Using horizontal curve speed reduction extracted from the naturalistic driving study to predict curve collision frequency

Accident Analysis and Prevention 123 (2019) 190–199

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Using horizontal curve speed reduction extracted from the naturalistic driving study to predict curve collision frequency☆ Bashar Dhahir, Yasser Hassan

T

⁎

Department of Civil Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S5B6, Canada

A R T I C LE I N FO

A B S T R A C T

Keywords: Naturalistic driving study Horizontal curves Design consistency Speed differential Speed reduction Safety performance Road safety

Many models have been developed to predict collision frequency and evaluate safety performance on horizontal curves. The approach used in data collection or some assumptions made in the analysis methodology might lead to inaccurate results. For example, manual data collection, equipment limitations, and field experiments involving monitoring driving behavior for a specific region for a short-term are potential sources of errors in data collection. This paper aims at overcoming some of these issues in developing models to evaluate safety performance of horizontal curves and predict the curve collision frequency. The developed models relate expected collision frequency on horizontal curves to the speed reduction from the approach tangent to the curve, which is commonly used as a major geometric design consistency measure. The methodology to achieve this objective included three tasks; data collection, evaluating and modeling the viable speed reduction parameters, and developing safety performance models to estimate collision frequency on horizontal curves. Individual drivers’ trips on 49 horizontal curves on rural two-lane highways in rolling and mountainous terrains in Washington State were extracted from the Naturalistic Driving Study (NDS) database. Models were developed to relate different speed reduction parameters to curve characteristics. These models were then applied to 1430 horizontal curves in Washington State to estimate the speed reduction parameters and relate them to collision frequency. Several safety performance models were developed which show that speed reduction, as a design consistency measure, is directly related to collision frequency on horizontal curves. Furthermore, the speed reduction parameters are more significant variables in predicting collision frequency than all curve geometric parameters.

1. Introduction Highway geometric design consistency (simply referred to as design consistency) is a concept that aims at coordinating the successive road elements to ensure harmonious travel along the roadway section and that the geometric design would conform to driver’s expectations (Gibreel et al., 1999). Different measures have been suggested to evaluate design consistency (Gibreel et al., 1999; Hassan, 2004). The most common design consistency measures have been based on operating speed, which is defined as the 85th percentile speed of free-flow vehicles at a specific location V85 , on horizontal curves (Himes et al., 2011). Several studies have shown that design consistency has a significant effect on safety of rural highways (Montella and Imbriani, 2015). From the designer’s perspective, design consistency can be proactive tool for roadway safety improvement. By establishing the

relationship between the expected safety performance and design consistency measures, these measures can be used as surrogate safety measures that help designers eliminate areas with expected higher collision frequencies in the design stage. For example, Lamm et al. (Lamm et al. (1999)) suggested a safety criterion, which is referred to as Safety Criterion II, based on the difference of operating speed ( ΔV85 ) between two successive curves or between a tangent and the following curve. Krammes et al. (Krammes et al. (1995)) used linear regression analysis and found collision rates on horizontal curves to be directly related to the mean speed reduction on the curve. Subsequent analysis by Anderson et al. (Anderson et al. (1999)) in the USA and Ng and Sayed (Ng and Sayed (2004)) in Canada, showed also that collision frequency on horizontal curves of two-lane rural highways can be related to ΔV85 . Montella and Imbriani (Montella and Imbriani (2015)) studied design consistency based on operating

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This paper is based on the RSS2017 paper no. 58 “Relationship between Traffic Safety and Speed Differential on Horizontal Curves Based on Naturalistic Driving Studies” but has been considerably revised and the database used in the analysis is considerably expanded. ⁎ Corresponding author. E-mail addresses: [email protected] (B. Dhahir), [email protected] (Y. Hassan). https://doi.org/10.1016/j.aap.2018.11.020 Received 18 February 2018; Received in revised form 7 October 2018; Accepted 26 November 2018 0001-4575/ © 2018 Elsevier Ltd. All rights reserved.

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sample of NDS speed data on rural arterial two-lane highways and the geometric characteristics of the corresponding horizontal curves. Second, models were developed to relate different speed differential (or speed reduction) parameters between tangents and curves to curve characteristics. Finally, safety performance data were collected for a larger sample of horizontal curves on rural arterial two-lane highways in the same geographic area where the NDS data were collected. The speed reduction models were then used to predict the speed reduction parameters as potential measures of design consistency and relate them to the safety performance of these horizontal curves.

speed consistency (the absolute value of ΔV85 ) and inertial speed consistency (difference between the operating speed on the curve and the average operating speed along the 5 km before the curve), and found that both parameters were significant variables in safety modeling. However, McFadden and Elefteriadou (McFadden and Elefteriadou (2000)) reported that ΔV85 does not necessarily represent the speed reduction experienced by drivers accurately. Rather, the 85th percentile maximum speed reduction (85MSR ) was examined using speed profile data on 21 horizontal curves and was found to be approximately two times higher than ΔV85 . Misaghi and Hassan (Misaghi and Hassan (2005)) proposed a similar speed reduction parameter, referred to as the 85th percentile speed differential Δ85 V . Rather than simple subtraction of 85th percentile speeds, Δ85 V is determined as the 85th percentile of the speed reduction experienced by each driver from approach tangent to curve. It was shown that ΔV85 underestimates the speed reduction compared to Δ85 V . Bella et al. (Bella et al. (2014)) used day and night speed data collected using a driving simulator to model three different speed differential parameters, namely ΔV85 , 85MSR , and Δ85 V . Further analysis confirmed that ΔV85 underestimates the speed reduction relative to both 85MSR and Δ85 V for both day and night driving conditions (Calvi and Bella, 2014). By examining speed and collision data in the same geographic area, Awatta et al. (Awatta et al. (2006)) found that horizontal curve safety is related to both ΔV85 and Δ85 V , but Δ85 V was more strongly related to geometric features. However, most of the design consistency research has been based on data collected using laser/Lidar or radar guns, electronic counters/ classifiers installed at different points on the approach tangent and curve, driving simulator, or an instrumented vehicle with continuous speed tracking (Hassan et al., 2011). Besides the short duration of data collection, several issues can be raised regarding the accuracy of collected data such as human errors and equipment limitations, and monitoring of unnatural driving behavior caused for example by driver’s perception of speed enforcement due to the presence of observers with measuring equipment (Hassan, 2004; Dhahir and Hassan, 2018a). Although studies have confirmed that operating speed differential between consecutive roadway elements is an important factor in evaluating design consistency, the speed differential is more accurately estimated using the complete speed profile of a representative sample of curve trips rather than the difference between speeds at mid-points of the curve and approach tangent. In addition, because of the normal fluctuations in instantaneous speed measurements, it is not clear whether the spot speeds at specified points, such as middle of curve, would provide an accurate estimate of the speed behavior on the curve. Therefore, a more accurate estimate of a driver speed behavior may be obtained using an average of the instantaneous measurements around the point of interest.

3. Speed reduction modeling As mentioned earlier, this paper used speed data from the NDS, which was a major part of SHRP 2 authorized in 2005. The NDS was conducted as the largest and most comprehensive study of the naturalistic driving behavior (Dingus et al., 2015). Data were collected in the period of 2011–2013 using instrumented vehicles in six states in the USA. For each trip, the collected data included vehicle location (GPS coordinates), headway, speed, longitudinal and lateral acceleration, brake conditions, and light intensity. Vehicle location was recorded every second while speed, lateral acceleration, and longitudinal acceleration, were measured by equipment installed in the vehicle at a rate of 10 Hz (10 measurements per second) (InSight Data Access Website, 2016). 3.1. NDS dataset To fulfill the objectives of the paper, driver speed behavior on horizontal curves on two-lane rural arterial highways was modeled using a subset of the NDS data. The NDS dataset used in this paper initially comprised 24 and 25 curves in Washington State in rolling and mountainous terrain, respectively. However, three curves in rolling terrain had small numbers of trips and were therefore excluded. Thus, the NDS dataset comprised 21 and 25 curves in rolling and mountainous terrain, respectively. The geometric data of the selected horizontal curves were obtained from Washington State Department of Transportation (WSDOT). Additional data were also obtained including roadway level of service, posted speed limit, design speed, terrain, and historical Annual Average Daily Traffic (AADT). The curves in the NDS dataset had radii ranging from 146 to 349 m (rolling terrain) or from 279 to 873 (mountainous terrain). In addition, the rolling terrain curves had five different posted speed limits ranging from 40 to 60 mi/h, while the mountainous terrain curves had only one posted speed limit of 60 mi/h. Each selected curve is preceded by an independent approach tangent, which is long enough to allow drivers to accelerate to their desired tangent speeds. As recommended by Polus et al. (Polus et al. (2000)), a tangent is considered to be independent if it is at least 200 m long. For the NDS dataset, 9 curves in rolling terrain and 21 curves in mountainous terrain were preceded by an independent tangent in one direction and were considered in that direction only. The remaining curves were preceded by independent tangents in both directions and were considered in the two directions. Thus, the total number of curves, considering each curve direction as a unique site, is 33 and 29 in rolling and mountainous terrain, respectively. The trips in the NDS dataset were driven by 178 test participants (98 males and 80 females) representing different professions and age groups. Each curve in the dataset had trips driven by at least 30 participants. The total number of trips on the NDS dataset curves in rolling and mountainous terrains, after removing trips that were missing essential information such as speed or GPS coordinates, was 2874 and 2,316, respectively. These trips were further divided into daytime (occurring after the civil twilight in the morning and before the civil twilight in the evening, where civil twilight is defined as the instant in the morning or evening when the center of the sun is at a depression angle of 6° below an ideal horizon (Geoscience Australia, 2017)) and

2. Objectives This paper aims at utilizing naturalistic driving behavior data to overcome the problems with design consistency research mentioned in the previous section. It should be noted that the literature has considerable research that involved using naturalistic driving data and speed profiles, modeled operating speed and speed reduction on horizontal curves, or examined the relationship between design consistency measures and expected safety performance. However, this paper presents two unique contributions: (1) speed data are extracted from the most comprehensive Naturalistic Driving Study (NDS) dataset available through the second phase of the Strategic Highway Research Program (SHRP 2); and (2) NDS and safety data for the same geographic data are used to develop the models to relate speed reduction to road characteristics and the models to relate safety performance to speed reduction. Thus, the results are expected to reflect the relationship between speed differential and safety more accurately The methodology followed to achieve the paper’s objective involved three main steps as shown in Fig. 1. The first step was the collection of a 191

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Fig. 1. Flow Chart of Analysis Methodology.

terrains was 1901 and 1,515, respectively. Fig. 2 shows the distribution of the NDS dataset trips by months of the year. It should be noted that only trips in daytime and favorable weather conditions were used in the analysis because the data sample for these conditions was larger than other conditions and should produce more representative modeling of the driver population.

3.2. Driver speed and speed reduction parameters Many studies developed models to predict operating speed on curves using the speed measurement at the mid-point of the curve (MC), thus assuming that speed of vehicles is constant throughout the horizontal curve. However, several other studies have suggested that collecting speed data at MC may affect the accuracy of the design consistency analysis (McFadden and Elefteriadou, 2000). In addition, because of fluctuations in vehicle speed, using one point of the entire speed profile of any trip might not be an accurate estimate of the speed at MC. For example, Fig. 3 shows the profiles of speed, longitudinal acceleration, and lateral acceleration for a vehicle on the approach tangent and curve from the NDS dataset. First, the figure shows that the speed on curve is not constant from beginning to end. Rather, the driver, who reached a maximum speed on the approach tangent,

Fig. 2. Distribution of Trips during Daytime and Favorable Weather Conditions by Month of the Year.

nighttime. In addition, the trip data were matched with weather conditions to identify the prevalent weather conditions during each trip. Daily weather conditions data for three years (2011–2013) of seven selected weather stations were obtained from the Office of Washington State Climatologist, where each curve was approximately 2 to 9 km away from at least one station. Accordingly, the total number of trips in daytime and favorable weather conditions in rolling and mountainous 192

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Fig. 3. Example Speed and Accelerations Profiles on Tangent and Curve.

(or speed reduction) parameters were evaluated as follows:

decelerates on approaching the curve and continues decelerating after the beginning of the curve (BC), which is often referred to also as point of curve (PC). In addition, the figure shows fluctuations in speed, confirming that using the speed measurement at one point may overestimate or underestimate the general driver’s speed behavior. On speed reduction, Fig. 3 also shows that the classical estimate of speed reduction as the difference between speed at a specific point on the tangent (normally taken as 200 m before BC) and the speed at MC can yield a different value from the difference between the maximum speed on the tangent and minimum speed on the curve. Subsequently, a Matlab script was developed to extract different speed parameters for each trip including the speed at different points on the curve and approach tangent. These speed parameters may be taken as the instantaneous value at the relevant point or an average value of the observations within a specific distance or time period of that point (Dhahir and Hassan, 2018a, b). Different parameters were extracted from the speed profile of each trip to evaluate the speed behavior of drivers as follows:

• Operating speed reduction: difference between the 85th percentile

speeds on the approach tangent and curve. Two types of operating speed reduction were considered in the analysis. First, the classical speed reduction ΔV85 is calculated as the difference between the 85th percentile Vap and 85th percentile Vcu . Second, the speed reduction ΔV85max is calculated as the difference between the 85th percentile Vapmax and the 85th percentile Vcumin . Thus,

•

(1)

ΔV85max = V85apmax − V85cumin

(2)

The 85th percentile of speed reduction: the 85th percentile speed reduction for the individual trips on the same curve. Therefore, the speed reduction for each trip on a specific curve from the approach tangent to the curve is calculated, then the 85th percentile value is calculated, and is referred to as Δ85 V .

• The 85th percentile of the maximum speed reduction (85MSR): the

• Speed

• •

ΔV85 = V85ap − V85cu

at approach tangent (Vap ): is the instantaneous approach tangent speed at 200 m before BC. Each tangent was checked in ArcMap to ensure that it is at least 450 m long. Thus, the tangent would be long enough for the speed measurements along a 50-m tangent section from 200 to 250 m before BC to be at least 200 m away from the upstream curve and not be affected by the characteristics of that curve. Maximum speed on approach tangent (Vapmax ): is the maximum vehicle speed recorded on the tangent within 200 m before BC. Speed on curve (Vcu ): is the average speed at the middle of the curve estimated by averaging speed observations within a short period of time of 1.0 s around MC. Specifically, the range used to average speed observations is (MC – 0.5 s) to (MC + 0.5 s). Minimum speed on curve (Vcumin ): is the minimum vehicle speed recorded along the curve.

85th percentile of maximum speed reduction for the individual trips on the same curve. To calculate this parameter, maximum speed reduction for each trip is calculated as the difference between Vapmax and Vcumin , then the 85th percentile value is calculated.

3.3. Preliminary analysis Design consistency evaluation is traditionally based on speed measures considering only free-flow speeds, where the driver’s speed is constrained only by the geometric features of the road. Previous studies have assumed that vehicles are in free flow conditions when having a headway of at least 5 s (Hassan et al., 2011). However, the relationship between headway (h) and driver’s speed choice was examined using the NDS dataset. The analysis did not show any specific trend between the mean or 85th percentile Vcu and headway threshold (h ≥ 1, h ≥ 2, …, h ≥ 5). However, the 85th percentile and mean Vcu for specific headway intervals (0 ≤ h < 1, 1 ≤ h < 2, …, h ≥ 5) showed a trend of increasing Vcu with increasing headway for h < 3 s for all posted speed limits. This trend indicates that a headway of 3 s can be a threshold for free-flow speed on curves for the collected data. Therefore, the speed and speed reduction parameters in this paper were calculated for trips that had a minimum headway of 3 s. More details on the analysis of headway threshold for free-flow speed conditions is presented elsewhere (Dhahir and Hassan, 2018b). A total of 1364 trips in rolling terrain and 1355 in mountainous terrain were used on the 62 NDS dataset curves, with a minimum of 30 trips on each curve. The bivariate (Pearson) correlation matrix among speed parameters and curve geometric characteristics in Table 1 was developed using

It should also be noted that One-Way-ANOVA was used to compare the speed parameters estimated by averaging speed measurements as explained above to the instantaneous speed at the corresponding curve point (e.g., Vcu is compared to instantaneous speed at MC). The outputs of statistical analysis revealed that there was no statistically significant difference between the average speed within a short time period or short distance from a specific point and the instantaneous speed at the same point (Dhahir and Hassan, 2018b). That is, there was no statistically significant difference between Vap and an average approach tangent speed along a 50 m distance between 200 m and 250 m before BC or between the instantaneous speed at MC and Vcu . Based on the extracted speed parameters, several speed differential

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Table 1 Bivariate (Pearson) Correlation Matrix. Vap

Vapmax

(km/h)

(km/h)

Vap Vapmax

1 0.946**

1

Vcu ΔV ΔVmax MSR R L e

0.839** 0.266** 0.183** 0.135** 0.403** 0.222** −0.241**

0.906** 0.055** 0.154** 0.120** 0.448** 0.234** −0.286**

Vcu (km/h)

ΔV (km/h)

MSR (km/h)

R (m)

L (m)

1 −0.297** −0.284** −0.265** 0.544** 0.300** −0.360**

1 0.818** 0.688** −0.230** −0.096** 0.195**

1 −0.156** 0.041* 0.132**

1 0.506** −0.688**

1 −0.372**

e

1

Vap = speed at approaching tangent, Vapmax = maximum speed along approach tangent, Vcu = average speed on middle section of the curve, ΔV = Vap - Vcu , MSR = Vapmax - Vcumin , R = curve radius, L = length of curve, e = superelevation rate. * Correlation is significant at 5% level of significance (2-tailed). ** Correlation is significant at 1% level of significance (2-tailed).

Q3 + 1.5(Q3 − Q1) , where Q1 is the first quartile (25th percentile), and Q3 is the third quartile (75th percentile). Thus, the extreme outliers in this analysis represent values more than three times (Q3 − Q1). The presence of extreme outliers that can affect the regression results was also confirmed using the Matlab function “fitlm” with the robust regression option on. The maximum number of curves that were identified as outliers in any model is three out of a total of 59 curves used in the regression analysis. Subsequently, stepwise regression was performed using SPSS after removing the extreme outliers with the potential independent variables listed earlier. As shown in Table 2, all models had the curve radius in the mathematical form (1/ R ) as a significant independent variable. In addition, V85ap was a significant independent variable in all four models and improved the model fit in three of them as reflected in the higher 2 values of Radj . This trend is consistent with previous research using speed data obtained from a driving simulator (Bella, 2007). In all of these four models, T was not a significant variable, which indicates that V85ap is highly dependent on the terrain type. It should be noted that V85ap is generally difficult to predict. Thus, the models that require V85ap to predict speed reduction will be difficult to apply unless the user has accurate estimates of the speed on tangents. However, these models are presented in this paper to provide a statistical evidence using NDS data that speed differential is correlated to speed at tangent, which can be considered a surrogate of the speed environment on a particular road. This concept was suggested by McLean (McLean (1981)) but is rarely considered in design consistency research where curve speed and speed reduction are assumed to depend only on the curve geometric characteristics. For the cases where the user cannot accurately estimate the approach tangent speed, another set of models was developed after excluding V85ap from the list of independent variables. This set is referred to as Models SR1b to SR4b in Table 2. As shown in the table, VL is a significant parameter in three models, which can be another surrogate of the road’s speed environment in the absence of an estimate of V85ap . The other significant parameters were R , T , L , and Δ ; all are relatively easy to evaluate. On the other hand, G was found to be an insignificant parameter and was removed from all models, which indicates that for this dataset the vertical grade did not affect any of the speed reduction 2 parameters. The relatively low values of Radj are consistent with models that try to predict human behavior (Collins et al., 1999). It should be noted however that the F-statistic indicates that all models are statistically significant at 5% level of significance, where all F-statistic values shown in the table produce p-values less than 0.001. The table also shows that Model SR3b (for the speed reduction parameters Δ85 V ) has 2 the highest Radj value, and thus provides better fit than the other models.

SPSS software. The correlations between most variables are significant at the 1% level of significance. As shown in the table, the correlation between each two variables is in the expected direction except for the correlation between e and all speed and speed reduction parameters. While a higher e should allow a higher Vcu and lower speed reduction for same R due to the lower unbalanced lateral forces, the design practices normally use lower e on flatter curves. This is evident in the relatively high negative correlation of -0.688 (significant at 1% level of significance) between R and e. As for the correlations with curve speed and speed reductions, expectedly Vcu has a strong direct relationship with Vap , Vapmax , R , and L . On the other hand, Vcu has an inverse relationship with the speed reduction parameters at the trip level (ΔV and MSR ), both of which have an inverse relationship with R . In other words, speed reduction decreases with the increase in R. Thus, design consistency of horizontal curves with a preceding independent approach tangent is expected to improve as R increases.

3.4. Speed reduction models As explained earlier, five curve speed reduction parameters were evaluated in this paper. But first, the speed at tangent (Vap ) was compared to the curve speed (Vcu ) to identify the trips where the vehicle accelerated on the curve (Vap < Vcu ). Such trips are not consistent with the expected driving behavior when the speed choice is constrained by the curve geometry. The speed differential between the approach tangent and curve in such trips may not accurately reflect the effect of curve geometric characteristics. Therefore, only vehicles that decelerated or maintained constant speed on the curve were considered in the analysis (Vap ≥ Vcu ), while trips where the driver accelerated on the curve were removed from the data (312 and 367 trips in rolling and mountainous terrain, respectively). As a result, three curves in rolling terrain ended up with a total number of trips less than 20 and were removed from the NDS dataset. Thus, the speed reduction parameters were evaluated using 59 curves (30 in rolling terrain and 29 in mountainous terrain). SPSS stepwise linear regression was utilized considering all potential independent variables, namely curve radius (R ), curve length (L ), deflection angle (Δ ), superelevation rate (e ), grade (G ), presence of spiral curves, 85th percentile speed on the approach tangent (V85ap ), and posted speed limit (VL ). Two additional dummy variables for terrain type (T ) and curve turning direction were also included as independent variables. Finally, mathematical variations of many variables (specifically, the inverse of the variable) were included as potential independent variables. The models initially exhibited very low adjusted coefficients of 2 determination (Radj ). Therefore, SPSS Analytic Catalyst was utilized to identify outliers based on the interquartile range (IQR). A value is considered as outlier if it lies outside the range Q1 − 1.5(Q3 − Q1) and 194

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Table 2 Speed Reduction Models. No.

Model

2 Radj

SEE

F-Statistic

0.357

2.523

17.085

0.400

2.046

10.491

0.630

2.269

33.886

0.416

2.457

14.510

0.190

2.331

7.464

0.270

2.165

7.896

0.557

2.418

18.256

0.457

2.360

17.247

(a) Models Including V85ap as Independent Variable. SR1a

ΔV85 = −26.344 +

2277.833 + 0.245 V85ap R 2097.871 63.292 1842.640 + − + Δ R V85ap

SR2a

ΔV85max = 23.520

SR3a

Δ85 V = 24.457 +

SR4a

85MSR = 16.911 +

4050.519 R

−

2856.696 R

507.762 L

−

−

1865.871 V85ap

− 56.737 e

2675.891 V85ap

+ 0.009 L

(b) Models Excluding V85ap from the Independent Variables. SR1b SR2b

ΔV85 = −7.934 + ΔV85max = 2.124

1496.105 + 0.091 VL R 1234.359 50.516 + − + 2.254 T R Δ 3685.591 520.896 667.118 − − + R L VL

SR3b

Δ85 V = 5.131 +

SR4b

85MSR = −17.155 +

2981.133 R

2.135 T

+ 0.169 VL + 0.009 L

R = radius of curve (m), L = length of curve (m), Δ = deflection angle (°), VL = posted speed limit (km/h), T= terrain ( = 0 for rolling and = 1 for mountainous), 2 Radj = adjusted coefficient of determination, and SEE = standard error of estimation.

4. Safety performance modeling

Akaike Information Criterion (AIC) and Bayesian information criterion (BIC) and Pearson dispersion statistic. The five-year collision frequency on horizontal curves corresponding to the period 2009–2013 was the dependent variable in the regression analysis while the main independent variables included AADT and L , in the form of natural logarithm, as traffic exposure parameters. In addition, curve geometric characteristics and speed reduction parameters ΔV85 , ΔV85max , Δ85 V , and 85MSR were considered as potential independent variables. It should be noted that for the speed reduction parameters to be useful design consistency and surrogate safety measures, they should correlate to total collision frequency on the horizontal curves rather than a subset of these collisions. Therefore, although the speed reduction parameters modeled in this paper correspond to day driving during favorable weather conditions, the dependent variable in the safety modeling was taken as the total collision frequency on each curve. The speed reduction parameters were predicted for the curves in the safety dataset using the models developed in the paper and shown in Table 2 as Models SR1b to SR4b. Therefore, only curves with radii ranging from 146 to 873 m were included in the safety modeling as the speed reduction parameters for curves outside this range would be an extrapolation of the developed models. The total number of curves used in the analysis was therefore reduced to 1430. The total number of collisions reported on these curves for five years was 1541. The mean and variance five-year collision frequency on the study curves were 1.077 and 4.525, respectively. Fig. 4 shows the frequency distribution of five-year collision frequency on the study curves. The figure shows that more than 50% of the curve experienced no collisions along the five years. A closer inspection showed that most of these curves had low

4.1. Safety dataset In addition to the NDS dataset, a larger database of horizontal curves was compiled to examine the relationship between design consistency and safety. For this dataset, referred to here as safety dataset, three subsets of data were required: geometric parameters, traffic characteristics, and historical safety data for five years. The safety dataset was also compiled for Washington State, which is the same geographic area for the NDS dataset used in this paper. Road geometry and traffic characteristics data for 2059 horizontal curves on two-lane rural arterial highways were obtained from WSDOT. Collision data on these curves were obtained from the Highway Safety Information System (HSIS) database for five years (2009 to 2013). The HSIS database had information of the location of each collision using Linear Referencing System, date, pavement surface conditions (dry, wet, snow, etc.), light condition (day light, dawn, dusk, etc.), collision severity, number of vehicles involved in collision, driver information (gender and age), and weather conditions (clear, raining, snowing, etc.). The location of the collision was reported to the nearest 0.10 mile for over 95% of the collisions in rural areas (Nujjetty et al., 2014). Collisions for each curve were considered if they occurred on the curve or if they occurred within a 50-m distance before BC or after end of curve (EC), which is often referred to also as point of tangent (PT). The total collisions on these curves for five years was 2223.

4.2. Modeling procedure Poisson and negative binomial (NB) regression models were used in developing safety models. The standard NB regression is an extension to Poisson regression by considering over or under-dispersion of the count data. Zero-inflated negative binomial (ZINB), zero-truncated negative binomial, and other enhancements have been developed to accommodate excessive zero counts in NB and to solve either distributional or variance problems (Hilbe, 2007). The zero-inflated models could be tested to determine whether they are different from their base model (Poisson or NB) using Vuong test. The zero-inflated model is generally preferred over the base model when having Vuong test value greater than +1.96, while the base model is preferred when the test value is less than -1.96. Neither model is preferred over the other if test value is between ± 1.96 (Hilbe, 2007). Regression analysis of the collision data was performed in this study using the Stata software. Selection of the best model that fit the data depends on several parameters such as

Fig. 4. Frequency Distribution of Five-Year Collision Frequency on Study Curves. 195

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traffic volumes and/or short length, where AADT and curve length for approximately half of these curves were less than 1600 veh/d and 150 m, respectively. Both conditions of low traffic volume and short curve length would translate to low traffic exposure and low probability of collision occurrence.

Table 3 Safety Performance Model Based on Traffic Exposure Parameters Only. Model

Likelihood-ratio test

AIC

BIC

Y = AADT 0.955L0.476e−9.847

237.52

3438.62

3459.681

Y = five-year collision frequency. The p-value for the likelihood-ratio test is less than 0.001.

4.3. Safety performance models 4.3.1. Poisson versus NB regression Several regression models were attempted to develop safety performance models to predict the five-year collision frequency on horizontal curves based on curve speed reduction as the major design consistency measure. First, it was observed that the data were overdispersed which caused the NB regression in general to provide better models than the Poisson regression. For example, when using only traffic exposure parameters (AADT and L ) as independent variables, the Poisson model had a Pearson dispersion statistic of 1.810 (Deviance = 1.427, AIC = 3674.136, and BIC = 3689.932), which means that some 81% extra dispersion existed in the data. On the other hand, the Pearson dispersion statistic in the NB model dropped to 1.185 (Deviance = 0.925). The AIC and BIC also dropped to 3436.620 and 3452.416, which means that the NB model fit the data better. Furthermore, Fig. 5 shows a comparison between the observed probabilities of different five-year collision frequencies and the predicted probabilities using NB and Poisson models. As shown in the figure, the NB regression provided a better agreement with the observed probabilities. Therefore, only NB models are presented in the paper.

Table 4 Safety Performance Models Based on Speed Reduction and Traffic Exposure (First Set). No.

Model

Likelihoodratio test

AIC

BIC

SP1a

Y = AADT 0.982L0.492e−10.417 + 0.061 Δ V85

240.84

3425.11

3451.44

SP2a

Y = AADT 0.967e−7.666 + 0.035

291.36

3499.90

3520.97

SP3a

Y = AADT 0.978L0.690e−11.531 + 0.031Δ85 V

232.05

3413.71

3440.04

SP4a

Y = AADT 0.983L0.446e−10.183 + 0.032 × 85MSR

240.50

3424.22

3450.55

Δ V85max

Y = five-year collision frequency. The p-value for the likelihood-ratio test is less than 0.001.

4.3.2. NB safety performance models The first step in the safety performance modeling was to relate the five-year collision frequency on horizontal curves to the traffic exposure parameters (AADT and L) only as independent variables. The developed model shown in Table 3 was used as the base model to compare other models incorporating speed reduction parameters and curve characteristics. Two sets of safety performance models were then developed using NB regression to estimate the five-year collision frequency on horizontal curves based on each speed reduction parameter. The first set was developed using only speed reduction and exposure parameters as independent variables as shown in Table 4. The table shows that all speed reduction parameters appear in their relevant models with a positive coefficient, which is the expected direction for the relationship between collision frequency and speed reduction as collision frequency is expected to increase with the increase of speed reduction. However, the exposure parameter L was not significant in Model SP2a, which is counterintuitive. The table also shows that Model SP3a had the lowest AIC and BIC values. On the other hand, only Model SP2a had higher AIC and BIC than the base model using traffic exposure parameters only. Thus, all models except for SP2a represent better fit to the data than the base model using traffic exposure only. To further compare these models, Fig. 6 shows the estimated five-

Fig. 6. Relationship between Collision Frequency and Speed Reduction Using First Set of Safety Performance Models.

year collision frequencies on horizontal curves with different speed reductions using this first set of models. The regions of speed reduction that can be classified as good, fair, or poor design according to Safety Criterion II (Lamm et al., 1999) are also shown in the figure. It should be noted that Safety Criterion II and the safety classification as good, fair, or poor were proposed based on ΔV85 only. Therefore, the figure presents a graphical illustration of the implications of extrapolating the application of this classification system using other speed reduction parameters. In addition, the figure illustrates the fundamental difference between the approach of design consistency evaluation using a continuous function, presented in more recent research and this paper, and the approach of design consistency evaluation using a step scale, as proposed in Safety Criterion II. As argued by Hassan (Hassan (2004)), the step scale is fundamentally flawed because safety performance does not suddenly change as the speed differential parameter crosses the boundary between two steps in the scale. The figure provides also a pictorial confirmation of the value of using speed reduction parameters in evaluating safety performance on horizontal curves. Expectedly, the base model using traffic exposure parameters only fails to differentiate between sites with different speed reductions. Another pictorial confirmation in the figure is the expected trend of a direct relationship between collision frequency and all speed reduction parameters as reported in all previous studies. Table 5 shows the second set of models developed considering curve geometric characteristics and posted speed limit as additional

Fig. 5. Probabilities of Observed and Predicted Collision Frequencies. 196

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B. Dhahir, Y. Hassan

Table 5 Safety Performance Models Based on Speed Reduction, Traffic Exposure, and Curve Characteristics (Second Set). No.

Model

Likelihood-ratio test

AIC

BIC

SP1b

Y = AADT 0.954L0.570e−9.675 + 0.085

Δ V85− 0.012 VL

215.48

3407.97

3439.56

SP2b

Y = AADT 0.928L0.694e−9.472 − 0.037

Δ V85max − 0.006 VL− 0.002 R

193.30

3405.44

3442.30

SP3b

Y = AADT 0.949L0.771e−10.943 + 0.034 Δ85 V − 0.010 VL

207.65

3402.09

3433.69

SP4b

Y = AADT 0.956L0.503e−9.370 + 0.043 × 85MSR − 0.012 VL

215.75

3408.49

3440.08

Y = five-year collision frequency. The p-value for the likelihood-ratio test is less than 0.001.

through the Naturalistic Driving Study (NDS) to improve design consistency evaluation of horizontal curves and model collision frequency on horizontal curves based on speed reduction from the approach tangent to the curve. A subset of NDS data in Washington State was used to investigate the relationship between design consistency and safety performance of horizontal curves on two-lane rural arterial highways while considering three major issues. First, the problems associated with data collection methods were addressed through using the NDS database, thus depicting drivers’ speed behavior at higher level of accuracy. Second, several parameters to quantify the 85th percentile speed reduction between the approach tangent and horizontal curve were evaluated using drivers’ speed profiles. Third, safety performance of horizontal curves was related to the speed reduction parameters and five-year collision frequency on 1430 horizontal curves in rolling and mountainous terrain in the same geographic area for the NDS data used in speed reduction modeling. The main findings of this study are:

independent variables. As shown in the table, all models in the second set have lower AIC and BIC values than the corresponding models in the first set and lower than the base model with traffic exposure parameters only. Therefore, the addition of these curve characteristics improves the safety prediction models. The table also shows that VL was the only additional significant variable in three models. This suggests that the speed reduction parameters are more significant variables in predicting collision frequency than all curve geometric characteristics. In all models, collision frequency decreases with the increase of VL . Normally, a higher speed limit should be a result of a higher design speed and more generous design in general, which should result in better safety performance. For Model SP2b, R was also a significant variable but the speed reduction parameter had a negative coefficient suggesting that collision frequency decreases with the increase in ΔV85max . Finally, Model SP3b had the lowest AIC and BIC values, followed closely by the other three models. The relationship between predicted collision frequency using the second set of models and speed reduction is shown graphically in Fig. 7. To apply these safety performance models, the speed reduction models in Table 2 were first applied to predict the different speed reduction parameters based on the curve geometric characteristics. It should be noted that the selected curve length of 200 m can produce unrealistically high deflection angles for sharp curves. These curves were kept in the figures only to illustrate the trends of speed reduction and collision frequency over a wide range of radius. As shown in Fig. 7a, expectedly, all speed reduction parameters decrease with the increase in R . However, the speed reduction parameter ΔV85 falls entirely within the good and fair regions according to Safety Criterion II. If this criterion is applied to the other speed reduction parameters, ΔV85max would also fall entirely in the good and fair design regions while Δ85 V and 85MSR would span over the three regions of good, fair, and poor design depending on the curve characteristics. The estimated speed reduction parameters were then used to estimate the collision frequency using the safety performance models in Table 5. Again, Fig. 7b shows that the base model using traffic exposure parameters only fails to account for the effect of R or speed reduction on the expected collision frequency. The figure also shows that for sharp curves, the predicted collision frequency using Model SP2b increases with the increase of. This is counterintuitive trend, which has likely resulted from the negative coefficient for ΔV85max in the model. Finally, although ΔV85 never fell in the poor design category, the predicted collision frequencies using Model SP1b were higher for example than Model SP3b.

• The NDS database is a useful resource to study driver behavior on

•

• • •

4.3.3. Zero-inflated models Because the collision dataset included excessive zero counts (784 out of 1430 curves had zero five-year collision frequency), zero-inflated negative binomial (ZINB) regression was examined. The values of the Vuong test statistic for all safety performance models to predict the fiveyear collision frequency were within ± 1.96. Thus, the ZINB models are equally preferred to the NB models and are not presented in this paper.

• •

5. Discussion and conclusions This paper presented an effort to exploit the data collection effort 197

horizontal curves on rural two-lane horizontal curves. The NDS data has been collected over a long time period (about three years). Thus, the developed speed reduction models should depict more realistic driving behavior compared to models that already exist in the literature. However, for the curves studied in this paper, the sample size of trips at night or during unfavorable weather conditions was much smaller than that during daytime and favorable weather conditions. Using a larger sample is recommended to model driving behavior during various weather and pavement conditions and daytime and nighttime. The data showed that a headway of 3 s can be set as the threshold for free-flow speed. Statistical testing showed no evidence that drivers’ speed behavior is affected by the headway beyond this threshold. The availability of speed profiles in the NDS data allows extraction of different parameters to estimate the speed reduction from an approach tangent to a horizontal curve. Approach tangent speed was a significant variable affecting all speed reduction parameters on horizontal curves, which confirms the concept that curve speed and speed reduction from tangent to curve depends on the road’s speed environment. However, the approach tangent speed can be difficult to estimate, and therefore alternative models were developed after excluding this variable. In this case, the posted speed limit became a significant variable in three speed reduction models, and thus it can be considered a surrogate measure of the road’s speed environment. In addition, curve radius, curve length, and deflection angle were also significant variables affecting speed reduction on horizontal curves. 2 Based on the values of Radj , the speed reduction models had poor to moderate fit. This is not uncommon in models that attempt to predict human behavior. Out of the four speed reduction parameters considered in the paper, the NDS data provided the best fit in predicting Δ85 V . Another advantage in this speed reduction parameter is that it can be evaluated without having to collect speed profile data. Thus, Model S3b is the recommended model in this paper to

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Fig. 7. Relationship between Collision Frequency and Speed Reduction Using Second Set of Safety Performance Models. (a) Relationship between Speed Reduction and Curve Radius. (b) Relationship between Collision Frequency and Curve Radius.

estimate speed reduction.

• The model to predict the traditional speed reduction ΔV

may underestimate speed reduction causing the estimated speed reduction to fall within the fair or good design region according to Safety Criterion II. In modeling the expected collision frequency on horizontal curves, the NB regression provided better results than Poisson regression because of the overdispersion in the collision data. Also, Vuong test showed that ZINB was not preferred over the NB. Therefore, only NB regression models are presented and discussed in the paper. The safety performance modeling was performed using three steps depending on the independent variables included in the regression: (1) base model using traffic exposure parameters only, (2) first models set using traffic exposure and speed reduction parameters, and (3) second models set traffic exposure, speed reduction, and curve characteristics. The first models set showed that all models except Models SP2a (speed reduction parameter is ΔV85max ) provided better fit than the base model as indicated by the AIC and BIC values. The same speed 85

• •

•

•

•

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reduction parameter in Model SP2b had a negative coefficient suggesting that collision frequency decreases with the increase in ΔV85max . Subsequently, based on the results of this paper, ΔV85max is not recommended as a speed reduction parameter for design consistency evaluation. For the three other speed reduction parameters, none of the curve geometric characteristics was an additional significant variable in the second set of safety performance models. This suggests that speed reduction parameters are more significant variables in predicting collision frequency than all curve geometric characteristics, and therefore safety performance on horizontal curves should account for the curve speed reduction. However, posted speed limit could be an additional significant variable, which confirms the importance of considering the road speed environment in design consistency evaluation. Based on the values of AIC and BIC, Models SP3a and SP3b using Δ85 V provided the best fit in the two sets of safety performance models, followed by Models SP4a and SP4b using 85MSR . It is also noted that the speed reduction models for these two parameters

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•

2 (Models SRb3 and SRb4) also had the Radj values. Subsequently, the results of this paper suggest that the Δ85 V is the best design consistency parameter followed closely by 85MSR . It is noted that previous research has also compared the speed reduction measures based on their relative values only; e.g. (Bella, 2007). However, the conclusion reached in this paper on the preference of Δ85 V as a speed reduction measure and design consistency parameter is based on two important criteria: goodness of fit of the speed reduction model and goodness of fit of the safety performance model. That is, based on the results of this study, Δ85 V is the parameter that is most accurately estimated using the road geometry and will most accurately estimate the expected safety performance.

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Disclaimer The findings and conclusions of this paper are those for the authors and do not necessarily represent the views of the VTTI, SHRP2, the Transportation Research Board, or the National Academy of Science. Acknowledgements The authors would like to express their gratitude to Anusha Nujjetty (HSIS lab manager) and Kevin Perkins (WSDOT-Multimodal Planning Division) for providing necessary data and information to conduct the analysis of this research. The authors are highly indebted to Miguel Perez, Christina Witcher, and Whitney Atkins (Virginia Tech Transportation Institute) for their help in providing the naturalistic driving behavior data. Financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. References Anderson, I., Bauer, K., Harwood, D., Fitzpatrick, K., 1999. Relationship to safety of geometric design consistency measures for rural two-lane highways. Transp. Res. Rec. 1658, 43–51. Awatta, M., Hassan, Y., Sayed, T., 2006. Quantitative evaluation of highway safety performance based on design consistency. Int. J. Adv. Transp. Stud. IX, 29–44. Bella, F., 2007. Parameters for evaluation of speed differential: contribution using driving simulator. Transp. Res. Rec. 2023, 37–43. Bella, F., Calvi, A., D’Amico, F., 2014. Analysis of driver speeds under night driving

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