1), 185–199.

ord, D., Park, P.Y-J., 2008. Investigating the effects of the fixed and varying dis-persion parameters of Poisson-gamma models on empirical Bayes estimates.Accident Analysis & Prevention 40 (4), 1441–1457.

ak, K., 1995. Safety effects of roadway design decisions-roadside. TransportationResearch Record 1512, 16–21.

ak, K.K., Sicking, D.L., Zimmerman, K., 1998. Roadside safety analysis program:a cost-effectiveness analysis procedure. Transportation Research Record 1647,67–74.

ensah, A., Hauer, E., 1998. Two problems of averaging arising in the estimationof the relationship between accidents and traffic flow. Transportation ResearchRecord 1635, 37–43.

iaou, S.P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersec-tions: dispersion parameter, functional form, and Bayes versus empirical Bayes.Transportation Research Record 1840, 31–40.

itra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehiclecrash prediction models. Accident Analysis & Prevention 39 (3), 459–468.

nd Prevention 63 (2014) 121– 132

Mountain, L., Fawaz, B., Jarrett, D., 1996. Accident prediction models for roads withminor junctions. Accident Analysis & Prevention 28, 695–707.

Pardillo-Mayora, J.M., Domínguez-Lira, C.A., Jurado-Pina, R., 2010. Empirical cali-bration of a roadside hazardousness index for Spanish two-lane rural roads.Accident Analysis & Prevention 42, 2018–2023.

Qin, X., Ivan, J.N., Ravishanker, N., Liu, J., Tepas, D., 2006. Bayesian estimation ofhourly exposure functions by crash type and time of day. Accident Analysis &Prevention 38, 1071–1080.

R Development Core Team, 2011. R: A Language and Environment for StatisticalComputing. R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0 http://www.R-project.org/

Ray, M.H., Carrigan, C.E., Plaxico, C.A., Miaou, S.P., Olaf Johnson, T., 2012. NCHRP 22-27 Roadside Safety Analysis Program (RSAP) Update. RoadSafe LLC, Final report:RSAP. Version 3.0.0.

Reurings, M., Janssen, T., Eenink, R., Elvik, R., Cardoso, J., Stefan, C., 2005. Acci-dent Prediction Models and Road Safety Impact Assessment: A State-of-the-art.RIPCORD-ISEREST, First deliverable of WP2 of RIPCORD-ISEREST.

Reurings, M., Janssen, T., 2007. Accident Prediction Models for Urban and Rural Car-riageways Based on Data from the Hague Region Haaglanden. SWOV, R-2006-14SWOV Project Number 39.150.

Ricker, W.E., 1954. Stock and recruitment. Journal of the Fisheries Research Boardof Canada 11, 559–623.

Roque, C., Cardoso, J.L., 2013. Observations on the relationship between Europeanstandards for safety barrier impact severity and the degree of injury sustained.IATSS Research 37 (1), 21–29.

Venables, W.N., Ripley, B.D., 2002. Modern Applied Statistics with S, 4th ed. Springer-Verlag, Nova Iorque.

Washington, S., Karlaftis, M., Mannering, F.L., 2011. Statistical and EconometricMethods for Transportation Data Analysis, 2nd ed. Chapman and Hall/CRC, ISBN:978-1-4200-8285-2.0.

Wood, G.R., 2005. Confidence and prediction intervals for generalized linear accidentmodels. Accident Analysis & Prevention 37 (2), 267–273.