Developing Safety Performance Functions for Commercial Motor Vehicle Crashes at Interchange Ramp Segments in Kentucky

Ramp-Related Safety Studies

Compared with roadway segments and intersections, ramp segments have not been thoroughly investigated in the literature. For example, Parajuli et al. ( 14 ) developed separate SPFs for different ramp classifications, including ramp segments, ramp terminals, and interchanges (i.e., the mainline portion of the freeway within the influence zone of the interchange). Crashes were gathered from different regions of Ontario between 1997 and 2003. For ramp segments, separate SPFs using the NB structure were developed for property damage only (PDO) and injury/fatal crashes. Two predictor variables were used to develop the ramp SPFs: annual average daily traffic (AADT) along the ramp section and ramp length. Lu et al. ( 15 ) developed a simple SPF model and a full SPF model, in which the former incorporated traffic volume only, whereas the latter included traffic volume and roadway characteristics. The two SPFs were separately developed using the NB model for total and fatal/injury crashes that occurred on urban four-lane freeway interchange influence areas in Florida between 2007 and 2010. The comparison results showed that the two models resulted in similar prediction performance and network screening. The empirical result supported the use of the flow-only SPF model, whose development requires much less effort compared with that of the full SPF.

Lee et al. ( 16 ) developed multi-level random-parameter models to predict traffic crashes along three different ramp types: direct, semi-direct, and loop ramps. Crash data were collected on 1,070 freeway ramps over a four-year period from 2007 to 2010 in South Korea. The results showed that crash frequency increased on ramp segments with higher AADT and short ramp lengths. Wang et al. ( 17 ) investigated the real-time crash risk for expressway ramps using different factors related to road geometric, traffic, socio-demographic, and trip generation characteristics. Detailed information on the aforementioned factors was gathered on three main expressways in Central Florida, including State Road 408, State Road 417, and State Road 528 from July 2013 to March 2014. The Bayesian logistic regression models were used to determine the effects of various factors on ramp crash risk. The results showed that these models with the socio-demographic and trip generation covariates had a better prediction performance than their counterparts without such parameters. Recently, Matarage and Dissanayake ( 18 ) estimated the Highway Safety Manual (HSM) calibration factors for freeway ramp segments in Kansas using ramp-related crash data collected over a three-year period from 2014 to 2016. For entrance ramps in Kansas, the results showed that the HSM methodology over-predicted single-vehicle crashes and multi-vehicle fatal and injury (FI) crashes. On the other hand, the HSM methodology under-predicted multi-vehicle crashes on exit ramps in Kansas.

Studies Applying CMP-Based and Heterogeneity-Based Models

Compared with the standard NB model, the CMP with varying dispersion parameter (i.e., the HTCMP model) has rarely been applied in safety research for predicting traffic crashes. For example, Lord et al. ( 12 ) examined the application of the CMP regression model for crash prediction and compared the CMP modeling results with those of the traditional NB model. Two different data sets were used in this study, the first set was collected at four-legged, signalized intersections in Toronto, Ontario, Canada, and the second set of data was obtained on rural, four-lane, undivided highways in Texas from 1997 to 2001. The results indicated that the CMP model outperformed the NB model in goodness-of-fit statistics and predictive performance. In another similar study, Geedipally and Lord ( 13 ) compared the performance of NB (with both fixed and varying dispersion parameters) and CMP (with both fixed and varying shape parameters) models in estimating the crash variance. The results indicated that the trend of crash variance predicted by the CMP model was similar to that predicted by the NB model.

Lu and Tolliver ( 19 ) developed and compared alternate crash prediction models, including Poisson, NB, gamma, CMP, Bernoulli, hurdle Poisson, and zero-inflated Poisson regression models, for predicting highway-rail grade crossing (RGC) crashes characterized by under-dispersion issues. Data on RGC-related crashes and highway-rail crossing characteristics were collected from public RGCs in North Dakota from 1996 to 2014. The results indicated that Bernoulli, CMP, and hurdle Poisson models handled under-dispersed RGC crash data properly. Abdella et al. ( 20 ) applied a penalized CMP regression model for predicting over-dispersed crash data. Crashes were collected at 868 signalized intersections in Toronto, Ontario, in 1995. The results demonstrated that the penalized CMP regression model was a promising approach in simultaneously accommodating both dispersion and collinearity issues. Most recently, Shirani-bidabadi et al. ( 8 ) used the CMP model to develop separate SPFs for five highway facilities in Alabama. The CMP models were compared with the multivariate adaptive regression splines (MARS) data mining technique. A total of 1,311 bicycle–vehicle crashes, collected from 2011 through 2015 in Alabama, were used. Overall, the MARS models outperformed the corresponding CMP models.

There are studies that have accounted for crash heterogeneity (e.g., 21 , 22 ). For example, Wali et al. ( 21 ) applied fixed- and random-parameter Poisson and Poisson-gamma count models on two-lane roads in Tennessee and found that the random-parameter Poisson models provided the best model fit. Another method of introducing heterogeneity in crash frequency estimations is the Bayesian approach ( 22 ). With this, Li et al. ( 22 ) applied the Bayesian generalized NB modeling approach while predicting crash frequencies on Georgia’s rural four-lane roadways and found this approach to be successful.

CMV-Related Crash Frequency Studies

Schneider et al. ( 23 ) adopted the NB model to analyze the impacts of horizontal curvature on truck crashes that occurred on rural two-lane collector and arterial horizontal curves in Ohio between 2002 and 2006. It was found that horizontal curvature, truck average daily traffic, and passenger car traffic increased the number of truck crashes. Vadlamani et al. ( 24 ) identified high-risk sites associated with large truck crashes in Arizona. Two high-crash identification approaches were used: the property damage only equivalents (PDOE) method and the NB model. Crash data were collected from Arizona over a six-year period (2001 to 2006). High-crash locations identified using the NB model had lesser severe-injury crashes and larger AADTs. By contrast, high-risk locations identified via the PDOE method exhibited relatively larger fractions of trucks and severe crashes, but smaller AADTs.

Dong et al. ( 25 ) applied a multivariate random-parameters zero-inflated negative binomial model to jointly model crash frequencies by vehicle type at urban signalized intersections. Information on intersection-specific and five-year (2001 to 2005) crash history were obtained for a sample of 603 intersections in Tennessee. It was found that left-turn number, international roughness index, exclusive right-turn number, and posted speed limit significantly contributed to crash frequencies for all crash types. In another similar study using the same database, Dong et al. ( 26 ) investigated the frequency of large truck–involved crashes that occurred on Tennessee state route roadways between 2006 and 2010 using the NB model. The study showed that AADT, truck percentage, driver condition, and weather conditions were significantly associated with crash frequency of large truck–involved crashes.

Summary of Literature Review and Study Contribution

The review of the literature has shown that relatively limited research has been conducted to investigate CMV crash frequencies along ramp sections. Therefore, this study aimed at fulfilling that gap while developing different CMV crash-specific SPFs (using each of the HTCMP, CMP, HTNB, and NB modeling approaches) on ramp segments using CMV-related crashes that occurred on 298 ramp segments in the state of Kentucky between 2015 and 2019. The previously proposed models (i.e., HTCMP, CMP, HTNB, and NB) were explored given their easy-to-apply capabilities compared with other sophisticated methods (e.g., Bayesian generalized NB model), as well as their accommodation of crash observation heterogeneity (specifically the HTCMP and HTNB models).

NB, HTNB, CMP-Based, and HTCMP Models

The standard NB (i.e., Poisson-gamma mixture) regression is the most frequent and popular model that has been used by safety researchers for predicting traffic crashes and developing SPFs. The NB model assumes that a dispersion parameter follows a gamma distribution ( 31 ). Therefore, the NB model captures the over-dispersion by incorporating an error term into Poisson means and allows the variance to differ from the mean, such that:

μ i = E [ y i ] = exp ( β X i + ε i ) = exp ( β X i ) . exp ( ε i )

(1)

Var [ y i ] = E [ y i ] . ( 1 + α . E [ y i ] ) = E [ y i ] + α . E [ y i ] 2

(2)

where

μ i is the expected number of crashes on segment i ,

X i is a vector of covariates,

β is a vector of unknown coefficients,

α denotes dispersion parameter, and

exp ( ε i ) is a gamma-distributed error term with mean one and variance α.

The probability density function (pdf) of the NB model is given as:

Pr ( Y = y i ) = Γ ( y i + θ ) Γ ( θ ) y i ! ( μ i μ i + θ ) y i ( θ μ i + θ ) θ

(3)

where θ is the inverse dispersion parameter (1/α) and Γ ( . ) is a value of the gamma function.

The standard NB model assumes the dispersion parameter (α) to be fixed across all sites. However, some previous studies have shown that this assumption does not hold in many cases where the dispersion parameter (α) of the NB model is not fixed ( 5 , 6 , 32 , 33 ). The heterogeneous negative binomial (HTNB) model (in some documents, referred to as generalized negative binomial (GNB) model) is an extension of the NB model, which allows “α” to vary as a function of site-specific characteristics, as follows ( 5 ):

α i = exp ( γ Z i )

(4)

where Z i is a vector of site-specific variables and γ is a vector of parameters to be estimated.

Compared with the traditional NB model, the HTNB model can properly handle the over-dispersion by allowing the dispersion parameter to vary across observations. However, both NB and HTNB models have been reported to exhibit some limitations. For example, these two models are unable to handle under-dispersion, where the variance of crashes is less than the mean ( 7 , 34 , 35 ).

A CMP (or Conway–Maxwell–Poisson) regression model can appropriately account for the aforementioned issues. An advantage of the CMP model over the NB model is that the former can model count data subjected to both over- and under-dispersion, while the latter only estimates over-dispersed count data ( 8 , 12 , 36 ). Like the HTNB model, the heterogeneous Conway–Maxwell–Poisson regression model (HTCMP) is a promising generalization of the standard CMP model that allows the dispersion parameter to vary across observations depending on site-specific characteristics ( 13 ). The probability mass function (pmf) of an HTCMP distribution is:

P ( Y = y ) = λ y ( y ! ) ν Z ( λ , ν ) , y = 0 , 1 , 2 , …

(5)

ln ( λ i ) = β X i

(6)

Z ( λ , ν ) = ∑ n = 0 ∞ λ n ( n ! ) ν

(7)

ln ( ν ) = γ ψ i

(8)

E ( Y ) ≈ λ 1 v − v − 1 2 v

(9)

where

λ is the centering parameter that approximates the mean of observations,

ν is the dispersion parameter (also known as shape parameter) of the HTCMP model (similar to the varying dispersion parameter of the HTNB model) (if ν = 1 , a CMP model reduces to the standard Poisson model, ν > 1 represents under-dispersion, ν < 1 represents over-dispersion),

ψ i is a vector of explanatory variables associated with the dispersion parameter,

γ is a vector of regression coefficients corresponding to covariates ψ i , and

Z ( λ , ν ) normalizes the distribution ( 37 ).

It should be noted that the CMP distribution was later reparametrized by Guikema and Goffelt ( 38 ) through substituting μ = λ 1 v , so that the mean was approximated as E ( Y ) ≈ λ + 1 2 v − 1 2 . However, this formulation is accurate when μ > 10 ( 12 , 13 ). In this study, E ( Y ) was estimated using Equation 9 as all the means estimated were less than 10.

Goodness-of-Fit (GOF) Measures

Overall, four count-response models, including NB, HTNB, CMP, and HCMP models, were fitted and compared to develop the CMV crash-specific SPFs along ramp segments. For each model, the overall GOF measure was determined by calculating the deviance statistic (D). This statistic is calculated using Equation 10, as follows:

D = − 2 ln ( L L 0 − L L β ) ≅ χ ( d . f . = p ) 2

(10)

where

L L β is the log-likelihood at convergence,

L L 0 is the log-likelihood of the null model (i.e., constant-only model), and

p is the number of predictors in the final model.

A significant value for the deviance statistic indicates a good statistical fit.

Another GOF measure used to assess the fitted models was the McFadden Pseudo R 2 statistic, and can be calculated as shown in Equation 11:

McFadden Pseudo R 2 = 1 − ( L L β L L 0 )

(11)

The Akaike information criterion (AIC) and Bayesian information criterion (BIC) were also used to select the best-fit model. The AIC and BIC are calculated as follows:

AIC = − 2 LL + 2 P

(12)

BIC = − 2 LL + P ( ln ( n ) )

(13)

where

LL is the model’s log-likelihood at convergence,

P is the number of model parameters, and

n is the number of observations.

In general, models with the least AIC and BIC values are preferred. All models were fitted using R software ( 39 ).

To assess the predictive performance of the four alternative models developed in this study (i.e., NB, HTNB, CMP, and HTCMP), three additional GOF measures were applied using the validation data set. These were the mean absolute deviance (MAD), the mean square prediction error (MSPE), and the cumulative residuals (CURE) plots. The MAD and MSPE statistics were computed using Equations 14 and 15, respectively:

MAD = 1 n ∑ i = 1 n | y ^ i − y i |

(14)

MSPE = 1 n ∑ i = 1 n ( y ^ i − y i ) 2

(15)

where y i and y ^ i are, respectively, the observed and predicted number of crashes on segment i and n is the sample size of the validation data set ( n  = 75). The model with lower MAD and MSPE values provides better prediction performance.

To visually assess how well the four developed SPFs fitted the crash data, the CURE plot versus AADT was developed for each model using out-of-sample observations (i.e., validation database). More details on the CURE method are found in previous studies (e.g., 40–42).

CMV Crash-Specific SPF Model Results Along Ramp Segments

The results of the four developed NB, HTNB, CMP, and HTCMP models and their associated GOF statistics are shown in Table 2. From Table 2, as indicated by the deviance statistic (D), all fitted models were statistically different from their constant-only counterparts, which rejects the null hypothesis that the models had the same prediction performance as their corresponding constant-only models. According to McFadden Pseudo R², AIC, and BIC statistics, the HTCMP model outperformed the other three models. The superiority of the HTCMP model was also supported by the MAD and MSPE measures (especially the MSPE measure) from the validation data set, where the HTCMP had the least MSPE and second least MAD.

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Table 2.

CMV Crash-Specific SPF Results for Ramp Segments

Variable a	NB			HTNB			CMP			HTCMP
	Coef.	Z-stat.	P-value	Coef.	Z-stat.	P-value	Coef.	Z-stat.	P-value	Coef.	Z-stat.	P-value
Mean part
Constant	−10.847	−14.00	0.000	−11.744	−18.08	0.000	−11.028	−13.35	0.000	−12.504	−15.01	0.000
Ln(Ramp AADT)	0.232	2.95	0.003	0.313	4.22	0.000	0.317	3.60	0.000	0.418	4.46	0.000
Ramp configuration:
Diamond	N/S b	N/S	N/S	N/S	N/S	N/S	N/S	N/S	N/S	1.474	4.69	0.000
Parclo/Free-flow loop	N/S	N/S	N/S	N/S	N/S	N/S	N/S	N/S	N/S	1.014	2.00	0.046
On-ramp	0.298	2.27	0.023	0.261	1.97	0.049	0.364	2.60	0.009	0.433	2.86	0.004
Left shoulder width (ft)	N/S	N/S	N/S	N/S	N/S	N/S	−0.067	−2.13	0.033	−0.093	−2.70	0.007
Absence of left roadside barrier	0.406	2.50	0.013	0.388	2.30	0.022	0.393	2.24	0.025	0.411	2.36	0.018
Presence of ramp(s) within ramp of interest	0.498	2.88	0.004	0.536	3.09	0.002	0.721	3.51	0.000	0.524	2.64	0.008
Ramp gore length (ft)	−0.965	−3.57	0.000	−1.055	−3.74	0.000	−0.872	−3.20	0.001	−1.143	−3.56	0.000
Mainline heavy vehicle percentage (HVP)	−0.015	−1.65	0.100	N/S	N/S	N/S	−0.019	−1.99	0.047	N/S	N/S	N/S
Dispersion part	Fixed dispersion parameter			Varying dispersion parameter			Fixed dispersion parameter			Varying dispersion parameter
Ln(Dispersion parameter)	−2.465	−4.08	0.000	−22.333	−2.45	0.014	0.376	2.98	0.003	2.062	9.42	0.000
Ramp configuration:
Parclo/Free-flow loop	na	na	na	N/S	N/S	N/S	na	na	na	−0.608	−1.81	0.070
Direct/Semi/Outer connection	na	na	na	N/S	N/S	N/S	na	na	na	−1.482	−7.22	0.000
Urban area type	na	na	na	N/S	N/S	N/S	na	na	na	−0.786	−3.77	0.000
Ln(Ramp AADT)	na	na	na	2.129	2.33	0.020	na	na	na	N/S	N/S	N/S
Goodness-of-fit (GOF) statistics	NB			HTNB			CMP			HTCMP
Number of observations	223			223			223			223
Number of parameters	8			8			9			13
Log-likelihood at convergence ( L L β )	−291.72			−288.08			−288.23			−247.35
Log-likelihood at zero ( L L 0 )	−314.33			−314.32			−321.98			−321.98
Deviance (=2 ×  ( L L β − L L 0 ) ) (d.f., P-value)	45.22 (d.f. = 7, P-value < 0.001)			52.5 (d.f. = 7, P-value < 0.001)			67.5 (d.f. = 8, P-value < 0.001)			149.3 (d.f. = 12, P-value < 0.001)
McFadden Pseudo R2	0.07			0.08			0.10			0.23
AIC	599.4			592.2			594.5			520.7
BIC	626.7			619.4			625.1			565.0
MAD (Validation dataset)	1.36			1.40			1.30			1.36
MSPE (Validation dataset)	9.94			10.39			9.36			9.22

Note: CMV = commercial motor vehicles; SPF = safety performance function; NB = negative binomial; HTNB = heterogeneous negative binomial; CMP = Conway–Maxwell–Poisson; HTCMP = heterogeneous Conway–Maxwell–Poisson; AADT = average annual daily traffic; AIC = Akaike information criterion; BIC = Bayesian information criterion; MAD = mean absolute deviance; MSPE = mean square prediction error; Coeff. = coefficient; Stat. = statistic; na = not applicable.

a

Ln(5 × ramp segment length) used as offset in all four models.

b

Not significant (N/S) at 10% significance level.

As seen in Table 2, ramp type (on-ramp), presence of merging or diverging ramp(s) within the ramp of interest, ramp gore length, absence of left roadside barrier, and ramp AADT (represented by “Ln(Ramp AADT)”) were found to significantly influence CMV crashes in all fitted models. For the HTCMP model, there were two additional variables contributing to crash frequency, ramp configuration and ramp left shoulder width. The former (diamond and parclo/free-flow loop ramps) were associated with higher CMV crashes compared with direct/semi/outer connection ramps, whereas the increase in the latter (left shoulder width) was associated with CMV crash reduction.

The results obtained in this study were intuitive and in line with prior expectations. For example, as ramp AADT increased, the exposure to crash risk increased. In other words, the higher the number of motor vehicles on ramp segments, the higher the risk of CMV crashes. This result was reported in previous studies ( 14 , 16 , 17 , 24 , 26 , 43 ). Not surprisingly, wider left shoulders reduced the number of CMV crashes by providing more recovery room for the driver of an errant vehicle to regain control of the vehicle when departing their intended travel lane. A similar finding was reached for the absence of left safety barrier on the ramp, where there was an increase in CMV-related crashes on ramp segments without left-side safety barrier. This is mainly since roadside safety barriers prevent errant vehicles from running off the roadway and colliding with roadside fixed hazards or rolling over on steep embankments ( 44 ). In addition, drivers tend to drive more attentively and at lower speeds on those road segments installed with roadside safety barriers, mainly a result of drivers’ sense of having less room to take corrective actions after leaving the roadway.

An interesting finding was that ramp gore length was negatively associated with CMV crashes, indicating that the longer the ramp gore length, the lower the number of CMV crashes was. This can be explained by a longer gore length reducing the risk of sudden lane changes or conflicts between the mainline through traffic and ramp-related traffic, where drivers have sufficient length to safely change their lane or adjust their speed before exiting or entering the freeway’s mainline toward a ramp. Another interesting finding was that the presence of ramp(s) merging into or diverging from the ramp of interest increased CMV crashes. This result is intuitive and can be explained by the risk of vehicle-to-vehicle collisions, especially same-direction sideswipe or rear-end crashes, increasing with the presence of merging/diverging ramp(s).

With regard to ramp configuration, the results showed that diamond and parclo/free-flow loop ramps experienced more crashes than other ramp categories. A possible explanation for this finding is that diamond and parclo ramps have relatively lower speed limits than other ramp types. Consequently, drivers may face more difficulties in adjusting their speed to the speed limits of these ramps, especially near ramp terminals, which require drivers to come to a complete stop at some instances. Moreover, parclo and free-flow loop ramps have smaller horizontal curve radii, which might result in loss of vehicle control and possible CMV-related crashes. Similar findings were reported by Bared et al. ( 43 ). With regard to ramp type, on-ramps were found to experience more crashes than off-ramps. One potential explanation is that on-ramps were more prone than off-ramps to riskier driving actions, such as lane changing or merging maneuvers. Specifically, this is mainly a result of the increasing weaving maneuvers at on-ramps as drivers tend to change lanes from the ramp to the mainline lanes. This result is consistent with those in previous studies ( 43 , 45 ).

Cumulative Residuals (or CURE) Plots for Model Assessment

Figures 4 and 5, respectively, show the CURE plots against ramp AADT and fitted crashes created for each SPF model using the validation data set. As seen in Figure 4, all the CURE plots stayed mostly within ± 2σ (or ± 2 crash standard deviation limits) boundaries and were almost below the zero line. This indicates that all the models overestimated the expected number of crashes for a wide range of AADT in the validation data set. A similar CURE plot trend was also reported by Lu et al. ( 42 ), where the CURE plots were consistently below the zero line. The NB, HTNB, and CMP models generated almost identical CURE plots. However, the HTCMP model provided the best fit since its CURE plot was closer to zero (horizontal axis), especially for ramp AADT ranging from 5,000 to 15,000.

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Figure 4.

CURE plots versus AADT for: (a) NB model, (b) HTNB model, (c) CMP model, and (d) HTCMP model.

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Figure 5.

CURE plots versus fitted crashes for: (a) NB model, (b) HTNB model, (c) CMP model, and (d) HTCMP model.

Similarly, Figure 5 demonstrates that the cumulative residual plots for the NB and HTNB models were quite similar, while that of the CMP model stayed beyond the −2σ boundary at the beginning of the curve for the fitted crashes ranging between 0.24 and 0.64. Nevertheless, the plot for the HTCMP model was less biased and oscillated more closely to zero (horizontal axis), indicating a better overall fit for the model. The CURE plots thus supported the conclusion that the HTCMP model was the best-fit approach compared with the other models.

Overall, almost all GOF measures favored the HTCMP model over the other fitted models, indicating that the HTCMP could be used as a promising SPF tool for effectively predicting CMV-related crashes on ramp segments. The superiority of the HTCMP model can be attributed to this model, compared with the other comparing models, capturing better unobserved heterogeneity by allowing the dispersion parameter to vary across the crash sites.

Proposed Countermeasures for CMV Safety Along Ramp Segments

Based on the HTCMP model results, potential engineering and vehicle-related countermeasures were proposed to improve CMV safety along ramp segments, as follows:

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Figure 6.

Commercial motor vehicles (CMV) crashes versus: (a) ramp left shoulder width (ft); and (b) ramp gore length (in 1,000 ft).