# Hamiltonian Monte Carlo with Random Effect for Analyzing Cyclist Crash Severity
###### Abstract  
Vulnerable traffic users, such as bikers and pedestrians, account for a significant number of fatalities on the roadways. Extensive research has been conducted in the literature review to identify factors to those crashes. Studying factors to those crashes is especially important in the Western state in the US, due to one of the highest fatality rates in the nation and its unique geographic conditions. The first step in identifying factors to the severity of cyclist crashes is to find the underlying factors to that type of crash, while accounting for the heterogeneity in the dataset. Various techniques such as mixed parameter or mixed effect models have been employed in the literature to account for the heterogeneity of the dataset. In the mixed effect model, often the random effect parameter has been assigned subjectively, and based on some attributes and engineering intuitions. Those assignments are expected to account for the heterogeneity in the dataset and enhancement of the model fit. However, a question might arise whether those factors could account for an optimum amount of the heterogeneity in the dataset. A more reasonable way might be to let the algorithm such as the finite mixture model (FMM) to identify those clusters based on parameters of the Gaussian model, means and covariance matrices of the dataset, and allocate each observation to the related clusters. Thus, in this study, to capture optimum amount of heterogeneity, first we implemented the finite mixture model in the context of maximum likelihood, due the label switching issue of the method in the context of the Bayesian method. After assignment of the parameters to the observation, the main method of Hamiltonian Monte Carlo (HMC) with random effect was implemented. The results highlighted a significant improvement in the model fit, in terms of Widely Applicable Information Criterion (WAIC). The results of this study highlighted factors such as older biker age, increased number of lanes, nighttime travelling, increased posted speed limit and driving while under emotional conditions are some factors contributing to an increased severity of bikers' crash severity. Extensive discussion has been made regarding the methodological algorithms and model parameters estimations. c 2021 2021 3 August 2021 17 August 2021 10 April 2021 3 August 2021 10 April 2021 3 August 2021 10 April 2021 3 August 2021  
It has been highlighted in the previous study that the proportion of biking and cycling are overrepresented in crash fatalities [6]. For instance, bicyclists are twelve times more likely to be killed in road crashes than motor vehicle drivers, and also in the US they are three times and twice as likely to be involved in fatal crashes compared with Dutch and Germany, respectively [7]. The following paragraphs would outlines few studies related to cyclists' crashes. The impact of the built environment was evaluated on bicyclist injury severity [8]. The generalized ordered logit and generalized additive models were employed. The results highlighted that employment density, reflective clothing, age and posted speed limit are some of the factors contributing to the severity of bikers' crash severity. The mixed logit analysis was used for analyzing bicyclist injury severity at intersection, and non-intersection locations [9]. The results highlighted that at the intersection when the bikers did not wear a helmet, or when the motorist was under the influence, the severity of crashes was increased. On the other hand, the results highlighted at non-intersection, when the motorist was under the influence, or a vehicle strikes the side of bicycle and the time that the heavy truck hit a bicycle, the severity of crashes increases. Bicyclist crash types were investigated on national and local levels [10]. The results highlighted that while the majority of injuries occurred at intersections, most of the fatal and disabling injury bicyclist crashes occurred at the non-intersection area.