2006-figueroa y tarko (2007)

MODELING THE ENDOGENOUS RELATIONSHIP BETWEEN DRIVER BEHAVIOR AND HIGHWAY SAFETY

by

Alberto M. Figueroa MedinaEmail: [email protected]

Civil Engineering and Surveying DepartmentUniversity of Puerto Rico

P.O. Box 9041Mayagüez, PR 00681-9041

Phone: (787) 832-4040 (ext. 3418)Fax: (765) 833-8260

and

Andrew P. TarkoEmail: [email protected]

School of Civil EngineeringPurdue University

550 Stadium Mall DriveWest Lafayette, IN 47907

Phone: (765) 494-5027Fax: (765) 496-1105

Submitted for presentation at the 85th Annual Meeting of the Transportation Research Board, January 22-26, 2006, Washington DC

Number of words in manuscript = 5736Tables and figures = 7 x 250 = 1750Total number of words = 7486

TRB 2006 Annual Meeting CD-ROM Paper revised from original submittal.

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ABSTRACT

Misinterpretation of the risk by drivers has been recognized as a significant factor in highway crashes. It is generally accepted that a link exists between objective risk and subjective risk thatis determinant in the likelihood of crash, but the mutual relationship between the driver behavior as influenced by the risk perceived and the crash rate on the road has not been fully studied.

The main objective of this paper is to introduce an exploratory analysis of the relationship between roadway characteristics, driver behavior as influenced by the perceived risk, and safety.An advanced econometric model was developed that uses a system of relationships approach to link driver behavior (free-flow speeds), objective risk (crash rate), and the roadway characteristics of four-lane highways. The proposed system of relationships is formulated using a simultaneous equations model that consists of two structural equations involving two endogenous variables, the objective risk and the speed selected by the drivers.

The interpretation of the interrelationship between the mean speed and the objective risk was straightforward from the system of equations. The results established that the mean speed decreases with an increasing crash rate and in the presence of restricted conditions as represented by the roadway characteristics. On the other hand, the crash rate equation included the effects of the mean speed and six different roadway characteristics. The equation established that the crash rate on the road increases with increasing mean speed and the presence of riskier conditions as represented by the roadway characteristics.

Key words: crash rate and mean speed factors, safety modeling, simultaneous equations, four-lane highways


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INTRODUCTIONHighway crashes are random events that are caused by many and very different contributing factors. Misinterpretation of the risk by drivers has been recognized as a significant factor in highway crashes. It is generally accepted that a link exists between objective risk and subjective risk that is determinant in the likelihood of crash, but the interrelationship between the risk perceived by drivers and the crash rate on the road has not been fully studied. A great amount of research has been dedicated to investigate and estimate the effects of the roadway characteristics on the crash frequency or crash rate, but there is still need for improved methodologies and prediction models. Two limitations of the single-equation approaches typically used in modeling safety is their lack of consideration of driver behavior as influenced by the physical roadway characteristics and their inadequacy in modeling the endogenous relationship between speeds and crashes.

There are two primary motifs of drivers’ selection of speed on a road: safety and mobility. Individual drivers use their perception of the risk on the road to carry out a trade-off between travel time and safety when selecting their speed on a trip. When the perceived risk on the road is low, a typical driver will increase the speed (reducing travel time), and when the perceived risk on the road is high, a typical driver will reduce the speed (increasing safety). This perception, when driving under free-flow conditions, is mainly influenced by the roadway physical characteristics, the driver’s skills, and the performance characteristics of the vehicle. Wright and Boyle (1) indicated that the number of crashes is mainly a function of the extent to which the risk associated with the roadway features of a specific location is greater than the perceived risk by drivers.

This paper seeks to provide additional insight on the association between the objective risk and driver behavior on the road by presenting the results from an exploratory analysis of the interrelationship between roadway characteristics, driver behavior as influenced by the perceived risk, and highway safety. The main objective of the analysis was the development of an advanced econometric model that uses a system of relationships to link driver behavior (free-flow mean speed), objective risk (crash rate), and the roadway characteristics of four-lane highways. DEVELOPMENT OF THE SYSTEM OF SIMULTANEOUS EQUATIONSTwo well-known theories of driver behavior under a risk of crash are the theory of homeostasis (2) and the theory of rational behavior (3, 4). The first theory claims that a driver continuously attempts to reach the target risk by modifying driving behavior. The second theory assumes that a driver minimizes the perceived cost of the trip (or maximizes the trip utility).

The model of rational driver behavior, composed of the perceived travel time cost and the perceived risk cost for an individual driver at a specific highway location, was used as basis to develop the proposed system of relationships. Figure 1 shows the proposed model. The fundamental assumption of the theory is that a driver “feels” the risk of crash and the value of time on the same scale and adjusts his/her behavior to minimize the total cost of “unsafety” and travel time. The perceived risk cost is a growing function of speed while the cost of time is a decreasing function of speed. The point where the sum of the two curves is minimal indicates the speed that is selected by a driver. The reduced form of the model postulates that the


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equilibrium speed is a linear function of the perceived risk by drivers based on the roadway conditions present on the road.

Figure 2a presents the postulated system of relationships. The five components of the system are the roadway characteristics, the speed, the travel time unit cost, the objective risk, and the perceived risk. The objective risk, represented with the crash rate, is assumed to be a function of the particular roadway characteristics and the speed selected by drivers. The crash rate, the number of crashes per vehicle-miles traveled, is widely recognized as a direct measure of the objective risk on the highway and is a more adequate measure of the crash risk faced and perceived by individual drivers than the crash frequency. The crash frequency is highly related to the traffic volume; consequently, it is very likely that a high-volume section will also have a high crash frequency even if the likelihood of crash faced by individual drivers is not particularly high. The crash-roadway relationship is substantiated with the results from a crash performance function (5) that identified three four-lane roadway characteristics in addition to the traffic volume and the length of the highway section as crash factors. Although the single-equation model failed to include speed as a significant parameter, it is also postulated that the speed plays a significant role in the likelihood of crash on the road. The influence of speed on crashes is definitely apparent as high speeds increase the complexity of the driving task, compromise the driver’s ability to avoid a potential crash, and increase the severity of a crash.

The perceived risk is assumed to be a function of the particular roadway characteristics and the objective risk on the road. This relationship is substantiated with the results obtained from a risk perception study (6) that identified seven four-lane roadway characteristics as risk perception factors. The risk perception study confirmed the significant role of the objective risk on the perceived risk of drivers by showing that drivers perceived more risk on highway sections with high crash rates.

The objective risk interacts with the perceived risk to alter the driver behavior on the road. The driver behavior is represented by the selection of speeds. The speed selected by drivers is believed to act as a measure of the driver’s attitude towards risk and willingness to be exposed to the risk of a crash (7). It is generally accepted that the perception of the risk of drivers is related to the hazardousness of a location (8-10). There is an obvious feedback between these two components as the perceived risk also depends of the speed selected by the drivers.

The mutual association between the speed and the travel time unit cost assumed in the system is understandable as changes in the speed selected by drivers affects their travel time cost, and conversely, changes in the travel time cost affects the speed selected by drivers. The relationship between speed and travel time unit cost is assumed fixed for the driver population for all highway conditions and therefore is neglected for the estimation of the system of relationships.

A main issue for researchers has been the identification of a suitable measure to estimate the perceived risk of drivers (1). The perceived risk of drivers cannot be measured directly on the road for obvious practical and operational issues. The minimization of the total disutility cost provided the equilibrium between the crash cost and the travel time cost functions that established a direct association between the perceived risk and the speed selected by drivers. By


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expressing the perceived risk as a function of the speed, the system of relationships can be simplified to three components as shown in Figure 2b.

The simplified system proposes that the relationship between the objective and perceived risks and the cyclic relationship between the objective risk, the perceived risk, and the speed be replaced by an endogenous relationship between the objective risk and speed. The relationship between the roadway characteristics and the perceived risk is accordingly replaced by an exogenous relationship between the roadway characteristics and the speed selected by the drivers. The exogenous relationship between the roadway characteristics and the speed is substantiated with the results obtained for a speed prediction model for four-lane highways (11) that included eight different roadway characteristics as speed factors.

The simplified system is formulated using a simultaneous equations model that consists of two structural equations involving two endogenous variables, the objective risk and the speed selected by the drivers, whose values are determined within the system. The values of both endogenous variables also depend on several exogenous variables that represent different roadway characteristics. The exogenous variables are assumed to be determined outside of the system and causal, characterizing the environment in which the endogenous variables are determined (12). Equation 1 presents the driver behavior as influenced by the perceived risk, represented with the free-flow mean speed, as a function of the objective risk and the roadway characteristics. Equation 2 presents the objective risk, represented with the crash rate, as a function of the mean free-flow speed and the roadway characteristics. The system of simultaneous equations is formulated as

VV εRλV +⋅+⋅= VV Xβ

RR εVλR +⋅+⋅= RR Xβ ,

(1)

(2)

where V is the free-flow mean speed, R is the objective risk, X represents the vectors of the exogenous roadway characteristics influencing the speed or the objective risk, β represents the vectors of estimable parameters, λ represents the estimable scalars, and ε represents the disturbance terms.

The free-flow mean speed was used to represent driver behavior on a specific location. The selected four-lane highway sections are located in suburban or rural areas that operate under non-congested traffic conditions on a regular basis. It is assumed that most of the drivers observed in the sections use them frequently and are familiar with the existing roadway characteristics. The mean free-flow speed characterizes the purest representation of the revealed preference of drivers on those sections and it was considered to be close to the upper-bound value of the annual average speed on the observed highway sections.

The system is complete as the number of equations equals the number of endogenous variables. The interrelationship between the two dependent variables in the system has an important implication in the interpretation and estimation of the equations (13). The behavior of the dependent variables is conditional on the rest of the model and the expression endogenous


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variables, given to the dependent variables in the system, is preferred to reflect the simultaneous characteristics of the system that determine these variables (12).

The endogenous relationship between the objective risk and speed creates a feedback into the system that cannot be captured and described adequately by calibrating each dependent variable separately using single-equation regression approaches. If the endogenous relationship between the dependent variables in the system is ignored and the system is solved using ordinary least squares (OLS) regression, the parameter estimates will be biased and inconsistent and the tests of hypothesis will be no longer valid (14). A key assumption of OLS regression to attain best linear unbiased estimators is violated because the regressors and the disturbances are no longer uncorrelated (15). For more details about structural equation modeling, readers are referred to the econometric theory books written by Rudd (12) and Greene (13).

DATA COLLECTIONThe first step in the collection process was the use of maps and aerial photos to identify clusters of four-lane highway sections based on their horizontal alignment (tangent or winding) and location with respect to developed areas (suburban or rural). Only U.S. and state highways in Indiana were considered. Interstate and local roads were excluded.

The following roadway characteristics were collected:• General characteristics: terrain type, rural/suburban area, pavement surface, and speed

limit• Access: density of intersections, driveways and median openings, and presence of

residential and commercial driveways• Tangents: grade, sight distance, cross-section dimensions, roadside obstruction, and

presence of sidewalks• Medians: median width, type, and surface, and presence of barrier or two-way left turn

(TWLT) lane• Horizontal curves: radius, maximum superelevation rate, and length

Free-flow speeds were measured on weekdays during daylight hours and favorable weather conditions (i.e., no heavy rain, no strong wind, and no fog). Time headways of five seconds or more were used to identify free-flow vehicles. At least a hundred speed observations were taken with a laser gun or with classifiers and rubber tubes at approximately the middle of each section. The measured speeds were not affected by traffic signals or stop signs. Crash counts from three years were collected to calculate the crash rate for each highway section.

SUMMARY OF COLLECTED DATAThe data set is composed of roadway characteristics, free-flow speeds and crash rates for 67 independent highway sections. The selected sections are not contiguous and do not share systematically crash rates, traffic volumes, or roadway characteristics. The average length of the highway sections was 1.2 miles. An effort was made to identify equal-length sections of one mile, but due to the location of the intersecting roads and to maintain consistency with the crash database, shorter and longer sections were identified.


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Table 1 presents descriptive statistics of some of the observed characteristics. The selected sections had posted speed limits from 40 to 55 mph (60 to 90 km/h). Nine sites were selected onsections located in rural areas and showing posted speed limits of 55 mph. Although a large range of 1,528 ft (465.9 m) in sight distance was observed, most sites provided sufficient sight distance, as indicated by the average sight distance. Most of the sites were located on a flat terrain. The roadway grade had a range of 12 percent, but the variability in grade values was relatively small. Only five sites were located on sections with grades of more than three percent.

The intersection density varied from zero to 14 intersections per mile; while the driveway densityvaried enormously from zero to 38 driveways per mile. As expected, most of the sections in suburban areas had high access densities. Seventeen sites were selected on sections with high density of residential driveways and eight sites were selected on sections with high density of commercial driveways. “High density” was assigned to sections having more than 10 driveways per mile. All the sections in suburban areas having high density of commercial driveways had speed limits lower than 55 mph. Lower speed limits are frequently requested by residents and business owners with the objective of controlling speeds and improving highway safety.

A diverse combination of cross-section configurations and dimensions was observed. Figure 3shows the six typical cross-section configurations observed in the selected highways. A largevariability in shoulder and median surface types and widths was observed. The wider cross-sections were observed on the sections having the higher posted speed limits. This trend isgenerated from design standards where increasing roadway widths are associated with increasing design speeds, as the width of the lanes and the shoulders is believed to influence the highway level of service, safety, and comfort of driving (16).

Two different cross-section configurations were observed in rural sections; one with narrow median and clear zones and frequent access points (Figure 3a) and another with median and clear zones of more than 40 ft (12 m) wide and full access control (Figure 3b). Median widths of 40 ft or more provides drivers with a higher sense of separation from the opposing traffic and greatly reduces headlight glare (16).

Four different cross-section configurations were observed in suburban sections. Eight sites were selected on undivided sections; while 13 sites were located on curbed sections (Figure 3c). All the undivided sections had 18 or more driveways per mile and roadside clear zones of less than 13 ft (4 m). Eight sites were selected in suburban sections having a concrete wall or guardrail as a median barrier (Figure 3d). The sections with median barriers provided better access control and wider roadside clear zones (around 30 ft) than the undivided sections.

The other two cross-section configurations in suburban sections had wider medians and roadside clear zones and different median types. Sixteen sites were selected in sections having TWLT median lanes (Figure 3e). The median lanes provide increased access to closely spaced commercial and residential driveways. Twelve sites having the TWLT median lane also had high density of driveways. Twenty-five sites were located on suburban sections with a depressed or flush median with a grass or paved surface (Figure 3f).


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High diversity in the design of horizontal curves was not observed. Typically, the design of the horizontal and vertical alignments of four-lane highways maintains an adequate consistency between adjacent segments; therefore, spots with adverse curvature conditions and sight distance restrictions that would force drivers to reduce their speeds are unusual. Sharp curves were only observed in suburban sections with posted speed limits lower than 40 mph (60 km/h) or located too close to traffic signals or stop signs and were therefore not included.

SIMULTANEOUS EQUATIONS MODEL The estimation of the system of simultaneous equations was performed using the three-stage least squares (3SLS) method. The 3SLS method is frequently used for the estimation of systems of simultaneous equations because it accounts for restrictions in over-identified equations and contemporaneous (cross-equation) disturbance term correlation (15). The 3SLS method identifies instrumental variables (IV) that are uncorrelated with the disturbance terms and highly correlated with the endogenous variables in the right-hand side of the structural equations. The IV then become a good substitute variable for the right-hand side endogenous variables. The parameter estimates obtained with the IV are consistent because they are uncorrelated with the disturbance term. Because of the consideration of the contemporaneous correlation of disturbances (i.e., correlation of disturbance terms across the equation system), 3SLS produces more efficient parameter estimates than single-equation estimation methods (15).

The system of simultaneous equations was calibrated using the sample of mean free-flow speeds, crash rates, and roadway characteristics for the 67 four-lane highway sections. The system was calibrated using the 3SLS procedure of the LIMDEP software (17). All the parameters included in the system of equations are significant at a 90 percent confidence level.

Table 2 presents the calibration results for the equation of the mean free-flow speed, as a function of the crash rate and the roadway characteristics. The best specification of the equation to estimate the mean free-flow speed, in mph, on four-lane highways is

HC.ECLR.CW.RA.

PSL.PSL.PSL.R..V

×−×+×+×+×−×−×−×−=

7711405204304173

640746057553413012252

20

404550(3)

whereR = objective risk, represented by the crash rate, in crashes per VMTPSL50 = 1 if the posted speed limit on the highway section is 50 mph; 0 otherwisePSL45 = 1 if the posted speed limit on the highway section is 45 mph; 0 otherwisePSL40 = 1 if the posted speed limit on the highway section is 40 mph; 0 otherwiseRA = 1 if the highway section is located in a rural area; 0 if the highway section is located in a suburban areaCW = cross-section width, lateral distance measured from the inside edge of the opposing traveled way or the median barrier face, if a barrier is present in the median, to the face of the roadside obstruction, feetECLR20 = 1 if the width of the external clear zone, the lateral clearance distance measured from the exterior edge of the traveled way to the face of the roadside obstruction, is 20 feet or more; 0 otherwiseHC = 1 if a horizontal curve is present; 0 otherwise.


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The adjusted R-squared statistic of the speed model, a measure of the overall fit of the model that takes into account the number of variables included in the model, shows that 75 percent of the variability in the mean speed variable is explained by the model. The standard error of the model estimates is 2.45 mph.

Table 3 presents the calibration results for the equation of objective risk, represented by the crash rate on the road, as a function of the mean speed and the roadway characteristics. The best specification of the equation to estimate the objective risk in four-lane highways is

PB.SW.TWLT.MOD.ID.

PSL.PSL.PSL.V..R

×+×+×−×+×+×+×+×+×+−=51214912926034403350

394446224851183044310 404550 (4)

whereID = intersection density, number of intersections per mileMOD = median opening density, number of median openings per mileTWLT = 1 if a two-way left-turn median lane is present; 0 otherwise SW = 1 if a curbed sidewalk is present; 0 otherwise PB = 1 if a pole line or an embankment is present on the roadside clear zone; 0 otherwise

The adjusted R-squared statistic of the objective risk equation shows that 42 percent of the variability in the crash rate variable is explained by the model. The standard error of the model estimates is 1.5 crashes per VMT.

The mean speed model in Equation 3 includes one endogenous variable, seven exogenous variables, and a constant term. The exogenous variables represent the effects of five different roadway characteristics and the endogenous variable represents the effect of the objective risk on the mean free-flow speed. The posted speed limit, the surrounding area type, the cross-section width, and the presence of a narrow roadside clear zone and a horizontal curve were identified as the exogenous variables in the mean speed model. The implication of the sign of each parameter in the model is the conventional one; a positive sign of a regression parameter indicates that the variable increases the mean speed, therefore reducing the perceived risk of drivers on the road.

The objective risk model in Equation 4 includes one endogenous variable, eight exogenous variables, and a constant term. The exogenous variables represent the effects of six different roadway characteristics and the endogenous variable represents the effect of the mean free-flow speed on the objective risk on the road. The posted speed limit, the intersection and median opening densities, and the presence of median TWLT lanes, pole lines, and embankments were identified as the exogenous variables in the objective risk model. A positive sign of a regression parameter in the model indicates that the variable increases the crash rate on the road.

BALANCE CONDITIONS OF THE SYSTEMThe results of the system of simultaneous equations were evaluated with an example set of values for the roadway characteristics included in the equations. The roadway design and thecombination of design elements affect driver speed selection by creating and defining the driving


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task and by guiding the expectation of drivers about the posted and comfortable speeds on the road (18). The following values for the roadway characteristics were used in Equations 3 and 4:

• 55-mph posted speed limit (PSL50 = PSL45 = PSL40 = 0)• Suburban area (RA = 0)• Cross section width of 75 ft (average value for suburban sections in the sample)• Roadside clear zone width of 20 ft or less (ECLR20 = 0) • Intersection density of five intersections per mile (average value for suburban sections in

the sample)• Median opening density of 0.5 openings per mile (average value for suburban sections in

the sample)• No TWLT median lanes, horizontal curves or roadside obstructions (TWLT = HC = SW

= PB = 0)

Figure 4 shows the mean speed as a function of the objective risk (negative slope) and the objective risk as a function of the mean speed (positive slope) using the example set of values. The outcomes of Equations 3 and 4 are shown in the figure with the points connected with the solid lines. The results of the system confirm the postulated inverse relationship between the perceived risk and the objective risk. As the crash rate in the road increases, the risk perceived by drivers also increases, resulting in a reduction in the mean speed. Simultaneously, as the mean speed increases, the crash rate also increases.

The impact of two different intersection densities was evaluated: a density of two intersections per mile and a density of 12 intersections per mile. The outcome of the system based on the two new intersection densities is illustrated in Figure 4 with the points connected with broken lines. The higher intersection density leads drivers, on average, to perceive more risk on the road and to reduce their speed. The reduction in the mean speed will then lead to a reduction in the objective risk (i.e., likelihood of crash) on the road. In contrast, the lower intersection density leads drivers, on average, to perceive less risk on the road (because of the smaller potential for vehicles entering and exiting the highway), and thereby to increase their mean speed. The higher mean speed will lead to a higher likelihood of crash on the road. The observed impact on speed and crash rates based on changes in the intersection density is consistent with previous results.

OBJECTIVE RISK AND SPEED FACTORSThe effects of the individual roadway characteristics on the mean speed and crash rates were explored by solving the simultaneous equations into their reduced form, which replaced the endogenous variable in each equation in the system with their predetermined variables and disturbances. The reduced form of the mean speed and objective risk models are, respectively:

PB.SW.TWLT.MOD.

ID.HC.ECLR.CW.

RA.PSL.PSL.PSL..V

×−×−×+×−×−×−×+×+

×+×−×−×−=

5820955035501320

1280646123620390

176378880206061445652

20

404550

(5)


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PB.SW.TWLT.MOD.

ID.HC.ECLR.CW.

RA.PSL.PSL.PSL..R

×+×+×−×+×+×−×+×+

×+×+×+×+−=

4101315286103200

311030204100007310

58207832358174008210

20

404550

(6)

Correlation between some of the roadway characteristics was expected due to the application of design standards and guidelines. In spite of the considerable correlation between model variables, there was no multicolinearity between the variables and no variables had to be removed to enable the model estimation.

The system of simultaneous equations showed its efficiency over the single-equation models in identifying the effects of 10 different roadway characteristics, in comparison with only three roadway characteristics in the crash frequency model and with seven roadway characteristics in the mean speed model. Due to the paper length limitations, only the effects of some of the roadway characteristics are discussed. A complete analysis is presented elsewhere (5).

The interpretation of the effects of the roadway characteristics on the mean speed isstraightforward. Mean speeds are reduced in the presence of restricted roadway conditions. The interpretation of some of the effects of the roadway characteristics on the crash rate needs to take attention to the speed-crash endogenous relationship. The effect on safety of the cross-section, curves and rural indicator variables is explained with the changes in mean speed attributed to those variables. No direct causal relationship must be attributed between the crash rates and changes in the values of these roadway variables in the sample.

As expected, the model suggests that a reduction in speed limit is associated with a reduction in mean speed. In addition, the model suggests that a reduction in speed limit is associated with higher crash rates. The effect of the speed limit variable in the simultaneous-equations model presents an interesting situation that needs special consideration. It is important to note that a mere reduction in the posted speed limit will not lead to an improvement in safety on a highway section, whereas the impact of the speed limit on the crash rate cannot be allocated directly to the selection of the posted speed limit on the road. This relationship may be the combined result of the tendency to set lower limits on high-crash segments (in order to calm drivers’ speeds) and the weak effectiveness of the limits in improving safety.

The posted speed limit is typically associated with many of the roadway characteristics. Amultinomial logit model of the selection of the 55-mph speed limit in the sample (5) confirmedthe typical practice of setting speed limits using the operating speed and the crash history of the section. Sections with lower speed limits in the sample were related to: suburban areas; higher intersection, median opening and driveway densities; narrower cross-section widths and roadsideclear zones; and the presence of median barriers (typically associated with narrow median widths). The presence of a speed limit lower than 55 mph on some of those sections might be the end result of an engineering study that identified a safety problem and attempted to reduce the crash rate of a hazardous highway section by reducing the speed of drivers on the road. The positive impact on the crash rate cannot be interpreted as a causal relationship between the posted speed limit and the crash rate. Speeding could be another factor to consider when attributing the increase in crash rates to the presence of lower posted speed limits on the road.


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CONCLUDING REMARKSThe existing safety evaluation methods make use of single-equation statistical models that only appraise the individual effects of the roadway characteristics on the crash frequency or the crash rate. Other models evaluate safety implicitly by evaluating the differences in speed induced by the roadway characteristics between adjacent highway sections. The main limitation of the existing methods and models is the lack of consideration of driver behavior on the road as influenced by the perceived risk, reflected in their selection of speeds, and its interrelationship with the roadway characteristics and crashes.

The system of simultaneous equations model provides a substantial improvement in theidentification of roadway characteristics as speed or crash rate factors, in comparison with the single-equation models. In addition, the system of equations was able to identify the extent of the endogenous relationship between the objective risk and the mean speed. The simultaneous-equations model did not provide a considerable improvement in the prediction of speed and crashrate estimates which might be attributed to the sample size used for the study and the potentially large variability of the measurement error of the crash rates.

The case of reversed, but not contradicting, relationships is the most appealing and valid finding, which further justifies estimating the two equations together. The relationship between the objective risk and the mean speed obtained from the system of equations was consistent with the proposed model of rational driver behavior. The single-equation speed and crash rate models failed to identify the endogenous relationship between the objective risk and the mean speed. By failing to identify the endogenous relationship between speed and crash rates, the results from single equation models developed using OLS regression provided parameter estimates that were biased and inconsistent, from which any inference made using the models will be invalid.

The interpretation of the relationship between the mean speed and the objective risk was straightforward from the results obtained from the system of simultaneous equations. The relationship established that the mean speed on the road decreases with an increasing crash rate and in the presence of restricted roadway conditions. On the other hand, the system crash rate equation included the effects of the mean speed and six different roadway characteristics. The relationship established that the crash rate on the road increases with an increasing mean speedand the presence of riskier conditions represented by the roadway characteristics included in the model. The first relationships can be convincingly explained on the ground of the theory of rational behavior (or homeostasis). The second relationship can be convincingly explained based on limited human and vehicle performance

REFERENCES1. Wright, C. and A. Boyle. Road Accident Causation and Engineering Treatment: A

Review of Some Current Issues. Traffic Engineering and Control, Vol. 28, 1987, pp. 475-483.

2. Wilde, G. Beyond the Concept of Risk Homeostasis: Suggestions for Research and Application Towards the Prevention of Accidents and Lifestyle-Related Disease. Accident Analysis and Prevention, Vol. 18, No. 5, 1986, pp. 377-401.


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3. O'Neil, B. (1977). A Decision Theory Model of Danger Compensation. Accident Analysis and Prevention. Vol. 9,157-165

4. Janssen, W. and E. Tenkink. Considerations on Speed Selection and Risk Homeostasis in Driving. Accident Analysis and Prevention, Vol. 20, No. 2, 1988, pp. 137-142.

5. Figueroa, A. Subjective and Objective Risks Consideration in Modeling Highway Safety. Ph.D. dissertation, Purdue University, 2005.

6. Figueroa, A., Kong, S., and A. Tarko. Roadway and Driver Factors of Risk Perception on Four-Lane Highways, International Conference: Safety on Four Continents, Warsaw, October 2005.

7. Wasielewski, P. Speed as a Measure of Driver Risk: Observed Speeds versus Driver and Vehicle Characteristics. Accident Analysis and Prevention, Vol. 16, No. 2, 1984, pp. 89-103.

8. Chipman, M., C. MacGregor, A. Smiley, and M. Lee-Gosselin. The Role of Exposure in Comparisons of Crash Risk among Different Drivers and Driving Environments. Accident Analysis and Prevention, Vol. 25, No. 2, 1993, pp. 207-211.

9. Renge, K. Drivers’ Hazard and Risk Perception, Confidence in Safe Driving, and Choice of Speed. IATSS Research, Vol. 22, No. 2, 1998, pp. 103-110.

10. DeSalle, B. and A. Tarko. Learning About Highway Hazards with Internet Assistance. Institute of Transportation Engineers Journal, Vol. 73 No 12, 2003, pp. 69-73.

11. Figueroa A. and A. Tarko. Speed Factors on Four-lane Highways in Free-flow Conditions. Conference proceedings of the 84th Annual Meeting of the Transportation Research Board, Washington D.C., January, 2005.

12. Rudd, P. An Introduction to Classical Econometric Theory. Oxford University Press, New York, 2000.

13. Greene, W. Econometric Analysis. Fourth edition. Prentice Hall, New Jersey, 2000.

14. Ramanathan, R. Introductory Econometrics with Applications. Second edition. The Dryden Press, Harcourt Brace Jovanovich College Publishers, New York, 1994.

15. Washington, S., M. Karlaftis, and F. Mannering. Statistical and Econometric Methods for Transportation Data Analysis. Chapman & Hall / CRC Press LLC, Boca Raton, Florida, 2003.

16. AASHTO. A Policy on Geometric Design of Highways and Streets. American Association of State Highway and Transportation Officials, Washington, D.C., 2001.


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17. Greene, W. LIMDEP User’s Manual: Version 7.0. Econometric Software, New York, USA, 1998.

18. Lerner, N., R. Llaneras, A. Smiley, and F. Hanscom. Comprehensive Human Factors Guidelines for Road Systems. NCHRP document 70, Project 17-18(08), National Cooperative Highway Research Program, Transportation Research Board, National Research Council, Washington, D.C., 2005.


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LIST OF TABLES

Table 1 Descriptive Statistics of Highway Section Characteristics

Table 2 Mean Speed System Model Estimation Results

Table 3 Crash Rate System Model Estimation Results

LIST OF FIGURES

Figure 1 Econometric Model of Rational Driver Behavior

Figure 2 Relationships between System Components

Figure 3 Observed cross-section configurations in four-lane highway sections.

Figure 4 Balance Conditions of the System of Equations


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Table 1 Descriptive Statistics of Highway Section Characteristics

Roadway characteristic Mean value

Standard deviation

Minimum value

Maximum value

Section length, mi 1.21 0.30 0.77 2.52Mean speed, mph 53.72 5.30 42.05 62.29Number of crashes 69.33 65.59 3 271Crash rate, crashes / VMT x 106 2.55 2.17 0.10 11.39Posted speed limit, mph 50.22 5.18 40 55Average annual daily traffic, vpd 21,734 11,614 3,550 58,580Sight distance, ft 1,424.00 408.70 549.45 2,078.00Roadway grade, percent 0.006 1.42 -6.20 6.00Intersection density, #/mi 4.60 3.53 0 14Driveway density, #/mi/direction 8.24 10.09 0 38Median opening density, #/mi 0.48 1.91 0 10Traveled way width, ft 23.46 0.88 21.41 25.83Pavement width, ft 34.73 5.76 24.33 47.17Roadside clear zone, ft 23.27 16.27 0 81.50Median width, ft 25.60 19.75 0 62.08Degree of curvature, degrees/100-ft 3.10 1.23 1.55 5.91Horizontal curve radius, ft 2,108.69 764.21 969.71 3,695.59Maximum superelevation, percent 3.22 1.81 1.18 6.65Horizontal curve length, ft 1,578.60 1,723.70 165.00 5,300.00


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TABLE 2 Mean Speed System Model Estimation Results

Parameter Estimate Standard error

b / std. error p-value

Constant 52.122 1.2572 41.46 0.0000Crash rate (endogenous variable) -0.4127 0.2016 -2.05 0.0407Posted speed limit indicatorsPSL50 -3.7555 0.8484 -4.43 0.0000PSL45 -5.4604 1.0130 -5.39 0.0000PSL40 -7.6396 1.4731 -5.19 0.0000Roadway characteristicsRural area indicator RA 3.4167 0.9883 3.46 0.0005Cross-section width CW 0.0428 0.0126 3.41 0.0006Clear zone indicator ECLR20 2.4048 0.9770 2.46 0.0138Horizontal curve indicator HC -1.7708 1.0311 -1.72 0.0859Fit statisticsR-squared 0.7836Adjusted R-squared 0.7537Root mean square error 2.446


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TABLE 3 Crash Rate System Model Estimation Results

Parameter Estimate Standard error

b / std. error p-value

Intercept -10.4427 5.7534 -1.82 0.0695Mean speed (endogenous variable) 0.1834 0.0957 1.92 0.0554Posted speed limit indicatorsPSL50 1.4853 0.7489 1.98 0.0473PSL45 2.4618 0.8956 2.75 0.0060PSL40 4.3946 1.2262 3.58 0.0003Roadway characteristicsIntersection density ID 0.3346 0.0728 4.60 0.0000Median opening density MOD 0.3441 0.1153 2.98 0.0029TWLT median lane indicator TWLT -0.9262 0.4785 -1.94 0.0529Presence of sidewalk indicator SW 2.4908 0.9962 2.50 0.0124Presence of roadside pole line or embankment indicator PB

1.5167 0.5385 2.82 0.0049

Fit statisticsR-squared 0.5024Adjusted R-squared 0.4239Root mean square error 1.522


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Speed, V

Cos

t, C

VA

Perceivedtravel time cost, CT

Perceived risk cost, CR

ACA = CT = CR

Total disutility cost

FIGURE 1 Econometric Model of Rational Driver Behavior


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ROADWAYCHARACTERISTICS

SPEED TRAVEL TIMEUNIT COST

OBJECTIVERISK

PERCEIVEDRISK

a) Complete system

ROADWAYCHARACTERISTICS SPEED

OBJECTIVERISK

EXOGENEOUSEXOGENEOUSRELATIONSHIPRELATIONSHIP

ENDOGENEOUSENDOGENEOUSRELATIONSHIPRELATIONSHIP

LEGEND:

b) Simplified system

FIGURE 2 Relationships between System Components


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a) Rural divided with narrow median b) Rural d ivided with wide median

c) Suburban undivided d) Suburban divided with median barrier

e) Suburban divided with TWLT median lane f) Suburban divided with grass median

FIGURE 3 Observed cross-section configurations in four-lane highway sections.


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0

1

2

3

4

5

50 51 52 53 54 55 56 57 58 59 60Mean speed (mph)

Cra

sh r

ate

(cra

shes

per

VM

T)

x10^

6 V ~ f (R)

R ~ f (V, ID)

ID = 12 int/mi

ID = 2 int/mi

ID = 5 int/mi

FIGURE 4 Balance Conditions of the System of Equations


2006-figueroa y tarko (2007)

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