# Temporal Instability Analysis of Injury Severities for Angle and Non-Angle Crashes at Roundabouts
## 1 Introduction  
Roundabouts are considered a safe intersection design due to slower speeds and fewer conflict points [1]. Numerous studies have found that installing roundabouts significantly reduces the total number of severe injury crashes while increasing crashes of other severity [2]. [PERSON] et al. [3] found that roundabout collision frequencies in the United States are still high compared to Australia, France, and the United Kingdom. These suggest that the safety benefits of roundabouts may not be satisfactory in all situations [4], and such gains may come at a cost. While reducing casualties is a positive step, there is room for improvement. Therefore, further research on roundabout safety is necessary. Traffic safety research at roundabouts has been carried out earlier [5, 6]. Among them, the literature on crash injury severity is a crucial part of the field. Based on three models of roundabouts, [PERSON] et al. [7] studied various factors impacting driver injury severity and their variations. [PERSON] and [PERSON] [8] investigated the crash rates and severityof injuries in single-lane and two-lane roundabouts, finding that the former had lower crash rates. By building a safety performance function, [PERSON] et al. [9] analyzed the impact of inlet design characteristics on injury severity. [PERSON] et al. [10] investigated the severity of traffic crashes at roundabouts in Flanders-Belgium, considering diverse road users. Conclusions suggest that vulnerable user groups are more susceptible to serious harm. Although many studies have been conducted on the factors influencing the injury severity of roundabout crashes, the unobserved heterogeneity and temporal instability have been ignored [11; 12]. More importantly, few studies focus on the collision modes of roundabout crashes. Due to unique geometry, collisions at roundabouts differ from regular road segments or intersections. Analysis shows that angle crashes are one of the leading collision modes for roundabouts [13]. According to statistics, angle crashes are also the collision type with the biggest share in the roundabout crash dataset used in the analysis. It can be seen that the angle crash has an essential impact on the traffic safety of the roundabout. According to the collision point of the vehicle involved in crashes, the collision mode is often divided into rear-end collision, head-on collision, sideswipe, angle collision, and so on [14]. The related research on collision modes generally includes two types: studies considering multiple collisions and studies only focusing on one single collision mode. Investigations considering multiple collision modes mainly aim to compare crash frequency, injury severity, and influencing factors of different modes [15; 16]. The studies on the single collision mode are mainly concerned with head-on collisions [17]. Multiple pieces of literature conduct separate studies on different collision modes, which can lead to more precise conclusions. For example, [PERSON] et al. [18] conduct research on head-on collisions to reduce injury severity effectively. [PERSON] and [PERSON] [19] point out that separation research based on collision mode can more clearly verify the improved safety performance by extending the left-turn lane. In addition, risk factor studies targeting collision modes are beneficial for detecting heterogeneity [20]. Nevertheless, few studies are devoted to angle collision, and its essential effect is often overlooked, especially at roundabouts. Furthermore, the heterogeneity and temporal instability of a single collision mode (such as an angle collision) have not received enough attention in existing studies. To bridge the aforementioned gaps, this study focuses on the temporal instability and variability of the factors influencing the injury severity of angle and non-angle crashes at roundabouts. To investigate unobserved heterogeneity, random parameter logit models with heterogeneity in the mean and variance were chosen from numerous statistical models.