predicting the frequency of median barrier crashes on pennsylvania interstate highways

Accident Analysis and Prevention 38 (2006) 590–599

Predicting the frequency of median barrier crashes onPennsylvania interstate highways

Eric T. Donnell a,∗, John M. Mason Jr. b

a Department of Civil and Environmental Engineering, 212 Sackett Building, The Pennsylvania State University, University Park, PA 16802, USAb College of Engineering, 101 Hammond Building, The Pennsylvania State University, University Park, PA 16802, USA

Received 4 March 2005; received in revised form 26 May 2005; accepted 5 December 2005

Abstract

Median barrier warrant criteria were developed in the 1970s and generally remain unchanged today. Vehicle travel, including both traffic volumesand operating speeds, have increased over this same time period. Encroachments into the median, and subsequent collisions with vehicles travelingin the opposite travel lanes, result in high severity crashes. Median barrier is typically used to prevent cross-median crashes; median barrier selectionis based on median width and traffic volumes. Quantifiable information regarding the effects of median barrier installation and its placement on crashfrequency is limited. This paper investigates median barrier crash frequency on Pennsylvania Interstate highways, including separate models for theTTma©

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urnpike and all other Interstate-designated highways. Negative binomial regression models were used to develop predictive crash frequency tools.raffic volume, horizontal alignment, interchange ramp presence, and median barrier offset distance from the travel lanes were used to estimateedian barrier crash frequency. The analytical methodology developed in this research can be used, in concert with other prediction models, to

ssess the consequences of median barrier placement decisions.2005 Elsevier Ltd. All rights reserved.

eywords: Median barrier; Negative binomial; Clear zone

. Introduction

For more than 30 years, the clear zone concept has been auiding principle for roadside safety. On high-speed, dividedighways with full control of access, median barriers are oftenroposed to prevent cross-median crashes. The guidelines usedo evaluate the need for longitudinal median barrier are basedrincipally on the clear zone concept and do not explicitly con-ider the trade-off between cross-median and median barrierrashes. This paper characterizes median barrier crashes onennsylvania Interstate highways, including separate models for

he Turnpike (toll road) and all other Interstate-designated high-ays with median barrier (non-toll). Included is a discussion of

he databases used to develop median barrier crash predictionodels, characteristics of median barrier crashes, and predic-

ive tools to determine the frequency of median barrier crashess a function of various geometric and cross-section elements.he median barrier crash frequency models developed herein

∗ Corresponding author. Tel.: +1 814 863 7053; fax: +1 814 863 7304.

could be used in concert with cross-median crash frequencymodels, median-related crash severity models, and a benefit-costevaluation to evaluate the need for median barrier on Interstatehighways in Pennsylvania.

2. Background

The clear zone concept provides errant vehicles with atraversable recovery area free of fixed objects. Since the early1970s, traversable slopes free of fixed objects have been usedin highway design to minimize the consequences of run-off-the-road. This principle has guided the development of designpolicies, such as the American Association of State Highway andTransportation Officials’ Roadside Design Guide (2002). Earlyresearchers suggested a 9.1 m (30 ft) clear zone for high-speedroadways (Garner and Deen, 1973; Hutchinson and Kennedy,1967). Additionally, highway embankment slopes were thoughtto be traversable as long as slopes were flatter than 3:1 (3 horizon-tal:1 vertical). Recoverable slopes were considered 4:1 or flatter.These policies, along with estimates of average daily traffic vol-umes (ADT), formed the basis for median safety on Interstates

E-mail address: [email protected] (E.T. Donnell). and expressways. Currently, median barriers are typically con-

001-4575/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.oi:10.1016/j.aap.2005.12.011

E.T. Donnell, J.M. Mason Jr. / Accident Analysis and Prevention 38 (2006) 590–599 591

Fig. 1. Median barrier warrant criteria (PENNDOT, 1998).

sidered on access-controlled highways if the median width isless than 10 m (32.8 ft) and the ADT is greater than 30,000 vehi-cles per day (AASHTO, 2002). Fig. 1 shows the PennsylvaniaDepartment of Transportation’s (PennDOT) median barrier war-rant criteria (PennDOT, 1998)—they match the AASHTO policyguidelines.

Pennsylvania has found that a proportion of fatal crashes onhigh-speed, divided highways are the result of vehicles crossingthe median and colliding with vehicles traveling in the oppositedirection. For instance, cross-median collisions represent only0.6% of all non-ramp-related crashes on Interstates and express-ways; however, they do account for nearly 15% of the fatalitieson these same roadways (Mason et al., 2001). A possible expla-nation for this occurrence may be that Interstate highway travelin the United States has increased more than twofold since1980 (FHWA, 1996, 2004). Posted speed limits, and as a resultoperating speeds, have also increased dramatically on Interstatehighways in the last 20 years (Insurance Institute for HighwaySafety, 1994, 2003). Increased speeds on Interstate highwaysdecrease the time in which errant vehicles can cross a medianon a divided highway without a longitudinal barrier. Elvik (1995)used meta-analysis to conclude that median barriers on dividedhighways will increase the total number of crashes by about 30%.However, the analysis showed that crash severity, in particularfatalities, is reduced after installing median barrier on dividedhighways. A detailed analysis of the trade-off, using statisticalmqb

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et al., 2004). Roadway inventory and crash data from Penn-sylvania Interstates and expressways were used to estimate anegative binomial regression model of cross-median collision(CMC) frequency using only median width and average dailytraffic (ADT) as explanatory variables. The results suggest thatas median width increases the CMC crash frequency decreasesand as the one-way ADT increases, the expected CMC crash fre-quency increases (Donnell et al., 2002). Roadway inventory andcrash data from Washington State was used to estimate a neg-ative multinomial regression model of CMC crash frequency.The statistically significant explanatory variables included in themodel were ADT, median width, section length, and number ofhorizontal curves/km (Ulfarsson and Shankar, 2004).

A Bayesian approach was used to estimate CMC and medianbarrier crashes on Texas Interstates, freeways, and expressways(Miaou et al., 2004). The statistically significant explanatoryvariables included in the CMC model include median width,number of lanes, and posted speed limit. Interpretation ofthe CMC model indicates that the expected CMC crash fre-quency decreases as the median width increases; the expectedCMC crash frequency decreases as the number of travel lanesincreases. Similar models were developed for median barriercrash frequency. The statistically significant explanatory vari-ables were median width and posted speed. The expected medianbarrier crash frequency was shown to decrease as the medianwidth increases.

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ethods, between cross-median and median barrier crash fre-uency and severity is needed to recommend revised medianarrier warrant criteria.

Specific research to quantify the expected frequency ofedian barrier crashes is limited. Several recent studies have

eveloped prediction models of various median-related crashrequency as a function of geometric variables and traffic vol-me (Donnell et al., 2001; Ulfarsson and Shankar, 2004; Miaou

In addition to the median-related crash models described pre-iously, several other researchers have advocated the use ofither Poisson or negative binomial regression models to predictrash frequency as a function of geometric design variables. Fornstance, Miaou and Lum (1993) demonstrated how the Pois-on regression model can be used to evaluate the effects ofighway geometric design on truck accident involvement. Rec-gnizing the limitation of the Poisson model, Miaou (1994) laternvestigated the performance of the negative binomial and zero-nflated Poisson regression models when estimating truck acci-ent involvement on Interstate highways in Utah. Other authorsKniuman et al., 1993; Bauer and Harwwod, 1997; Milton and

annering, 1998) have also supported the use of Poisson or neg-tive binomial regression models when crash frequency is useds the dependent variable in regression models. Predictor vari-bles such as average daily traffic (ADT), horizontal curvature,ertical grades, vehicle operating speeds, cross-section width,nd environmental conditions have been considered in roadwayesign-safety models.

In summary, count regression models are appropriate toolso predict the expected frequency of crashes given a set of road-ay geometric design and traffic operational variables. The most

ommon examples include the Poisson and negative binomialodels. Few median barrier crash prediction models have been

eveloped, and none consider a range of geometric design vari-bles. Presented in this paper are predictive models of medianarrier crash frequency that include various geometric designariables. The results could be used in combination with CMCrash frequency models, crash severity models, and benefit-costvaluations to ultimately develop revised median barrier warrantriteria.

592 E.T. Donnell, J.M. Mason Jr. / Accident Analysis and Prevention 38 (2006) 590–599

3. Study methodology

Developing a relationship between geometric design vari-ables and safety involves modeling crash occurrence using oneof the statistical analysis techniques described above. All ofthese model types are appropriate for non-negative, discretecount data such as vehicle crashes. The Poisson regression modelwas used initially to develop relationships between median bar-rier crashes and geometric, traffic operations, and other physicalroadway variables. The Poisson model is central to the discus-sion of count regression, but the model rarely provides betterestimation results than other count models. A limitation of thePoisson regression model is that the variance of the crash data isrestrained to equal the mean. Negative binomial regression wasthen used to account for the overdispersed (variance is greaterthan mean) nature of crash count data. The main impetus ofthis particular research stems from a previous effort (Mason etal., 2001) where the frequency of CMC crashes were predictedon Pennsylvania Interstates using count regression. There wasa need to develop median barrier crash frequency models tobe used in concert with the CMC crash frequency models sothat designers can evaluate the safety, and design tradeoffs ofdivided highway sections with and without longitudinal medianbarrier.

The dependent variable in the median barrier crash mod-els developed is the number of crashes per year per directionoPsPtssmasits

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The dispersion parameter is not known before estimating themodel. Maximum likelihood estimation is used to determine theregression coefficients.

The Pearson and deviance statistics are used to assess themodel goodness-of-fit and a general discussion is provided later.

4. Description of data

Electronic data used in the modeling effort were obtainedfrom Pennsylvania’s Roadway Management System (RMS),Highway Performance Monitoring System (HPMS), and CrashReporting System (CRS) databases. The RMS contains geomet-ric design, traffic control, and roadside features of PennsylvaniaInterstate highways. The following data were extracted fromthe RMS database for the purpose of this research: averagedaily traffic (ADT), number of travel lanes, pavement width (ft),left shoulder width (ft), divisor width (ft), divisor type (earth-divided, fixed barrier, natural barrier), and speed limit (mph).These variables may not necessarily be constant over an entiresegment length, but they are typically an average value or thevalue that is most predominant within a segment.

The RMS database is organized into segments, each nomi-nally 1/2-mile long. Segments are generally homogeneous withrespect to traffic volume and geometric design elements. Eachsegment is then identified by county, state route number, andsegment number. Within each segment are offset distances ref-eatTswfi

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f travel. Data from the entire Interstate highway system inennsylvania was aggregated to form datasets for use in thetatistical evaluation of median barrier crashes. Because theennsylvania Turnpike is a toll road, it functions differently

han the rest of the Interstate system. In particular, interchangepacing is further apart on the Turnpike than on other Inter-tate sections; the Turnpike was constructed many years beforeost of the other Interstate system in Pennsylvania, and nearly

ll of the Turnpike contains longitudinal median barrier. Asuch, data from the Turnpike were analyzed separately. Thendependent variables considered were various geometric andraffic operational variables that are described in the followingections.

The Poisson and negative binomial models were developedsing the SAS GENMOD (SAS Publishing, 1999) procedure.he basic model for count data is the Poisson distribution and

he Poisson regression model was tried first with the given crashnd roadway inventory data. The Poisson distribution is com-only referred to as the “law of rare events” that states the total

umber of events will follow the Poisson distribution if an eventay occur in any of a large number of trials but the probability of

ccurrence in any given trial is small. This is true of crashes alongighway sections where the events may be considered random.n the case of the Poisson distribution, the model coefficientsere estimated by the maximum likelihood method. Addition-

lly, the Pearson and deviance statistics were used to assess theodel goodness-of-fit. In the case of crash data, overdispersion

ommonly exists as variance exceeds the mean of the data dis-ribution. In such a case, the negative binomial distribution isn appropriate alternative model to the Poisson. The negativeinomial regression model contains a dispersion parameter, a.

renced to the beginning of the segment (0 offset). Segmentsre generally even in the north- and eastbound directions, whilehey are odd-numbered in the south- and westbound directions.he last digit in the segment number can be used to distinguisheparate directions of travel for a roadway. The divisor type codeas used to identify divided Interstate highway sections with axed longitudinal median barrier.

The HPMS reflects the extent, condition, performance, use,nd operating characteristics of the Pennsylvania Turnpike. ThePMS database contains specific geometric, traffic control,

nd roadside variables, including the following used for thisesearch: area type (rural or urban), section length (miles), aver-ge annual daily traffic, number of through lanes, lane widthft), median type (fixed barrier or natural barrier), shoulderidth and type, horizontal alignment adequacy code, terrain

ype, speed limit (mph), and% heavy vehicles. It should be notedhat the HPMS segments are not uniform in length. They rangerom 0.5 to 10.75 miles (0.8–17.2 km), and are not necessarilyomogeneous over their entire length. Section lengths on theennsylvania Turnpike are generally defined based on nodeslong the network. Nodes can be intersections with other roads,ounty lines, state boundaries, or some other physical feature.s such, section lengths may not be homogeneous with respect

o geometric features or traffic volumes.The horizontal alignment adequacy code in the HPMS

atabase provides information regarding highway geometry.ategories include: (a) all curves meet appropriate design stan-ards, (b) some curves may be below appropriate design stan-ards, but can be safety negotiated at the prevailing speed limit,c) infrequent curves with design speeds less than the prevail-ng speed limit, and (d) several curves unsafe when traveling


at the prevailing speed limit. This code is calculated from thehorizontal alignment class data.

To simplify the statistical modeling approach, a binary cat-egorical variable was created. The two classes for the variableare: (a) all curves meet the design standards within the section,and (b) some curves do not meet the design standards withinthe highway section. The primary reason for creating these twolevels (as opposed for four levels) was because a quantifiable dis-tinction (e.g., curve radius or degree of curvature) between theoriginal four categories is not provided in the HPMS databasewithin the alignment adequacy code. The two categories chosen(0 = all curves in section are adequate; 1 = some curves in sec-tion are inadequate) makes the distinction clearer and relates toa 3.5◦ of curve, the maximum current standard for design.

The location of interchange entrance ramps were identifiedusing the video photologs based on the milepost location ofthe ramp with respect to each roadway section. Median barrieroffset distance was determined via a combination of field visitsand from video photologs.

The CRS contains crash history data for Pennsylvania’s Inter-state and Turnpike highways. The CRS database has three levelsof data—event, vehicle, and persons involved in crashes. Onlyevent-level data were used for this analysis. Specific informationincluded in the CRS include: crash severity, number of vehiclesinvolved, crash time of day and day of week, harmful eventindicator (collision with median barrier, collision with vehicletweotlhetwhc

appended to the RMS and HPMS databases to create the analysisdatasets.

In addition to the existing PENNDOT databases that wereused for the study, other data were obtained from police reports,highway construction plans, and video photologs.

All electronic data were obtained for the years 1994 through1998, inclusive. The entire population of median barrier crashesfor this time period was used to estimate the frequency mod-els. Approximately 1180 km (738 miles) on the Interstate systemcontain fixed median barrier while about 750 km (465 miles) onthe Pennsylvania Turnpike contain fixed median barrier. Theentire Pennsylvania Interstate highway system covers nearly4120 km (2560 miles) while the Interstate-designated portionof the Turnpike measures nearly 750 km (470 miles). Separatemodels for the Turnpike and the remainder of the Interstate high-way system were developed because the available data are main-tained in different electronic systems. Over the 5-year period,4416 median barrier crashes occurred on the Interstate systemwhile 2858 median barrier crashes occurred on the Turnpike.

Tables 1 and 2 provide a description of the data used tomodel both Interstate and Turnpike median barrier crash fre-quency. Included in the tables are the variable name, type, andrange of values. The data in Table 1 are from PENNDOT’s RMSdatabase, while the data in Table 2 are from PENNDOT’s HPMSdatabase. While the RMS database does not contain informa-tion on horizontal curve data, field reviews and video photologoiseNlitttTi

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raveling in opposing direction, etc.), weather conditions (dry,et, snow/ice, etc.), light condition (light, dark but lighted, dark,

tc.), pavement surface condition (dry, wet, snow, etc.), locationf crash (left of travel lane, right of travel lane, etc.). Providedhat road sections contained a fixed barrier, median barrier col-isions could be readily identified using a combination of thearmful event and location of crash indicators. If the harmfulvent included a collision with a fixed barrier, and the loca-ion of the crash was to the left of the travel lanes, the crashas coded as a median barrier crash. Review of a sample ofardcopy police accident reports confirmed that this identifi-ation methodology was correct. The CRS database had to be

able 1escription of interstate median barrier crash data

ariable name Variable description

arrier freq (response) Interstate median barrier collision frequeneglen (predictor) Segment length (km or miles)DT (predictor) Average directional daily

traffic volume within segment(vehicles/day)

peed (predictor)Speed limit within roadwaysegment (km/h or mph)

arrierOffset (predictor) Distance between left-edge of travel wayto face of median barrier (m or ft)

urve (predictor)Indicator of horizontalcurve direction insegment

nterchangepredictor)

Indicator of interchange entranceramp within within segment

bservations permitted the creation of a categorical variable tondicate curve direction. Similarly, the RMS electronic data wasupplemented with an indicator variable describing interchangentrance ramp locations. Video photologs, maintained by PEN-DOT for all Interstate highways, were used to determine their

ocation within each roadway segment. The ADT variable werencluded in the electronic RMS database. The barrier offset dis-ance from the left edge of the traveled way was measured inhe field or observed from the photologs. It is worth noting thathe range of median barrier offset in the Interstate dataset (seeable 1) is (0–6.6 m) 0–24 ft. The range of median barrier offset

n the Turnpike dataset is (0–1.2 m) 0–4 ft. The distribution of

Variable type Range of values

Count Range: 0–9Continuous Range: 0.3–1.6 km (0.2–1.0 miles)Continuous Range: 1578–53,833

Categorical0 = 88 km/h (55 mph)1 = 104 km/h (65 mph)

Continuous Range: 0–7.3 m (0–24 ft)

Categorical0 = No curve1 = Curve to the right2 = Curve to the left

Categorical0 = No ramp1 = Ramp within segment


Table 2Description of Pennsylvania turnpike median barrier crash data

Variable name Variable description Variable type Range of values

BarrierFreq (response) Turnpike median barrier collisionfrequency

Count Range: 0–15

SectionLength (predictor) Section length (km or miles) Continuous Range: 0.8–17.2 km (0.5–10.8 miles)ADT (predictor) Average daily traffic volume per

direction (vpd)Continuous Range: 1297–50,229

BarrierOffset (predictor) Distance between left-edge of travelway to face of barrier (m or ft)

Continuous Range: 0.3–1.2 m (1–4 ft)

AlignAd (predictor)Indicator of horizontalalignment adequacy

Categorical0 = Adequate (all curves in section with R > 514 m)1 = Not adequate (at least one curve in section with R < 514 m)

Ramp (predictor)Indicator of interchange entranceramp within segment

Categorical0 = No ramp1 = Ramp within segment

Trucks (predictor) Proportion of trucks in traffic stream Continuous Range: 0–35

RuralUrban(predictor)

Indicator of environmental setting Categorical0 = Rural1 = Urban

Terrain (predictor)Indicator of terrain type withinTurnpike section

Categorical0 = Level1 = Rolling or mountainous

the median barrier offset on the Interstate-designated system isas follows:

• (0–1.2 m) (0–4 ft): 38.4%• (1.5–2.4 m) (5–8 ft): 42.9%• (2.7–3.6 m) (9–12 ft): 16.7%• (>3.6 m) (>12 ft): 2.0%

Nearly 16.3% of the Pennsylvania Turnpike mileage containslongitudinal median barrier that is offset 2 ft (0.6 m) or less fromthe left-edge of the traveled way. The remaining 83.7% of theTurnpike contains median barrier that is offset between 2.0 and4.0 ft (0.6–1.2 m) from the left-edge of the traveled way.

An issue referred to earlier in the paper is one of developingrevised median barrier warrant criteria. To do so, it is proposedthat only the Interstate median barrier crash frequency modelsdeveloped in this paper be used in concert with the CMC crashfrequency models (see Donnell et al., 2002), crash severity mod-els (see Donnell and Mason, 2005), and an economic evaluation.

5. Descriptive measures of median barrier crashes

Prior to developing models of median barrier crash frequency,several descriptive measures of median barrier crashes wereinvestigated. First, Table 3 shows the severity distribution ofmrpaimlpwdw

respectively. The severity distribution reported by Donnell et al.(2002) for cross-median crashes (17% fatal and 67% injury) onPennsylvania Interstates is significantly different from that ofmedian barrier crashes.

The contributory factor in median barrier crashes on the Penn-sylvania Interstate system and Turnpike differ slightly. Therewere more than 125 potential contributory factors available inthe computerized crash data. The five most common contribu-tory factors in median barrier crashes occurring on the Interstatesystem were: (1) careless lane change, (2) driver loss of con-trol, (3) traveling too fast for inclement weather conditions, (4)traveling over posted speed limit, and (5) forced vehicle move-ment. These five factors, all shown in Table 4 , contributed to2088 of 4416 (47.2%) of median barrier crashes on the Interstatesystem. The five most common contributory factors in Turnpikemedian barrier crashes are also shown in Table 4. These factorsaccounted for 1,439 of 2,858 (50.4%) median barrier crashes.

Median barrier crashes occur about 52% of the time duringdaylight conditions on Interstate system. Approximately 43%of median barrier crashes occur during dark conditions with anearly equal split between lighted and unlighted highway con-

Table 3Crash severity levels

Severity level Number (%) of crashes

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A

edian barrier crashes on the Interstate system and Turnpike,espectively. The crash severity of all Interstate system and Turn-ike crashes is also shown in Table 3. All crash types includell reported single- and multi-vehicle crashes that occurred dur-ng the analysis time period. When comparing all crashes to the

edian barrier crashes, the severity distributions are very simi-ar. Of the 4416 median barrier crashes occurring on the non-tollortion of the Pennsylvania Interstate highway system, 0.7%ere fatal, 56.0% were injury crashes, and 43.3% were property-amage only (PDO) crashes. On the Turnpike, the distributionas 0.9%, 52.5%, and 46.6% for fatal, injury, and PDO crashes,

Interstate Turnpike Total

edian barrier collisionsFatal 31 0.7 26 0.9 57 0.8Injury 2471 56.0 1500 52.5 3971 54.6Property damage only 1914 43.3 1332 46.6 3246 44.6Total 4416 100.0 2858 100.0 7274 100.0

ll crash types combinedFatal 412 1.3 86 0.9 498 1.3Injury 15827 51.7 4177 45.8 20004 50.3Property damage only 14415 47.0 4862 53.3 19277 48.5Total 30654 100.0 9125 100.0 39779 100.0


Table 4Median barrier crash contributory factors

Crash contributory factor Number (%) of crashes

Turnpike Interstate

Number % Number %

Traveling too fast for inclementweather conditions

465 16.3 351 7.9

Over posted speed limit 273 9.6 318 7.2Driver lost control 262 9.2 562 12.7Careless lane change 250 8.7 639 14.5Drowsy/fatigued driver (Turnpike)

or forced movement (Interstate)189 6.6 218 4.9

Total 1439 50.4 2088 47.2

ditions on Interstate highways. On the Pennsylvania Turnpikeabout 63% of median barrier crashes occur during daylight peri-ods and nearly 30% of median barrier crashes in dark, unlightedconditions.

Median barrier crashes on dry pavements were more frequenton the Interstate system (61%) when compared the Turnpike(47%). Wet pavement condition crashes on the Turnpike (36%)were more prevalent than similar types on the Interstate system(22%). Drug or alcohol involvement was less frequently cited forTurnpike median barrier crashes (2%) than for Interstate systemcrashes (9%).

6. Predictive model estimation

A traffic crash is an event count, or a non-negative integer-based random variable. Traditionally, models used to predicttraffic safety counts include Poisson or negative binomial regres-sion. The theory associated with both distributions is well-documented and can be found elsewhere (refer to Miaou,1994; Miaou and Lum, 1993; Kniuman et al., 1993; Bauerand Harwwod, 1997; Cameron and Trivedi, 1998; Milton andMannering, 1998). A limitation of the Poisson distribution is theequidispersion property, or equality of the mean and variance.

This property is often violated when modeling event counts.In the case of traffic crashes, overdispersion commonly existswhere the variance exceeds the mean. To account for overdisper-sion, the negative binomial distribution is often used to modelcrash data. The negative binomial often arises when the dataare Poisson, but there is gamma-distributed unobserved individ-ual heterogeneity reflecting an imperfect true mean observation.Both the Poission and negative binomial regression models wereused in this research; however, only the negative binomial regres-sion results are presented herein.

Common goodness of fit measures for Poisson and nega-tive binomial regression models are the Pearson chi-square anddeviance statistics. The deviance compares a given model toa full model—in this paper a given model is one in which allstatistically significant independent variables are included anda full model has one parameter for each observation, and canreproduce the observed data (Long, 1997). The deviance is thestatistic for testing the hypothesis that all parameters in the fullmodel but not in the given model equal 0. It is defined as twicethe difference between the maximum log-likelihood achievableand the log-likelihood of the given (or fitted) model (Cameronand Trivedi, 1998). Overdispersion commonly occurs in countdata, and thus, a dispersion parameter is used to scale the vari-ance. A scaled deviance is the deviance statistic divided by thedispersion parameter. The scaled deviance can be used as anapproximate guide to assess the goodness of fit of a given model(

doWffiscfot

rier c
Fig. 2. Interstate median bar
SAS Publishing, 1999).For count models, the deviance is approximately chi-square

istributed with n–k degrees of freedom, where n is the numberf observations and k the number of parameters in the model.hen dividing the scaled deviance statistic by its degrees of

reedom, a resulting value near 1.0 may indicate a good modelt. The Pearson chi-square statistic is similar to the deviancetatistic, but it compares the observed count to the expectedount. The chi-square distribution is used with the degrees ofreedom equal to the number of observations minus the numberf model parameters. Pearson chi-square statistics divided byheir degrees of freedom near 1.0 may indicate a good model fit.

rash frequency distribution.


Fig. 3. Turnpike median barrier crash frequency distribution.

The general shape of each crash frequency distribution wasassessed prior to developing the statistical models. The plotsshown in Figs. 2 and 3 show the crash frequency distributionsfor Interstate median barrier and Turnpike median barrier col-lisions, respectively, based on data from 1994 through 1998.The mean and variance for the data presented in Fig. 2 were0.703 and 1.020, respectively. The mean and variance of thedata presented in Fig. 3 were 1.181 and 3.213, respectively.This initially indicated that overdispersion may exist in the data.To further investigate this issue, the deviance and Pearson chi-square statistics, both divided by their respective degrees offreedom were examined by first estimating a Poission regressionmodel. The deviance and Pearson chi-square statistics, dividedby their respective degrees of freedom, were 1.093 and 1.135for the Interstate and Turnpike datasets, which again suggeststhat the data were overdispersed. As such, the prediction modelsdiscussed below are based on the negative binomial distributionwhich can better account for count data that are overdispersed.

7. Modeling results

Using the data presented in Tables 1 and 2, both Poissonand negative binomial regression models were developed forInterstate and Pennsylvania Turnpike median barrier crashes.

For illustrative purposes, only the negative binomial regressionmodels are shown in Tables 5 and 6 for the Interstate systemand Turnpike median barrier crash data. Interpretation of thePoisson regression model is very similar to the negative bino-mial model. Likelihood ratio tests were used to determine ifoverdispersion existed in the median barrier crash frequencydata. The test is performed using information from both the Pois-son and negative binomial regression models. This test examinesthe equality of the mean and variance imposed by the Poissondistribution against the alternative that the variance exceeds themean. The test statistic is −2 times the difference between thelog-likehood of the Poisson model and the log-likehood of thenegative binomial model. The likelihood ratio statistic for theInterstate median barrier crash frequency model was 73.12, andthe likelihood ratio statistic for the Turnpike median barrier crashfrequency model was 150.04. Both test statistics greatly exceedthe 1% critical value of χ2

0.98(1d.f.) = 5.41. Therefore, the nullhypothesis of a Poisson model is rejected, indicating the pres-ence of overdispersion.

For Interstate median barrier crashes (Table 5), the ADT,speed limit, horizontal alignment, presence of an interchangeentrance ramp, and median barrier offset from the edge of thetravel way were all statistically significant predictors of the varia-tion in median barrier crash frequency. The deviance and Pearson

Table 5I

V Chi-s

I 556.2A 574.5R 3.7S 85.7C 17.2C 5.7B 56.4D

G hi-squn

nterstate negative binomial regression model

ariable Coefficient Standard error

ntercept −6.8673 0.2912DT (log) 0.7553 0.0315amp −0.0664 0.0344peed −0.4810 0.0519urve 0 −0.1726 0.0415urve 1 0.0923 0.0344arrier offset −0.0352 0.0047ispersion parameter 0.1802 0.0253

oodness of fit measures: deviance (value/d.f.) = 6097.01 (0.9741); Pearson cumber of observations = 6280.

quare statistic Pr > Chi-square Wald 90% confidence limits

Lower Upper

3 <0.0001 −7.346 −6.3881 <0.0001 0.704 0.8074 0.0532 −0.123 −0.0108 <0.0001 −0.566 −0.3969 <0.0001 −0.241 −0.1047 0.0163 0.029 0.1561 <0.0001 −0.043 −0.028

0.143 0.227

are (value/d.f.) = 6324.18 (1.0104); Log-likelihood = −5396.38; R2α = 0.583;


Table 6Turnpike negative binomial regression model

Variable Coefficient Standard error Chi-square statistic Pr > Chi-square Wald 90% confidence limits

Lower Upper

Intercept −4.2298 0.5844 52.39 <0.0001 −5.191 −3.269ADT (log) 0.4180 0.0567 54.31 <0.0001 0.325 0.511Section length (log) 1.0281 0.0310 1101.89 <0.0001 0.978 1.080Rural/urban indicator −0.2406 0.0547 19.34 <0.0001 −0.331 −0.151Barrier offset −0.0187 0.0244 0.59 0.4428 −0.059 0.021Alignment adequacy indicator −0.1050 0.0721 2.12 0.1450 −0.224 0.014Terrain indicator −0.1042 0.0638 2.67 0.1024 −0.209 0.001Trucks −0.0001 0.0035 0.00 0.9665 −0.006 0.006Ramp indicator 0.1866 0.0539 11.98 0.0005 0.098 0.275Dispersion 0.2831 0.0336 0.233 0.344

Goodness of fit measures: deviance (value/d.f.) = 2277.1274 (0.9449); Pearson chi-square (value/d.f.) = 2564.4788 (1.0641); Log-likelihood = −1127.8716; R2α =

0.798; number of observations = 2420.

chi-square statistics, divided by the degrees of freedom, are bothnear 1.0.

Interpretation of the parameter estimates shown in Table 5is straightforward. The parameter estimates reported in Table 5are for the lower level (zero) of each categorical variable shownin Table 1. The speed parameter estimate is −0.4810 (standarderror = 0.0519). Alternatively stated, the relative effect of thespeed limit variable is exp(−0.4810) = 0.6182. This indicatesthat posted speed limits of 88 km/h (55 mph) lower the expectedmedian barrier crash frequency by nearly 39% when comparedto a posted speed limit of 105 km/h (65 mph), holding all othervariables constant. The absence of an interchange entranceramp decreases the expected median barrier crash frequencyby approximately 6.5% (β = −0.0644; S.E. = 0.0344). Increas-ing the offset of a longitudinal barrier by 1 unit will decreasethe expected median barrier crash frequency by approximately3.5% (β = −0.0352; S.E. = 0.0047). Lastly, tangent sections ofInterstate highway decrease the expected median barrier crashfrequency by nearly 16% (β = −0.1726; S.E. = 0.0415), and sec-tions curved to the right increase the expected crash frequencyby more than 9% (β = 0.0923; S.E. = 0.0344), when comparedto sections curved to the left. It should be noted that a curveto the left would have a reported coefficient of 0.0865 whichis the opposite of the sum of the reported classes of the curvecategorical variable in Table 5.

Table 6 is the Turnpike median barrier crash model using neg-aTscacItcevfT

adequacy. When traveling within a section with all curves hav-ing adequate radii (greater than 514 m or 1685 ft) as opposed tosections where one or more curves have inadequate radii (lessthan 514 m or 1685 ft) decreases the expected crash frequencyby about 10% (β = −0.1050; S.E. = 0.0721). Longitudinal bar-rier offset has little relative effect on the expected median barriercrash frequency on the Pennsylvania Turnpike. Lastly, sectionsof roadway without interchange entrance ramps were foundto have 21% (β = 0.1866; S.E. = 0.0539) more median barriercrashes than sections with interchange entrance ramps. Table 6shows the negative binomial goodness-of-fit measures, both arenear 1.0. The recommended form of the Interstate and Turnpikemedian barrier crash frequency models are shown in Eqs. (1)and (2), respectively.

NIntBarrier = e−6.867L(X1)0.755exp(−0.066X2) exp(−0.481X3)

×exp(−0.173X4) exp(0.092X5) exp(−0.035X6)

(1)

where, NIntBarrier = expected number of Interstate median bar-rier crashes per year for a highway segment, L = segment length(miles), X1 = average daily traffic volume for one direction oftravel (vehicles/day), X2 = interchange entrance ramp indicator(1 if no ramp present, 0 otherwise), X3 = posted speed limit (1if 55 miles/h, 0 otherwise), X4 = horizontal curve indicator (1 iftif

N

wb(te

tive binomial regression. The parameter estimates reported inable 6 are for the lower level (zero) of each categorical variablehown in Table 2. Interpretation of the chi-square statistics indi-ates that the ADT, rural/urban location indicator, horizontallignment adequacy indicator, and ramp indicator are statisti-ally significant predictor variables at the 10% significance level.nterpretation of the parameter estimates provides insight abouthe relative effects that each predictor has on median barrierrash frequency on the Turnpike. For instance, traveling in a ruralnvironment as opposed to an urban environment (with all otherariables constant) decreases the expected median barrier crashrequency by approximately 21% (β = −0.2406; S.E. = 0.0547).he same can be stated with regard to the horizontal alignment

angent section, 0 otherwise), X5 = horizontal curve indicator (1f curved to right, 0 otherwise), and X6 = barrier offset distancerom edge of travel way (ft).

TurnpikeBarrier = e−4.430L1.028(X1)0.418exp(−0.241X2)

×exp(−0.105X3) exp(−0.104X4)

×exp(0.188X5) (2)

here, NTurnpikeBarrier = expected number of Turnpike medianarrier crashes/year for a highway segment, L = section lengthmiles), X1 = average daily traffic volume for one direction ofravel (vehicles/day), X2 = location indicator (1 if rural, 0 oth-rwise), X3 = horizontal alignment indicator (1 if all curve radii


in section >514 m or 1685 ft, 0 otherwise), X4 = terrain indica-tor (1 if level; 0 otherwise), and X5 = interchange entrance rampindicator (1 if ramp not present, 0 otherwise).

When comparing Eqs. (1) and (2), some commonalities exist.For instance, the existence of horizontal curvature and aver-age daily traffic volumes play an important role in explainingthe variability in median barrier crash frequency. The oppositesign for the interchange entrance ramp indicator suggests thatmedian barrier crash frequency increases on Interstate highwayswhere one is present yet the expected frequency decreases onthe Turnpike when a ramp is present. For the Interstate model,the median barrier offset distance from the travel way explainssome of the median barrier crash frequency variability. In bothmodels, the average daily traffic volume accounted for mostof the variability in the median barrier crash data—the othertraffic operational and geometric variables explained only asmall amount of the variability. The difference in magnitudeof the traffic volume variables between the two models couldbe explained by the fact that the Turnpike is a toll road that isused by travelers for long-distance trips (average interchangespacing ∼11.5 miles). The non-toll portion of the PennsylvaniaInterstate system is typically used for various reasons, but caninclude shorter trips (average Interchange spacing ∼3.0 milesbased on review of Interstates 79, 80, and 81). For 99% of itstotal length of 469 miles, the Turnpike cross-section containstwo travel lanes per direction, a narrow inside shoulder, and lon-gcssmtl

8

bctfrt

(

(

between adjacent sections. Further research is needed todetermine median-related crash causation in the presenceof interchange ramps.

(c) Increasing the median barrier offset from the left-edge of thetravel way decreases median barrier crash frequency.

(d) Curved horizontal alignment and curves with less than mini-mum curve radii increase the median barrier crash frequency.

(e) A rural travel environment exhibits lower median barriercrash frequencies than an urban environment on the Turnpikeroad network.

(f) Increased traffic volumes increase the frequency of medianbarrier crashes. The modeling efforts indicate that medianbarrier crash frequency can be modeled using the negativebinomial distribution.

Prior research has quantified safety on earth-divided,traversable medians without barrier. This particular researchcould be used in concert with previous efforts to revise medianbarrier warrant criteria. Using both cross-median and medianbarrier crash models could provide useful insights about thetrade-offs in crash experience and severity when deciding toinstall longitudinal median barrier.

Several issues require further research. Namely, the design(including length, grade, and horizontal curvature) of inter-change ramps should be considered. The notion that median-riieHsemdelb

A

pitmm

R

A

B

C

itudinal median barrier. It has a near uniform and consistentross-section. In contrast, the remaining Interstate-designatedystem contains a significant amount of variability in the cross-ection. There can be sections with an earth-divided, traversableedian that are 30 ft or more wide adjacent to sections that con-

ain longitudinal barrier in the median that is 4-ft offset from theeft-edge of the traveled way.

. Conclusion

The crash frequency models estimate the number of medianarrier crashes for a particular set of geometric element andross-section element dimensions. From the models developed,raffic volumes, horizontal alignment, barrier offset distancerom the travel lanes, and the presence of interchange entranceamps all influence median barrier crash frequency. Generalrends of crash frequency indicate the following:

a) As the speed limit increases on the non-toll portion of theInterstate highway, the frequency of median barrier crashesincreases.

b) The presence of interchange entrance ramps increases thefrequency of median barrier crashes on the Interstate sys-tem, but decreases the expected frequency on the Turnpiketoll road. A possible explanation for this is driver trip-making characteristics on the toll versus non-toll roads. Asstated previously, the Turnpike is a toll road with signifi-cant distances between interchanges. It typically containsa uniform, yet consistent cross-section. The non-toll por-tion of the Interstate includes interchanges that have a closerspacing; however, the cross-section can vary considerably

elated crashes are frequent at such locations requires furthernvestigation to understand why such maneuvers occur. Also,t would be useful to evaluate the transferability of these mod-ls to other states. Simulation packages such as the Interactiveighway Safety Design Model (IHSDM), or AASHTO’s Road-

ide Safety Analysis Program (RSAP), do not contain explicitvaluation procedures for median design. Models of CMC andedian barrier crashes, combined with an economic evaluation

ecision-making tool, may prove beneficial to geometric designngineers. However, it is important that models from severalocations be developed and consider a wide-range of medianarrier types and placement locations.

cknowledgments

The assistance of the Pennsylvania Department of Trans-ortation, Bureau of Highway Safety and Traffic Engineerings acknowledged for providing the data necessary to completehe study. The authors would also like to thank the three anony-

ous reviewers whose insights and comments improved theanuscript significantly.

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