SAFETY PERFORMANCE FUNCTIONS FOR CRASH SEVERITY ON UNDIVIDED RURAL ROADS

7/26/2019 SAFETY PERFORMANCE FUNCTIONS FOR CRASH SEVERITY ON UNDIVIDED RURAL ROADS

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Accident Analysis and Prevention 93 (2016) 75–91

Contents lists available at ScienceDirect

Accident Analysis and Prevention

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

Safety performance functions for crash severity on undivided rural

roads

Francesca Russo∗, Mariarosaria Busiello, Gianluca Dell’Acqua

Department of Civil,Architectural andEnvironmental Engineering,Federico II University of Naples, ViaClaudio 21, 80125 Naples, Italy

a r t i c l e i n f o

Article history:

Received 8 January 2016

Received in revised form 1 April 2016

Accepted 12 April 2016

Keywords:

Undivided rural roads

Safety performance functions

Injury and fatality frequency

Countermeasures

a b s t r a c t

The objective of this paper is to explore the effect of the road features of two-lane rural road networks

on crash severity. One of the main goals is to calibrate Safety Performance Functions (SPFs) that can

predict the frequency per year of injuries and fatalities on homogeneous road segments. It was found

that on more than 2000 km of study-road network that annual average daily traffic, lane width, curvature

change rate, length, and vertical grade are important variables in explaining the severity of crashes. A

crash database covering a 5-year period was examined to achieve the goals (1295 injurious crashes that

included 2089injuriesand 235 fatalities). A total of 1000 km were used to calibrate SPFs andthe remaining

1000 km reflecting the traffic, geometric, functional features of the preceding one were used to validate

their effectiveness. A negative binomial regression model was used. Reflecting the crash configurations

of the dataset and maximizing the validation outcomes, four main sets of SPFs were developed as follows:

(a) one equation to predict only injury frequency per year for the subset where only non-fatal injuries

occurred, (b) two different equations to predict injury frequency and fatality frequency per year per sub-

set where at least one fa tality occurred together with one injury, and (c) only one equation to predict the

total frequency per year of total casualties correlating accurate percentages to obtain the final expected

frequency of injuries and fatalities per year on homogeneous road segments. Residual analysis confirms

the effectiveness of the SPFs.© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

The relationships between crashes or crash victims and road

scenario can be represented by many confounding variables that

can influence crash occurrence or injury severity. Some available

research in the scientific literature has focused more on effects

of demographic, psychological, situational, behavioral factors on

the crashes (Norris et al., 2000; Abbas and Al-Hossieny, 2004;

Chandraratnaet al., 2006; Bouaoun et al., 2015) than others focused

on therelationships between thedriver perception of theevolution

of the geometric road features of a traveled path and risk of a crash

as well as number of crashes during a study period (AASHTO, 2010;Aarts and Van Schagen, 2006; Elvik et al., 2004).

Itseems, therefore, that onewayto powerfully help reducecrash

casualties and the number of the crashes is to design roadway that

meets driver expectations improving road alignment consistency.

∗ Corresponding author.

E-mail addresses: [email protected] (F. Russo),

[email protected] (M. Busiello), [email protected]

(G. Dell’Acqua).

Hadi and Aruldhas (1995), f or example, estimated the effects of

cross-section design elements on total, fatality, and injurious crash

rates forvarious types of rural andurbanhighwaysat differenttraf-

fic levels by adopting a Negative binomial (NB) regression analysis.

The results showed that, depending on the highway type investi-

gated, increasing lane width, median width, inside shoulder width,

and/or outside shoulder width, the number of crashes decreased.

The research presented here is part of a wider research pro-

gram that deals with theanalysis of therelationships between road

consistency and the crash phenomenon for two-lane rural roads in

Southern Italy (Dell’Acqua et al., 2012, 2013; Russo et al., 2014).

The main purpose of the research presented here is to make a con-tribution for bridging the gap existing in the literature where more

road crash frequency prediction models exist than specific equa-

tions focused on the prediction of road crash casualties. We should

point outhere that no crash that can be associated with the driver’s

physiological causes have been investigated; namely, crashes due

to drowsiness, drunkenness, or distraction. Of course crashes tak-

ing place at the intersections segments were not included in the

study. By carefully analyzing the crash reports during the study

period that have been made available by the qualified authorities,

only crashes due to the unsafe maneuvers of drivers were filtered

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76 F. Russo et al./ Accident Analysis and Prevention 93 (2016) 75–91

and analyzed in order to study the effects of the road features on

driver behavior on undivided two-lane rural roads.

The research was carried out through several steps as follows:

• investigating preliminary correlations between independent

variables (geometric features and traffic measurement) and

dependent variable (crash frequency per year of casualties) by

adopting statistical processing such as the Classification and

Regression Tree and Pearson coefficient;• identifying two different road networks to calibrate and validate

the SPFs;• removing anomalouscrash frequenciesper yearfor injuries, fatal-

ities andall casualties on thehomogeneousroad segments before

moving to the calibration phase by using the 3 method;• calibrating SPFs by following Highway Safety Manual (HSM) pro-

cedures (AASHTO, 2010) that can predict, under specific road

geometricconditions, roadcrash frequenciesper yearfor injuries,

fatalities, and all casualties;• carrying out a validation procedure of the SPFs consisting in the

evaluation of the residuals by calculating some main synthetic

statistical parameters and plotting the diagrams of the cumula-

tive squared residuals for each SPF on the basis of an increasing

order ofAnnual Average DailyTraffic(AADT) tocheck theabsenceof vertical jumps;

Themain goal wasto assesswhether only oneSPF able topredict

thetotal casualties crash frequencyper year is reliable andwhether

it is able to return values not statistically different from those we

could have obtained if we had used ad-hoc SPF calibrated for only

injuries and for only fatalities when injuries+ fatalities happened

on the homogeneous road segments during the study period.

The research also focused on identifying road strategies to

improve road safety conditions.

2. Literature review

Several studies have been presented in the scientific literaturefor the prediction of the crashes where linear, non-linear, general-

ized linear models (GLM), generalized estimating equations (GEE),

NB or Poisson regression models were used in turn and were used

in the development of the models.

Abbas (2004) provided, for example, an assessment of traffic

safety conditions for rural roads in Egypt. A number of statistical

models that can be used to predict the expected numberof crashes,

injuries, fatalities and casualties was developed. Time series data

fortrafficand crashes forthe considered roadsovera 10-year period

were used to calibrate these predictive models. Several functional

forms were explored and tested in the calibration process; these

include linear, exponential, power, logarithmic and polynomial

functions.

Elvik et al. (2004) investigated power functions that describethe relationships between speed and road safety in terms of six

equations; in particular the power model was calibrated chang-

ing the exponent for different equations able to predict fatalities,

seriously injured road users, slightly injured road users, all injured

roadusers (severitynot stated),fatal crashes, serious injury crashes,

slight injury crashes, all injury crashes (severity not stated), and

property-damage-only (PDO) crashes.

Shively et al. (2010) implemented a semi-parametric Poisson-

gamma model to estimate the relationships between crash counts

and various roadway characteristics, including curvature, traffic

levels, speed limit and surface width. A Bayesian nonparametric

estimation procedure was employed for the model’s link function,

substantially reducing the risk of a mis-specified model. Results

suggest that the key factors explaining crash rate variability across

roadways were the amount and density of traffic, the presence and

degree of a horizontal curve, and road classification.

Hosseinpouret al.(2014) identifiedthe factors affecting boththe

frequency and severity of head-on crashes occurring on 448 seg-

ments of five federal roads in Malaysia. The frequencies of head-on

crashes were fitted by developing and comparing seven count-

data models including Poisson, standard NB random-effect NB,

hurdle Poisson, hurdle NB, zero-inflated Poisson, and zero-inflated

NB models. On the basis of the results of the model, the horizon-

tal curvature, terrain type, heavy-vehicle traffic, and access point

variables were found to be positively related to the frequency of

head-on crashes, while posted speed limits and shoulder width

decreased the crash frequency. As for crash severity, it emerged

that horizontal curvature, paved shoulder width, terrain type, and

side friction were associated with more severe crashes, whereas

land use, access points, and the presence of a median reduced the

probability of severe crashes.

Mohammadi et al. (2014) developed an NB model using a GEE

procedure to analyze interstate highways in Missouri. The signif-

icant explanatory variables were area type, lane width and the

presence of heavy vehicles involved in the crash. The longitudinal

model employs an autoregressive correlation structure to provide

more accurate crash frequency models so that safety policies and

crash countermeasures will be more efficient at saving lives and

resources.

Russo et al. (2014) calibrated and validated six models on two-

lane rural roads to predict the number of injurious crashes per

year per 108 vehicles/km on the road segment using a study of

the influence of the human factors (gender/age/number-of-drivers)

and road scenario (combination of infrastructure and environmen-

talconditionsfoundat thesiteat thetimeof thecrash) onthe effects

of a crash by varying the dynamic. GEE and additional log linkage

equations were adopted to calibrate SPFs. The Akaike information

criterion (AIC) and the Bayesian information criterion (BIC) were

used to check the goodness of fit and reliability of the models.

Models in the scientific literature were studied where base-

line equations were calibrated as a function of a small number

of explanatory variables (i.e. AADT, length of the road segment)and graduallyadjusted to take into account the road geometric and

non-geometric features of the local context.

Part C ofthe HSM (AASHTO, 2010) provides, for example, a pre-

dictive method of estimating the expected average crash frequency

of a network, facility or individual site on undivided rural two-lane

highways. Because SPFs were developed on the basis of data from

a subset of states, the HSM recommends that local agencies either

(a) develop SPFs for their local conditions or (b) use a calibration

procedure to adjust SPFs to reflect local conditions.

Brimley et al.(2012) calibratedHSM SPFsfor rural two-lane two-

wayroadway segments in Utah using NB regression. Thesignificant

variables were as follows: AADT, segment length, speed limit, and

the percentage of AADT made up of multiple-unit trucks. The new

specific models show that the relationships between crashes androadway characteristics in Utah may be different from those pre-

sented inthe HSM (AASHTO, 2010). Thecalibrationfactor of theSPF

forruraltwo-lanetwo-way roads in Utah wasfoundto be 1.16. This

indicates that more crashes occur on rural two-lane two-way roads

in Utah than those predicted by HSM models where the calibration

factor is 1.10.

Bauer and Harwood (2013) evaluated the safety effects of the

combination of horizontal curvature and longitudinal grade on

rural two-lane highways. Safety prediction models for fatal-and-

injury and PDO crashes were evaluated, and crash modification

factors (CMFs) representing safety performance on tangents were

developed from these models.

Table 1 shows an overview of the predictive models introduced

in the literaturereview section of thispaper comparing explanatory



F. Russo et al./ Accident Analysis andPrevention93 (2016) 75–91 77

Table 1

A comparison of some of thecrash predictive models found in theliterature.

Reference Predicted variables Explanatory variables Calibration method of the

function

Purpose of theutilization of

such models

Abbas (2004) Expected number of crashes,

injuries, fatalities and

casualties

AADT, Annual Average Vehicle Kilometers

(AAVK)

Linear, exponential, power,

logarithmic, polynomial

functions

Models to be usedfor 5

considered rural roads as well

as for thenetwork composed

of these 5 roads.

Elvik et al. (2004) Ratio before/after fatalities,

seriously injured road user,slightly injured road user, all

injured road users (severity not

stated), fatal crashes, serious

injury crashes, slight injury

crashes, all injury crashes

(severity not stated), PDO

crashes

Speed after over speed before Power model by means of

traditional techniques as wellas techniques for

meta-regression (multivariate

models) in which a variable is

raised to a certain exponent

Assessment of theeffects of

changes in speed on thenumberof crashes and the

severity of injuries.

Shively et al.(2010) Expected number of crashes Horizontal curvature degree, Horizontal and

Vertical Curve Length, vertical curve grade,

average shoulder width, AADT, speed limit,

type of terrain, surface width, road

classification

Bayesian semi-parametric

estimation procedure

(semi-parametric

Poisson-gamma model)

Relationships between crash

counts and various roadway

characteristics.

Hosseinpour et al.

(2014)

Head-on (HO) crash frequency Heavy vehicle traffic, horizontal curvature,

accesspoints, and terrain type, posted speed

limit, paved and unpaved shoulder width

Poisson, standard NB,

random-effect NB, hurdle

Poisson, hurdle NB,

zero-inflated Poisson,

zero-inflated NB;

random-effect generalized

ordered probit model

Evaluating the effects of

various roadway geometric

designs, the environment, and

traffic characteristics on the

frequency and theseverity of

head-on collisions.

Mohammadi et al.

(2014)

Crash f requency AADT o f vehicles, a nnual a verage p ercentage o f

trucks or heavy vehicles

(Percent commericial), area type, number of

lanes, interaction between Area type and Ln

AADT and between Area type and

percent commericial

GEEmethod to develop a

longitudinal NB model

Exploringthe effects of

temporal correlation in crash

frequency modelsat the

highway segment level.

Russo et al. (2014) Number of injurious crashes

per year per 108 vehicles/km

Gender/age/number-of-drivers, light

conditions, wet/dry road surface, geometric

element, crash dynamic

GEEwith additional loglinkage Defining countermeasuresby

working on explanatory

variables value.

HSM (AASHTO

(2010)

Expected average crash

frequency per year of a

network, facility or individual

site

Annual average Daily traffic volume, Road

length, Lane width, -Shoulder width, -Shoulder

type Paved, Roadside hazard rating, Driveways

density per mile, Horizontal curvature, Vertical

curvature, Centerline rumble strips, Passing

lanes, Two-way left-turn lanes, Lighting,

Automated speed enforcement, Grade Level

EmpiricalB aye s Met ho d Quant if ying t he safet y e ff ects

of decisionsin planning,

design, operations, and

maintenance through the use

of analytical methods.

Brimley et al.

(2012)

Numberof predicted crashes AADT, segment Length, driveway density,

shoulder rumble strip, passing lanes,

multiple-unit trucks, speed limit, shoulder

width,

NB Examining the relationships

between crashes and roadway

characteristics.

Bauer and

Harwood (2013)

Fatal-and-injury and PDO

crashes/mi/yr

AADT, Segment Length; Horizontal curve

radius; Absolute value of percent grade;

Horizontal curve length; Vertical curve length;

algebraic difference between initial and final

grades; Measure of thesharpness of vertical

curvature

Cross-sectional analysis using a

GLM approach with a NB

distributionand a loglink

setof safety prediction models

for fatal-and-injury and PDO

crashes for quantifying the

safetyeffects of five types of

horizontal and vertical

alignment combinations

variables, dependent variables to be predicted, calibration meth-

ods, and the purpose of the models’ use.

3. Overview of the HSM procedure adopted as amethodology for predicting road crash casualties

Part C of the HSM (AASHTO, 2010) provides a structured

methodology for estimating the expected average crash frequency

of a site, facility or roadway network over a given time period, the

geometric design and traffic control features, and AADT. The pre-

dictive models in Part C of the HSM for Rural Two-Lane Two-Way

Roads can be undertaken for a roadway network, a facility, or an

individual site. A site is either an intersection or a homogeneous

roadway segment. Although models can vary by facility and site

type, all have the same basic elements as follows:

• Statistical “base” models: SPFs are used to estimate the average

crash frequency for a facility type with specified base conditions,

typically functions of only a few variables, primarily the AADT

and length of the road segments;• CMFs: CMFs are multiplied by the crash frequency predicted by

thestatistical base modelsto account forthe difference between a

site under base conditions anda site notmeetingbase conditions;• Calibration factor (Cx): this is multiplied by the crash frequency

predicted by the statistical base model to account for the differ-

ences between the general local conditions and HSM conditions.

The predictive models in the HSM (AASHTO, 2010) useful for

estimatingthe expected average crash frequencyN predicted, are gen-

erally calculated using Eq. (1):

N predicted = N SPFx ×

CMF 1 x × CMF 2 x × ... × CMF yx

× C x (1)

where

Npredicted = predictive model estimate of crash frequency for a

specific year on site type x (crashes/year);




NSPFx = predicted average crash frequency determined for base

conditions with the SPF representing site type x (crashes/year);

CMFyx = Crash Modification Factors specific to site type x and

variable y to study that can affect crash frequency;

Cx = Calibration Factor to adjust the site type x for local condi-

tions.

The SPFs that were developed and shown in the HSM refer

to specific “base conditions” and reflect explicit geometric design

and traffic control features. The HSM base conditions for roadway

segments on rural two-lane two-way roads are as follows: Lane

width (LW) of 12ft, Shoulder width (SW) of 6ft, Paved Shoulder

type, Roadside hazard rating (RHR) equal to 3, Five driveways per

mile, No Horizontal curvature, No Vertical curvature, No Center-

line rumble strips, No Passing lanes, No Two-way left-turn lanes,

No Lighting, No Automated speed enforcement, Grade Level less of

3%.

The SPFs are developed by means of statistical multiple regres-

sion techniques by assuming that crash frequencies follow an NB

distribution. An example of an SPF for rural two-way roadway seg-

ments under “base geometric conditions” from Chapter 10 of the

HSM (AASHTO, 2010) is shown in Eq. (2):

N SPFx = AADT × L × 365× 10−6× e−0.312 (2)

where

AADT= annual average daily traffic volume on a roadway seg-

ment whose range is from 0 to 17,800 vehicles per day;

L = length of roadway segment (miles).

Adjustment to the prediction made by an SPF for road segments

under “base” geometric conditions is required to account for the

difference between base conditions (condition ‘a’) of the model in

Eq. (2) and local/state conditions (condition ‘b’) of the site under

consideration. The CMFs are multiplicative because the effects of

the features they represent are presumed to be independent. CMFs

areusedto correct thecrashfrequency of road segments that donot

meet the base conditions. Eq. (3) shows the calculation of a CMF.

CMF =expected average crash frequency with conditionb

expected average crash frequency with condition

a(3)

Under base conditions, the value of a CMF is 1.00. CMF values

less than 1.00 indicate that alternative treatment reduces the esti-

mated average crash frequency compared to the base condition.

CMF values greater than 1.00 indicate that alternative treatment

increases the estimated average crash frequency compared with

the base condition.

CMF values in the HSM are either presented textually (typi-

cally where there are a limited range of options for a particular

treatment), in formulae(typically wheretreatment options arecon-

tinuous variables) or in tabular form (where the CMF values vary

by facility type, or are in discrete categories).

A calibration factor (Cx) is also used to account for the overall

variations in the geographical conditions of the areas of passage

from those reflecting base geometric conditions to non-base ones.Cx (see Eq. (4)) is equal to the ratio between the weighted aver-

age of the observed number of crashes at all the segments studied,

where theweighting refersto thelength of theroadway segment in

relation to the weighted total of the predicted number of crashes.

The default value of Cx suggested by the HSM procedure is 1.10.

C x =

allcasualties

observed crashes

allcasualties

predicted crashes(4)

While the SPFs estimate the average crash frequency for all

crashes, HSM provides procedures to separate the estimated crash

frequency into components by crash severity levels and collision

Table 2

HSM Default Distribution for Crash Severity on Two-Lane Rural Segments (see

Exhibit 10–6).

Crash severity level Percentage of total

roadway segment

crashes

Fatal 1.3

Incapacitating Injury 5.4

Nonincapacitating injury 10.9

Possible injury 14.5Total fatal plus injury 32.1

PDO 67.9

TOTAL 100.0

types (such as run-off-roador rear-end crashes). In most instances,

this is accomplished using default distributions of crash severity

level and/or collisiontype.As these distributions will vary between

sites and facilities, the estimations will benefit from updates based

on local crash severity and collision type data. Table 2 shows

the HSM default proportions based on HSIS data for Washington

(2002–2006) for severity levels used to estimate crash frequen-

cies by crash severity level. The HSM suggests that Table 2 may be

updated on the basis of local data as part of the calibration process.While theHSM provides thepercentage distributionof road seg-

ments by severity crash level in Table 2, the goal of the research

presented here focuses mainly on the calibration and validation of

functions that can predict the frequency per year of injuries and

fatalities, and all casualties (injuries+ fatalities) on homogeneous

road segments.

4. Data collection

This study is part of a wider research program to analyze the

effects of roadalignment consistencyon crashfrequencyand sever-

ity above all for two-lane rural roads (Dell’Acqua et al., 2012, 2013;

Russo et al., 2014).

Crash data were collected on two-lane rural roads in SouthernItaly covering more than 2000km. The overall procedure consisted

of two main steps: SPFs calibration and then SPFs validation. The

road network used to calibrate the SPF does not include road seg-

ments from the validation sub-set. During the first step, more than

1000km were used, as in step second, to check the reliability of the

SPFs.A 5-year period (2006–2010) was selected to carefullyanalyze

the crash reports held by the competent authorities. The road net-

work of the calibration procedure counted 693 crashes during the

5-year study period: 583 crashes with only injuries that included a

total amount of 668 injuries was observed and 110 crashes with

both injuries and fatalities that included a total amount of 399

injuries + 140fatalitieswas observed.The road network used toval-

idate the SPFs counted 602crashes during the 5-year study period:

512 crashes with only injuries (580 injuries) and 90 crashes withboth injuries and fatalities (442 injuries + 95 fatalities).

In terms of the crash data, when we refer to the term “injuries”,

this expression involves both slight injuries only and serious

injuries only that call for an emergency vehicle in situ and the ride

to the hospital without death in the hospital. When we refer to

fatality, this expression means death in situ after the crash hap-

pened and also the ride to the hospital where the patient dies in

the hospital within 48h.

A preliminary analysis was carried out to identify the relation-

ships between the possible explanatory variables using a Pearson

coefficient and the dependent variables (frequency of the injuries

per year, frequency of the fatalities per year, total frequency per

year of injuries + fatalities) used to calibratethe SPF. The covariance

matrix indicatedthat the AADT, LW, horizontal curvature indicator




(CI), vertical grade (VG) and segment length (L) of the homo-

geneous road segments can be considered to be significant, and

consistent variables can be used to explain the crash phenomenon.

In particular, the horizontal curvature indicator CI reflects the

grade of non-tortuosity of the horizontal alignment of a geomet-

ric road in terms of the horizontal curvature change rate (CCR

in gon/km) defined as the sum of the absolute values of angu-

lar changes in horizontal alignment divided by total road segment

length. The unit “gon” is particularly used in specialized areas such

as surveying, mining and geology, and is equivalent to 1/400 of a

turn, 9/10 of a degree or /2 00 of a radian. In order to identify

homogenous segments, we referred to indications from the Ger-

man standard (Richtlinien für die Anlage von Strassen, 1995). A

diagram was plotted: the x-axis shows the road distance expressed

in km, and the y-axis shows the cumulative of the absolute value

of the angular changes. The slope of each fitting line with the high-

est coefficient of determination calculated using the ordinary least

squares method shows the CCR of a homogeneous road segment.

No more than three homogeneous road segments were identi-

fied on the investigated rural roads. Some levels of the CI variable

are suggested here on the basis of the available data reflecting

variations of the CCR: (a) CI from 0 to 0.5 means high curvature

(CCR> 400 gon/km); (b) CI from 0.5 to 0.8 means medium curva-

ture (50gon/km≤CCR ≤400gon/km); (c) CI from 0.8 to 1 means

low road horizontal curvature (CCR < 50gon/km).

Additional statistical processing was carried out such as the

Classification and Regression Tree (CART) for a preliminary exam-

ination of the relationships between risk factors and crash road

casualties (Chang and Chen, 2005; Chang and Wang, 2006). CART,

a non-parametric model with no pre-defined underlying relation-

ship between the target variable (in this case coinciding with crash

road casualties) and the predictors have been used to identify the

risk factors affecting crash casualties levels in traffic crashes. With

the capacity of automatically searching for the best predictors and

the best threshold values for all predictors to classify the target

variable, CART has been shown to be a powerful tool, particularly

for dealing with prediction and classification problems. Thresholds

of six main predictor variables were identified. These six predic-tor variables include highway characteristics (AADT, road segment

length, travel lane width per direction, curvature change rate, ver-

tical grade), and crash variables (crash casualties). Table 3 gives the

definition of the predictor variables. The Gini splitting criterion,

the CART default, is used in this study. Fig. 1 shows the classifi-

cation tree reproduced by CART. The tree has 15 terminal nodes.

This implies that these variables, and their thresholds, are critical

in classifying injury severity in traffic crashes.

Despite the advantages, the CART model also has its disad-

vantages (Chang and Wang, 2006). Firstly, CART analysis does

not provide a probability level or confidence interval for risk fac-

tors (splitters) and predictions. In addition, the CART method has

difficulty in conducting elasticity analysis or sensitivity analysis.

Elasticity analysis (or sensitivity analysis) allows an examinationof the marginal effects of the variables on injury severity likeli-

hood, and provides valuable information for traffic authorities to

evaluate the greatest risk factors and establish priorities for miti-

gation. The final disadvantage is that the classification tree models

are generally very “unstable”. For example, the tree structure and

classification accuracy might alter significantly if different strate-

gies such as stratified randomsampling (with injury severity as the

stratification variable) are applied for creating learning and testing

datasets.This is the reason whythe tree models areoften used only

to identify important variables and some other flexible modeling

techniques are used to develop final models.

From a careful analysis of the correlations between crash casu-

alties and road features, some specific geometric combinations

have been found where no fatalities occurred. Fig. 2 shows some

diagrams indicating identified regions where no fatal crash fre-

quency per year was recorded: VG< 5%+L>3 km+CI> 0.8. This is

the reason why the SPF that can predict annual injuries only

frequency per year on homogeneous road segments has been cali-

brated to maximize the results of the validation procedure.

Four main subsetof differentSPFs were developedreflecting the

facets of the crash dataset and maximizing the validation results:

• one equation that predicts only frequency of injuries per yearon homogeneous road segments where injuries but no fatali-

ties were observed during the study period on the selected road

network according to previous geometric combinations;• one equation that predicts frequency of injuries per year on

homogeneous road segments where both injuries and fatalities

were observed at the same time during the study period;• one equation that predicts frequency of fatalities per year on

homogeneous road segments where both injuries and fatalities

were observed at the same time during the study period;• oneequationthatpredictsthe total frequencyper year of allcasu-

alties, correlating accurate percentages to calculate the expected

number of injuries and fatalities per year from the total number

of casualties per year on homogeneous road segments.

Part I of Table 3 shows a statistical overview of the main

explanatory geometric features to study the severity of the crash

phenomenon. Part II of Table 3 shows a statistical summary of

crash severity for two previous listed cases. It should be recalled

that this research does not include PDO crashes or crashes due to

physiological causes (i.e. sleepiness, drunkenness).

5. Data analysis

The filtering procedure structure and data processing to create

SPFs are summarized in summary in Fig. 3.

Two differentroad networks were used to calibrateand validate

the SPFs for checking their effectiveness when the frequency of

injuries and fatalities per year on homogeneous road segments ispredicted. One of the first steps to begin to adjust or to calibrate

SPFs according to HSM procedure is to divide the roadway into

individual sites consisting of homogenous roadway segments and

intersections. Predictivemodels can be developed to estimate crash

frequencies separately for roadway segments and intersections.

According to HSM procedure, roadway segments begin at the

center of an intersection and end at either the center of the next

intersection, or where there is a change from one homogeneous

roadway segment to another homogenoussegment.The segmenta-

tion process produces a set of roadway segments of varying length,

each of which is homogeneous with respect to characteristics such

as traffic volumes, roadway design characteristics, and traffic con-

trol features.

In light of a preliminary analysis to identify the relationshipsbetween the possible explanatory variables and the variables to

be predicted, the total length of the road network studied was

segmented by using a change in the CCR. There are instances

in the literature where CCR is used to divide the sample into

homogeneous segments, and investigate road consistency (Federal

Highway Administration, 2000).

Before moving to the calibration phase according to the road-

crash scenarios listed in the previous paragraph, two steps have

been carried out:

• converting the measure units of the explanatory and dependent

variables from metric (SI) units to US Customaryunits for reflect-

ing the same measure units system that was adopted in the HSM

procedure to get comparable results and carry out proper and




Table 3

Descriptive statistics of the total road network and crash road casualties.

AADT, veh/day VG, % L, km LW per travel direction, m CCR, gon/km

Part I Min 400 0.12 1.42 2.50 30.05

Mean 3490 2.25 4.16 3.00 479.14

Max 12000 6.95 8.37 5.50 562.67

St. Dev. 2648 1.46 1.17 1.64 148.27

Part II Injuries Frequency, injuries/year Injuries + Fatalities Frequency, injuries +fatalities/year

Calibration Sample 668 399 injuries + 140 fatalities

Min 0.20 0.80

Mean 0.80 2.03

Max 3.60 5.00

St. Dev. 0.92 2.45

Validation Sample 580 442 injuries + 95 fatalities

Min 0.20 0.40

Mean 0.98 2.01

Max 5.00 5.20

St. Dev. 1.04 2.30

Node 1

N=187

Node 2 N= 132

AADT ≤

4098veh/day

Node 4

N= 124

L ≤ 2.84 km

Node 6

N= 85

CI ≤ 0.802

Node 8

N= 55

l ≤ 3.6 m

Node 10

N= 49

AADT ≤ 3927veh/day

Node 12 N= 39

VG ≤ 3.9%

Node 14 N= 20

VG ≤ 1.5%

Node 15 N= 19VG > 1.5%

Node 13 N= 10

VG > 3.9%

Node 11

N= 6

AADT > 3927veh/day

Node 9

N= 30

l > 3.6 m

Node 16

N= 23

L ≤ 1.89 km

Node 18

N= 10


Node 19

N= 13

AADT > 3881veh/day

Node 17

N= 7

L > 1.89 km

Node 7

N= 39

CI > 0.802

Node 20 N= 9

VG ≤ 1.2%

Node 21

N= 30

VG > 1.2%

Node 22

N= 24

L ≤ 1.79 km

Node 24 N= 9


Node 25 N= 15

AADT > 2237veh/day

Node 23

N= 6

L > 1.79 km

Node 5

N= 8

L > 2.84 km

Node 3 N= 55

AADT > 4098veh/day

Node 27

N= 12

VG > 0.8%

Node 26

N= 43

VG ≤ 0.8%

Node 28

N= 36

CI ≤ 0.75

Node 30

N= 15

L ≤ 2.40 km

Node 31

N= 21

L > 2.40 km

Node 32

N= 5


Node 33

N= 16

AADT > 5035veh/day

Node 29

N= 7

CI > 0.75

Fig. 1. Theoutput resultingfrom Cart Datamining for a sample calibration.

balanced comparisonsbetween available HSM SPFs and new SPFs

developed;• filtering each study dataset by using the 3 standard deviation

(3) method for removing anomalous crash frequencies per year

for injuries, fatalities and all casualties on the homogeneous

road segments. The 3 method makes it possible to check the

homogeneity distribution around the mean, and the maximum

deviation of frequency distribution is equal to 3. The sever-

ity frequencies falling outside mean± 3 × std.dev . ( ± 3 × ) are

removed from each of the four datasets both under base condi-

tions and non-base conditions before moving to the calibration

step.

HSM base conditions for rural two-lane two-way roads are

defined in Section 3. According to the data available in the crash

database, it was observed that new base conditions should be

defined according to the functional meaning of CMF parame-

ters and the experimental relationships found among explanatory




Fig. 2. Fatality frequency profile with changing explanatory variables: a—Segment Length and vertical grade, b—Curvature indicator and segment length, c—Curvature

indicator and vertical grade.

Fig. 3. Overview of thedata analysis procedure.




Fig. 4. Casualty frequency observed as a function of AADT and LW.

Table 4

Overview of HSM CMFs.

LW (CMF1r ), ft AADT, v eh/day

<400 400–2000 >20009 or less 1.05 1.05 + 2.81×10−4 (AADT-400) 1.50

10 1.02 1.02 + 1.75×10−4 (AADT-400) 1.30

11 1.01 1.01 + 2.5×10−5 (AADT-400) 1.05

12 or more 1.00 1.00 1.00

Table 5

Results of filtering technique.

Geometric Conditions injuryfrequency per year

involving homogenous road

segments where only

injuries were recorded [%]

injury+ fatality frequency per

year involving homogenous

road segments where at least

onefatality was recorded

with at least oneinjury [%]

Base VG≤1%+CI≥0.8 ±=70% ±=100%

±2=23%

±3= 7%

No base ±=86% ±=100%

±2=12%

±3= 2%

variables and the crash frequency of casualties. The new base con-

ditions differ partly from those defined by the HSM because the

crash frequency of casualties is not constant with a lane width of

12ft, varying the AADT, as shown in Fig. 4, as opposed to what is

described in the HSM where it can be observed that the CMF = 1 for

LWgreater than 12ft (see Table 4). The legend of the colors in Fig. 4

refers to thevariableto be predictedon the vertical axis “frequency

of the all casualties per year”.

The new geometric base conditions for each of the 4 main sub-sets converged on a VG≤1%+CI ≥0.8 where the crash frequency

per year of injuries and injuries+ fatalities (all casualties), when

this last situation happened on homogeneous road segments, was

actually lower than the general mean observed value inside thefull

database.

As shown in Fig. 3, a total of 187 homogeneous road segments

were used to calibrate SPFs, and the 3 method filtering technique,

as shown below, was applied to remove anomalous frequencies

at each of the four datasets. Table 5 shows the results of the 3

method where at least 93% of the values per dataset involve seg-

ments where only injuries were collected falling within the range

× ± 2× both under base and non-base geometric conditions,

while for the remaining three sub-sets, all frequencies fall within

the corresponding range× ± ×.

According to the HSM methodology and to the data available

in the crash database, for each of the four subsets in Fig. 3, several

CMFs were calculated: CMFs for lane width (CMFLW), vertical grade

(CMFVG) and curvature indicator (CMFCI). For the remaining vari-

ables that the HSM suggested be checked in order verify whether a

road segment can reflect “base” geometric conditions, the authors

clarify some features of the road network investigated:

• road surface and shoulders are paved;• no spiral transition curves exist between tangent segments and

circular curves;• the radii of the horizontal curves were not available, so the CI

measurement of the CCR was used (CMFCI);• driveway density is less than 5 driveways per mile and default

value of CMF is 1.00 accordingto thevaluesuggested by theHSM;• no centerline rumble strips are installed on undivided highways

along the centerline of the roadway which can divide oppos-

ing directions of traffic flow, and default value of CMF is 1.00

according to the value suggested by the HSM;• no passing lanes exist,and thedefaultvalueof CMFis 1.00accord-

ing to the value suggested by the HSM;• no center two-way left-turn lanes exist, and the default value of

CMF is 1.00 according to the value suggested by the HSM;• no automated speed enforcement exists, and the default value of

CMF is 1.00 according to the value suggested by the HSM;• shoulder width is usually between 3.28 ft and 6.56 ft, but no

properly documented details are currently available, and future

developments will be addressed to investigate the effects of this

variable on the crash frequency of injuries and fatalities per year;• super elevationof the horizontal curves is not availableand same

considerations clarified for the shoulder parameter need to be

kept;• no info about lighting of the road segments, and some consider-

ations clarified for previous parameters are to be kept.

5.1. SPFs predicting frequency of injuries only per yr on road

segments where no fatalities occurred during the period of study

5.1.1. Base geometric conditions

A total of 30 road segments (17% of the total road network

lengths involved in the calibration phase) met the base conditions.

Thus,the“N base” was calibratedunder NB distribution forpredicting

frequency of the injuries only per year on homogeneous road seg-

ments where no fatalities were observed during the study period

on the investigated road segments as shown in Eq. (5) in Table 6.

Eq. (5) reflects its original form shown in Eq. (2) with L variable

in mile and new coefficient of the negative exponential function

equal to −2.13 instead of −0.312 in the baseline SPF: this means

that values of frequency predicted by using SPF in Table 6 are

lower than those returned by HSM SPFs. We point out here that the

coefficient of variation C.V. (standard deviation divided by meanvalue) has been calculated both to calibrate Nbase for this subset

(injury frequency per year for homogenous road segments where

only injuries were collected) and for the remaining SPFs related to

other recognized datasets. It is an efficient indicator of measure-

ment dispersion for comparing different subsets and ensuring a

homogeneous sample. It was noted every time for each sample of

explanatory variables and variables to be predicted, that the C.V.

was less than 1: this means that homogeneous dataset exists to

be used for modeling. Table 6 also shows the CV for each variable

adopted in the performing model. The reliability of CMFs will be

confirmed during the validation process of each function through

an analysis of theresiduals that areestimates of experimentalerror

obtained by subtracting the observed responses from the predicted

responses. The predicted response is calculated from the chosen




Table 6

TheSPF on Road Segmentsreflecting newbase geometric conditions where no fatalities occurred.

SPF Std. Error p-level Lo. Conf. Limit Up. Conf. Limit 2

N no−base = AADT × L × 365 × 10−6× e−2.13 (5) 0.16 0.000002 −2.44 −1.81 0.87

L, mile AADT, vehicles/day Injurious crash frequency per year

C.V. 0.57 0.64 0.98

model built from the experimental data. Examining residuals is a

key part of all statistical modeling; carefully looking at residuals

can tell us whether our assumptions are reasonable and our choice

of function is appropriate.

5.1.2. Road segments that do not meet base geometric conditions

A total of 132 road segments (68% of the total road network

length involved in the calibration phase) did not meet local base

conditions, so new CMFs following HSM methodology for lane

width (CMFLW), curvature indicator (CMFCI) and vertical grade

(CMFVG) were calculated as mentioned in Section 3. The CMFs

are the ratio of the mean value of the estimated average crash

frequencies for specific sites under non-base conditions over the

mean value of the estimated average crash frequencies for spe-

cific site base conditions. Under base conditions, the CMF is 1.00.

As explained in Section 3, we should point out here the procedure

that has been used through the manuscript when road segments

did not meet “base” geometric conditions. For each class that has

beenformedto properlyreflect the observedyearly crashfrequency

of injuries, and fatalities where applicable, a C.V. has been calcu-

lated, to make consistent and reliable values for CMFs by varying

the geometric explanatory variables investigated.

5.1.2.1. Lanewidth (LW)CMF. Three classes of LW along with three

AADT classes were found and suggested by the HSM procedure

following an iterative process for maximizing the validation pro-

cedure’s results (see Table 7).

Table 7 shows a mean value for the crash frequency per year of

the injuries only on homogeneous road segments reflecting base

(f b) and non-base conditions (f nb) at each combination of the AADTandLW. No severity frequencywas observed forthe last class (a not

available n.a. expression was used); thus, the potential CMF value

for the applications should be the mean of the closest classes.

5.1.2.2. Curvature indicator (CI) CMF. On the basis of the two

preliminary analyses discussed in Section 4, constraints were

recognized for maximizing results during the calibration and vali-

dation phases of the SPFs:

• no fatalities when VG< 5%+ L >3 km+ CI> 0.8;• base geometric conditions when VG≤1%+CI≥0.8.

Thus, for this specific subset (frequency of the injuries only per

year on road segments where no fatalities occurred during theperiod of study), the CMF for the CI is 1.00.

5.1.2.3. Vertical grade (VG) CMF. Three classes of VG parameter

were suggested by an iterative process for maximizing the vali-

dation procedure results (lower than 1%, from 1% to 4%, from 4% to

5%). All segments whose VG fell outside the base conditions were

modified by a CMF value as shown in Table 8.

5.1.2.4. Calibration factorC x. A calibration factor Cx was developed

to adjust the SPFs to local conditions as suggested by the HSM

method. Andit hasbeen calculated by using Eq. (4) equal,according

toHSM procedure, to theratio between theweightedaverage of the

observed number of injuries at all the segments studied, where the

weighting refers to the length of the roadway segment, in relation

to the weighted total of the predicted number of injuries (see Eq.

(6))

C x =

all road segments

observed injuries

all road segments

predicte dinjuries(6)

Cx equals 0.587 for the prediction of the frequency per year of

the onlyinjuries on homogeneousroad segments reflecting specific

geometric conditions where no fatalities occurred.

In conclusion, the final complete predictive model of only fre-

quency of injuries per year on homogeneous road segments under

non-base conditions for which no fatalities were recorded reflects

Eq. (1), asmay beseen in Eq. (7):

N no−base = N base × (CMF LW × CMF CI × CMF VG) × C x (7)

that can be rewritten more explicitly as follows in Eq. (8)

N no−base = AADT × L × 365× 10−6

× e−2.13

× (CMF LW × CMF CI × CMF VG) × 0.587 (8)

where

• CMFLW is available in Table 7;• CMFCI is 1.00 for this subset of homogeneous road segments

where no fatality was collected for the specific geometric con-

ditions related to the CI parameter to be adjusted (no fatality was

observed when VG< 5%+ L> 3km + CI> 0.8 that overlaps, focus-

ing attention only on the CI to be analyzed here, with a part

of the new local base geometric conditions corresponding to

VG≤1%+CI≥0.8);• CMFVG is available in Table 8.

5.2. SPFs on homogenous road segments with fatalities and

injuries

5.2.1. Base geometric conditions

The same methodology applied in Section 5.1 was used to

calibrate SPFs when fatalities occurred on homogeneous road seg-

mentsunder base geometric conditions“N base” (see Table 9). A total

of 8 road segments (3% of the total road network length involved in

the calibration phase) met the new base conditions shown above.

Table 9 shows several equations as follows:

• Eq. (9) that predicts only the portion of injuries on segments

where both injuries and fatalities were recorded in the period

throughout the duration of the study;• Eq. (10) that predicts only the portion of fatalities on segments

where both injuries and fatalities were recorded throughout the

duration of the study;• Eq. (11) that predicts the total casualties per year.

To assess only the frequency of injurious crashes and fatali-

ties per year from Eq. (11), it is necessary to apply 79% and 21%

respectively as shown in Eqs. (12) and (13).




Table 7

LW CMFs.

LW, ft AADT, veh/day

<3000 3000≤AADT≤7500 >7500

3.2 or less f b =0.59, f nb = 1.06 f b =0.63, f nb = 1.13 f b =0.70, f nb = 1.23

CMF = 1.80 CMF = 1.79 CMF = 1.76

3.2 < l≤4.5 f b =0.57, f nb = 0.75 f b =0.62, f nb = 0.80 f b =0.68, f nb = 0.82

CMF = 1.32 CMF = 1.29 CMF = 1.21

4.5 or more f b =0.55, f nb = 0.71 f b =0.60, f nb = 0.72 f b = n.a.

CMF = 1.29 CMF = 1.20 f nb = n.a.

3.2 or less C.V. of “f b” sample: 0.69 C.V. of “f b” sample: 0.70 C.V. of “f b” sample: 0.66

C.V. of“f nb” sample: 0.98 C.V. of “f nb” sample: 0.89 C.V. of “f nb” sample: 0.84

3.2 < l≤4.5 C.V. of “f b” sample: 0.86 C.V. of “f b” sample: 0.64 C.V. of “f b” sample: 0.81

C.V. of“f nb” sample: 0.60 C.V. of “f nb” sample: 0.65 C.V. of “f nb” sample: 0.52

4.5 or more C.V. of “f b” sample: 0.71 C.V. of “f b” sample: 0.83 n.a.

C.V. of “f nb” sample: 0.70 C.V. of “f nb” sample: 0.87 n.a.

Table 8

VG CMFs.

VG, %

≤1 1 < CI≤4 4 < CI≤5

f b =0.70, f nb = 0.70 f b =0.70, f nb = 0.86 f b =0.70, f nb = 0.94

CMF = 1.00 CMF = 1.23 CMF =1.34

C.V. of“f b” sample: 0.98 (see Table 6) C.V.of “fnb” sample: 0.96 C.V. of “fnb” sample: 0.95

Table 9

SPFs on Road Segments with injuries and fatalities reflecting base geometric conditions.

SPFS Std. Error p-level Lo. Conf. Limit Up. Conf. Limit 2

N base injuries = AADT × L × 365 × 10−6× e−1.93 (9) 0.17 0.000008 −2.32 −1.54 0.85

N base fatalities = AADT × L × 365 × 10−6× e−3.55 (10) 0.46 0.000011 −4.64 −2.47 0.82

N base allcasualties = AADT × L × 365 × 10−6× e−1.75 (11) 0.16 0.000009 −2.12 −1.39 0.87

N baseinjuries from allcasualties = 0.79 ×N baseall casualties (12)

N base fatalities fromall casualties = 0

.21 ×

N baseall casuaties (13)

5.2.2. Road segments that do not meet base geometric conditions

A total of 17roadsegments(12%of thetotal road network length

involved in the calibration phase) did not meet local base condi-

tions, so new CMFs for LW (CMFLW), CI (CMFCI) and VG (CMFVG)

were calculated as mentioned in Section 3.

5.2.2.1. LW CMF. Table 10 shows CMF values for adjusting the

effect of LW.

5.2.2.2. CI CMF. The curvature indicator is different from 1.00, so

three classes of curvature indicator were suggested to maximize

the validation results. The curvature indicators that did not reflect

the base settings were modified by the CMF values in Table 11.

5.2.2.3. VG CMF. Three VG classes were suggested on the basis of

a correlation analysis for three of the datasets. All segments whose

VG fell outside the base conditions were modified by a CMF value

as shown in Table 12.

5.2.2.4. Calibration factor C x. Cx was developed to adjust SPFs to

local conditions as suggested by the HSM method. It is a multi-

plicative factor and is applied to the equations in Table 9 and their

CMFs (from Tables 10–12) for homogenous segments that do not

meet base geometric conditions.

Calibration factor Cx for SPF that predicts the frequency of only

injuries per year when injuries+ fatalities happened is 0.224 as a

result of Eq. (14).

Cx =

all road segments

observedinjuries

all road segments

predictedinjuries(14)

Calibration factor Cx for SPF that predicts the frequency only of

fatalities per year when injuries+ fatalities happened is 0.292 as a

result of Eq. (15).

Cx =

all road segments

observed fatalities

all road segments

predicted fatalities(15)

Calibration factor Cx for the SPF that predicts the yearly fre-

quency of total casualties (injuries + fatalities) is 0.68 as a result of

Eq. (16).

Cx =

all roadsegments

observed all casualties

allroad segments

predicted all casualties(16)




Table 10

Lane Width CMFs.

LW, ft AADT, veh/day

<3000 3000≤AADT≤7500 >7500

(a) all casualties

3.2 or less f b = n.a. f b =1.60, f nb = 2.27 n.a.

f nb = 2.17 CMF =1.42

3.2 < l≤4.5 f b = 1.00 f b =1.20, f nb = 1.62 f b =1.54, f nb = 1.97

f nb = n.a CMF = 1.35 CMF =1.28

4.5 or more f b = n.a. f b = n.a. f b =1.40, f nb = 1.58

f nb = 1.43 f nb = 1.46 CMF =1.13

(b) injuries

3.2 or less f b = n.a. f b =1.30, f nb = 1.80 n.a.

f nb = 1.78 CMF = 1.38

3.2 < l≤4.5 f b = 0.80 f b =1.00, f nb = 1.31 f b =1.27, f nb = 1.56

f nb = n.a. CMF = 1.31 CMF =1.23


f nb = 1.15 f nb = 1.17 CMF =1.07

(c) fatalities

3.2 or less f b = n.a. f b =0.30, f b = 0.47 n.a.

f nb = 0.39 CMF =1.57

3.2 < l≤4.5 f b = 0.20 f b =0.20, f nb = 0.31 f b =0.27, f nb = 0.41

f nb = n.a. CMF = 1.55 CMF = 1.52


f nb = 0.28 f nb = 0.29 CMF =1.50

frequency of crash injuries

3.2 or less C.V. of “f nb” sample: 0.39 C.V. of “f b” sample: 0.85 n.a.

C.V. of “f nb” sample: 0.47


C.V. of “f nb” sample: 0.31 C.V. of “f nb” sample: 0.41

4.5 or more C.V. of “f nb” sample: 0.35 C.V. of “f nb” sample: 0.29 C.V. of “f b” sample: 0.02


frequency of crash fatalities

3.2 or less C.V. of “f nb” sample: 0.58 C.V. of “f b” sample: 0.85 n.a.



C.V. of “f nb” sample: 0.20 C.V. of “f nb” sample: 0.74

4.5 or more C.V. of “f nb” sample: 0.28 C.V. of “f nb” sample: 0.29 C.V. of “f b” sample: 0.02

C.V. of “f nb” sample:0.20

Themeaningof thefactors in Table 10 is as follows:(a) frequencyof allcasualties peryear; (b)frequency of injuriesper year; (c)frequencyof fatalitiesper year; notavailable

(n.a). No severity frequency wasdetected forsome traffic/lane width combinationsand consequently, potential CMFvalues may be themean of theclosest classes.

Table 11

CI CMFs.

Curvature Indicator [−]

<0.623 0.623≤CI≤0.8 >0.8

all casualties


CMF = 1.81 CMF = 1.42 CMF =1.00

injuries


CMF = 1.84 CMF = 1.49 CMF =1.00

fatalities

f b =0.23, f nb = 0.37 f b =0.23, f nb = 0.34 f b =0.23, f nb = 0.23CMF = 1.61 CMF = 1.48 CMF =1.00


<0.623 0.623≤CI≤0.8 >0.8

C.V. of“f nb” sample: 0.91 C.V. of “f nb” sample: 0.95 C.V. of “f b” sample: 0.86


<0.623 0.623≤CI≤0.8 >0.8

C.V. of“f nb” sample: 0.89 C.V. of “f nb” sample: 0.98 C.V. of “f b” sample: 0.84

In conclusion, to predict thefrequencyof injuries peryear or the

frequency of fatalities per year, or the total frequency of casualties

per year on homogeneous road segments that do not meet local

base conditions on two-lane rural roads, the equations in Table 9

should be used by adopting and differentiating the values of the

CMFs and Calibration factor Cx for each model that is needed to

predict one of the three defined variables.




Table 12

Vertical Grade CMFs.

Vertical Grade,%

≤1 1 < VG≤4 >4

all casualties


CMF = 1.00 CMF = 1.21 CMF = 1.74

injuries

f b =1.22, f nb = 1.22 f b =1.22, f nb = 1.45 f b =1.22, f nb = 2.16CMF = 1.00 CMF = 1.19 CMF = 1.36

fatalities


CMF = 1.00 CMF = 1.35 CMF = 1.61


<0.623 0.623≤CI≤0.8 >0.8

C.V. of“f b” sample: 0.87 C.V. of “f nb” sample: 0.91 C.V. of “f nb” sample: 0.96


<0.623 0.623≤CI≤0.8 > 0.8

C.V. of“f b” sample: 0.94 C.V. of “f nb” sample: 0.91 C.V. of “f nb” sample: 0.83

Table 13

Overviewof the SPFs for each of the 4 main defined subsets.

Study severity crash

frequency per year

Local geometric c onditions Equation

frequency of injuries per

year on road segments

where no fatalities

occurred

Base geometric conditions N base = AADT × L × 365 × 10−6× e−2.13 (5)

No base geometric

conditions

N nobase = ( AADT × L × 365 × 10−6× e−2.13) × (CMF LW × CMF CI × CMF VG) × 0.587 (8)

CMFLW see Table 7, CMFCI =1, CMFVG see Table 8

frequency of injuries and

fatalities peryear on road

segments

Base geometric conditions N baseinjuries = AADT × L×365×10-6 × e−1.93 (9)

N basefatalities = AADT × L×365×10-6 × e−3.55 (10)

N baseallcasualties = AADT × L×365×10-6 × e−1.75 (11)

N base injuries from N base all casualties = 0.79 ×N baseall casualties (12)

N base fatalities from N base all casualties = 0.21 ×N base all casualties (13)No base geometric

conditions

N nob aseinjuries = ( AADT × L×365×10-6 × e−1.93 )× (CMF LW ×CMF CI ×CMF VG)×0.224 (17)

CMFLW see Table 10, CMFCI see Table 11, CMFVG see Table 12

N nob asefatalities = ( AADT × L×365×10-6 × e−3.55 )× ( AMF LW × AMF CI × AMF VG)×0.292 (18)


N nob aseallcasualties = ( AADT × L×365×10-6 × e−1.75 )× (CMF LW × CMF CI ×CMF VG)×0.68 (19)

N no baseinjuries fromN no base all casualties = 0.79× N no base allcasualties (20)

N no basefatalitiesfrom N no base all casualties = 0.21 × N no base allcasualties × 0.49 (21)


5.3. Overview of calibrated SPFs to predict the yearly crash

frequency of injuries and fatalities on homogeneousroad segments

Table 13 brings together all the SPFs that have been calibratedand presented in this manuscript for predicting crash severity for

4 main study subsets.

The SPFs in Table 13 are good from the point of view of sta-

tistical significance, taking into consideration the p-value of the

coefficients, the CV of each investigated explanatory variable, and

the results of the validation procedure as will be discussed later on.

6. Validation phase

Almost1000km were used totest thereliabilityof previous SPFs

covering the fiveyears of crash data from 2006 to 2010 to a total of

442 injuries and 95 fatalities. The validation procedure focused on

residuals analysis; the residual is the difference between observed

andpredictedinjuries, fatalities or allcasualties crashfrequencyper

year at each homogeneous road segment depending on the study

variable to be predicted and the road geometric context. Resid-

uals are estimates of experimental error obtained by subtracting

the observed responses from the predicted responses. In particular,several parameter evaluations were considered:

• MAD (Mean Absolute Deviation) equals the sum of the absolute

values of the difference between predicted and observed crash

severity frequency per year over the number of segments reflect-

ing the specific geometric conditions;• MSE equals the Mean Square Error;• CV has been assessed as the square root of MSE over the mean

value of the predicted variable in question according to the mod-

els in Table 13.

Residual analysis is an essential tool in this process since it

makes it possible to identify where the predictive models may

miss the mark, over- or underestimating the real predicted crash




Table 14

Descriptive statistics of the validation procedure results.

Study severity crash frequency per year Equation MAD MSE C.V.

frequency of injuries per

year on road segments

where no fatalities

occurred

N base = AADT × L × 365 × 10−6× e−2.13 (5) 0.24 0.146 0.62

N nobase = ( AADT × L × 365 × 10−6× e−2.13 ) × (CMF LW × CMF CI × CMF VG) × 0.587 (8) 0.15 0.09 0.50

CMFLW see Table 7, CMFCI =1, CMFVG see Table8

frequency of injuries andfatalitiesper year on road

segments

N baseinjuries = AADT × L×365×10-6

× e−1.93

(9) 0.28 0.17 0.16

N basefatalities = AADT × L×365×10-6 × e−3.55 (10) 0.15 0.04 0.75

N baseallcasualties = AADT × L×365×10-6 × e−1.75 (11) 0.26 0.19 0.18

N base injuries from N base all casualties = 0.79 × N baseall casualties (12) 0.23 0.19 0.80

N base fatalities from N base all casualties = 0.21 × N base all casualties (13) 0.21 0.06 0.91

N nob aseinjuries = ( AADT × L×365×10-6 ×e−1.93 )× (CMF LW × CMF CI ×CMF VG)×0.224 (17) 0.11 0.03 0.07


N nob asefatalities = ( AADT × L×365×10-6 × e−3.55 )× ( AMF LW × AMF CI × AMF VG)×0.292 (18) 0.17 0.04 0.77


N nob aseallcasualties = ( AADT × L×365×10-6 × e−1.75 )× (CMF LW × CMF CI ×CMF VG)×0.68 (19) 0.15 0.09 0.13

N no baseinjuries fromN no base all casualties = 0.79 ×N no base allcasualties (20) 0.10 0.12 0.08

N no basefatalitiesfrom N no base all casualties = 0.21 × N no base allcasualties × 0.49 (21) 0.16 0.04 0.69


frequency. It emerges as shown in Table 14, that MAD was lowerthan 0.3, MSE lower than 0.10 and C.V. was lower than 1 at each

subset, which means models reflect properly observed crash phe-

nomena investigated on the ruralroad segments and return reliable

responses.

The validation procedure also consisted in the construction of

the diagrams of the Cumulative squared Residuals plotted for each

SPF in Fig.5 on the basis ofan increasing order of AADT tocheck the

absence of vertical jumps. A vertical jump reflects the lack of flexi-

bilityin thefunctionalform in themodel. SPFthatpredicts thecrash

casualties frequencyper year canbe reliable when no vertical drops

that can be called “outliers” exist in the CURE profile. An outlier is

an observation point that is distant from other observations.

The presence of outliers means that the residuals contain struc-

tures that are not accounted for in the model. The CURE plot showshow well or how poorly the SPF would predict for various values

of the independent variable. The residuals analysis is one of sev-

eral basic methods availablein the scientific literature to assess the

goodness offit ofa model.Thisprocedureconsistsof plottingCumu-

lated squared Residuals for study-specific independent variable in

ascending order. The CumulatedSquared Residuals plot shouldnot

have long increasing or decreasing runs because they correspond

to regions of consistent over andunderestimation. . . (Hauer, 2004;

Hauer, 2015).

Fig. 5 proves the absence of huge jumps in the diagram of cumu-

lated squared residuals for each predictive function, confirming

the reliability of the SPFs in crash frequency prediction per year of

injuries,fatalities andall casualtieson homogeneous roadsegments

on the basis of an increasing order of AADT.

A Kruskal-Wallis (KW) test (Kruskal and Wallis, 1952) was car-

ried out based on Eq. (22) to check the following questions:

• Are the crash frequencies per year for injuries only (when fatal-

ities also happened) returned by Eqs. (9) and (17) statistically

equal to the values returned by Eqs. (11) and (19) revised by a

percentage equaling 79% of the total casualties crashes (see Eqs.

(12) and (20))?• Are the crash frequencies per year for fatalities only (when

injuries also happened) returned by Eqs. (10) and (18) statisti-

cally equal to the values returned by Eqs. (11) and (19) revised

by a percentage equaling 21% of the total casualties crashes (see

Eqs. (12) and (21))?

KW =

12

N (N + 1)

ki=1

R2i

ni

− 3 × (N + 1) (22)

where:

ni is the number of observations at each subset i

R i is the sum of the rank of observation at each subset i

N is the total number of observations.

The KW test is a one-way analysis of variance by ranks and it

is a non-parametric method for testing whether samples originate

from the same sample. The assumptions of the test are that data

are not normally distributed and that there is no heteroscedastic-

ity. The Levene test was used to verify heteroscedasticity. The null

hypothesis ofthe KW test is that themeanvalueof each group is not

statistically different from that relating to the sample with which

it is compared. The level of significance is 0.05.

Table 15 shows the results of the KW test where it is confirmed

that only equations 11 for base conditions and 19 for no-base con-

ditions, can be used to predict the frequency of crash frequency

per year of injuries and fatalities on homogeneous road segments

by using accurate distribution percentages for injuries and fatali-

ties instead of two SPFs calibrated ad hoc. This makes applications

easier and faster.

7. Results

Treasuring the careful study of the crash database and on the

basis of the results achieved, the factors having greatest impact

on the yearly frequency of crash injuries and fatalities to be pre-dictedwere recognized as follows: L, AADT, LW, CI,and VG.Definite

combinations of road geometric features which influence the con-

sequences of a crash in terms of number of injuries and fatalities

were taken into account for the calibration of the SPFs. The effects

of VG,LW andCI on thecrashfrequency of injuries andfatalities per

year was assessed using CMFs reflecting HSM procedure (AASHTO,

2010).

Fig. 6 is an example of the graphical profile of the SPF calibrated

to predict the yearly frequency of all casualties on homogeneous

road segments, taking into consideration the results of the vali-

dation procedure and the KW test. In particular, considering that

when homogeneous road segments do not meet the base geomet-

ric conditions defined on the study road network, three additional

explanatory variables are to be included (VG, CI, Lane width) with




c. See Eqs 10 -18 predicting freq. for only fatalities d. See Eqs 13-19 predicting freq. for the total casualties

per yr when injuries and fatalities happened on segments per yr on road segments

e. See Eqs. 12-20 predicting freq. for only injuries from an f. See Equ. 13-21 predicting freq. for only fatalities from

SPF predicting total casualties frequency per year an SPF predicting total casualties frequency per year

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000 12000 C u m u l a t e d

S q u a r e d

R e s i d u a l s

AADT

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2000 4000 6000 8000 10000 12000

C u m u l a t e d

S q u a r e d

R e s i d

u a l s

AADT

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2000 4000 6000 8000 10000 12000

C u m u l a t e d

S q u a r e d

R e s i d u a l s

AADT

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2000 4000 6000 8000 10000 12000

C u m u l a t e d

S q u a r e d R e s i d u a l s

AADT

a.See Eqs 5-8 predicting freq. for injuries per yr on road b. See Eqs 9-17 predicting freq. for only injuries

segments where no fatalities happened per yr on segments when fatalities and injuries happened

0

2

4

6

8

10

12

14

16

18

20

0

2000

4000

6000

8000

10000

12000

C u m u l a t e d

S q u a r e d

R e s i d u a l s

AADT

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 2000 4000 6000 8000 10000 12000

C u m u l a t e d S q u a r e d R e s i d u a l s

AADT

Fig. 5. Cumulated Squared Residuals diagrams for the SPFs.

Table 15

Results of theKW test.

Npredicted Degree of freedom Number of groupings p-value KW KWthreshold

N injuries 2 3 0.919 0.169 5.99

Nfatalities 2 3 0.881 0.253




Fig. 6. SPF Trend based on Eq. (10), changing AADT and setting some geometric

parameters.

respect to the “base” conditions (LW, AADT), some explanatory

variables have been fixed to simplify the construction of the aba-

cus, keeping constant the remaining. Of course, the frequency per

year for injuries and the frequency per year for fatalities are imme-

diately consequential and can be calculated by applying specific

percentages as described in Section 5.3, Section 6 in the validation

procedure and as shown in Table 13. It is possible to identify all the

potentialcombinations of existing explicative variables thathelp to

reduce the dangerof each identifiedscenario.The numberof possi-

ble structural road interventions to reduce the crash frequency per

year of injuries, fatalities, and all casualties, on homogeneous road

segments is equal to the number of availablevariables employed in

the model on which it is actually possible to work to improve road

safety conditions. In this way, working on more than one variable,

the crash cost can be reduced, represented graphically by a shift

from the upper lines of the graphic profile to lower lines even if

costly intervention is required. The degree of effectiveness of per-

forming an intervention on a road, assuming that one is acting only

on one variable, cannot be separated from the analysis of the com-

bined effects produced by the remaining variables as a result of an adjustment made on only explanatory variables. Fig. 6 refers

to some established values for the explanatory variables defined

in Eqs. (11) and (19): LW (3.45 m ), VG (2.25% CMF: 1.21), and L

(4.16 m). Other values are variable, such as LW CMFs, that change

according to LW and AADT, as well as three CI CMF values that

change according to the CI (CI = 0.1, CCR:10gon/km CMF: 1.81; CI:

0.7, CCR: 289 gon/km CMF: 1.63; CI:1, CCR: 847 gon/km CMF:1). In

Fig.6, the y-axis shows thetotal frequencyper year forall casualties

on a homogeneous road segment, while the x-axis shows an inde-

pendent variable of the predictive model, for example, the AADT

whose range of values reflects those available in Table 3. Fig. 6

presents a series of straight lines with a constant value for the CI

that reflects the degree of non-tortuosity of the horizontal align-

ment geometrically evaluated through the CCR parameter. It canbe observed that moving from the lower line associated with the

lowest CCR to the upper lines associated with the highest CCR, the

predicted crash frequency of the all casualties per year increases.

For example, having fixed a value for the AADT on the x-axis (AADT

of 3350 vehicles per day) and the mean tortuosity of the road seg-

ment (CI of 0.7), the predicted crash frequency per year of the

all casualties on the homogeneous segment can be assessed: the

blue circle in Fig. 6 represents condition to be investigated and the

predicted crash frequency per year for all casualties is 1.98.

Looking at Fig. 6, which is a graphical interpretation of Eqs.

(11) and (19), strategies can be suggested to improve safety condi-

tionsworking,for example,on the tortuosity parameter. A potential

approach to reducing the crash frequency per year for the total

number of the casualties corresponds to a decrease in the CCR,

namely an increase in the CI parameter that reflects the non-

tortuosity of the horizontal alignment. Moving from the blue circle

along the red arrow towards red circle where the AADT is kept

while the CCR is reduced, the predicted frequency per year of the

all casualties on a homogeneous segment reduces from 1.98 to

1.11. This type of strategy is a structural road adjustment of the

road horizontal alignment: working on the deviation angles of the

existing horizontalprofileby varying the features of the existing cir-

cular curves, tangent segment length and spiral transition curves

between circular curves and tangents that belong to the studied

road alignment, and keeping proper juxtaposition with a vertical

alignment, a new CCR would have to be calculated. Both consis-

tency and safety can improve and, consequently, new CI parameter

can be defined. The Italian Standard requires specific attention to

the design of the single tangent segment and circular curve, some

of which are as follows:

• tangentsegmentlength must be less than 22timesthe maximum

design speed and greater than the minimum length associated

with the maximum design speed;• the horizontal curve radius should be greater than the near tan-

gent segment lengths if they are less than 300 m , otherwise

greater than 400m;• the horizontal minimum radius of a circular curve should satisfy

stability criteria with regard to heeling and overturning of vehi-

cles travelling along the circular curve taking into account the

minimum values of design speed, cross slope, and cross friction;• the circular curve should be traveled in more than 2.5 s;• the maximum vertical grade is established on the basis of the

functional classification of the roadway and not less of 1%;• the radius of the vertical curve should be designed to ensure a

maximum valueof the lateral acceleration of 0.6m/s2 withavalue

that should ensure a satisfactory space to stop vehicles, with a

minimum value for the curve radius of 40m sacs and 20m crests.

By adopting a new value for CI in the abacus of Fig. 6, or by

applying specific SPFs in Table 13, it becomes possible to check

new predicted total casualty frequencies per year and to verify thegoodness and effectiveness of the suggested strategies.

It can work in the same way with LW. Looking at the CMF val-

ues in Tables 10–12, it is obvious how CMF values decrease with

increasing LW. Thus, other types of road adjustment to reduce

casualty frequency includes amplifying LW on the basis of Road

Geometric Design Standard requirements. As stated in Section

5.2.2.1, having identified the new value of the variable LW, it is

possible to assess a new CMF value and to adopt a specific SPF in

Table 10 f or the prediction of the new casualty frequency per year.

In the same way, it is possible to work on the variable VGby chang-

ingthe vertical profileof theroadalignment in full compliance with

horizontal alignment.

The observations that have been formulated to reduce the pre-

dictedcasualtyfrequency peryear canalso be repeated andappliedto the remaining SPFs available in Table 13 keeping in mind the

premises on geometric roadfeature combinations whereno fatality

happened during the study period.

In conclusion, the number of possible profiles for the SPFs is

equal to the number of available variables employed in the model

on which it can actually work to improve road safety conditions.

According to the intent of the HSM, the predictive models for

estimating potential crash frequency and severity on highway net-

works can be used to identify sites with the greatest potential

for crash frequency or severity reduction, and factors contribut-

ing to crashes together with associated potential countermeasures

to address these issues. They can thus be used to conduct eco-

nomicappraisals of improvements, prioritize projects,and evaluate

the crash reduction benefits of the treatment implemented. These




applications are used to consider projects and activities related not

only to safety, but also those intended to improve other aspects of

the roadway, such as capacity and the transit service.

8. Conclusions

The main objective of the research presented here is to make a

contribution for bridging the gap existing in the literature where

more road crash frequency prediction models exist than spe-cific functions focused on the prediction of road crash casualties.

No crash that can be associated with the driver’s physiological

causes has been investigated; namely, crashes due to drowsiness,

drunkenness, or distraction. Of course crashes taking place at the

intersections segments were not included in the study. The road

network used to calibrate the SPF does not include road segments

from the validation sub-set. A 5-year period was selected to care-

fully analyze the crash reports. A negative binomial regression

model was used. Reflecting the crash configurations of the dataset

and maximizing the validation outcomes, four main groups of SPFs

were developed by using the HSM procedure for road segments

that meet base and non-base geometric conditions as follows:

• one equation to predict only injury frequency per year for thesubset where only non-fatal injuries occurred;

• two different equations to predict injury frequency and fatal-

ity frequency per year per sub-set where at least one fatality

occurred together with one injury,• only one equation to predict the total frequency per year of total

casualties correlating accurate percentages to obtain the final

expected frequency of injuries and fatalities per year on homo-

geneous road segments.

Themain goal wasto assesswhether only oneSPF able topredict

thetotal casualties crash frequencyper year is reliable andwhether

it is able to return values not statistically different from those we

could have obtained if we had used ad-hoc SPF calibrated for only

injuries and for only fatalities when injuries+ fatalities happened

on the homogeneous road segments during the study period.

The factors having greatest impact on the yearly frequency of

crash injuries and fatalities to be predicted were recognized as fol-

lows: L, AADT, LW, CI, and VG. The effects of VG, LW and CI on

the crash frequency of injuries and fatalities per year was assessed

using CMFs reflecting HSM procedure.

The achieved results confirmed that it is possible to meet the

initial goal on theinvestigatedtwo-laneruralroads, andthis makes

it possible to save time when predicting crash casualties.

The research confirms the effectiveness of the HSM procedure

on the one hand, but on the other it sheds light on the model sug-

gested by HSM for rural undivided roads but which overestimates

the observed crash frequency; in fact the following issues were

observed:

• the negative exponential function of the SPF for road segments

that meet base geometric conditions is always lower in the HSM

procedure than those suggested by the SPFs presented here;• Cx is always greater in the HSM procedure than that suggested

by the SPFs presented here.

Some strategies can be suggested by using and working on the

explanatory variable of the SPFs.

The effect of the geometric variations by changing the values

of the variables numerically makes it possible to quantify the ben-

efit for the studied network in terms of a reduction in the yearly

frequency of injuries and fatalities, and all casualties, depending

on whether the investigated road segment comes under base or

non-base geometric basic conditions.

These functions can help experts in their road maintenance

and construction safety strategies to improve safety conditions

reducing the frequency of injurious and fatal and crashes. An easy

application is to suggest structural countermeasures by acting on

the values of some available explanatory variables whose real and

predicted effects are well known.

Future development will be addressed to the investigation of

the effects of other geometric and non-geometric variables in the

prediction of thefrequency of injuries andfatalities peryear as well

as an examination of the crash data base sourceto carefullyanalyze

how to change the SPFs when only slight injuries may be predicted

from the prediction of only serious crashes.

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SAFETY PERFORMANCE FUNCTIONS FOR CRASH SEVERITY ON UNDIVIDED RURAL ROADS