Temporal Instability Analysis of Injury Severities for Angle and Non-Angle Crashes at Roundabouts

Abstract

Roundabout crashes are common worldwide but have received little attention. In particular, the investigation remains relatively understudied in distinct collision modes of roundabout crashes, including angle and non-angle crashes. This study investigates differences in factors affecting injury severity between angle and non-angle crashes and how these differences change over time. Random parameter logit models with heterogeneity in mean and variance were estimated using 2012–2019 Florida crash data. Variables considered for the modeling include temporal characteristics, environmental characteristics, road characteristics, spatial characteristics, vehicle characteristics, crash characteristics, and human characteristics. In addition, likelihood ratio tests were used to investigate the temporal instabilities of the models and differences in collision modes. The results showed that only a few variables demonstrated temporally stable effects for angle collisions (weekend and dark-lighted) and non-angle collisions (paved shoulders and cyclists), respectively. Unique influencing factors of injury severity were identified for different collision modes, such as dark-lighted, straight alignment, merge, lane departure, the disabled driver for angle collisions, and clear, fog, commercial vehicles, and aggressive driving for non-angle collisions. The results of the out-of-sample prediction simulations further demonstrate the difference in injury severity over time for angle and non-angle collisions. Overall results suggest that countermeasures can be implemented to reduce the injury severity of angle crashes/non-angle crashes based on the significant variables identified. This study may be used to improve roundabout safety by considering different collision modes.

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]. Robinson 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, Zubaidi et al. [7] studied various factors impacting driver injury severity and their variations. Mamlouk and Souliman [8] investigated the crash rates and severity of injuries in single-lane and two-lane roundabouts, finding that the former had lower crash rates. By building a safety performance function, Novák et al. [9] analyzed the impact of inlet design characteristics on injury severity. Daniels 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, Smith et al. [18] conduct research on head-on collisions to reduce injury severity effectively. Guo and Sayed [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. The contributions of this study are as follows: (1) Investigating whether the determinants of injury severity are transferable across angle and non-angle collisions, and (2) revealing how the factors influencing the severity of roundabout collisions vary over time. The findings can not only assist in identifying temporal instability and differences in injury severity across different collision modes but also serve as a guide for regulators to develop effective countermeasures for roundabout crashes.

The remainder of the paper is organized as follows: Section 2 provides a literature review of relevant points. Section 3 details the crash dataset and related variables. Section 4 presents the theoretical framework for the method used. Section 5 provides the rationale and computational results of the likelihood ratio tests. Section 6 presents and discusses the model results categorically. Section 7 summarizes the main findings and prospects.

2. Literature Review

2.1. Review of Temporal Instability Research

For traffic safety research, it is crucial to explore the temporal stability of variables. Suppose the effects of factors are temporarily unstable. In that case, it cannot be determined to what extent changes in injury severity are due to temporal changes and how much depends on the implementation of specific safety countermeasures [21]. Ignoring the time factor can lead to false conclusions and ineffective or even dangerous safety policies [22]. The effectiveness of various data analysis methods is also potentially affected by temporal instability. When modeling crash data, numerous road safety studies have proposed the assumption that the impacts of influencing factors do not vary over time. However, there is growing evidence that explanatory factors’ influence on crash injury outcomes is temporally unstable.

Through research in psychology, neuroscience, economics, and cognitive science, Mannering [22] found that the instability of the influence of explanatory variables over time exists due to some basic behaviors. Subsequently, Alnawmasi and Mannering [11] tested the difference in injury severity of motorcycle drivers at different times based on two crash datasets. As expected, the results indicated temporal instabilities in the statistical model. Temporal instability of factors affecting injury severity was also found in truck crashes [12]. In addition, the study by Xiong et al. [23] showed that crash injury severity was unstable over time by estimating a Markov transformation model. The findings of Wu et al. [24] and Venkataraman et al. [25] further confirm this conclusion. The non-negligible effect of temporal instability on model estimation is shown in the above studies, which are increasingly becoming one of the focuses of traffic safety research [26].

2.2. Review of Modeling Approaches to Injury Severity

Discrete-choice models have been widely used in the study of crash injury severity. According to whether or not the sequence is considered, analysis methods can be divided into ordered and unordered models. Specifically, commonly used methods include the ordered logit/probit model, generalized ordered logit/probit model, binary logit model, multinomial logit model, and mixed logit model. For example, the ordered logistic model was applied by Rezapour et al. [27] to develop the injury-severity model of single and multiple-vehicle downgrade crashes. Using the same approach, Song and Fan [28] analyzed truck driver injury severity, considering heterogeneity in latent classes. Generalized ordered logit models were further implemented due to relaxing the proportional odds or parallel-regression assumption of variables [29]. Regarding the disordered model, the multinomial logit model is the most basic statistical model in injury severity analysis. A multinomial logit model was developed to examine the effect of road characteristics on crash injury severity [30].

However, the previously mentioned traditional ordered methods and multinomial logit models cannot capture the unobserved heterogeneity. Unobserved heterogeneity can lead to biased model estimates [31]. To solve the problem, methods such as random parameter models have been developed. Using an ordered logit model with random parameters, Naik et al. [32] studied crash injury severity under different weather conditions. Coincidentally, similar studies by Haleem [33] show that the mixed logit model outperforms the basic logit model. Many researchers believe that a random parameter logit model with heterogeneity in mean and variance is an approach to tapping more latent factors [34,35]. Of course, some rare statistical methods are used to analyze the severity of crash injuries. Patil et al. [36] used a nested logit model for analysis. A random parameter Tobit model was introduced to analyze accident rates by crash injury severity [37].

2.3. Contributing Factors to Crash Injury Severity

The severity of crash injuries results from the interaction of environmental characteristics, vehicle characteristics, traffic participants, and road characteristics [38]. Understanding contributing factors that impact injury severity is the key to improving traffic safety.

Environmental characteristics include weather (clear, rain, ice, etc.) and lighting conditions (daylight, dawn, dark, dark-lighted, etc.). Crashes in clear weather were more likely to cause severe injuries, possibly because of higher driving speeds than in other weather conditions [36,39]. Dark lighting was statistically significant in multiple studies; however, its effect on injury severity varied [36,40]. In addition, individual studies have found that factors such as ‘snow’ and ‘dawn’ also affect the severity of injuries [21].

As for vehicle characteristics, vehicle type is the focus of existing literature research. Sport utility vehicles and pickup trucks are more likely to be involved in no-injury crashes [26,41], while motorcycles and public transport are more likely to be involved in severe crashes [29]. The researchers believe it has to do with vehicle usage and the physical design of the vehicle. Specifically, pickup trucks have better shock resistance to protect their occupants. Public transport drivers are prone to fatigue and severe injury due to long-distance, long-term driving tasks.

Gender, age, and poor driving behavior are the main contributing factors for traffic participants. However, for the same affecting factors, the findings reached by different researchers vary greatly. Al-Bdairi and Hernandez [42] found that male drivers were more likely to be seriously injured than females, while Yu et al. [26] came to the opposite conclusion. Some age-related studies argue that older drivers are more vulnerable to severe injury due to their more fragile physiological abilities [43]. Others believe that younger drivers are more likely to be seriously injured due to aggressive driving [39]. In addition, poor driving behaviors such as alcohol involvement, distraction, and drowsy driving were associated with a higher probability of severe injury in most studies [26,44].

The roadway characteristic appears in the literature as one of the most common influencing factors of injury severity. Studies have suggested that roadway surface, alignment, and road class are noteworthy. Poor road conditions significantly impact injury severity [21]. Curved roads are often more likely to cause severe and fatal injuries than straight roads [39]. In different grades of roadway, secondary roads and dedicated public transport roadways increase accident injury severity [40].

In light of the above, this study considers factors including temporal, environmental, road, spatial, vehicle, crash, and human characteristics, which are commonly used in previous studies.

2.4. Review of Roundabout Crashes

As shown in

Table 1. Statistics on roundabout accidents by region [45,46,47,48,49,50,51].

Region	Periods	Number of Roundabouts	Number of Accidents	Accident Rate
Wisconsin	2009–2014	53	1467	-
Korea	2007–2010	93	137	-
Erftkreis	-	27	335	-
Washington	2001–2013	-	3067	-
Oregon	2011–2015	-	1006	-
UK	1999–2003	1162	-	1.87%
Arizona	-	17	-	2~6%
Germany	-	-	-	162/a million vehicles

Note: Reprinted/adapted with permission from Ref. [45]. Burdett, B., Bill, A., and Noyce, D., Evaluation of roundabout-related single-vehicle crashes; published by Transportation Research Record, 2017, Ref. [47]. Erftkreis District Police Authority, data from a study of 27 roundabouts in the Erftkreis, 2002, Ref. [48]. Steyn, H., Griffin, A., and Rodegerdts, L., Accelerating Roundabouts in the United States: Volume IV of VII-A Review of Fatal and Severe Injury Crashes at Roundabouts, 2015., Ref. [50]. Kennedy, J., Peirce, J., and Summersgill, I., Review of accident research at roundabouts; published by Transportation Research Circular, 2005, Ref. [51]. Souliman, B., Effect of Roundabouts on Accident Rate and Severity in Arizona, published by Arizona State University, 2016.

, crashes involving roundabouts are still prevalent throughout the globe. Therefore, roundabouts deserve attention as a crucial aspect of enhancing traffic safety.

3. Data Description

Data available for this study were roundabout crashes provided by Florida’s Crash Analysis Reporting Data System, which were gathered over eight years from 1 January 2012 to 31 December 2019. For this study, every two years is regarded as a period, including the 2012–2013, 2014–2015, 2016–2017, and 2018–2019 time periods in the dataset.

In the dataset, crashes are classified into seven categories according to the collision modes: front to rear, front to front, angle, sideswipe-same direction, sideswipe-opposite direction, rear to side, and rear to rear. This study maintains the original angle crashes while taking all other collision modes as non-angle crashes. The number of angle collisions at roundabouts totaled 1351 over the eight years. In comparison, 3288 non-angle collisions occurred during the period. The information in the dataset covers crashes, roads, crash-involved people, vehicles, and other aspects. Details include the severity of the injury, the type of collision, the influence of alcohol, weather, work zone conditions, and more. Three injury severities were identified as outcome variables: no injury, minor injury (possible and non-incapacitating injury), and severe injury (incapacitating injury and fatality) [31]. Follow-up studies only presented and discussed those variables that were statistically significant parameters in the model estimation. Figure 1 shows the percentage distribution of no injury, minor injury, and severe injury for angle and non-angle crashes during the four analysis periods (2012–2013, 2014–2015, 2016–2017, 2018–2019).

Table 2. Angle collisions descriptive statistics for significant variables (non-angle collisions statistics in parentheses).

Variable	2012–2013		2014–2015		2016–2017		2018–2019
	Mean	Std. Dev	Mean	Std. Dev	Mean	Std. Dev	Mean	Std. Dev
Temporal characteristics
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise)	0.227	0.420	0.226	0.418	0.213	0.410	0.233	0.423
	(0.301)	(0.459)	(0.298)	(0.458)	(0.263)	(0.440)	(0.294)	(0.456)
Environmental characteristics
Daylight indicator	0.630	0.466	0.717	0.450	0.722	0.448	0.679	0.467
(1 if the light condition was daylight, 0 otherwise)	(0.499)	(0.500)	(0.525)	(0.499)	(0.536)	(0.499)	(0.580)	(0.494)
Dawn indicator	0.009	0.097	0.012	0.108	0.019	0.136	0.013	0.111
(1 if the light condition was dawn, 0 otherwise)	(0.019)	(0.137)	(0.012)	(0.109)	(0.019)	(0.135)	(0.021)	(0.143)
Dark-lighted indicator (1 if the light condition was dark-lighted, 0 otherwise)	0.242	0.428	0.209	0.407	0.205	0.404	0.204	0.403
	(0.382)	(0.486)	(0.362)	(0.481)	(0.351)	(0.477)	(0.303)	(0.460)
Dark-Not Lighted-indicator (1 if the light condition was dark-not-lighted, 0 otherwise)	0.090	0.191	0.043	0.202	0.029	0.169	0.042	0.200
	(0.062)	(0.241)	(0.057)	(0.231)	(0.059)	(0.236)	(0.063)	(0.243)
Clear indicator	0.810	0.392	0.817	0.387	0.791	0.407	0.837	0.369
(1 if the weather was clear, 0 otherwise)	(0.772)	(0.420)	(0.796)	(0.403)	(0.789)	(0.408)	(0.775)	(0.418)
Cloudy indicator	0.104	0.306	0.100	0.300	0.121	0.326	0.079	0.270
(1 if the weather was cloudy, 0 otherwise)	(0.122)	(0.327)	(0.118)	(0.323)	(0.118)	(0.323)	(0.135)	(0.342)
Rain indicator	0.081	0.272	0.081	0.273	0.086	0.280	0.079	0.270
(1 if the weather was rainy, 0 otherwise)	(0.092)	(0.289)	(0.072)	(0.258)	(0.080)	(0.271	(0.074)	(0.263)
Roadway characteristics
Paved shoulder indicator	0.199	0.400	0.178	0.383	0.296	0.457	0.313	0.464
(1 if the type of shoulder was paved, 0 otherwise)	(0.174)	(0.203)	(0.025)	(0.032)	(0.218)	(0.395)	(0.301)	(0.399)
Unpaved shoulder indicator (1 if the type of the shoulder was unpaved, 0 otherwise)	0.118	0.323	0.105	0.306	0.077	0.267	0.079	0.270
	(0.177)	(0.382)	(0.123)	(0.329)	(0.098)	(0.298)	(0.113)	(0.316)
Dry surface indicator	0.967	0.179	0.969	0.173	0.952	0.214	0.975	0.156
(1 if road the surface condition was dry, 0 otherwise)	(0.964)	(0.187)	(0.954)	(0.209)	(0.939)	(0.240)	(0.912)	(0.283)
Wet surface indicator	0.003	0.023	0.002	0.049	0.002	0.041	0.005	0.037
(1 if the road surface condition was wet, 0 otherwise)	(0.005)	(0.039)	(0.002)	(0.045)	(0.003)	(0.052)	(0.008)	(0.043)
Ice/Frost surface indicator (1 if the road surface condition was ice or frost, 0 otherwise)	0.014	0.118	0.005	0.069	0.013	0.111	0.004	0.064
	(0.014)	(0.119)	(0.007)	(0.083)	(0.013)	(0.115)	(0.036)	(0.187)
Straight indicator (1 if road roadway alignment was straight, 0 otherwise)	0.521	0.500	0.523	0.500	0.526	0.499	0.500	0.500
	(0.450)	(0.498)	(0.458)	(0.498)	(0.477)	(0.500)	(0.490)	(0.500)
Curve right indicator (1 if road roadway alignment was curve right, 0 otherwise)	0.152	0.359	0.204	0.403	0.219	0.414	0.192	0.394
	(0.242)	(0.429)	(0.234)	(0.423)	(0.216)	(0.411)	(0.197)	(0.398)
Curve left indicator (1 if road roadway alignment was curve left, 0 otherwise)	0.313	0.464	0.254	0.436	0.219	0.414	0.292	0.455
	(0.298)	(0.457)	(0.278)	(0.448)	(0.269)	(0.443)	(0.271)	(0.445)
Spatial characteristics
Town indicator	0.687	0.464	0.713	0.453	0.708	0.455	0.708	0.455
(1 if a crash occurred in town, 0 otherwise)	(0.724)	(0.447)	(0.709)	(0.454)	(0.703)	(0.457)	(0.712)	(0.453)
Merge indicator	0.583	0.493	0.603	0.489	0.522	0.500	0.638	0.481
(1 if a crash occurred in the merging area, 0 otherwise)	(0.388)	(0.487)	(0.398)	(0.489)	(0.408)	(0.492)	(0.574)	(0.495)
Vehicle characteristics
School bus indicator (1 if a school bus was indirectly	0.009	0.097	0.005	0.069	0.006	0.079	0.004	0.064
involved in the crash, 0 otherwise)	(0.010)	(0.097)	(0.008)	(0.089)	(0.008)	(0.089)	(0.004)	(0.062)
Commercial vehicle indicator (1 if commercial vehicles were involved in the crash, 0 otherwise)	0.038	0.191	0.033	0.179	0.035	0.185	0.042	0.200
	(0.082)	(0.275)	(0.071)	(0.256)	(0.063)	(0.243)	(0.061)	(0.240)
Crash characteristics
Wrongway indicator (1 if using the wrong roadway involved in the crash, 0 otherwise)	0.009	0.097	0.005	0.069	0.003	0.062	0.008	0.088
	(0.005)	(0.069)	(0.006)	(0.077)	(0.006)	(0.079)	(0.010)	(0.097)
Lane departure indicator (1 if lane departure is involved in the crash, 0 otherwise)	0.109	0.312	0.081	0.273	0.067	0.250	0.067	0.250
	(0.377)	(0.485)	(0.372)	(0.483)	(0.335)	(0.472)	(0.275)	(0.447)
Speeding indicator (1 if speeding is involved in the crash, 0 otherwise)	0.005	0.069	0.045	0.137	0.042	0.150	0.088	0.111
	(0.063)	(0.244)	(0.058)	(0.233)	(0.058)	(0.233)	(0.063)	(0.243)
Aggressive driving indicator (1 if aggressive driving is involved in the crash, 0 otherwise)	0.024	0.152	0.026	0.160	0.031	0.174	0.021	0.143
	(0.035)	(0.183)	(0.033)	(0.178)	(0.036)	(0.187)	(0.046)	(0.209)
Human characteristics
Disabled driver indicator (1 if a disabled driver is involved in the crash, 0 otherwise)	0.085	0.280	0.088	0.283	0.106	0.309	0.138	0.345
	(0.138)	(0.345)	(0.130)	(0.337)	(0.125)	(0.331)	(0.149)	(0.356)
Pedestrian indicator	0.009	0.097	0.007	0.084	0.004	0.065	0.010	0.073
(1 if pedestrians were involved in the crash, 0 otherwise)	(0.013)	(0.112)	(0.012)	(0.109)	(0.011)	(0.103)	(0.025)	(0.156)
Cyclist indicator	0.043	0.202	0.052	0.202	0.044	0.190	0.088	0.190
(1 if cyclists were involved in the crash, 0 otherwise)	(0.033)	(0.152)	(0.020)	(0.140)	(0.022)	(0.147)	(0.044)	(0.137)

shows the descriptive statistics of the variables retained after the correlation analysis.

4. Methodology

To study crash injury severity, various discrete models have been adopted, including ordered logit models, multinomial logit models, nested logit models, latent-class logit models, mixed logit models, and others [31,44,52]. In current studies, possible unobserved heterogeneity and temporal instabilities are often explained by random parameter approaches. This study uses random parameter logit models with possible heterogeneity in means and variances, which aims to discover potential heterogeneity and temporal instabilities in angle and non-angle crashes at roundabouts. Experiments were carried out in the statistical software package NLOGIT V5.0. The framework of the methodology section is shown in Figure 2.

First, a function that determines the injury severity of crashes is defined as,

$$S_{kn}={\mathit{\beta }}_k{\mathbf{X}}_{kn}+{\epsilon }_{kn}$$

(1)

where $S_{kn}$ is an injury severity function determining the probability of injury severity $k$ in crash $n$ at roundabouts. ${\mathit{\beta }}_k$ is the vector of estimated parameters, ${\mathbf{X}}_{kn}$ is a vector of explanatory variables affecting the injury severity $k$ in angle and non-angle crashes, and ${\epsilon }_{kn}$ is a disturbance term assumed to be a generalized extreme value distribution.

Then, the multinomial logit model is expressed as follows [52]:

$${\mathrm{P}}_{\mathrm{n}}(k)=\frac{\mathrm{EXP}\left[{\mathit{\beta }}_k{\mathbf{X}}_{kn}\right]}{\sum _{\forall \mathrm{K}} \mathrm{E}\mathrm{X}\mathrm{P}\left({\mathit{\beta }}_k{\mathbf{X}}_{kn}\right)}$$

(2)

where ${\mathrm{P}}_{\mathrm{n}}(k)$ represents the probability of injury severity $k$ in an angle or non-angle crash $n$ at a roundabout, $k$ is a set of three possible injury severities. Considering possible parameter fluctuations, the injury severity probabilities for angle and non-angle collisions are rewritten as [31]:

$${\mathrm{P}}_{\mathrm{n}}(k)=\int \frac{\mathrm{EXP}\left({\mathit{\beta }}_k{\mathbf{X}}_{kn}\right)}{\sum _{\forall K} \mathrm{E}\mathrm{X}\mathrm{P}\left({\mathit{\beta }}_k{\mathbf{X}}_{kn}\right)}f\left({\mathit{\beta }}_k\mid {\mathit{\phi }}_{\mathit{k}}\right)d{\mathit{\beta }}_k$$

(3)

where $f\left({\mathit{\beta }}_k\mid {\mathit{\phi }}_{\mathit{k}}\right)$ is the density function of a vector ${\mathit{\beta }}_k$, ${\mathit{\phi }}_{\mathit{k}}$ is a vector of parameters characterizing the density function, other terms are consistent with previous definitions.

Given the possibility of unobserved heterogeneity in the means and variances of the parameters, ${\mathit{\beta }}_{kn}$ is allowed to be a vector of estimated parameters that vary across crashes. The vector is defined as [31]:

$${\mathit{\beta }}_{kn}={\mathit{\beta }}_k+{\mathsf{\Theta }}_{kn}{\mathbf{Z}}_{kn}+{\sigma }_{kn}EXP\left({\mathsf{\Psi }}_{kn}{\mathbf{W}}_{kn}\right)v_{kn}$$

(4)

where ${\mathit{\beta }}_k$ is the mean parameter estimate for all angle and non-angle crashes at the roundabout, ${\mathsf{\Theta }}_{kn}$ is a vector related to estimated parameters, and ${\mathbf{Z}}_{kn}$ is a vector of explanatory variables that discovers heterogeneity in the mean that determines the injury severity $k$ for angle and non-angle crashes at roundabouts. The heterogeneity in the standard deviation ${\sigma }_{kn}$ of random parameters is captured by ${\mathbf{W}}_{kn}$, with estimable parameters in vector ${\mathsf{\Psi }}_{kn}$, and $v_{kn}$ is the error term.

We used the simulated maximum likelihood method with 1000 Halton draws to estimate model parameters [31]. As for the possible distribution of parameters, the normal distribution, logarithmic distribution, and uniform distribution are tested, and the results show that the normal distribution gives the best statistical fit. Marginal effects are used to describe the changes in the likelihood of injury severity due to a one-unit increase in the explanatory variable $X_{knj}$ [53],

$$\partial P_n(k)/\partial X_{knj}=\overline{P_n\left(k\right)}\left[given{X}_{knj}=1\right]-\overline{P_n\left(k\right)}\left[given{X}_{knj}=0\right]$$

(5)

Several evaluation indexes are applied to evaluate and compare model performance, including the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and likelihood ratio. The formulas for AIC and BIC are as follows [31]:

$$\mathrm{A}\mathrm{I}\mathrm{C}=-2\mathrm{l}\mathrm{n}\left(\mathrm{M}\mathrm{L}\right)+2\mathrm{K}$$

(6)

$$\mathrm{B}\mathrm{I}\mathrm{C}=-2\mathrm{l}\mathrm{n}\left(\mathrm{M}\mathrm{L}\right)+\mathrm{l}\mathrm{n}\left(\mathrm{m}\right)·\mathrm{K}$$

(7)

where $ML$ denotes the maximum value of the likelihood function for the model, $K$ denotes the number of parameters, and $m$ represents the number of observations.

5. Temporal Instability and Transferability Tests

Among the temporal instability studies [11,12], the likelihood ratio test is one of the most commonly used methods. This study conducted two sets of likelihood ratio tests. One group addressed the differences between angle and non-angle crashes. The other aimed to explore the changes in influencing factors over time.

Each period (2012–2013, 2014–2015, 2016–2017, and 2018–2019) was tested to compare the injury severity of angle and non-angle crashes. The equation of the likelihood ratio test is:

$$X_t^2=-2\left[LL\left({\mathit{\beta }}_{\mathrm{all},t}\right)-LL\left({\mathit{\beta }}_{\mathrm{non}-\mathrm{angle},t}\right)-LL\left({\mathit{\beta }}_{\mathrm{angle},t}\right)\right]$$

(8)

where $LL\left({\mathit{\beta }}_{\mathrm{all},t}\right)$ is the log-likelihood when the model converges using all angle crashes and non-angle crashes data in period $t$, $LL\left({\mathit{\beta }}_{\mathrm{non}-\mathrm{angle},t}\right)$ is the log-likelihood when the model converges using the non-angle crash data in period $t$, and $LL\left({\mathit{\beta }}_{\mathrm{angle},t}\right)$ is the log-likelihood when the model converges using the angle crash data in period $t$. The $X^2$ estimates for the four periods 2012–2013, 2014–2015, 2016–2017, and 2018–2019 are 54.18, 22.05, 30.61, and 23.56, respectively. These estimates are distributed with 10, 12, 9, and 10 degrees of freedom, indicating a confidence level above 90%. Then, the assumption that the parameters of angle and non-angle crashes are equal can be rejected.

To validate the two-year combinations, the likelihood test is further applied:

$${\chi }_c^2=-2[LL\left({\mathit{\beta }}_{t_i{t}_{i+1}}\right)-LL\left({\mathit{\beta }}_{t_i}\right)-LL\left({\mathit{\beta }}_{t_{i+1}}\right)]$$

(9)

where $LL\left({\mathit{\beta }}_{t_i{t}_{i+1}}\right)$ represents the log-likelihood at convergence when using both year $i$ and year $i+1$ data, $LL\left({\mathit{\beta }}_{t_i}\right)$ and $LL\left({\mathit{\beta }}_{t_{i+1}}\right)$ represent the log-likelihood at convergence when using only year $i$ and year $i+1$ data, respectively. The results in

Table 3. Likelihood ratio test results between consecutive years (Angle crash).

Year Group	2012–2013	2014–2015	2016–2017	2018–2019
$LL\left({\mathit{\beta }}_{t_i{t}_{i+1}}\right)$	−174.39	−293.34	−259.37	−187.49
Sub-group 1	2012	2014	2016	2018
$LL\left({\mathit{\beta }}_{t_i}\right)$	−79.97	−130.95	−127.31	−144.67
Sub-group 2	2013	2015	2017	2019
$LL\left({\mathit{\beta }}_{t_{i+1}}\right)$	−85.45	−152.97	−125.98	−37.94
${\chi }^2$ value	17.94	18.84	12.16	9.76
Degrees of freedom	12	13	14	12
Confidence level	88.25%	87.19%	40.65%	36.30%

and

Table 4. Likelihood ratio test results between consecutive years (Non-angle crash).

Year Group	2012–2013	2014–2015	2016–2017	2018–2019
$LL\left({\mathit{\beta }}_{t_i{t}_{i+1}}\right)$	−447.59	−687.43	−759.74	−344.63
Sub-group 1	2012	2014	2016	2018
$LL\left({\mathit{\beta }}_{t_i}\right)$	−215.38	−315.24	−378.12	−257.97
Sub-group 2	2013	2015	2017	2019
$LL\left({\mathit{\beta }}_{t_{i+1}}\right)$	−225.54	−362.71	−373.87	−81.57
${\chi }^2$ value	13.34	18.96	15.50	10.18
Degrees of freedom	10	13	16	11
Confidence level	79.47%	87.57%	51.16%	48.57%

show that the test confidence level is much lower than 90%. That is, for angle and non-angle collisions, the null hypothesis that the parameters are equal between two test years cannot be rejected. The results support the rationality of combining the data from two years in this study. (Due to convergence problems caused by a small number of observations, the likelihood ratio tests for one-year groups were inconclusive and unable to validate one-year groups. Therefore, this study considered two-year or multi-year groupings. In detail, two-year groupings from even to odd years (such as 2012–2013) in this study were based on likelihood ratio tests for potential temporal stability (

Table A1. Likelihood ratio tests for different years grouping (Angle crash).

Year Group	2012–2013	2012–2014	2014–2015	2014–2016	2016–2017	2016–2018	2018–2019
$LL\left({\beta }_{all}\right)$	−174.39	−278.93	−293.34	−447.92	−259.37	−382.95	−187.49
Sub-group 1	2012	2012	2014	2014	2016	2016	2018
$LL\left({\beta }_{t_i}\right)$	−79.97	−66.88	−130.95	−129.33	−127.31	−137.46	−144.67
Sub-group 2	2013	2013	2015	2015	2017	2017	2019
$LL\left({\beta }_{t_{i+1}}\right)$	−85.45	−71.77	−152.97	−153.85	−125.98	−127.45	−37.94
Sub-group 3	-	2014	-	2016	-	2018	-
$LL\left({\beta }_{t_{i+2}}\right)$	-	−123.75	-	−146.92	-	−106.59	-
${\chi }^2$ value	17.94	33.06	18.84	35.64	12.16	22.90	9.76
Degrees of freedom	12	11	13	12	14	10	12
Confidence level	88.25%	99.95%	87.19%	99.96%	40.65%	98.89%	36.30%

and

Table A2. Likelihood ratio tests for different years grouping (Non-angle crash).

Year Group	2012–2013	2012–2014	2014–2015	2014–2016	2016–2017	2016–2018	2018–2019
$LL\left({\beta }_{all}\right)$	−447.59	−769.56	−687.43	−1071.85	−759.74	−1025.10	−344.63
Sub-group 1	2012	2012	2014	2014	2016	2016	2018
$LL\left({\beta }_{t_i}\right)$	−215.38	−219.78	−315.24	−313.88	−378.12	−378.12	−257.97
Sub-group 2	2013	2013	2015	2015	2017	2017	2019
$LL\left({\beta }_{t_{i+1}}\right)$	−225.54	−223.22	−362.71	−358.91	−373.87	−373.87	−81.57
Sub-group 3	-	2014	-	2016	-	2018	-
$LL\left({\beta }_{t_{i+2}}\right)$	-	−313.42	-	−378.85	-	−259.52	-
${\chi }^2$ value	13.34	26.28	18.96	40.42	15.50	27.18	10.18
Degrees of freedom	10	11	13	16	16	16	11
Confidence level	79.47%	99.41%	87.57%	99.93%	51.16%	96.05%	48.57%

in Appendix A). If the confidence level obtained by the test is above 90%, the tested models can be considered to be statistically different; otherwise, they are considered to be the same. Starting with 2012 and 2013, the chronological assessment tests were performed year by year for consecutive years as follows: if the two periods were different, they would be modeled separately; if there was no significant difference between the two periods, they would be combined into one group; and then conduct likelihood ratio tests to validate the combination of the previously combined group and its subsequent consecutive years. Following these steps, we tested the yearly data sequentially, searching for a next-year combination with the current group until the end of the data were reached.

The formula for the difference analysis of any two periods is as follows:

$$X^2=-2[LL\left({\beta }_{t_2{t}_1}\right)-LL\left({\beta }_{t_1}\right)]$$

(10)

where $LL\left({\beta }_{t_2{t}_1}\right)$ is the log-likelihood of the model when using the significant variables of period t₂ and using the data of period t₁, $LL\left({\beta }_{t_1}\right)$ is the log-likelihood of the model taking the data from period t₁, using the same variables but not limited to the convergence parameters of period t₂. To ensure the completeness of the test, the periods t₁ and t₂ are exchanged, and the same calculation is performed again. The temporal stability test results for the pairwise periods are shown in

Table 5. Likelihood ratio test results between different periods (Angle crash).

t1	t2
	2012–2013	2014–2015	2016–2017	2018–2019
2012–2013	-	49.04 (13) [>99.99%]	52.19 (14) [>99.99%]	44.81 (12) [>99.99%]
2014–2015	61.98 (12) [>99.99%]	-	68.57 (14) [>99.99%]	68.41 (12) [>99.99%]
2016–2017	98.57 (12) [>99.99%]	34.79 (13) [>99.90%]	-	80.63 (12) [>99.99%]
2018–2019	31.68 (12) [>99.84%]	21.17 (13) [>93.03%]	54.48 (14) [>99.99%]	-

and

Table 6. Likelihood ratio test results between different periods (Non-angle crash).

t1	t2
	2012–2013	2014–2015	2016–2017	2018–2019
2012–2013	-	28.13 (13) [>99.13%]	61.40 (16) [>99.99%]	36.54 (11) [>99.98%]
2014–2015	32.35 (10) [>99.96%]	-	42.30 (16) [>99.96%]	23.07 (11) [>98.27%]
2016–2017	40.37 (10) [>99.99%]	25.02 (13) [>97.70%]	-	31.17 (11) [>99.89%]
2018–2019	20.71 (10) [>97.67%]	21.25 (13) [>93.19%]	24.16 (16) [>91.39%]	-

according to angle and non-angle crashes, respectively.

Subsequently, the temporal instabilities of injury severity in angle and non-angle crashes were clarified using the likelihood ratio test.

$$X_g^2=-2\left[LL\left({\mathit{\beta }}_{2012–2019g}\right)-LL\left({\mathit{\beta }}_{2012–2013g}\right)-LL\left({\mathit{\beta }}_{2014–2015g}\right)-LL\left({\mathit{\beta }}_{2016–2017g}\right)-LL\left({\mathit{\beta }}_{2018–2019g}\right)\right]$$

(11)

where $LL\left({\mathit{\beta }}_{2012–19g}\right)$ is the log-likelihood at the convergence of the corresponding model using the data $g$ for all angle crashes or all non-angle crashes for the period 2012–2019, $LL\left({\mathit{\beta }}_{2012–13g}\right)$ is the log-likelihood when the model converges using the crash data $g$ in 2012–2013, $LL\left({\mathit{\beta }}_{2014–15g}\right)$ is the log-likelihood when the model converges using the crash data $g$ in 2014–2015, $LL\left({\mathit{\beta }}_{2016–17g}\right)$ is the log-likelihood when the model converges using the crash data $g$ in 2016–2017, $LL\left({\mathit{\beta }}_{2018–19g}\right)$ is the log-likelihood when the model converges using the crash data $g$ in 2018–2019.

For angle crashes, the $X^2$ estimate is 90.58, corresponding to the $X^2$ distribution with 33 degrees of freedom. This result provides 99.99% confidence that the null hypothesis that the parameters are equal across four periods (2012–2013, 2014–2015, 2016–2017, and 2018–2019) can be rejected. For non-angle collision crashes, the $X^2$ estimate is 83.76, corresponding to an $X^2$ distribution with 26 degrees of freedom. This result also provides 99.99% confidence that the null hypothesis that the parameters are equal across four periods can be rejected. The test results show temporal instabilities in both angle and non-angle collisions. The explanatory variables of the two types of crashes also differ in their impact on injury severity.

6. Results and Discussion

Table A3. Model results of mixed logit with heterogeneity in means for angle crashes in Florida 2012–2013.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−0.59	−1.66
Random parameter (normally distributed)
Unpaved shoulder indicator (1 if the type of shoulder was unpaved, 0 otherwise) [MI]	2.05	1.90	−0.0432	0.0477	−0.0045
	(2.28)	(1.83)
Temporal characteristics
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise) [SI]	−1.27	−2.04	0.0096	0.0067	−0.0163
Environmental characteristics
Dark-Lighted indicator (1 if the light condition was dark-lighted, 0 otherwise) [MI]	−1.50	−3.38	0.0435	−0.0459	0.0024
Dark-Lighted indicator (1 if the light condition was dark-lighted, 0 otherwise) [SI]	−2.36	−3.20	0.0173	0.0037	−0.0210
Dark-Not lighted indicator (1 if the light condition was dark- not lighted, 0 otherwise) [MI]	2.33	2.53	−0.0118	0.0145	−0.0027
Cloudy indicator (1 if the weather was cloudy, 0 otherwise) [NI]	1.22	2.48	0.0260	−0.0153	−0.0107
Roadway characteristics
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [NI]	1.48	3.59	0.0488	−0.0314	−0.0174
Unpaved shoulder indicator (1 if the type of shoulder was unpaved, 0 otherwise) [NI]	1.75	1.90	0.0408	−0.0368	−0.0040
Spatial characteristics
Town indicator (1 if a crash occurred in town, 0 otherwise) [MI]	0.93	2.40	−0.0888	0.1223	−0.0335
Crash characteristics
Lane departure indicator (1 if lane departure involved in the crash, 0 otherwise) [NI]	1.22	2.19	0.0203	−0.0143	−0.0060
Number of observations			211
Number of estimated parameters			12
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−231.81
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−174.39
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.25
AIC			372.78
BIC			413.00

Note: NI = no injury; MI = minor injury; SI = severe injury.

,

Table A4. Model results of mixed logit with heterogeneity in means for angle crashes in Florida 2014–2015.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−0.51	−2.06
Random parameter (normally distributed)
Cloudy indicator (1 if the weather was cloudy, 0 otherwise) [MI]	−1.11	−1.98	0.0053	−0.0068	0.0015
	(2.45)	(2.72)
Heterogeneity in the mean of random parameter
Cloudy indicator (1 if the weather was cloudy, 0 otherwise) [MI]: Disabled driver indicator (1 if a disabled driver involved in the crash, 0 otherwise)	2.52	1.78
Temporal characteristics
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise) [SI]	−1.77	−2.74	0.0053	0.0053	−0.0106
Roadway characteristics
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [NI]	2.18	2.11	0.0711	−0.0679	−0.0032
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [MI]	2.19	2.08	−0.0679	0.0697	−0.0018
Curve left indicator (1 if road roadway alignment was curve left, 0 otherwise) [NI]	0.69	2.65	0.0281	−0.0187	−0.0094
Spatial characteristics
Town indicator (1 if a crash occurred in town, 0 otherwise) [MI]	−0.50	−1.84	0.0550	−0.0578	0.0028
Town indicator (1 if a crash occurred in town, 0 otherwise) [SI]	−2.81	−7.57	0.0353	0.0156	−0.0509
Crash characteristics
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [MI]	1.23	1.69	−0.0051	0.0057	−0.0006
Human characteristics
Cyclist indicator (1 if cyclists involved in the crash, 0 otherwise) [NI]	−4.07	−3.82	−0.0089	0.0075	0.0014
Disabled driver indicator (1 if a disabled driver involved in the crash, 0 otherwise) [NI]	1.16	2.40	0.0141	−0.0098	−0.0043
Number of observations			421
Number of estimated parameters			13
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−462.52
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−293.34
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.37
AIC			614.70
BIC			671.28

Note: NI = no injury; MI = minor injury; SI = severe injury.

,

Table A5. Model results of mixed logit with heterogeneity in means for angle crashes in Florida 2016–2017.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−3.67	−3.09
Random parameter (normally distributed)
Daylight indicator (1 if the light condition was daylight, 0 otherwise) [NI]	1.56	2.03	−0.0062	0.0104	−0.0042
	(2.03)	(1.91)
Heterogeneity in the mean of random parameter
Daylight indicator (1 if the light condition was daylight, 0 otherwise) [NI]: Merge indicator (1 if a crash occurred in the merging area, 0 otherwise)	−0.92	−1.95
Temporal characteristics
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise) [SI]	−2.09	−1.77	0.0023	0.0017	−0.0040
Environmental characteristics
Cloudy indicator (1 if the weather was cloudy, 0 otherwise) [SI]	2.35	2.18	−0.0065	−0.0028	0.0093
Roadway characteristics
Straight indicator (1 if road roadway alignment was straight, 0 otherwise) [NI]	0.84	2.04	0.0472	−0.0445	−0.0027
Straight indicator (1 if road roadway alignment was straight, 0 otherwise) [SI]	−1.40	−1.90	0.0046	0.0026	−0.0072
Dry surface indicator (1 if road surface condition was dry, 0 otherwise) [MI]	2.89	2.40	−0.3075	0.3275	−0.0200
Spatial characteristics
Town indicator (1 if a crash occurred in town, 0 otherwise) [SI]	−3.28	−4.44	0.0109	0.0068	−0.0177
Merge indicator (1 if a crash occurred in the merging area, 0 otherwise) [SI]	−4.21	−3.54	0.0047	0.0039	−0.0086
Crash characteristics
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [MI]	2.78	3.06	−0.0083	0.0107	−0.0024
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [SI]	3.54	2.41	−0.0018	−0.0031	0.0049
Human characteristics
Cyclist indicator (1 if cyclists involved in the crash, 0 otherwise) [NI]	−3.79	−3.57	−0.0149	0.0136	0.0013
Number of observations			479
Number of estimated parameters			14
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−526.24
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−259.37
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.51
AIC			546.74
BIC			605.14

Note: NI = no injury; MI = minor injury; SI = severe injury.

and

Table A6. Model results of mixed logit with heterogeneity in means for angle crashes in Florida 2018–2019.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−0.88	−3.15
Random parameter (normally distributed)
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise) [SI]	−3.23	−2.54	0.0044	0.0026	−0.0070
	(4.16)	(2.07)
Heterogeneity in the mean of random parameter
Weekend indicator (1 if the crash occurred during the weekend, 0 otherwise) [SI]:	4.32	2.18
Rain indicator (1 if the weather was rainy, 0 otherwise)
Environmental characteristics
Dark-Lighted indicator (1 if the light condition was dark-lighted, 0 otherwise) [NI]	1.25	3.06	0.0343	−0.0189	−0.0154
Rain indicator (1 if the weather was rainy, 0 otherwise) [NI]	2.20	2.08	0.0147	−0.0040	−0.0107
Roadway characteristics
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [NI]	4.13	3.08	0.2070	−0.1975	−0.0095
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [MI]	3.59	2.60	−0.1714	0.1738	−0.0024
Unpaved shoulder indicator (1 if the type of shoulder was unpaved, 0 otherwise) [NI]	0.89	1.69	0.0139	−0.0062	−0.0077
Straight indicator (1 if road roadway alignment was straight, 0 otherwise) [MI]	0.64	1.86	−0.0426	0.0560	−0.0134
Crash characteristics
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [SI]	4.48	2.57	−0.0078	−0.0018	0.0096
Human characteristics
Cyclist indicator (1 if cyclists involved in the crash, 0 otherwise) [MI]	2.03	2.73	−0.0107	0.0166	−0.0059
Number of observations			240
Number of estimated parameters			12
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−263.67
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−187.49
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.29
AIC			398.98
BIC			440.75

Note: NI = no injury; MI = minor injury; SI = severe injury.

show the model estimation results of angle collisions in 2012–2013, 2014–2015, 2016–2017, and 2018–2019, respectively. Four models fit well with ${\rho }^2$ values of 0.25, 0.37, 0.51, and 0.29, respectively. The results of non-angle collisions are displayed in

Table A7. Model results of mixed logit with heterogeneity in means for non-angle crashes in Florida 2012–2013.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−2.09	−3.42
Random parameter (normally distributed)
Constant [SI]	−4.17	−6.43
(Standard deviation of parameter distribution)	(4.02)	(2.81)
Environmental characteristics
Fog indicator (1 if the weather was foggy, 0 otherwise) [NI]	−2.03	−1.73	−0.0024	0.0019	0.0005
Roadway characteristics
Unpaved shoulder indicator (1 if the type of shoulder was unpaved, 0 otherwise) [SI]	0.92	2.12	−0.0078	−0.0036	0.0114
Dry surface indicator (1 if road surface condition was dry, 0 otherwise) [NI]	−1.02	−1.73	−0.2050	0.1787	0.0263
Spatial characteristics
Town indicator (1 if a crash occurred in town, 0 otherwise) [MI]	0.38	1.77	−0.0510	0.0543	−0.0033
Vehicle characteristics
Commercial vehicle indicator (1 if commercial vehicles involved in the crash, 0 otherwise) [NI]	2.52	3.46	0.0076	−0.0068	−0.0008
Crash characteristics
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [SI]	1.74	3.61	−0.0108	−0.0047	0.0155
Human characteristics
Cyclist indicator (1 if cyclists involved in the crash, 0 otherwise) [NI]	−1.45	−2.59	−0.0076	0.0068	0.0008
Number of observations			631
Number of estimated parameters			10
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−693.22
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−447.59
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.35
AIC			915.18
BIC			959.65

Note: NI = no injury; MI = minor injury; SI = severe injury.

,

Table A8. Model results of mixed logit with heterogeneity in means for non-angle crashes in Florida 2014–2015.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [SI]	−2.91	−4.67
Random parameter (normally distributed)
Daylight indicator (1 if the light condition was daylight, 0 otherwise) [NI]	0.92	1.69	−0.0151	0.0111	0.0040
	(3.06)	(2.00)
Heterogeneity in the mean of random parameter
Daylight indicator (1 if the light condition was daylight, 0 otherwise) [NI]: Cyclist indicator (1 if cyclists involved in the crash, 0 otherwise)	−5.87	−2.31
Temporal characteristics
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise) [NI]	0.59	2.34	0.0200	−0.0175	−0.0025
Environmental characteristics
Clear indicator (1 if the weather was clear, 0 otherwise) [NI]	−1.24	−1.82	−0.1312	0.1104	0.0208
Clear indicator (1 if the weather was clear, 0 otherwise) [MI]	−1.52	−2.47	0.1493	−0.1722	0.0229
Roadway characteristics
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [NI]	0.38	2.05	0.0157	−0.0136	−0.0021
Spatial characteristics
Town indicator (1 if a crash occurred in town, 0 otherwise) [NI]	0.49	2.15	0.0382	−0.0339	−0.0043
Town indicator (1 if a crash occurred in town, 0 otherwise) [SI]	−0.71	−2.05	0.0081	0.0063	−0.0144
Vehicle characteristics
Commercial vehicle indicator (1 if commercial vehicles involved in the crash, 0 otherwise) [NI]	2.12	2.94	0.0104	−0.0054	−0.0050
Commercial vehicle indicator (1 if commercial vehicles involved in the crash, 0 otherwise) [SI]	2.07	2.39	−0.0031	−0.0016	0.0047
Crash characteristics
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [SI]	0.12	2.12	−0.0029	−0.0016	0.0045
Number of observations			1006
Number of estimated parameters			13
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−1105.20
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−687.43
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.38
AIC			1400.86
BIC			1464.74

Note: NI = no injury; MI = minor injury; SI = severe injury.

,

Table A9. Model results of mixed logit with heterogeneity in means for non-angle crashes in Florida 2016–2017.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−0.92	−7.25
Random parameter (normally distributed)
Town indicator (1 if a crash occurred in town, 0 otherwise) [SI]	−8.17	−1.77	−0.0032	−0.0012	0.0044
	(4.58)	(1.86)
Heterogeneity in the mean of random parameter
Town indicator (1 if the crash occurred in town, 0 otherwise) [SI]:  Speeding indicator (1 if speeding involved in the crash, 0 otherwise)	5.18	1.84
Temporal characteristics
Weekend indicator (1 if a crash occurred during the weekend, 0 otherwise) [SI]	−1.15	−3.01	0.0060	0.0025	−0.0085
Environmental characteristics
Daylight indicator (1 if the light condition was daylight, 0 otherwise) [MI]	−0.27	−1.79	0.0229	−0.0236	0.0007
Daylight indicator (1 if the light condition was daylight, 0 otherwise) [SI]	−1.94	−5.91	0.0147	0.0052	−0.0199
Dark-Not Lighted indicator (1 if the light condition was dark-not lighted, 0 otherwise) [NI]	0.62	2.05	0.0067	−0.0047	−0.0020
Cloudy indicator (1 if the weather was cloudy, 0 otherwise) [SI]	−1.60	−2.14	0.0022	0.0009	−0.0031
Rain indicator (1 if the weather was rain, 0 otherwise) [SI]	−3.97	−1.65	0.0009	0.0005	−0.0014
Roadway characteristics
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [MI]	−0.35	−2.21	0.0178	−0.0182	0.0004
Paved shoulder indicator (1 if the type of shoulder was paved, 0 otherwise) [SI]	−1.74	−3.30	0.0071	0.0022	−0.0093
Curve left indicator (1 if road roadway alignment was curve left, 0 otherwise) [MI]	0.42	2.49	−0.0169	0.0181	−0.0012
Vehicle characteristics
Commercial vehicle indicator (1 if commercial vehicles involved in the crash, 0 otherwise) [MI]	−1.10	−2.51	0.0674	−0.0994	0.0320
Crash characteristics
Speeding indicator (1 if speeding involved in the crash, 0 otherwise) [NI]	−0.57	−1.98	−0.0072	0.0061	0.0011
Human characteristics
Cyclist indicator (1 if cyclists involved in the crash, 0 otherwise) [NI]	−2.58	−5.02	−0.0089	0.0086	0.0003
Number of observations			1127
Number of estimated parameters			16
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−1238.14
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−759.74
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.39
AIC			1551.48
BIC			1631.92

Note: NI = no injury; MI = minor injury; SI = severe injury.

and

Table A10. Model results of mixed logit with heterogeneity in means for non-angle crashes in Florida 2018–2019.

Variable	Parameter Estimates	z-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [MI]	−1.25	−9.68
Constant [SI]	−3.21	−7.14
Random parameter (normally distributed)
Cyclist indicator	−2.11	−2.92	−0.0079	0.0071	0.0008
(1 if cyclists involved in the crash, 0 otherwise) [NI]	(2.87)	(3.09)
Curve left indicator	−0.38	−1.74	−0.0215	0.0186	0.0029
(1 if road roadway alignment was curve left, 0 otherwise) [NI]	(2.55)	(1.82)
Temporal characteristics
Weekend indicator  (1 if a crash occurred during the weekend, 0 otherwise) [SI]	1.21	2.48	−0.0152	−0.0058	0.0210
Spatial characteristics
Town indicator  (1 if a crash occurred in town, 0 otherwise) [SI]	−0.96	−1.97	0.0113	0.0046	−0.0159
Vehicle characteristics
Commercial vehicle indicator  (1 if commercial vehicles involved in the crash, 0 otherwise) [MI]	−1.72	−2.29	0.2132	−0.2377	0.0245
Crash characteristics
Aggressive driving indicator  (1 if aggressive driving involved in the crash, 0 otherwise) [MI]	1.54	3.43	−0.0144	0.0174	−0.0030
Aggressive driving indicator  (1 if aggressive driving involved in the crash, 0 otherwise) [SI]	1.77	2.07	−0.0023	−0.0034	0.0057
Number of observations			524
Number of estimated parameters			11
Log-likelihood at zero $\left(LL\left(0\right)\right)$			−575.67
Log-likelihood at convergence $\left(LL\left(\beta \right)\right)$			−344.63
${\rho }^2=1-LL\left(\beta \right)/LL\left(0\right)$			0.40
AIC			711.26
BIC			758.13

Note: NI = no injury; MI = minor injury; SI = severe injury.

, which also have considerable statistical fit effects with ${\rho }^2$ values of 0.35, 0.38, 0.39, and 0.40, respectively.

The factors affecting the injury severity in each model are shown in Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9 and Table A10 in Appendix A, involving time, environment, road, space, vehicle, crash, and human characteristics. To compare possible differences, the marginal effects of all statistically significant variables in angle and non-angle crashes are presented in

Table A11. Comparison of marginal effects between angle and non-angle crashes over the years (marginal effects for non-angle crashes in italics).

Variable	No Injury				Minor Injury				Severe Injury
	2012–2013	2014–2015	2016–2017	2018–2019	2012–2013	2014–2015	2016–2017	2018–2019	2012–2013	2014–2015	2016–2017	2018–2019
Temporal characteristics
Weekend indicator	0.0096	0.0053	0.0023	0.0044	0.0067	0.0053	0.0017	0.0026	−0.0163	−0.0106	−0.0040	−0.0070
(1 if the crash occurred during the weekend, 0 otherwise)	-	0.0200	0.0060	−0.0152	-	−0.0175	0.0025	−0.0058	-	−0.0025	−0.0085	0.0210
Environmental characteristics
Daylight indicator	-	-	−0.0062	-	-	-	0.0104	-	-	-	−0.0042	-
(1 if the light condition was daylight, 0 otherwise)	-	−0.0151	0.0376	-	-	0.0111	−0.0184	-	-	0.0040	−0.0192	-
Dark-lighted indicator	0.0608	-	-	0.0343	−0.0422	-	-	−0.0189	−0.0186	-	-	−0.0154
(1 if the light condition was dark-lighted, 0 otherwise)	-	-	-	-	-	-	-	-	-	-	-	-
Dark-Not Lighted indicator	−0.0118	-	-	-	0.0145	-	-	-	−0.0027	-	-	-
(1 if the light condition was dark-not lighted, 0 otherwise)	-	-	0.0067	-	-	-	−0.0047	-	-	-	−0.0020	-
Clear indicator	-	-	-	-	-	-	-	-	-	-	-	-
(1 if the weather was clear, 0 otherwise)	-	0.0181	-	-	-	−0.0618	-	-	-	0.0437	-	-
Cloudy indicator	0.0260	0.0053	−0.0065	-	−0.0153	−0.0068	−0.0028	-	−0.0107	0.0015	0.0093	-
(1 if the weather was cloudy, 0 otherwise)	-	-	0.0022	-	-	-	0.0009	-	-	-	−0.0031	-
Rain indicator	-	-	-	0.0147	-	-	-	−0.0040	-	-	-	−0.0107
(1 if the weather was rainy, 0 otherwise)	-	-	0.0009	-	-	-	0.0005	-	-	-	−0.0014	-
Fog indicator	-	-	-	-	-	-	-	-	-	-	-	-
(1 if the weather was foggy, 0 otherwise)	−0.0024	-	-	-	0.0019	-	-	-	0.0005	-	-	-
Roadway characteristics
Paved shoulder indicator	0.0488	0.0032	-	0.0356	−0.0314	0.0018	-	−0.0237	−0.0174	−0.0050	-	−0.0119
(1 if the type of shoulder was paved, 0 otherwise)	-	0.0157	0.0249	-	-	−0.0136	−0.0160	-	-	−0.0021	−0.0089	-
Unpaved shoulder indicator	−0.0024	-	-	0.0139	0.0109	-	-	−0.0062	−0.0085	-	-	−0.0077
(1 if the type of shoulder was unpaved, 0 otherwise)	−0.0078	-	-	-	-0.0036	-	-	-	0.0114	-	-	-
Dry surface indicator	-	-	−0.3075	-	-	-	0.3275	-	-	-	−0.0200	-
(1 if road surface condition was dry, 0 otherwise)	−0.2050	-	-	-	0.1787	-	-	-	0.0263	-	-	-
Straight indicator	-	-	0.0518	−0.0426	-	-	−0.0419	0.0560	-	-	−0.0099	−0.0134
(1 if roadway alignment was straight, 0 otherwise)	-	-	-	-	-	-	-	-	-	-	-	-
Curve left indicator	-	0.0281	-	-	-	−0.0187	-	-	-	−0.0094	-	-
(1 if roadway alignment was curve left, 0 otherwise)	-	-	−0.0169	−0.0215	-	-	0.0181	0.0186	-	-	−0.0012	0.0029
Spatial characteristics
Town indicator	−0.0888	0.0903	0.0109	-	0.1223	−0.0422	0.0068	-	−0.0335	−0.0481	−0.0177	-
(1 if a crash occurred in town, 0 otherwise)	−0.0510	0.0463	−0.0032	0.0113	0.0543	−0.0276	−0.0012	0.0046	−0.0033	−0.0187	0.0044	−0.0159
Merge indicator	-	-	0.0047	-	-	-	0.0039	-	-	-	−0.0086	-
(1 if the crash occurred in a merging area, 0 otherwise)	-	-	-	-	-	-	-	-	-	-	-	-
Vehicle characteristics
Commercial vehicle indicator	-	-	-	-	-	-	-	-	-	-	-	-
(1 if commercial vehicles involved in the crash, 0 otherwise)	0.0076	0.0073	0.0674	0.2132	−0.0068	−0.0070	−0.0994	−0.2377	−0.0008	−0.0003	0.0320	0.0245
Crash characteristics
Lane departure indicator	0.0203	-	-	-	−0.0143	-	-	-	−0.0060	-	-	-
(1 if lane departure involved in the crash, 0 otherwise)	-	-	-	-	-	-	-	-	-	-	-	-
Speeding indicator	-	−0.0051	−0.0101	−0.0078	-	0.0057	0.0076	−0.0018	-	−0.0006	0.0025	0.0096
(1 if speeding involved in the crash, 0 otherwise)	−0.0108	−0.0029	−0.0072	-	−0.0047	−0.0016	0.0061	-	0.0155	0.0045	0.0011	-
Aggressive driving indicator	-	-	-	-	-	-	-	-	-	-	-	-
(1 if aggressive driving involved in the crash, 0 otherwise)	-	-	-	−0.0167	-	-	-	0.0140	-	-	-	0.0027
Human characteristics
Disabled driver indicator	-	0.0141	-	-	-	−0.0098	-	-	-	−0.0043	-	-
(1 if a disabled driver involved in the crash, 0 otherwise)	-	-	-	-	-	-	-	-	-	-	-	-
Cyclist indicator	-	−0.0089	−0.0149	−0.0107	-	0.0075	0.0136	0.0166	-	0.0014	0.0013	−0.0059
(1 if cyclists involved in the crash, 0 otherwise)	−0.0076	-	−0.0089	−0.0079	0.0068	-	0.0086	0.0071	0.0008	-	0.0003	0.0008

. It is worth noting that several factors were found to have impacts on crash injury severity in all periods. For non-angle collisions, the town and commercial vehicle indicators were statistically significant across all periods. In contrast, only the weekend indicator in angle collisions had sustained impacts over the four periods. Moreover, unique influencing factors were identified in angle and non-angle crash models, respectively. Interestingly, the same indicator may have opposite effects on collisions of different modes. These statistically significant variables are analyzed in detail below.

6.1. Analysis of the Characteristics’ Effects

6.1.1. Temporal Characteristics

As shown in Table A11, the weekend indicator had a significant effect on both angle and non-angle collisions for almost all the analyzed periods (excluding non-angle collisions in 2012–2013). Regarding angle collisions, a time-stable effect of this variable was found. Specifically, crashes occurring on weekends increased the likelihood of no injury or minor injury while decreasing the likelihood of severe injury. Nevertheless, previous research has shown that weekends are more likely to lead to severe crashes [53]. This discrepancy may be since the areas where roundabouts are located typically have lower traffic volumes, especially on weekends. More specifically, there are fewer commuting vehicles, and drivers prefer steady and slow speeds since they are more relaxed on weekends. As a result, weekend crashes in roundabouts are less likely to cause serious injuries.

For non-angle collisions, there was considerable temporal instability in the impacts of the weekend indicator. Nonetheless, according to the marginal effects in Table A11, non-angle collisions occurring over the weekend showed an overall trend toward more serious injuries over time. Between 2012 and 2019, Florida’s total number of tourists increased annually [54]. Especially on weekends, the enormous number of tourists who are unfamiliar with the road will raise the driving risks at the roundabout.

6.1.2. Environmental Characteristics

Environmental characteristics that were statistically significant include daylight, dark-lighted, dark-not-lighted, clear, cloudy, rain, and fog. Particularly, some variables only impact a single collision mode (either angle or non-angle collisions) at specific periods. For instance, the dark-lighted indicator was associated with reducing the injury severity of angle collisions in 2012–2013 and 2018–2019, while the indicator was not statistically significant for non-angle collision models.

As shown in Table A11, both the clear indicator in 2014–2015 and the fog indicator in 2012–2013 increased the probability of severe injury in non-angle collisions (0.0437 and 0.0005, respectively). Similar to findings from previous studies [55], increased risky driving behavior in clear weather may be responsible for the higher likelihood of severe injury. However, neither of the indicators (fog or clear) had a significant effect on angle collisions. In general, it appears that angle collisions tend to be affected by lighting characteristics, while non-angle collisions are more sensitive to weather characteristics. One of the reasons may be that lighting conditions directly affect the driver’s field of view [56], which is particularly pronounced for angle collisions.

The daylight indicator was positively correlated with minor injury (0.0104, see Table A11) while negatively correlated with no injury (−0.0062) and severe injury (−0.0042) for angle collisions in the 2016–2017 model. Turning to the impact of daylight on non-angle collisions, injury severity increased in 2014–2015 and reversed in 2016–2017, indicating significant temporal instability. The marginal effects in Table A11 revealed that crashes occurring in dark, not-lighted environments are less likely to result in severe injury. This finding contradicts previous conclusions in general road sections that dark, not-lighted areas can exacerbate the severity of crash injuries [57]. Given this difference, a reasonable understanding is that roundabouts involve more complex geometric dimensions and require greater driver concentration than straight lanes [58]. In this complex traffic environment, ‘dark-not lighted’ as an additional disadvantage might make the driver more focused and cautious. Therefore, ‘dark-not-lighted’ plays a role in reducing crash injury severity at roundabouts.

As for ‘cloudy’, it had significant impacts on the probabilities of injury severity outcomes. For angle collisions, inconsistent effects of cloudy weather were observed in 2012–2013, 2014–2015, and 2016–2017, whereas the overall trend showed an increase in the chance of severe crashes over time. Perhaps it can be attributed to the fact that drivers gradually adjusted to the roundabout and improved their driving skills, mentally reducing vigilance against the poor external environment and causing more severe crashes. Furthermore, in the 2016–2017 model, cloudy weather affects the injury severity of both angle and non-angle collisions. However, angle collisions had a higher probability of severe injury and a lower probability of no injury or minor injury, while non-angle collisions were just the opposite. The finding suggests that collision modes may be related to driver heterogeneity, such as diverse responses to environmental factors. Under the same weather conditions, some drivers feel pleasant while others suffer discomfort, which leads to different driving behaviors and varying injury severity in collisions. Previous studies have found that cloudy weather is beneficial for reducing injury severity [11], which is consistent with the effect on non-angle collisions in this study.

Rain was a non-negligible factor for both modes of collisions at roundabouts, although the effect was only statistically significant in some periods. As shown in Table A11, positive marginal effects for no injury and negative marginal effects for severe injury resulted in fewer serious crashes. It is inconsistent with previous research [12]. Discrepancies between roundabouts and other road segments may explain the inconsistent finding. For basic road segments, drivers are usually in a relatively relaxed driving state and keep a higher speed, making it easy to cause serious injuries on rainy days. In contrast, the roundabout can catch more driver attention and elevate their alertness because of the special geometry and complex traffic [58]. Hence, drivers are more likely to slow down and adopt more conservative driving behavior when entering the roundabout on rainy days. This could explain why injuries in roundabouts are less severe on rainy days.

6.1.3. Roadway Characteristics

Road characteristics showed significant differences across periods and collision modes. Considering the shoulder type, the paved shoulder was found to be statistically significant for non-angle collision models in 2014–2015 and 2016–2017, which steadily reduced the likelihood of minor and severe injuries while increasing the possibility of no injury. Concerning the angle collisions, the same indicator had similar effects on all models except 2016–2017. During the analysis periods, the paved shoulder contributed to a lower probability of severe injuries and a higher probability of no injuries (see Table A11). However, unstable effects were observed for minor injuries. In addition, the unpaved shoulder significantly affected the injury severity of angle and non-angle collisions but did not produce stable results. It can be seen that a paved shoulder is beneficial in reducing the injury severity of crashes compared to an unpaved shoulder. The possible reason is that the paved shoulder provides a broader and more reliable safety redundancy zone for vehicles entering and exiting the roundabout. In contrast, Gong and Fan [43] reported that the unpaved shoulder had varied influences on the injury severity for drivers of different ages involved in a run-off-road collision. This suggests a strong distinction in the role of shoulder type in different sorts of crashes, which has not been discussed in detail in previous studies.

Regarding surface indicators, Table A11 showed that a dry surface was associated with lowering the probability of no injury and increasing the probability of minor injury for two collision modes. In contrast, the marginal effects of severe injury were inconsistent, with a lower probability for angle collisions (−0.0200) and a higher probability for non-angle collisions (0.0263). Interestingly, research findings on the effect of the dry surface on injury severity are inconsistent [44,59]. Driving around roundabouts requires more steering due to circular lanes and counter-clockwise driving rules, which can potentially lead to accidents on slippery surfaces, such as sideslips. In contrast, the dry surface has higher ground friction, improving the effectiveness of emergency braking and reducing the risk of collision during steering. Consequently, the dry surface indicator could reduce the frequency and severity of angle collisions, which are prone to occur during steering maneuvers.

Straight road alignment only affected angle collisions, which reduced the probability of severe injury in both 2016–2017 and 2018–2019 (−0.0099 and −0.0134, respectively). For no injury and minor injury, the straight indicator was found to be temporally unstable during the two periods (see Table A11). The left-curved road alignment influenced the injury severity outcomes for both angle and non-angle collisions. In the 2014–2015 angle collision model, ‘curved left’ resulted in more no injury, less minor injury, and more severe injury (0.0281, −0.0187, and −0.0094, respectively). However, non-angle collisions occurring on the left curved road had a lower probability of no injuries and a higher probability of minor injuries, while the marginal effect of severe injuries was unstable across 2016–2017 and 2018–2019.

Roads within the roundabout have different alignments, resulting in differentiated driving behaviors. There are straight roadways before entering or just after exiting the roundabout. Roads in the intersection area curved right, while loops within the roundabout curved left. Curved right is consistent with regular driving steering habits, which may have less impact on injury severity. In contrast, straight roads can reduce injury severity due to the simple traffic environment and operational requirements. Similar results can be found in Al-Bdairi and Hernandez [42] and Yu et al. [39]. The curved left road makes drivers more cautious because of frequent steering operations when driving around the loop; however, not all drivers are alert to the situation. Therefore, the impact of the curved left indicator on injury severity varied.

6.1.4. Spatial Characteristics

The model estimated results showed that, compared with rural areas, crashes that occurred in towns played a significant role in injury severity outcomes. Interestingly, the town indicator had a complex effect on the two collision modes over the four periods. For example, the effects on angle and non-angle collisions were the same in 2012–2013, as well as in 2014–2015. There was a stable negative correlation between the town indicator and severe injuries during the two periods. However, opposite effects were observed for angle and non-angle collisions in 2016–2017. Angle collisions in town had a lower probability of severe injury and a higher probability of no injury and minor injury, as opposed to non-angle collisions. One probable explanation is that, compared to drivers in rural areas, town drivers with higher levels of education adhere to roundabout driving rules and are more modest to each other when approaching or departing roundabouts, hence preventing severe angle collisions. From the perspective of all analyzed periods, the town indicator was temporally unstable, which may be related to the town’s nature [21]. Towns have more vehicles and better traffic management than rural areas. Orderly traffic management in towns may reduce the risk of severe injury. However, due to volatile elements such as traffic flow in different periods, the effects still demonstrate temporal instability.

Another indicator, merging area, was found to be associated with a decrease in the probability of severe injury (−0.0086) and an increase in the probability of minor injury (0.0039) only for the 2016–2017 angle collision model, which may be attributed to the slower speed when driving in the merging area.

6.1.5. Vehicle Characteristics

The commercial vehicle indicator had a significant impact on non-angle collisions in all periods but did not affect angle collisions. More specifically, there was a stable positive correlation with no injuries and a stable negative correlation with minor injuries. However, the indicator showed unstable impacts on severe injury, as evidenced by negative coefficients for 2012–2013 and 2014–2015 and positive coefficients for 2016–2017 and 2018–2019. The results indicated that the injury severity of crashes involving commercial vehicles in roundabouts has increased in recent years. This may be due to the growing demands on the efficiency of commercial vehicle transportation in a competitive economic climate, which has led to higher speeds and more severe collisions involving commercial vehicles. The findings on the commercial vehicle are similar to the results of a previous study [21]. Notably, commercial vehicle involvement only affected injury severity in non-angle collisions. This may be attributed to the fact that drivers are less likely to perform cut-in maneuvers when the other vehicle is a large commercial vehicle, which affects the probability of angle crashes. Therefore, the commercial vehicle indicator only contributes to different injury severity outcomes for non-angle collisions, while no significant impacts were found for angle crashes.

6.1.6. Crash Characteristics

Among the crash-related factors, several variables were found to significantly affect the injury severity in roundabout crashes. As shown in Table A11, the lane departure indicator was less likely to result in severe injury from angle crashes during 2012–2013, with probabilities of 0.0203, −0.0143, and −0.0060 for no injury, minor injury, and severe injury, respectively. However, the same indicator did not have a significant impact on non-angle collisions. Unlike driving in the wrong lane, lane departure can be observed intuitively without knowing the driver’s true driving intentions. Lane departure makes surrounding drivers alert, thereby reducing the crash injury severity. Previous research [24] has also revealed that only specific lane departures, such as left-side deviations, are associated with severe injuries.

In addition, speeding had a persistent negative effect on injuries in angle and non-angle collisions. According to the results in Table A11, speeding was more and more likely to cause severe injuries in angle collisions over time. Similarly, the same indicator was consistently associated with a higher probability of severe injury in non-angle collisions. The findings were supported by Waseem et al. [60], who argued that speeding is associated with more serious crash injuries. Therefore, it is necessary for traffic managers to take measures such as speed limits in roundabouts to prevent severe crashes caused by speeding.

Turning to aggressive driving, the indicator increased injury severity in non-angle collisions (−0.0167, 0.0140, and 0.0027 for no injury, minor injury, and severe injury, respectively), which is consistent with previous studies [61]. In addition, it is worth noting that aggressive driving did not significantly impact the injury severity outcomes of angle crashes.

6.1.7. Human Characteristics

Crashes involving disabled drivers were less likely to result in severe injuries (0.0141, −0.0098, and −0.0043 for no injury, minor injury, and severe injury, respectively). Understandably, drivers with disabilities may be more conservative than normal drivers, especially in roundabouts with complex traffic situations. According to the characteristics of different collision modes, angle collisions also usually occur in complex traffic involving frequent interactions, such as entering or exiting roundabouts. Therefore, the disabled driver indicator is more prominent in reducing the injury severity of angle collisions.

Regardless of the angle or non-angle collisions, the cyclist indicator steadily reduced the probability of no injury and increased the probability of a minor injury. In terms of severe injury, the effects of the cyclists showed slight temporal instability as the 2018–2019 angle collision model produced a negative effect (−0.0059, as shown in Table A11) that was different from other periods. In general, cyclist participation significantly increased the injury severity at roundabouts, as mentioned in previous studies [62]. This study further highlights that this variable is non-negligible for both angle and non-angle collisions.

6.1.8. Heterogeneity in Means and Variances of Random Parameters

Each model had variables identified as statistically significant and normally distributed random parameters, as shown in Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10 and Table A11. And all explanatory variables were checked for potential heterogeneity by means of random parameters. However, no statistically significant heterogeneity in variance was observed for any of the models.

For the angle collision models of 2012–2013, 2014–2015, 2016–2017, and 2018–2019, the corresponding variables of random parameters were the unpaved shoulder of minor injury, cloudy of minor injury, daylight of no injury, and the weekend indicator of severe injury, respectively. However, no statistically significant heterogeneity in mean or variance was observed for the unpaved shoulder indicator in 2012–2013 (Table A3). In the model of 2014–2015 (Table A4), the indicator of cloudy produced heterogeneity in the mean. When a disabled driver was involved in the crash, the mean of the cloudy increased, indicating a rise in the probability of minor injury. As indicated in Table A5, there was statistically significant heterogeneity in the mean of daylight specific to no injury in 2016–2017. In addition, the indicator merge decreased the mean of the random parameter, which means no injury is less likely for the crash that occurred in the merging area. For the 2018–2019 model (Table A6), the rainy indicator increased the mean of heterogeneity on the weekend of severe injury, raising the chance of severe injury.

Regarding the 2012–2013 non-angle collision model (Table A7), the constant term specific to severe injury was observed to be a random parameter. There was no significant heterogeneity in the mean or variance of this parameter. In the 2014–2015 model (Table A8), the daylight indicator of no injury produced a statistically significant random parameter with heterogeneity in the mean. Notably, the cyclist indicator had an influence on decreasing the mean of the parameter, suggesting a lower probability of no injury. As for the 2016–2017 model (Table A9), the town indicator of severe injury was identified as a random parameter. The results exhibited that speeding was related to an increased mean of the town indicator, resulting in a higher probability of severe injury. In particular, both the cyclist indicator and the curved left indicator specific to no injury had statistically significant random parameters in the 2018–2019 model (Table A10). However, neither was found to have heterogeneity in mean or variance.

6.1.9. Temporal Instability

Both angle and non-angle crashes showed temporal instabilities during the analysis periods. As Mannering [22] indicated, the causes of instability are diverse, particularly those involving changes in driver behavior. On the one hand, changes in the external environment (road, weather, etc.) can impact driver behavior. On the other hand, injury severity changes as drivers become more experienced and familiar with the road. Of course, part of the instability may come from accidental factors such as deviations in data records.

6.2. Predictive Comparisons (Prediction Simulations)

The summary of marginal effects in Table A11 showed a significant shift in the impacts of explanatory variables on injury severity across periods. Furthermore, it would be interesting to understand the effect of observed changes on the probability of injury severity. For example, use the parameters estimated from the model based on 2012–2013 to estimate the probability of a crash injury in 2018–2019 and compare the differences between predicted results and actual observations. Out-of-sample prediction is an effective way to achieve the goal [63]. It is important to note that variance must be considered for random parameter models. Otherwise, it will lead to biased estimates [31].

This study used different periods as the base year for prediction.

Table 7. Prediction results of angle and non-angle crashes for no injury, minor injury (in parentheses), and severe injury (in brackets).

Base Year	Target Year
	2014–2015		2016–2017		2018–2019
	Angle Crash	Non-Angle Crash	Angle Crash	Non-Angle Crash	Angle Crash	Non-Angle Crash
2012–2013	−0.0147	−0.0203	−0.0155	−0.0210	−0.0489	−0.0340
	(0.0069)	(0.0214)	(0.0062)	(0.0383)	(0.0721)	(0.0300)
	[0.0078]	[−0.0011]	[0.0093]	[−0.0173]	[−0.0232]	[0.0040]
2014–2015	-	-	−0.0078	−0.0013	0.0088	−0.0099
	-	-	(0.0026)	(0.0175)	(0.0016)	(0.0051)
	-	-	[0.0052]	[−0.0162]	[−0.0104]	[0.0048]
2016–2017	-	-	-	-	0.0149	−0.0098
	-	-	-	-	(0.0021)	(−0.0090)
	-	-	-	-	[−0.0170]	[0.0188]

Note: The results in the table are the difference between predicted and ‘observed’ probabilities.

presents the results of the comparative analysis. The outcomes indicated that the probabilities of minor and severe injuries were overestimated when using the angle collision parameters of 2012–2013 to predict the injury probabilities of 2014–2015 and 2016–2017. However, severe injuries were underestimated by 0.0232 in the 2018–2019 prediction model. Regarding the out-of-sample prediction based on the 2014–2015 model, the probability of severe injuries was overestimated, while no injuries were underestimated for target periods 2016–2017. The predicted results for 2018–2019 were just the opposite. As for the 2018–2019 prediction model based on 2016–2017, the probability of severe injury was underestimated by 0.0170, while the probability of no and minor injuries was overestimated by 0.0149 and 0.0021, respectively. That is, if the explanatory variables were the same for every crash that occurred in 2018–2019, the 2016–2017 estimated parameter predicted crash injury severity would be less severe than it actually was.

Figure 3 exemplifies the distribution of predicted probability differences for non-angle collisions (2012–2013 model for 2014–2015 prediction and 2014–2015 model for 2016–2017 prediction). According to Table 7, the model based on 2012–2013 underestimated the severe injury and no injury while overestimating the minor injury for target periods 2014–2015 and 2016–2017. In contrast, the probability of severe injury was overestimated by 0.0040 in the predicted period of 2018–2019. The estimated results of the 2014–2015 model for the subsequent two target periods were different. Severe injuries were underestimated in the 2016–2017 period (−0.0162), while the opposite was true in the 2018–2019 period (0.0048). The 2016–2017 model underestimated no injuries and minor injuries in the target period prediction (−0.0098, −0.0090) but overestimated the likelihood of severe injuries by 0.0188. Overall, the results of the predictive simulations provide strong support for the temporal instability of angle and non-angle collisions.

The differences in temporal shifts between the beginning and ending years are of particular interest [64]. Comparing 2012–2013 to 2018–2019, the distribution of injury severity for both angle and non-angle collisions showed a shift from minor injuries to no injuries and severe injuries (Angle collisions: no injury from 68.72% to 75.41%, minor injury from 28.91% to 21.67%, severe injury from 2.37% to 2.92%; non-angle collisions: no injury from 65.67% to 71.94%, minor injury from 30.05% to 23.62%, severe injury from 4.28% to 4.44%). For angle collisions, the predicted comparison trends of injury severity outcomes are consistent with the shift in statistical proportions. However, the two shifts are not exactly the same for non-angle collisions. As shown in Table 7, the 2012–2013 non-angle collision model predicted a decrease in no injury and an increase in minor and severe injuries relative to the 2018–2019 ‘observed’ values. Both the temporal shift of the parameters and the specific characteristics may play a role in the temporal shift affecting injury severity, which deserves the attention of future studies.

7. Summary and Conclusions

To investigate the heterogeneity and temporal instability of factors affecting the injury severity for roundabout angle and non-angle collisions, this study applied the 2012–2019 Florida roundabout crash data to estimate the random parameter logit models with heterogeneity in means and variances. With possible injury severity outcomes of no injury, minor injury, and severe injury, several potential factors are considered in the analysis, including time, environment, roadways, spaces, vehicles, crashes, and humans.

Overall temporal instability was captured by a series of likelihood ratio tests, whereas some variables still exhibited relative temporal stability. For instance, weekend and dark-lighted were found to be consistently associated with lower severe injury probability in angle collisions, while the paved shoulder and cyclist indicators had a stable effect on non-angle collisions. The findings can provide valuable insights for decision-makers to specify effective strategies from a long-term perspective. This paper also reveals the non-transferability between angle and non-angle crashes. Some factors were only statistically significant in the non-angle crash model, including clear, fog, commercial vehicles, and aggressive driving. In contrast, there are several contributing factors specific to the injury severity of angle crashes, such as dark-lighted, straight alignment, merge, lane departure, and disabled drivers. It is worth noting that certain indicators, cloudy and town, had opposite effects on the injury severity of angle and non-angle collisions even within the same period.

According to the divergence, some differentiated countermeasures should be developed targeting different stakeholders. First, a straight roadway is less likely to cause serious injuries in angle crashes. In contrast, it is necessary for traffic managers to place signs on the curved roadway after entering the roundabout to raise the driver’s awareness of driving carefully in this relatively unsafe situation. Furthermore, it is crucial to enhance the supervision of roundabouts in rural areas, while effective administration in town requires specialized strategies considering the specific traffic volume and local characteristics. Road engineering professionals are advised to prioritize the installation of shoulders over curbs, particularly emphasizing the implementation of paved shoulders to improve safety at roundabouts. In addition, dark-lighted crashes were found to be positively associated with no-injury-angle crashes, indicating a favorable effect of reducing injury severity. Therefore, lighting infrastructure should be regularly maintained or upgraded in roundabouts where natural light is poor. The findings of this study also provide valuable perspectives for distinct groups of drivers. The importance of enhancing skills training and traffic safety education for commercial vehicle drivers was highlighted due to the statistical significance of commercial vehicle indicators in all the non-angle crash models. There is another need for educational outreach and safety training targeting bicycles, as indicated by the unstable effects on roundabout safety. Further, additional enforcement countermeasures should be implemented to prevent aggressive driving, thereby decreasing the chance of serious injury in non-angle crashes.

It should be acknowledged that this study still has certain limitations. Research could consider more roundabout types and more relevant variables. In addition, the causes of temporal instability are not fully understood. The discrepancies and instabilities may be due to factors not considered in the analysis—these need to be further studied in future work.