Analysis of Injury Severity of Drivers Involved Different Types of Two-Vehicle Crashes Using Random-Parameters Logit Models with Heterogeneity in Means and Variances

Tis study proposes random-parameters multinomial logit models, with heterogeneity in means and variances, to explore the diferences in the factors infuencing injury severities of drivers involved in diferent types of two-vehicle crashes. Te models are verifed using crash data from the United Kingdom (UK) over three years (2016–2018). Tree types of crashes are separately identifed (car-car, car-truck, and truck-truck crashes). In this study, a wide variety of potential variables, including the driver, vehicle, road


Introduction
Trafc crashes, particularly those involving severe and fatal injuries, have resulted in an enormous loss in terms of human, economic, and social aspects.Worldwide, almost 1.3 million deaths were caused yearly by road trafc crashes [1].Furthermore, 62% of the crashes across six years (2012-2018) reported by the police were two-vehicle crashes [2,3].Te Crash Report Sampling System (CRSS) analysis also illustrated that the drivers are the most afected group injured or killed in trafc crashes.Specifcally, over 75% of all drivers in the US were injured or killed in 2018.
In addition, the injury-severity levels of both parties are expected to diferentiate involving diferent vehicle-type crashes (such as car-car, car-truck, and truck-truck crashes).In 2017, among the 4,761 people killed in crashes with the involvement of large trucks, 72% were occupants of other vehicles [3].Moreover, the drivers of passenger cars tend to sustain serious injuries when colliding with trucks.Furthermore, almost half of the motor vehicle crashes that occurred on I-80 from Wyoming were crashes involving trucks, with a remarkable proportion of fatalities caused by car-truck crashes [4].Concerning the considerable variations in injury outcomes among vehicles of light and heavy weights, the type of vehicles is suggested to be a critical element in injury modeling analysis.
Overwhelming evidence illustrated the unobserved heterogeneity in current trafc safety literature.Numerous economic modeling approaches were recently used to analyze the injury severities of crashes on highways to address this critical challenge.Examples include the latent class models [5], random-parameter ordered models [6], and random-parameters logit models [7][8][9].By accommodating variations of the explanatory variables across the observations and factors afecting the means and variances of the parameter density functions of the random parameters, the random-parameters logit models with heterogeneity in means and variances approaches are supposed to be more fexible in accounting for the unobserved heterogeneity, specifed by the statistical superiority in terms of accuracy and reduced heterogeneity [10][11][12].
Given this, this study comprehensively estimates the injury severities of two-vehicle crashes using advanced statistical models given the types of vehicles involved in the crash.Te study methodology is shown in Figure 1.Te methodology involves a literature review of the injury severities involving two-vehicle crashes.Ten, the data used for this study are described, followed by an introduction to the methodological approach.Ten, a detailed discussion of the model estimation results is presented.Finally, the last section summarizes the fndings of this study and discusses potential future directions.As shown in Figure 1, by collecting the characteristics of driver, vehicle, roadway, and environment among the car-car, truck-truck, and car-truck crashes, a series of models will be estimated based on the random-parameters multinomial logit models.Ten, based on the random parameters, signifcant factors, and marginal efects, some valuable fndings can be revealed, and practical applications can be implemented.

Literature Review
Table 1 briefy summarizes the signifcant factors determining driver's injury-severity outcomes in previous twovehicle crash studies.Varieties of explanatory factors regarding driver, vehicle, roadway, and environmental characteristics have been found to afect the injury severity in two-vehicle crashes.As for vehicle characteristics, most relevant studies have investigated the infuence of vehicle types on injury severity.Few studies have focused on comparing injury severity between diferent vehicle crashes to identify diferential efects of the same explanatory variables.However, the authors in [28] used binary logistic modeling with a Bayesian inference approach to investigate occupant injury severity of truck-involved crashes based on vehicle types on a mountainous freeway.However, their study was constrained to truck-involved crashes.Furthermore, the authors in [29] developed the heteroscedastic ordered logit models to analyze driver's injury severity in single-and two-vehicle crashes and compare how the efects of explanatory variables vary across various types of crashes.However, traditional logit and probit models assume that the estimated parameters are fxed for all observations, causing biased parameter estimates and erroneous inferences [30].Previous reviews summarized by several researchers have presented comprehensive studies of unobserved heterogeneity [30][31][32].
Recently, a more advanced statistical method has been used to fully address possible unobserved heterogeneity in the means and variances of the random parameters [33].Te application of this advanced modeling framework enables capturing the multilayered unobserved heterogeneity of the crash data in terms of (a) estimated parameters varying across the observations; (b) factors afecting the mean of the parameter density function of the random parameters (and thus shifts in the peak of the distribution of the betas); and (c) factors afecting the variance of the parameter density function of the random parameters (and thus changes in the tails of the distribution of the betas).However, research utilizing random-parameters logit models with heterogeneity in means and variances models in the context of diferent types of vehicle crashes is limited.
To fll this research gap, a random-parameters logit model with heterogeneity in means and variances model is estimated to examine the diference in contributing factors of injury severity of drivers involved in diferent types of vehicle crashes.

Data Description
Tree-year crash data from the United Kingdom (UK) were drawn from the STATS19 dataset, the most comprehensive and publicly available crash database in the UK containing information obtained from police crash reports [34].Te dataset comprises three fles: accident, vehicle, and casualty.We used this study's accident and vehicle reference numbers to merge the three subsets.After merging, the analysis unit in the current paper is the accident.Each case contains the time/date of accident occurrence, weather, road, light conditions, posted speed limit, road type, driver's age and gender, vehicle type, frst impact point of the vehicle, vehicles' maneuvers, and injury-severity level.A total of 8,373 two-vehicle crashes were extracted (missing and unreliable data were removed): 4,992 crashes involving car-car crashes, 2,770 crashes involving car-truck crashes, and 681 crashes involving truck-truck crashes.
Te dependent variable of the models is the "injuryseverity level."Following the STATS19 injury classifcation, three injury-severity outcomes are considered in this study: slight injury, severe injury, and fatality.Crashes resulting in no injuries are not recorded in the dataset [35].Because of the relatively low number of fatality crashes (1.33% of total two-vehicle crashes), fatal injuries and serious injuries were combined into one category of injury severity.Terefore, this study reclassifes cases using two severity levels: slight injury and severe injury (including serious injury and fatality) (86.02% and 13.98% for slight injury and severe injury in car-car crashes, note that 89.56% and 10.44% for slight

Gender
Female drivers are more likely to be injured [13][14][15].In contrast, the authors in [16] found that male drivers are more likely to sustain the probability of severe injury.

Seatbelt used
Drivers without use of safety equipment are more likely to be injured [13,20,21].

Alcohol involved
Te alcohol involvement variable is more likely to sustain the probability of severe injury outcomes [16,19,20,22,23].

Fault status
Te authors in [13] found that not-at-fault drivers are more likely to be injured.Furthermore, the authors in [18] examined and compared the efect of selected variables on driver-injury severities of both at-fault and not-at-fault drivers.

Vehicle type
Te involvement of trucks tends to increase the probability of more severe injuries [13,22,24].In contrast, the authors in [14,23] found that passenger cars are usually associated with increased injury severity.

Age of vehicle
Te increased age of a vehicle is found to increase the injury severity of drivers [13][14][15].

Road grade
Te author in [20] found that road grade variable imposes signifcantly infuences driver-injury severities.

Logit Model
Separate random-parameter logit models with heterogeneity in means and variances (RPLHMV) were estimated to identify the factors infuencing the driver's injury and severity involved in diferent vehicle crashes.To begin with, an injury-severity function, Y in , that determines the driverinjury-severity level i in crash n, is specifed as follows [36][37][38]: where X in are vectors of explanatory variables that afect driver-injury-severity level i (slight injury or severe injury) in crash n, β i is a vector of corresponding estimable parameters, and ε in is an error term assumed to follow an independent and identical distribution with zero mean and variance σ 2 .To account for unobserved heterogeneity, random parameters with heterogeneity in means and variances are introduced as follows [30,38,39]: where β i is the mean parameter estimate across all crashes, Z in are vectors of explanatory variables that infuence the mean, Θ in are vectors of corresponding estimable parameters, W in are vectors of explanatory variables that capture heterogeneity in variances, σ in , ψ in is a vector of corresponding estimable parameters, and υ in is a disturbance term.Ten, the outcome probability of the RPLHMV model formulation can be expressed as follows [26]: where p n (i|φ) is the probability of injury-severity level i conditional on f(β i |φ) and f(β i |φ) is the density function of β i , with φ referring to a vector of parameters (means and variances).Te RPLHMV model is estimated with a simulated maximum likelihood method, and 1,000 Halton draws are used to achieve stable parameter estimates [40].Regarding the distribution of the random parameters, the normal distribution is used to achieve the best goodness of ft [7,[41][42][43][44].
Te pseudo-elasticities are computed to quantitatively describe the impact of explanatory variables on the driver-injury severities.In this paper, all variables used in the estimated models are binary indicator variables.Terefore, the pseudo-elasticities quantify the change in outcome probability when an explanatory variable changes from "0" to "1" [37,43].
We conducted chi-square distributed likelihood ratio tests to determine whether there is any diference between injury-severity outcomes for vehicle types.To begin, likelihood ratio tests were conducted to compare injury-severity outcomes for vehicle types.Te test statistic is [45][46][47][48] where LL(β combined,t ) is the log-likelihood at the convergence of the model using all of the available two-vehicle crash data in the year t, LL(β car−car,t ) is the log-likelihood at the convergence of the model using car-car crash data only in year t, LL(β car−truck,t ) is the log-likelihood at the convergence of the model using car-truck crash data only in year t, and LL(β truck−truck,t ) is the log-likelihood at the convergence of the model using truck-truck crash data only in year t.Te model estimate gained from the test gave an χ 2 values of 61.64 with 12 degrees of freedom.Te modeling approach specifed the null hypothesis that statistically signifcant parameters in truck-car crash models are stable and can be rejected at 99.99% confdence level.

Results
Table 3 presents the model estimation results based on the random-parameter logit models with heterogeneity in means and variances, indicating a very good overall model ft with McFadden R-Squared between 0.374 and 0.478.Based on the estimated results, a detailed discussion is shown as follows.

Random-Parameters Insights.
For the car-car model, there are two statistically signifcant variables as random parameters (see Table 3), including young car driver (between 26-45 years) and the 30 mph speed limit.Among them, the young car-driver (between 26-45 years) indicator is signifcant as a normally distributed random parameter, wherein 92.44% of the observations increase the probability    Journal of Advanced Transportation of severe injury (and in a reduction in the rest 7.56%).Furthermore, the male indicators increase the mean of this young car-driver indicator, thus increasing the likelihood of severe injuries.Most young male car drivers tend to be overconfdent in their driving skills and are more likely to exhibit improper actions.However, they lack enough emergency response capabilities, resulting in severe and fatal injuries.Terefore, more enforcement and education programs about young male car drivers should be enhanced.Te 30 mph speed limit indicator is signifcant, with a lower probability of severe injury for 99.22% of the observations.Tere are two statistically signifcant variables as random parameters for the truck-truck model (see Table 3).Te 30 mph speed limit is also signifcant, with a low probability of severe injury for 80.62% of the observations.Te young truck driver (between 26 and 45 years) indicator is signifcant as a normally distributed random parameter, wherein 74.49% of the observations increase the probability of severe injury (and a reduction in the rest of 25.51%).Furthermore, the dark-lighted indicator decreases the mean of this young truck driver indicator, thus decreasing the likelihood of severe injuries.Driving at night, truck drivers are more likely to be tired, leading to misjudgment of the driving speed, thereby inducing severe crashes.Terefore, when making the nighttime scheduling plan, the truck drivers should be arranged reasonably to prevent fatigued driving.To that end, highly efcient street lighting over road segments with a high proportion of large trucks should be considered to improve visibility during nighttime conditions.Also, drivers should be careful with relatively lower speeds when driving on artifcially lit road sections.
For the car-truck model, there is only one statistically signifcant variable (see Table 3).However, the 30 mph speed limit indicator is also signifcant, where 70.19% of the observations decrease the probability of severe injury.Note that the weekday indicator increases the mean of the 30 mph speed limit indicator, making severe injuries more likely.Our fnding seems to be consistent with the literature in which the author clarifed that lower proportions of trucks characterize weekends compared to weekdays [28,49].Again, note that the variance of the speed limit indicator is afected by the rural area indicator, which increases its variance (it makes the distribution tail of the parameter density function fatter and thus ofers a more uniformly shaped distribution of the betas), refecting higher variability.

Car-Car Crashes (Car Driver Model).
As shown in Table 4, there are three statistically signifcant driver-related variables in the car-car crash model.Female drivers are associated with a 15.56% risk of severe injury compared to male drivers.A possible explanation for these results may be the lack of enough emergency response capabilities, resulting in serious and even fatal injuries.Young drivers are linked to a 10.71% increased risk of severe injury.Most young drivers (between 26 and 45 years) tend to be overconfdent in their driving skills and are more likely to exhibit improper actions.However, they lack enough emergency response capabilities, resulting in severe injury.More enforcement and education programs about young drivers should be enhanced.In addition, old drivers (65 years above) are associated with a 2.76% increased risk of severe injury.Te physical function of old drivers is programmed to diminish with age.And the reaction lag is also more likely to lead to severe injury crashes.Consequently, old drivers should attend regular physical check-ups and driving safety education; if failing a driving test, they could be considered to take the initiative to return their driver's license or take compulsory license suspension.
Four vehicle-related variables, speeding, turning, headon collision, and sideswipes collision, all have statistically signifcant efects on the injury severity of drivers in the carcar crash model.Among them, speeding is associated with a 3.54% increased risk of severe injury.When speeding while driving, the view of drivers becomes narrower.Furthermore, more kinetic energy after the collision is more likely to result in severe and fatal injuries.Turning is associated with a 6.01% increased risk of severe injury.When a vehicle turns, the view of drivers also becomes narrower, and they are more inattentive.Notably, the study found that both side and frontal crashes increased the risk of severe injury in carcar crashes and that side crashes are more likely to result in severe injury than frontal crashes (38.46% vs. 4.27%).Te main reason may be that the impact of the hit vehicle was on the side, meaning that at least one driver did not pay attention to oncoming trafc in the other direction.Hence, a high-speed collision is more likely to result in severe injury.More importantly, the vehicle's protection does not even work for the driver in the side direction.
Four roadway-related variables, including roundabouts, dual carriageway, 30 mph speed limit, and urban areas, are statistically signifcant in afecting the driver-injury severities in the car-car crash model.Among them, crashes that occurred at roundabouts are associated with a 1.96% increased risk of severe injury.Te speed of vehicles is generally slower at roundabouts, reducing the likelihood of serious crashes.However, due to improper actions (such as speeding or running a red light), they often collide with the other vehicles at the roundabout, thereby increasing the probability of slight crashes.Compared with single lanes, automotive vehicles driving in dual lanes are prone to frequent lane change behavior, increasing the risk of severe injury by 4.20%.Te 30 mph speed limit indicator decreases the probability of severe injury by 23.18%.Finally, compared to rural roads, the overall lower vehicle speeds on urban roads reduce the risk of severe injury by 14.33%.
Five environment-related variables, including dark lighted, dark no light, sunny, of-peak crash time, and summer, have a statistically signifcant efect on the injury severity of drivers in the car-car crash model.It is highlighted that under the dark lights, the perception of the external environment weakened the perception abilities of hazardous situations.When driving at night after working for long hours during the daytime, car drivers are more likely to be tired, leading to misjudgment of the driving speed, 8 Journal of Advanced Transportation  Journal of Advanced Transportation thereby inducing severe crashes.A sunny environment decreases the likelihood of severe injury by 14.59%.Te ofpeak time and summer indicators increase the estimated odds of a severe injury by 5.30% and 10.32%, respectively.

Truck-Truck Crashes (Truck Driver Model).
As shown in Table 4, there are three statistically signifcant driverrelated variables in the truck-truck crash model.Male drivers are associated with a 79.50% increased risk of severe injury compared with female drivers.On the one hand, due to the particularity of the truck occupation, the proportion of male drivers is more than females.On the other hand, compared to car drivers, truck drivers often need to drive longer to reach their destination.Drivers' fatigue driving is a signifcant cause of trafc accidents.Terefore, truck drivers should ensure sufcient sleep before driving, and if tired when driving, they should immediately go to the nearest service area and rest to ensure safe driving.Young drivers (26-45 years) and older drivers (65 years above) increase the odds of a severe injury by 16.05% and 1.85%, respectively.Two vehicle-related variables, including speeding and sideswipes collision, have a statistically signifcant efect on the injury severity of drivers in the truck-truck crash model.Among them, speeding contributes to more severe injuries.Te reason could be a signifcant speed diference between trucks based on their loads and weight-to-power ratio.It is also a common occurrence that trucks pass each other using the left lane.Lower severe injuries are found during sideswipes compared to diferent collision types.It seems possible that sideswipes contribute to more slight injury crashes compared to severe injuries.A recent study by the authors in [50] reported that sideswipes have diferent levels of impact on injury severity under other weather conditions.
Tree roadway-related variables, including junctions, 30 mph speed limit, and rural areas, have a statistically signifcant efect on the injury severity of drivers in the truck-truck crashes model.Among them, the presence of junctions increases severe injuries by an estimated odds of 7.52% on average, compared to no junction conditions, respectively.Te results are consistent with the previous studies where higher injury severity was reported for the presence of freeway merging and diverging segments [28,51].Te speed limit indicator decreases the probability of severe injury.Finally, drivers drive faster on rural roads, which are mostly single lanes (such as no clear division of trafc direction, no intermediate guardrail, shoulder width is limited, etc.), resulting in severe injury or fatality crashes.Terefore, truck drivers should look at the rural road before starting the heterogeneous section and slow down.In addition, advance warning signs should be implemented ahead of the heterogeneous section to warn drivers.
Tree environment-related variables, including dark, no light, weekdays, and rainy, have a statistically signifcant efect on the injury severity of drivers in the truck-truck crash model.Te inclement weather conditions increase the severe injuries by an estimated odd of 49.58% on average compared to clear weather, which is opposite to the previous truck occupant model.Unlit conditions increase severe injuries by an estimated odd of 4.04% on average compared to light.Te results are consistent with the previous studies where higher injury severity was reported for the presence of unlit conditions [28,50].Finally, among the crash characteristics, severe injury is increased during weekdays.Tis result complies with other studies where the author clarifed the presence of a lower percentage of trucks on weekends compared to weekdays as the possible reason behind this [48,50,52,53].

Car-Truck Crashes
(1) Car Driver Model.As shown in Table 4, there are three statistically signifcant driver-related variables in the cartruck crash model.Among them, the estimated odds of a severe injury increased by 6.55% and 26.95% on average, with the car drivers being old and female, respectively.Te previous study on driver-injury severities at various locations also reported that older and female drivers have a higher probability of more severe injuries [28].Note that young male truck drivers increase their risk of serious injury or fatality crashes and increase the probability of severe injury crashes for car drivers.Young male truck drivers are often overconfdent in their driving skills.As a result, they are more likely to engage in dangerous driving behaviors, leading to serious and fatal injuries while hindering other vehicles from driving normally on the road.Terefore, it is necessary to strengthen safety education for young male truck drivers and, at the same time, increase the punishment for the corresponding improper driving behavior.However, some studies have also shown that young drivers are less likely to be involved in serious injuries and fatal injuries in car-truck crashes due to their physical strengths and emergency response capabilities [10].
Five vehicle-related variables, including speeding car, changing truck, age of the car (3-10 years), head-on collision, and sideswipes collision, have a statistically signifcant efect on injury severity of car drivers in the car-truck crashes model.While investigating driving errors, car drivers' improper actions signifcantly increase the estimated odds of car and truck driver severe injuries by 11.80% and 0.60% on average, respectively.Te results also show that car drivers are more responsible than truck drivers for contributing more severe injuries to truck drivers in car-truck crashes [28].More enforcement and education programs about car drivers should be enhanced.Truck changing lane behavior signifcantly increases the probability of severe injury and fatality crashes for both car and truck drivers.Due to the large size of trucks, there is a blind-vision zone when changing or turning, leading to serious injury crashes.Terefore, some interventions should be implemented, such as reminding the rear car to pay attention to avoid, but also through the vehicle's advanced equipment to increase the back view of the truck driver.Older cars between 3 and 10 years reduce the probability of severe injury for drivers of cars.It is recommended that vehicles need regular servicing (3-10 years) and reach a longer service life (such as more 10 Journal of Advanced Transportation than ten years) to consider scrap processing and replacing the new car to get safer driving.Notably, the study found that both side and frontal crashes increased the risk of severe injury of car drivers in car-truck crashes and that side crashes are more likely to result in severe injury than frontal crashes.
Tree roadway-related variables, including dual carriageway, 30 mph speed limit, and rural areas, have a statistically signifcant efect on the injury severity of car drivers in the car-truck crashes model.Among them, the 30 mph speed limit indicator decreases car drivers' probability of severe injury.Te result is easy to understand intuitively, and a higher speed limit has been found in the literature to be related to severe injury crashes [50,54].Te rural roadway indicator decreases the car drivers' probability of severe injuries [52].Compared with single lanes, automotive vehicles driving in dual lanes are associated with a risk of severe injury by 3.81%.Te authors in [55] found that a car driver has a greater response to stimulus than a truck driver and maintains as small a front gap as possible.In addition, trucks generally have fewer braking capabilities than cars.Te authors in [28] show that car drivers are more responsible than truck drivers for contributing to more severe injuries to truck drivers in car-truck crashes.Terefore, when the road conditions allow, it is appropriate to set up separate lanes for passengers and trucks or large trucks' special lanes.Still, on the other hand, we must also prevent other vehicles from occupying the truck lane.
Tree environment-related variables, including dark no light, of-peak time, and rainy, have a statistically signifcant efect on the injury severity of car drivers in the car-truck crashes model.Te unlit conditions, rain conditions, and of-peak times increase the estimated odds of a severe injury of car drivers by 0.07%, 2.80%, and 15.43% on average, respectively.
(2) Truck Driver Model.As shown in Table 4, there are three statistically signifcant driver-related variables in the cartruck crash model.Among them, the estimated odds of a severe injury increased by 6.66%, 0.34%, and 21.46% on average, with the truck drivers being young, old, and male, respectively.
Four vehicle-related variables, including speeding car, changing truck, age of truck (10 years above), and head-on collision, have a statistically signifcant efect on injury severity of truck drivers in the car-truck crash model.While Journal of Advanced Transportation investigating driving errors, truck drivers' improper actions have no statistically signifcant impact on the injury severity of car drivers.Terefore, it can be concluded that car drivers are more responsible for severe injuries in car-truck crashes.Similar results were found in previous studies [28].Likewise, truck changing, head-on collisions, and older trucks (10 years above) increase the estimated odds of a severe injury of truck drivers by 1.03%, 9.80%, and 5.66% on average, respectively.Two roadway-related variables, including the 30 mph speed limit and rural areas, have a statistically signifcant efect on the injury severity of truck drivers in the car-truck crashes model.Among them, the speed limit indicator decreases truck drivers' probability of severe injury.In contrast, the rural roadways indicator increases truck drivers' likelihood of severe injuries.
Two environment-related variables, including dark, no light, and weekdays, have a statistically signifcant efect on the injury severity of truck drivers in the car-truck crash model.Unlit conditions and weekdays increase the truck drivers' estimated odds of a severe injury by 22.31% and 9.20% on average, respectively.Drivers are driving at night with inadequate road lighting and a limited range of vehicle lights, making it more difcult for drivers to judge the road conditions and speed.Terefore, it is appropriate to increase the lighting at night and the length of lighting, especially for some roadways sections with a high proportion of large trucks.Among the crash characteristics, the severe injury of truck drivers also increased during weekdays.Tis result complies with other studies [50,55].
To that end, Table 5 summarizes the efects of statistically signifcant variables on injury severity by vehicle and driver type.

Conclusions
Using three-year crash data from the UK, this study develops a random parameters logit model with heterogeneity in means and variances to explore the injury severity of drivers.Te estimated models reveal that varieties of drivers, vehicles, roads, and environment attributes afect drivers' injury severities.Te main conclusions are summarized as follows: ( Additionally, drivers should take precautions while traveling at a comparatively slower speed on the areas of the road that are artifcially lit.(e) Among the crash characteristics, severe injury is increased during weekdays for truck-involved crashes.Terefore, warning signs should be implemented to warn car drivers, especially for some roadways sections with a high proportion of large trucks during weekdays.
(4 )Tis study also has some limitations.Firstly, it is noted that certain data-specifc biases, particularly resulting from the shortcomings of the crash reporting system, may afect the empirical conclusions of the analysis (for example, the omission of no-injury crashes).In addition, the injury severity of both drivers might be correlated with being involved in the same crash.Another natural extension would be to examine the potential correlation considering the crash injury-severity levels of both parties.Lastly, more two-vehicle crash datasets should be included in the future to investigate the temporal stability and then to help policy-makers to take necessary measures in reducing motorcycle-involved crashes by forming appropriate and time-efcient strategies.

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Standard deviation values for 26-45 years car drivers (normally distributed)Standard deviation values for 26-45 years truck drivers (normally distributed) Heterogeneity in the mean of random parameter: darkness-lights litStandard deviation values for the 30 mph speed limit (normally distributed) Age of truck vehicle (1 if above ten years, 0 otherwise) driver, CD: car driver, SI: severe injury, and MI: minor injury.

Table 2
presents the descriptive statistics of these variables in injury-severity models.

Table 1 :
A summary of studies on the severity of two-vehicle crashes.

Table 2 :
Descriptive statistics of signifcant variables in the injury-severity model.

Table 3 :
Model results of injury severity of drivers involved in diferent types of vehicle crashes.
a TD: truck driver and CD: car driver.

Table 4 :
Marginal efects result of injury severity of drivers involved in diferent types of vehicle crashes.
↑indicates an increase in the likelihood of severe injuries; ↓ indicates a decrease in the likelihood of severe injuries.TD: truck driver and CD: car driver.
1) Te random parameters logit with heterogeneity in the means and variances (RPLHMV) model provides a superior statistical ft.It ofers additional insights compared to the traditional lower-order logit model counterparts by accommodating variations of the explanatory variables across the observations and factors afecting the means and variances of the parameter density functions of the random parameters.Tis allows us to identify additional factors that may play a role in determining a parameter's true efect on injury severity.(2) More importantly, concerning the contributing factors afecting the driver's severe injuries, separate injury-severity models based on vehicle and driver types ofer valuable insights.However, inconsistency exists in the determinants of each model.For example, only the speed limit is statistically signifcant in all the models, while others show signifcance, not in all models.In addition, young car drivers, car turning, roundabouts, urban areas, dark lights lit, fne conditions, and summer indicators have a signifcant efect only in car-car crash models.Similarly, some variables, including the age of drivers, gender, speeding, sideswipes, and weekdays, are signifcant in one driver model (truck driver or car driver) but not in other driver models.(3) Te fndings ofer numerous practical implications: (a) More enforcement and education programs about young and male drivers should be enhanced.Old truck drivers should attend regular physical check-ups and driving safety education; if failing a driving test, they could be considered to take the initiative to return their driver's license.And car drivers are more responsible for respective severe injuries in car-truck crashes.(b) It is recommended that vehicles need regular servicing (3-10 years) and reach a longer service life (such as more than ten years) to consider doing vehicle scrap processing and replace the new truck to get safer driving.Due to the large size of trucks, there is a blind-vision zone when changing or turning, leading to serious injury crashes.Terefore, some interventions should be implemented, such as reminding the rear car to pay attention to avoid, but also through the vehicle's advanced equipment to increase the back view of the truck driver.(c) Our fndings indicate that young drivers either lack adequate knowledge or experience complying with the highway code at junctions or violate trafc laws, as this results in severe crashes.Tus, more educational programs should be implemented to prevent young drivers from illegally driving, such as disobeying rightof-way, stop signs, and road markings.(d) Serious injury crashes are more likely to occur when driving at night.Terefore, truck service time must be carefully considered while creating the evening schedule plan to prevent drowsiness.