###### Abstract  
The effect of risk factors on crash severity varies across vehicle types. The objective of this study was to explore the risk factors associated with the severity of rural single-vehicle (SV) crashes. Four vehicle types including passenger car, motorcycle, pickup, and truck were considered. To synthetically accommodate unobserved heterogeneity and spatial correlation in crash data, a novel Bayesian spatial random parameters logit (SRP-logit) model is proposed. Rural SV crash data in Shandong Province were extracted to calibrate the model. Three traditional logit approaches--multinomial logit model, random parameter logit model, and random intercept logit model--were also established and compared with the proposed model. The results indicated that the SRP-logit model exhibits the best fit performance compared with other models, highlighting that simultaneously accommodating unobserved heterogeneity and spatial correlation is a promising modeling approach. Further, there is a significant positive correlation between weekend, dark (without street lighting) conditions, and collision with fixed object and severe crashes and a significant negative correlation between collision with pedestrians and severe crashes. The findings can provide valuable information for policy makers to improve traffic safety performance in rural areas.  
traffic safety; single-vehicle crash; spatial effect; Bayesian estimation +
Footnote †: journal: MedicalThere are many types of motor vehicles, such as passenger car, motorcycle, pickup, and truck, each with its own characteristics. It is expected that the influence of risk factors on the severity of rural SV crashes may vary across different vehicle types. If the same SPFs were used for different vehicle types, the variability of crash mechanisms will be ignored and will also lead to unsatisfactory prediction performance. Conversely, modeling of rural SV crashes based on different vehicle types is expected to overcome these deficiencies and can help to visualize the variant impacts of risk factors on crash severity.  
In addition, as a crash is the result of many factors, there are always some unobserved or unobservable factors that will affect crash severity, named unobserved heterogeneity [4,5]. Ignoring unobserved heterogeneity may lead to erroneous parameter inference [6].  
To capture the effects of unobserved heterogeneity on the severity of rural SV crashes, several heterogeneity models have been constructed [7,8], such as random parameters logit model and random intercept logit model. These models allow parameters to vary across observations, which provides a flexible framework. Theoretically, they can simulate all selection patterns based on probabilistic propensities [9,10]. However, such approaches cannot capture the spatial correlation across crashes, which will limit the promotion of the model. Importantly, the spatial correlation of adjacent crashes has been commonly recognized in crash rates, crash counts, and traffic conflict analyses [2,11], but this issue is often overlooked in crash severity analyses, especially for rural SV crashes, which may lead to the prediction accuracy being substantially undermined.  
Based on the above analysis, it is meaningful to establish statistical models across motor vehicle types for rural SV crash severity and to simultaneously accommodate the unobserved heterogeneity and spatial correlation. The research results are helpful to clarify the variable influences of risk factors on rural SV crash severity and create a safer transportation environment in rural areas.  
The rest of this paper is organized as follows. A comprehensive literature review for rural SV crash analysis is given in Section 2. Section 3 provides detailed information on the studied crash dataset. The methodological framework in this research is presented in Section 4, and analysis and discussion of the estimation results are given in Section 5. Conclusions and potential limitations of this research are described in Sections 6 and 7, respectively.