Crash Prediction Models for Rural Motorways

Transcription

1 Crash Prediction Models for Rural Motorways Alfonso Montella, Lucio Colantuoni, and Renato Lamberti In this paper, crash prediction models for estimating the safety of rural motorways are presented. Separate models were developed for total crashes and severe (fatal plus all injury) crashes. Generalized linear modeling techniques were used to fit the models, and a negative binomial distribution error structure was assumed. The study used a sample of 2,245 crashes (728 severe crashes) that occurred from 2001 to 2005 on Motorway A16 between Naples and Canosa in Italy. Many characteristics of the motorway were substandard. The motorway allowed investigation of a wide spectrum of geometric configurations. The models were developed by the stepwise-forward procedure with explanatory variables related to traffic volume and composition, horizontal alignment, vertical alignment, design consistency, sight distance, roadside context, cross section, speed limits, and interchange ramps. The decision to keep a variable in the model was based on two criteria. The first was whether the t-ratio of the variable s estimated coefficient was significant at the 5% level. The second criterion was related to the improvement of goodness-of-fit measures of the model that includes that variable. Goodness-of-fit measures were the parameter R 2 and Akaike s information criterion. All the parameters have a logical and expected sign. The most important result was that design consistency measures significantly affected road safety, not only on two-lane rural highways, but also on motorways. The success of safety improvement programs in reducing crash occurrence depends on the availability of methods that give reliable estimates of the safety level associated with existing road locations or proposed designs (1). Other than approaches based on safety inspection methodologies (2 4), which apply only to existing roads, several others exist for estimating safety. These range from simply using crash rates to crash prediction models, which relate the expected crash frequency at a road location to its traffic and geometric characteristics. Several researchers (5) have shown that the relationship between crash frequency and exposure is frequently nonlinear, which indicates that crash rates are not appropriate representatives of safety. This finding has led most safety researchers to use predictive models in which the dependent variable is the crash frequency. There are several regression techniques to develop crash prediction models. The earlier models were developed by using ordinary or normal linear regression. These models assume a normal error structure for the response variable, a constant variance for the residuals, and the existence of a linear relationship between the response and explana- Department of Transportation Engineering Luigi Tocchetti, University of Naples Federico II, Via Claudio 21, Naples, Italy. Corresponding author: A. Montella, alfonso.montella@unina.it. Transportation Research Record: Journal of the Transportation Research Board, No. 2083, Transportation Research Board of the National Academies, Washington, D.C., 2008, pp DOI: / tory variables. Many studies (6, 7) have demonstrated the inappropriateness of conventional linear regression in modeling discrete, nonnegative, and rare events such as traffic crashes due to the nonlinear relationship with traffic volume. Such limitations led to the use of a Poisson or a negative binomial error structure for modeling the occurrence of traffic crashes. The main advantage of the Poisson error structure is the simplicity of the calculations. However, most accident data are likely to be overdispersed (the variance is greater than the mean), which indicates that the negative binomial distribution is usually the more realistic assumption. Many independent variables affect crash frequency. However, due to the difficulty in estimating their value, researchers generally extract a reduced number of variables for inclusion in the proposed model. Generally, traffic flow and road geometry have been used as explanatory variables. Most prediction models refer to two-lane rural highways, whereas motorways (freeways) have been investigated to a lesser degree (8). Persaud and Dzbik (9) developed a prediction model with data for urban freeways in Ontario, Canada. Explanatory variables were traffic flow and section length. Hadi et al. (10) calibrated a set of safety prediction models with data from Florida roadways. The models were categorized by crash severity, area type (i.e., urban or rural), and number of through lanes. The freeway models addressed the following facility types: rural freeways with four or six lanes, urban four-lane freeways, and urban six-lane freeways. On rural freeways, explanatory variables were traffic flow, section length, inside shoulder width, and number of interchanges on freeway segment. On urban freeways, more variables were used: lane width, outside shoulder width, speed limit, and median width. Lord et al. (11) developed predictive models for rural and urban freeway segments located in Montreal, Quebec, Canada. Traffic flow characteristics such as traffic volume, vehicle density, and volume-to-capacity ratio were used as explanatory variables. Caliendo et al. (8) developed crash prediction models with data from a motorway in Italy. The models applied separately to tangents and curves. For curves, it was shown that significant variables were section length, traffic flow, and curvature, while, for tangents, they were section length, traffic flow. and presence of junctions. Only recently, design consistency measures, which quantify interaction between driver behavior and road geometry, have been successfully used as crash explanatory variables (12 14). Yet, the latter studies referred only to two-lane rural highways. The objective of this study is to develop crash prediction models for estimating the safety of rural motorways (freeways) as a function of explanatory variables related to traffic volume and composition, horizontal alignment, vertical alignment, design consistency, sight distance, roadside context, cross section, speed limits, and interchange ramps. Separate models were developed for total crashes and severe (fatal plus all injury) crashes. Generalized linear modeling techniques were used to fit the models, and a negative binomial distribution error 180

2 Montella, Colantuoni, and Lamberti 181 structure was assumed. Geometric data, traffic volumes, and crash records were collected for Motorway A16 from Naples to Canosa in Italy. Motorway A16 between Naples and Canosa in Italy is part of the Trans-European road network (Road E841). It is located in the south of Italy and links the west coast (Motorway A1) and the east coast (Motorway A14). It is a divided highway with two lanes for each direction (lane width of 3.75 m, right shoulder width of 0.50 to 3.50 m, and median width of 2.00 m), access control, and interchanges. Many characteristics of the motorway are substandard. This motorway allowed investigation of a wide spectrum of geometric configurations (Table 1). The section between Naples and Candela, with a length of km, was studied. The corridor is connected to the road network by 11 interchanges, with 21 exit ramps (13 on curves, eight on tangents) and 20 entrance ramps (11 on curves, nine on tangents). There are eight rest areas. Part of the route is in mountainous terrain, with 11 tunnels (total length of km) and 38 bridges (total length of km). Two climbing lanes are located in the east direction (total length of km), and one is located in the west direction (length of km). The general speed limit is 130 km/h, the maximum legal speed in Italy. Local speed limits of 80 km/h are present in both travel directions. Sections with local speed limits have a total length of km in the east direction and km in the west direction. Horizontal alignment characteristics and traffic flow volumes were used to divide the study section into 646 homogeneous segments (343 for each direction of the motorway). Longitudinal grade was not considered as a parameter for road segmentation because crash data, even if accurate in relation to horizontal alignment, do not report reliable longitudinal grade information. Length of segments varies between 62 and 3,509 m. The radius of horizontal curves varies between 245 and 4,000 m. Spiral transitions are not present. The deflection angle varies between 5 and 109 gon. The superelevation mean is equal to 3.25%. The radius of vertical curves varies between 3,000 m (sag curve) and 30,000 m (crest curve). The maximum longitudinal grade is equal to 6.35%. Sight distance is often less than the stopping sight distance and ranges between 86 and 840 m. DATA DESCRIPTION Geometric Data Traffic and Crash Data Average annual daily traffic (AADT) data were provided from the motorway management agency for 2001 to They were disaggregated for each carriageway and for each section between successive interchanges. AADT ranged between 7,266 and 16,000 vehicles per day. Weighted AADTs were computed for each year (Table 2). Heavy-vehicle traffic ranged between 1.70% and 16.48% of the total traffic (mean = 11.10%, standard deviation = 4.23%). Two-wheeler traffic is not relevant (mean = 0.24%, standard deviation = 0.14%, minimum = 0.00%, maximum = 0.64%). Crash data were collected through analysis of police reports and integrated with detailed site inspections. Crash data covered 2001 to 2005 (Table 2). Crashes at the interchanges ramps, at rest areas, and at tollbooths were excluded because of the location descriptions of the police reports. In the analysis period, 2,245 crashes occurred; 1,085 crashes took place in the east carriageway and 1,160 in the west carriageway. Severe crashes (fatal plus all injury) were 728, with 305 occurring in the east carriageway and 423 in the west carriageway. The most frequent crash type was run off the road (N = 1,092, 49.6% of the total). MODEL DEVELOPMENT Model Description Generalized linear modeling techniques were used to fit the models, and a negative binomial distribution error structure was assumed. The approach had the following theoretical basis (15, 1). Let Y be the random variable that represents the accident frequency at a given site during a specific period, and let y be a certain realization of Y. The mean of Y, denoted by Λ, is itself a random variable (8). For Λ =λ, Y is Poisson distributed with parameter λ: ( ) = PY= yλ = λ λ e y! y λ () 1 TABLE 1 Summary Statistics of Geometric Data Parameter Mean Standard Deviation Minimum Maximum Length a (m) , Radius of horizontal curves b (m) , Deflection angle (gon) c Superelevation (%) Radius of sag vertical curves d (m) 9, , , , Radius of crest vertical curves e (m) 10, , , , Longitudinal grade (%) Sight distance (m) Right shoulder width (m) a 646 segments. b 326 horizontal curves. c The dimension gon corresponds to 400 angle units in a circle instead of 360 degrees. d 182 sag curves. e 36 crest curves.

3 182 Transportation Research Record 2083 TABLE 2 Summary Statistics of Traffic and Crash Data East Direction West Direction Total Total Severe Weighted Total Severe Weighted Total Severe Year Crashes Crashes AADT Crashes Crashes AADT Crashes Crashes AADT Range , , ,266 13, , , ,534 14, , , ,679 14, , , ,971 15, , , ,834 16,000 Total 1, , , EY ( Λ= λ) = λ ( 2) In the model, segment length is an offset variable. It captures the Var ( Y Λ= λ) = λ () 3 Usual practice is to assume that the distribution of Λ follows a gamma probability density function (15). If Λ is described by a gamma distribution with shape parameter κ and scale parameter κ/µ, then its density function is f Λ κμ λ e ( λ) ( ) = 1 Γ κ E( Λ) =μ () 5 Var ( Λ) = μ + μ 2 κ κ κ ( κμ) λ ( ) ( 4) () 6 and the distribution of Y around E(Λ) = µ is a negative binomial with an expected value and variance of logical requirement that if N crashes are expected to occur on 1 km of road, 2N crashes should be expected to occur on an identical road that is 2 km long. The model parameters and the dispersion parameter were estimated by the maximum likelihood method by using the GENMOD procedure in SAS. Separate models were developed for all crashes and severe (fatal plus all injury) crashes. First, to determine which explanatory variables (in addition to exposure variables) significantly affect crash frequency alone, simple models were estimated with one independent variable in each. Nonsignificant variables were excluded from further investigation. Then, the models were developed by the stepwise forward procedure, with one explanatory variable being added at each step. The decision on whether to keep a variable in the model was based on two criteria. The first was whether the t-ratio of the variable s estimated coefficient was significant at the 5% level. The second criterion was related to the improvement in the goodness-of-fit measures of the model that includes that variable. EY ( ) =μ ( 7) Var ( Y ) = μ + μ 2 κ The shape parameter κ is also called the dispersion parameter of the negative binomial distribution. It is estimated iteratively in the modeling procedure from the model residuals, with the method of maximum likelihood being the most widely used. Because the variance decreases as κ increases, the value of κ can also be used to compare the goodness of fit of various models fitted to the same data, in that the larger the value of κ is, the smaller the variance and the better the model will be (16). The dispersion parameter plays an important role in the empirical Bayes method that represents the best available approach for estimating the safety of a highway entity (17). The selected model form is as follows: ( ) = a + a ln( AADT) E Λ L e e n bi xi 0 1 i= 1 9 where E(Λ) = predicted annual crash frequency, L = segment length (m), a i, b i = model parameters, and x i = explanatory variables additional to L and AADT. () 8 () Measuring Goodness of Fit Several measures can be used to assess the goodness of fit of the models. To measure the goodness of fit in linear regression models, the coefficient of determination R 2 is often used. Miaou (18) used simulation to demonstrate the pitfalls of using the traditional R 2 goodnessof-fit measure in crash prediction models and suggested the use of R 2 α, a dispersion parameter based R 2, which is calculated as follows: 2 R α κ = 1 10 min κ ( ) where κ min is the smallest possible dispersion parameter that is obtained by having no covariates in the model (by assuming that all sites have an identical prediction estimate equal to the mean overall sites) and κ is the dispersion parameter for the calibrated model. The main advantage of this measure is its simplicity in addition to its being bound between 0 (when no covariate is included) and 1 (when covariates are perfectly specified). Another goodness-of-fit measure used was the Akaike information criterion (AIC) (19). The AIC value is calculated as follows: AIC = 2 ML + 2 p ( 11) where ML is the maximum log likelihood of the fitted model and P is the number of parameters in the model.

4 Montella, Colantuoni, and Lamberti 183 The smaller the value of AIC is, the better the model is. The first term in the AIC equation measures the badness of fit, or bias, when the maximum likelihood estimates of the parameters are used. The second term measures the complexity of the model, thus penalizing the model for using more parameters. The goal for selecting the best model is to choose the best fit with the least complexity. Selection of Variables A brief description of the explanatory variables selected in the study follows. Traffic Volume and Composition Relationship between crashes and traffic volume is frequently nonlinear. Thus, traffic volume was not considered as an offset variable. The natural logarithm of AADT, an aggregate measure of traffic volume, was considered. Traffic composition was evaluated against the variables percentage of heavy traffic (P hv ) and percentage of two wheelers (P tw ). Horizontal Alignment Literature suggests that the geometric design parameter that most affects road safety is the horizontal curvature or curvature change rate. Because spiral transitions are not present, curvature (1/R) was selected. Studies on crash modification factors (CMFs) show that for a given radius larger deflection angles (central angles of the curves) correspond to lower CMF values (20). This trend suggests that most curve crashes are associated with the curve entry maneuver such that curves with a larger deflection angle (for a given radius) are easier for the driver to detect in advance and thereby allow safe negotiation of the entry maneuver. Because deflection angle (def) in the study sites varies in a considerable range (5 to 109 gon), it was selected as an explanatory variable so as to verify whether the crash data confirm literature findings. Vertical Alignment Longitudinal slope is not always constant in the segments. To take this circumstance into account, for each segment, an equivalent downgrade (G d ) and upgrade (G u ) were assessed. Equivalent grade was obtained by weighing each gradient in relation to the segment length with the following formula: G j = i Gij L L i i ij ( 12) where G j = equivalent grade, with j equal to d (downgrade) and u (upgrade), G ij = grade of downgrade (upgrade) subsegment i, L ij = length of downgrade (upgrade) subsegment i, and L i = length of subsegments that make up segment under consideration. In the vertical curves, two situations occur. If the different grades have the same direction, half of the vertical curve is added to each grade. If the grades have opposite directions, a quarter of the vertical curve is added to each grade. Research carried out in Germany showed that upgrade sections generally experience fewer crashes than downgrade sections (21). The importance of the driver s receiving precise visual cues from the road environment is well recognized. If the visual cues are confusing or in any way cause the driver to assess incorrectly the approaching road environment, the driver s crash risk may increase. Experiments in simulated road environment have shown that overlapping crest curves made the horizontal curvature appear sharper and that overlapping sag curves made the horizontal curvature appear less sharp (22). Moreover, crash prediction models for two-lane rural highways (23) showed that crest-curve grade rate affects accident frequency. Variables included in the model are the curvature of sag (1/R sag ) and crest (1/R crest ) vertical curves. Moreover, a binary variable (Crest c ) was introduced and assumes a value of one if there is a crest curve with sight distance shorter than the stopping sight distance before the entrance in horizontal curve. Design Consistency Design consistency refers to a highway geometry s conformance with driver expectancy. A consistent highway design ensures that successive elements are coordinated in such a way as to produce harmonious and homogeneous driver performances along the road. Practice emphasizes that an alignment with inconsistencies requires drivers to handle speed gradients to drive safely on certain alignment elements. On this basis, the importance of identifying inconsistencies on highways is emerging as an important feature in highway design. Three consistency measures, introduced by Lamm et al. (24, 25) are included in the model: (a) ΔV d, design speed consistency (absolute value of the difference between design speed and operating speed), (b) ΔV 85, operating speed consistency (absolute value of the 85thpercentile speed reduction through successive elements of the road), and (c) Δf r, consistency in driving dynamics (difference between side friction assumed with respect to the design speed and side friction demanded at the 85th-percentile speed). The variables are assessed only on curves. V d is set equal to the average 85th percentile of the speed along the road (weighted to the element length). If the curve requires an acceleration, ΔV 85 is set to 0. The variable Δf r is computed by the formula Δfr = fra frd = V V 5 2 ( d d ) 2 V R e ( 13) where f ra = side friction assumed with respect to the design speed, f rd = side friction demanded at 85th-percentile speed, V d = design speed (km/h), V 85 = operating speed [85th percentile of the speed distribution (km/h)], R = radius of horizontal curve (m), and e = curve superelevation.

5 184 Transportation Research Record 2083 Operating speed (V 85 ) was assessed by using the following predictive model (R 2 =.95) calibrated in Italy on Motorway A3 (26): V def = g ( 14) R 2 where def 2 is the total deflection angle in the 2 km before the end of the segment (gon) and g is the absolute value of longitudinal grade (%). Brenac (27) reported that the increase in the length of the straight alignment preceding the curve has a positive effect on crashes. The explanatory variable length of the tangent preceding the curve (Lt) was thus introduced. Moreover, to take into account the effect of short-term drivers expectancy, the variable second curve (Sc) was introduced. It is a binary variable, which assumes a value of 1 if the curve is preceded by another curve or by a tangent shorter than 250 m. According to the Italian Geometric Design Standards (28), tangents shorter than 250 m are not considered as elements interrupting the curve-to-curve sequence. Sight Distance Inadequate sight distance on horizontal and vertical curves is a common contributory factor to crashes. Previous research showed that, below certain threshold values, restrictions to sight distance increase crash frequency (24). The literature has reported significant safety benefits related to measures that improve sight distance (29). Sight distance evaluations were performed by a three-dimensional digital reconstruction of the route with the software Civil Design 6.0. In the sections without obstructions, available sight distance was set equal to 840 m (6 V dmax, where V dmax is the maximum design speed according to the Italian standards). Planimetric obstructions to sight distance were safety barriers, retaining walls, bridge abutments, and tunnel walls. Sight distance was evaluated in both the inside and the outside lanes. According to the criteria in the Italian standards (28), driver s eye height was set equal to 1.1 m, whereas obstacle height was set equal to 0.1 m. On the vertical profile, the road surface of crest vertical curves was assumed as an obstruction limiting the driver s sight distance. Sight distance evaluations were performed with a step equal to 20 m. Section sight distance was set equal to the average sight distance along the segment. Roadside Context To take into account the safety effect of the roadside context, the following binary variables were introduced: bridge, equal to 1 if the section was on a bridge; tunnel, equal to 1 if the section was in a tunnel; embankment, equal to 1 if after the right shoulder there was an embankment; cut, equal to 1 if after the right shoulder there was a cut. Cross Section The following variables related to cross section were introduced: right-shoulder width (RSW); presence of a climbing lane, binary variable equal to 1 if a climbing lane was present; median, binary variable equal to 1 if a concrete barrier was present in the median (0 if a steel barrier was present). According Hadi et al. (10), RSW is a significant variable on urban freeways. On two-lane rural highways, significant crash reductions were observed following installation of a climbing lane (20). On French motorways, presence of median concrete barriers increased crash severity compared with steel barriers. In run-off-the-road single-vehicle crashes against concrete barriers, the ratio of injury crashes to total crashes was 1.9 greater than in crashes against steel barriers (30). Speed Limits Hadi et al. (10) found that speed limit is a significant variable on urban freeways. The binary variable speed limit was introduced. Interchange Ramps Interchange ramps may have a noteworthy effect on crash frequency. Caliendo et al. (8) found that the presence of interchange ramps was a significant variable in tangent sections, whereas it was not significant on horizontal curves. Abdel-Aty and Pemmanaboina (31) found a different effect for on-ramps and off-ramps. To test the results reported in the literature, the effect of entrance and exit ramps on both tangents and curves were studied by introducing the following binary variables: Ramp, equal to 1 if there is a ramp in the section Ramp c, equal to 1 if there is a ramp in a curve Ramp t, equal to 1 if there is a ramp in a tangent Exit, equal to 1 if there is an exit ramp in the section Exit c, equal to 1 if there is an exit in a curve Exit t, equal to 1 if there is an exit in a tangent Ent, equal to 1 if there is an entrance ramp in the section Ent c, equal to 1 if there is an entrance in a curve Ent t, equal to 1 if there is an entrance in a tangent Rest, equal to 1 if there is a rest area entrance or exit in the section Year Effect Crash numbers during 2002 were greater than crashes during the remaining years for both severe and all crashes and in both carriageways. Because a specific cause for this phenomenon was not found, dummy variable Yr02 was introduced to capture the potential nonrandom year effect. Correlations Between Variables The Pearson product moment correlation (Pearson s correlation) was assessed to identify correlations between explanatory variables. Pearson s correlation reflects the degree of linear relationship between two variables. ESTIMATION RESULTS Single-Variable Models Simple models were estimated with one independent variable (additional to exposure variables) in each. Separate models were developed for all crashes and severe crashes (fatal plus all injury). Regression results are reported in Table 3. Variables not significant at the 5%

6 Montella, Colantuoni, and Lamberti 185 TABLE 3 Parameters Estimates and Goodness of Fit for Single Variable Models Total Crashes Severe Crashes Variable Estimate p-value R 2 α AIC Estimate p-value R 2 α AIC Ln(AADT) only < , < ,372.0 Δf r < , < , /R < , < ,733.0 ΔV < , < ,948.1 Def < , < ,046.4 ΔV d < , < ,111.6 Sc < , ,344.0 SL < , < ,310.5 SD < , ,360.3 L t < , < ,345.3 G d < , < ,295.5 G u < , < , /R crest < , ,350.9 Yr , < ,333.6 Ramp t , < ,344.4 Exit t , ,351.0 Median , ,361.7 Tunnel , ,372.9 Exit c , ,373.5 P tw , ,360.8 Ent , ,365.6 Ent t , ,372.0 Ramp , ,355.4 Embankment , ,366.4 Cut , ,352.0 Rest , ,370.2 CL , , /R sag , ,381.0 Ent c , ,372.0 RSW , ,357.9 Crest c , ,372.4 P hv , ,360.8 Bridge , ,374.6 Exit , ,361.2 Ramp c , ,372.6 level of significance were excluded from further analysis. Significant variables were ranked according decreasing value of R 2 α. Horizontal curvature and design consistency measures are the parameters that give rise to the models with the best goodness of fit (greatest R 2 α and smallest AIC). As expected, crash frequency increased with a decrease in Δf r and an increase in 1/R, ΔV 85, and ΔV d. The aforementioned variables are strongly correlated with each other. In addition, the two consistency measures Sc and Lt were positively related to crash frequency. The variable def was positively related to crash frequency, in contrast with literature results (28, 29). It mainly depended on the positive correlation (CP =.87) between def and curvature. Longitudinal slope had a positive sign for downgrade (G d ) and a negative sign for upgrade (G u ). This result was consistent with literature findings (21). Curvature of sag vertical curves (1/R sag ) was not significant, whereas curvature of crest vertical curves (1/R crest ) was significant with a negative sign, which was an effect opposite to the expected one. This result depended on the negative correlation with curvature and design consistency variables. Year effect (Yr2002) was significant and had a positive effect on crashes. Complete Models The models were developed by the stepwise-forward procedure, with one explanatory variable added at each step. Variables were added according to the descending order of R 2 α. Results of the stepwise procedure for total crashes are reported in Table 4. Results for severe crashes are reported in Table 5.

7 186 Transportation Research Record 2083 TABLE 4 Stepwise Procedure: Parameter Estimates and Goodness-of-Fit Measures, Total Crashes Step k Constant Ln(AADT) Δf r 1/R ΔV 85 Def ΔV d Sc SL SD L t n.s TABLE 5 Stepwise Procedure: Parameter Estimates and Goodness-of-Fit Measures, Severe Crashes Step k Constant Ln(AADT) 1/R Δf r ΔV 85 Def ΔV d SL G d L t G u Yr02 Sc Ramp t n.s n.s

8 Montella, Colantuoni, and Lamberti 187 G d G u 1/R crest Yr02 Ramp t Exit t Median Tunnel Exit c p-value R 2 α AIC < ,762.1 < ,508.3 < ,906.7 < ,877.7 < ,877.6 < , , , , ,861.3 < , ,835.2 n.s < , , , , , , , ,799.9 Exit t 1/R crest Cut RSW Ramp Ent t P hv SD CL Ent Bridge Median p-value R 2 α AIC < ,716.2 < ,372.0 < ,732.6 < ,711.5 < ,711.4 < , , , ,692.5 < ,674.5 < ,586.3 < , , , , , , , , , , , , , , ,501.9

9 188 Transportation Research Record 2083 For total crashes, the best model has the following equation: ( ) = E Λ L e where E(Λ) = predicted annual total crash frequency, Yr02 = binary variable, equal to 1 in 2002, and G u = upgrade (%). All the parameters have a logical and expected sign. The most important result is that measures of design consistency significantly affect road safety not only on two-lane rural highways but also on motorways. As the difference between friction demand and friction supply on horizontal curves (Δf r ) decreases, crash frequency is expected to increase. Given that existing curve superelevation is frequently less than the superelevation required by the standards and that superelevation adjustment is a feasible and quick measure, this result has a relevant practical effect on the selection of safety measures. Furthermore, an increase in operating speed reduction (ΔV 85 ) on horizontal curves has a positive effect on crash frequency. Even Lt is positively related to crash frequency, as reported in the literature (27). The negative sign of the estimated coefficient for def confirmed the hypothesis that curves with a larger deflection angle (for a given radius) are easier for the driver to detect in advance and thereby allow safe negotiation of the entry maneuver. The negative sign of upgrade parameters suggests that upgrade sections generally experience fewer crashes than downgrade sections, as reported in the literature (21). The model s AIC and R 2 α values were equal respectively to 4,792.6 and 54.2%, indicating an acceptable fit. For severe crashes, the best model has the following equation: E Λ L e s ( ) = e R ΔV Lt Yr Δf r def Gu e ln( AADT) ln( AADT) ΔV Median Yr Δf R r 034. Sc def Gu Phv bridge ( 15) ( 16) where E s (Λ) = predicted annual severe crash (fatal plus all injury) frequency, median = binary variable, equal to 1 if a concrete barrier is present in the median, Sc = binary variable, equal to 1 if the curve is preceded by another curve or by a tangent shorter than 250 m, P hv = heavy traffic (%), and bridge = binary variable, equal to 1 if the section is on a bridge. The best model for severe crashes was similar to the all-crashes model. Significant variables of the total crashes model were significant also for severe crashes and had the same sign. An exception was the variable length of the tangent preceding the curve (Lt), which was not significant in the severe-crashes model. Four variables not significant in the previous model became significant in the severe-crashes model: median (positive sign), Sc, P hv, and bridge (negative sign). The significance and the positive sign of the parameter median indicate that the type of median safety barriers does not affect crash frequency but has a significant effect on crash severity. Total crashes against a median with steel safety barriers were 360 (86 severe crashes), with a ratio between severe and total crashes of 23.9%. Total crashes against a median with concrete barriers were 59 (29 severe crashes), with a ratio between severe and total crashes of 49.2%. For concrete barriers, the percentage of severe crashes was 2.06 times that for steel safety barriers. This result was consistent with the findings of a crash study performed on medians in French motorways (30), which found that the percentage of severe crashes for concrete barriers is 1.9 times that for steel barriers. The model s finding suggests the installation of median steel safety barriers in place of concrete barriers. The negative sign of the second curve (Sc) parameter highlights the role of short-term drivers expectancy because it shows that, all else being equal, a reduction in severe crashes on a curve is expected if the curve is preceded by another curve. The negative sign of the parameters P hv and bridge suggests that drivers might increase their risk perception in heavy traffic and on bridges and thus decrease their speed and then thereby crash severity. The AIC and R 2 α values of the severe-crashes model were, respectively, 5,501.9 and 65.8%, indicating a reasonable goodness of fit. For AIC, the severe-crashes model performed better than the all-crashes model, but for R 2 α, it performed worse. When no covariates were added to the model, the overdispersion parameter of the all-crashes model was greater than the one for severe crashes (0.477 versus 0.264), indicating a smaller variance for the total-crashes data. This difference explains the smaller value of AIC. In contrast, covariates explain a larger portion of the systematic variability in severe-crash frequency, giving rise to the greater R 2 value. α CONCLUSIONS For total crashes, the best model includes curvature (1/R), operating speed reduction (ΔV 85 ), length of the tangent preceding the curve (Lt), and year effect (Yr02), all with a positive sign; the difference between the friction demand and supply (Δf r ), deflection (def), and upgrade (G u ), all with a negative sign. All the parameters had a logical and expected sign. The most important result was that measures of design consistency significantly affected road safety, not only on two-lane rural highways but also on motorways. As Δf r decreases, crash frequency is expected to increase. Given that existing curve superelevation is frequently smaller than the superelevation required by the standards and that superelevation adjustment is a feasible and quick measure, this result has a relevant practical effect for the selection of safety measures. The severe-crashes best model is similar to the all-crashes model. The significant variables of the total crashes model are significant also for severe crashes and have the same sign. The exception is the variable length of the tangent preceding the curve (Lt), which is not significant in the severe-crashes model. Four variables not significant in the previous model become significant in the severe crashes model: median (positive sign); second curve (Sc), percentage of heavy traffic (P hv ), and bridge (negative sign). The significance and the positive sign of the parameter median indicate that the type of median safety barriers has a significant effect on crash severity. Indeed, for concrete barriers the percentage of severe crashes is 2.06 times that for steel safety barriers. This result, which is consistent with the literature findings, suggests the installation of median steel safety barriers in place of concrete barriers. The model s AIC and R 2 α values were, respectively, 4,792.6 and 54.2% for all crashes and 5,501.9 and 65.8% for severe crashes, indicating a reasonable goodness of fit of the models.

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