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category: literaturenote citekey: chenresamplingapproachdisaggregateanalysis2022 title: A resampling approach to disaggregate analysis of bus-involved crashes using panel data with excessive zeros authors: "Chen, Tiantian; Lu, Yuhuan; Fu, Xiaowen; Sze, N.N.; Ding, Hongliang" year: 2022 date: 2022-00-00 2022 doi: 10.1016/j.aap.2021.106496 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2021.106496" zotero_key: R3BSZDJX zotero_storage: A9VIHDZE collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Chen, Tiantian"

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Contents lists available at ScienceDirect

Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

A resampling approach to disaggregate analysis of bus-involved crashes using panel data with excessive zeros

Tiantian Chen a,b , Yuhuan Lu c , Xiaowen Fu a,d , N.N. Sze b,* , Hongliang Ding b

  • a Department of Industrial and System Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
  • b Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
  • c Department of Computer and Information Science, State Key Laboratory of Internet of Things for Smart City, University of Macau, Taipa, Macao
  • d Knowledge Management and Innovation Research Centre, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

ARTICLE INFO

Keywords: Bus safety Crash frequency model Excessive zeros Resampling approach

ABSTRACT

Public bus constitutes more than 70% of the overall road-based public transport patronage in Hong Kong, and its crash involvement rate has been the highest among all public transport modes. Though previous studies had identified explanatory factors that affect the crash risk of buses, use of considerably imbalanced crash data with excessive zero observations could lead to inaccurate parameter estimation. This study aims to resolve the excess zero problem of disaggregate analysis of bus-involved crashes based on synthetic data using a Synthetic Minority Over-Sampling Technique for panel data (SMOTE-P). Dataset comprising crash, traffic, and road inventory data of 88 road segments in Hong Kong during the period from 2014 to 2017 is used. To assess the data balancing performance, other common data generation approaches such as Random Under-sampling of the Majority Class (RUMC) technique, Cluster-Based Under-Sampling (CBUS), and mixed resampling, are also considered. Random effect Poisson (REP) models based on synthetic data and random effect zero-inflated Poisson (REZIP) model based on original data are estimated. Results indicate that REP model based on synthetic data using SMOTE-P outperforms REZIP model based on original data and REP models based on synthetic data using RUMC, CBUS and mixed approaches, in terms of statistical fit, prediction error, and explanatory factors identified. Results of model estimation based on SMOTE-P suggest that factors including morning peak, evening peak, hourly traffic flow, average lane width, road length, bus stop density, percentage of bus in the traffic stream, and presence of bus priority lane all affect the bus-involved crash frequency. More importantly, this study provides a feasible solution for disaggregate crash analysis with imbalanced panel data.

1. Introduction

Hong Kong is a densely populated city with limited road space, where public transport modes dominate 90% of work commute and 46% of non-work commute. Every day, over 60% of total passenger trips are made by road-based public transport (Transport and Housing Bureau, 2017). Specifically, public buses constitute more than 70% of the overall road-based public transport patronage (Transport Department, 2014). Despite that bus drivers are considered relatively cautious and less aggressive owing to their higher sense of social responsibility (Chen et al., 2020a; Paleti et al., 2010), crash involvement rate of public buses in Hong Kong has been the highest among different vehicle types including private cars, light goods vehicles, heavy goods vehicles, and

taxis, in the period from 2009 to 2019 (Transport Department, 2020a). In the past five years, there were a total of 13,642 bus-involved crashes. In addition, likelihood of bus passengers incurring fatality and severe injury was two times higher than that of private car passengers (Transport Department, 2020b). This could be attributed to the structural design and safety features of public buses (Chimba et al., 2010; Feng et al., 2016; Zhou et al., 2020). In particular, majority of public buses in Hong Kong are double-deckers. Standing passengers are considerable. Also, seats of majority of double-deckers in Hong Kong are currently not fitted with safety belts (Lunn et al., 2018). Therefore, it is of paramount importance to identify the factors that contribute to an increase in bus-involved crashes.

To improve bus safety, previous studies have identified factors

E-mail addresses: tt-nicole.chen@connect.polyu.hk (T. Chen), lu.yuhuan@connect.umac.mo (Y. Lu), xiaowen.fu@polyu.edu.hk (X. Fu), tony.nn.sze@polyu.edu. hk (N.N. Sze), hongliang.ding@connect.polyu.hk (H. Ding).

* Corresponding author.

including operational attributes, roadway profiles and traffic characteristics that affect the risk of bus-involved crashes. For example, Chimba et al. (2010) found that wider traffic lanes and shoulders are associated with a decrease in bus-related crashes, while higher traffic volume per lane, higher speed limit, presence of on-street parking and increase in the number of lanes all contribute to an increase in bus crashes. Moreover, factors including season of the year, day of the week, hour of the day, road surface condition, and characteristics of bus drivers that affect the risk of fatal bus crashes were identified (Feng et al., 2016; Tamakloe et al., 2020). Results indicated that the likelihood of more severe bus crashes was higher in the autumn and winter seasons, weekends, and evening time. To better capture the space–time variations in crash risk at the microscopic level, disaggregate crash frequency models of individual entities (e.g., intersections, and road segments) and/or short time periods (e.g., month, day, and hour) are often established. However, these models are often subject to excessive zero crash observations (see Shankar et al., 2003; Lord et al., 2005, 2007; Anastasopoulos, 2016; Pei et al., 2016; Raihan et al., 2019). In particular, imbalanced crash data with excess zeros is prevalent when modeling crashes of specific collision types, crash severity levels (e.g., fatal and severe injury crashes) and transportation modes (e.g., bicycles, and public transport) (Tsui et al., 2009; Pei et al., 2012; Gu et al., 2020). It can in turn bias the parameter estimation. Hence, engineering solutions based on incorrect identification of explanatory factors (e.g., road geometry and traffic control) could be ineffective and even have adverse impact on overall safety.

To address the issue of excessive zero crash counts, earlier studies applied the zero-inflated Poisson and negative binomial (ZIP and ZINB) models, with which a dual-state generating process of crash data was assumed (Shankar et al., 2003; Qin et al., 2004). Mannering and Bhat (2014) indicated that zero-inflated models have been the most popular, to handle excessive zero counts, among other statistical approaches. Despite that zero-inflated models are still frequently used in recent research (Raihan et al., 2019; Gu et al., 2020), some researchers have criticized on the theoretical ground of zero-inflated models with respect to the source of excess zeros (Lord et al., 2005; Lord et al., 2007; Lord and Mannering, 2010). For example, excessive zeros can be attributed to the low vehicle exposure and high heterogeneity of road entities, short observation period, small spatial scale, and under-reporting of crashes rather than inherently safe sites as suggested by zero-inflated models. Though excessive zeros can be mitigated by accumulating more cases in an extended time period, it can result in the ignorance of time-varying factors (Behnood and Mannering, 2019; Mannering, 2018). In this context, Pei et al. (2016) developed a highly disaggregate model that incorporates the variables including year, day of the week, and time of day. To accommodate the predominance of zero-crash observations, a bootstrap resampling approach was applied to generate the data. Then, a conventional negative binomial model was established. Nonetheless, the computation time would increase remarkably when the bootstrap sample is large (Efron and Tibshirani, 1994; Efron, 2014).

In recent years, there is a body of literature throwing light on the application of resampling algorithm to deal with imbalanced datasets. In particular, there are two commonly applied resampling approaches (e. g., under-sampling and oversampling techniques) for the synthesis of imbalanced data (Basso et al., 2018; Cai et al., 2020; Fiorentini and Losa, 2020; Yin et al., 2019; Yuan et al., 2019). For instance, under-sampling technique reduces the number of observations of the majority class, while oversampling increases the number of observations of the minority class. For example, Yin et al. (2019) used the random undersampling of the majority class (RUMC) technique to eliminate excessive zero-crash observations from the raw dataset, while Morris and Yang (2021) used the cluster-based under-sampling approach (CBUS), respectively. The latter may have better data representation than random sampling. On the other hand, Basso et al. (2018) applied the synthetic minority over-sampling technique (SMOTE) to generate new crash dataset using k-nearest neighbors within a bootstrapping procedure based on the original crash dataset. Moreover, the mixed resampling strategy that combines oversampling of minority classes and under-sampling of majority classes is increasingly popular (Al-Radaideh and Daoud, 2018; Morris and Yang, 2021; Schlogl ¨ et al.,2019) To summarize, these referenced studies all suggest that synthesizing the imbalanced crash data using resampling approaches have promising prediction performances.

Previous studies applying zero-inflated models, including those aforementioned investigations, suggest that such application provides a better statistical fit. However, most of them did not delve into the assumption that the truly safe sites have a long-term mean of zero. In this context, Lord and Geedipally (2011) and Geedipally et al., (2012) argued that advanced statistical models are urgently needed since it is very difficult to change the characteristics of the dataset. Yet, with the rapid development of resampling approach, it is now plausible to apply data-balancing techniques to resolve the problem of excessive zeros. From the methodological perspective, data-balancing techniques including SMOTE, RUMC, and CBUS are not completely new. In fact, there are some recent studies that apply such techniques for the realtime crash prediction (using case-control technique, see Cai et al., 2020; Yin et al., 2019; Yuan et al., 2019; Schlogl ¨ et al., 2019). However, among those referenced literatures that deals with imbalanced datasets for crash analysis using resampling approach, it is rare that crashes involving buses, with which the excess zero problem is prevalent and crash consequences are more serious, are attempted.

In addressing this gap in the literature, this study aims to propose an alternative approach of dealing with the dataset characterized by a large number of zeros, that is, analyzing the new balanced dataset rather than formulating a sophisticated statistical model. In other words, the objective of this study is to document the application of conventional crash count models using a balanced dataset for the identification of crash explanatory factors. Bus-involved crashes are studied since they have relatively high injury likelihood, and the problem of excessive zeros is prevalent. A modified SMOTE for panel data (discussed later in Section 2.3) and RUMC are applied in this study to balance the bus crash dataset. Our results confirm that the model estimated based on the balanced data outperform the zero-inflated model. Also, explanatory factors identified in this study help provide government agencies and transport operators with fresh insights and useful suggestions for traffic management and control measures that can improve overall bus safety.

The remainder of this paper is organized as follows. Section 2 describes the methods of analysis and the resampling approaches. Section 3 provides a description of the data used in the analysis. Section 4 presents the parameter estimation results. Section 5 discusses the interpretations. Finally, Section 6 concludes the paper with a summary of the findings, and future research directions.

2. Method

2.1. Crash frequency model

Count data models including Poisson regression model have been widely used to estimate the effects of explanatory factors that affect the crash frequency (Chen et al., 2020b; Lord and Mannering, 2010; Sze et al., 2019). In a Poisson process, the probability of road segment i having Yit bus-involved crashes in time period t is specified as,

$$P(Y{ii}) = \frac{EXP(-\mu{ii})\mu{ii}^{Y{ii}}}{Y_{ii}!}$$ (1)

$$\mu{it} = EXP(\beta X{it}^m) \tag{2}$$

where i = 1, 2, …, N; t = 1, 2, …, T; μit denotes the expected number of bus-involved crashes at road segment i in time period t (μit = E [Yit]); and N and T refer to the total number of road segments and time periods, respectively. Xm it denotes the value of mth explanatory variables at road segment i in time period t and β is a vector of parameters.

However, it is not uncommon for the prevalent of zero-crash observations since crashes are rare events. To this end, zero-inflated distribution has been adopted by some road safety researchers (Anastasopoulos, 2016; Gu et al., 2020; Huang and Chin, 2010; Lord et al., 2005; Mannering and Bhat, 2014; Qin et al., 2004; Shankar et al., 2003). The zero-inflated model is developed based on the concept of a mixed Poisson process. It is proposed that excessive zero crash cases are generated by a dual-state process, with which the zero observations can either come from an inherently safe state or an imperfect state. Probability that road segment i is belonged to the zero state is denoted by Pit. Therefore, (1 − Pit) represents the probability that road segment i is classified as unsafe (non-zero state). Assuming a Poisson1 distribution, probability of a road segment having zero crashes is EXP(-μit). Therefore, overall probability of a road segment having zero-crash observations is the summation of Pi and *EXP(*− μit). Probability density function of the zero-inflated Poisson distribution is given by,

$$P(Y{ii}) = \begin{cases} P{ii} + (1 - P{ii})EXP(-\mu{ii}), (Y{ii} = 0) \ (1 - P{ii})\frac{EXP(-\mu{ii})\mu{ii}^{Y{ii}}}{Y{ii}}, (Y_{ii} \ge 1) \end{cases}$$ (3)

Then, a logit function is established for Pit as follows,

$$logit(P{ii}) = log(\frac{P{ii}}{1 - P{ii}}) = Z{ii}^{q} \gamma$$ (4)

where Zq it denotes the explanatory variables that may affect the value of Pit and γ is a vector of corresponding parameters.

There can be temporal correlation between observations since the data collected from the same road segment in consecutive time periods could share the same (unobserved) effects (more details about the data are provided in Section 3). Therefore, a random effect term is incorporated into the count model as (Huang and Chin, 2010; Lord and Mannering, 2010; Gu et al., 2020),

$$\log(\mu it) = X_{it}^m \beta + \omega i$$

where ωi is invariant within the observations of the same road segment and independent between different road segments, following a normal distribution, ωi ~ (0, σ2 ).

To assess the goodness-of-fit of the proposed model, a likelihood ratio test is given by,

$$LR = -2 \left[ LL(\beta_{H0}) - LL(\beta_n) \right]$$

where LL(βH0) is the log likelihood with the intercept only and LL(βn) is the log likelihood at convergence, respectively. Under the null hypothesis, LR is chi-square distributed with the degree of freedom equals to the difference in the number of parameters between the restricted and unrestricted models. A statistically significant LR indicates a good model fit. In this study, we applied the statistical package NLOGIT 5.0 to estimate the random effect Poisson (REP) and random effect zero-inflated Poisson (REZIP) regression models.

2.2. Marginal effects of explanatory variables

To extend the analysis, marginal effect is computed as the change in predicted frequency of bus-involved crashes μit given one unit increase in an explanatory variable. Average marginal effect of a continuous variable is estimated using Eq. (7), and that of a discrete variable is calculated using Eq. (8) (see Greene, 2012; Huo et al., 2020; Washington et al., 2020),

$$\frac{\partial E(Y{it}|X{it})}{\partial X_q} = \beta_q \cdot EXP(\beta \overline{X}) \tag{7}$$

$$\begin{split} E[Y{tt}|X{tt},\ d=1] - E[Y{tt}|X{tt},\ d=0] &= \textit{EXP}(\beta\overline{X}|\ X_d=1) - \textit{EXP}(\beta\overline{X}|\ X_d\ &= 0) \end{split}$$

(8)

where Xq denotes a continuous variable, *β*q denotes the coefficient of explanatory variable Xq, β is a corresponding (M × 1)-column vector of the coefficients of Xm it , X is a (M × 1)-column vector of the means of explanatory variables, and Xd denotes a discrete variable respectively.

2.3. Data balancing method

In this study, rather than developing a sophisticated statistical model, problem of excessive zero observations is addressed using data synthesis approach. To handle the imbalanced crash dataset and better predict the minority class (i.e., non-zero crash cases), oversampling and under-sampling techniques (i.e., SMOTE, random under-sampling of the majority class (RUMC), cluster-based under-sampling (CBUS), and mixed resampling method) are used to eliminate the excessive zerocrash observations (Japkowicz, 2000; Cai et al., 2020; Morris and Yang, 2021; Yin et al., 2019). In particular, the cluster-based undersampling process can minimize information loss (Morris and Yang, 2021). Clusters were generated from unsupervised stratified sampling process using the proportional k-means clustering technique. Then, the data was sampled in accordance with the cluster size. In this study, the cluster size is set to be 20, minimizing the within-cluster sum-of-squares.

SMOTE was first proposed to synthesize imbalanced data (Chawla et al., 2002). It has been a popular technique because of the capability of feature-specific synthesis and high computation efficiency (Abou Elassad et al., 2020; Basso et al., 2018; Li et al., 2020a,b; Wang et al., 2019; Yuan et al., 2019). To improve the data generation performance, alternate approaches including Safe-Level-SMOTE (Bunkhumpornpat et al., 2009), Local-Neighborhood-SMOTE (Maciejewski and Stefanowski, 2011), and Iterative-Partitioning Filter (IPF) SMOTE (Saez ´ et al., 2015) are proposed. Therefore, higher degree of consistencies between the original and synthetic data can be achieved.

Yet, the aforementioned SMOTE algorithms were specified for continuous variables of cross-sectional model only. They might not be capable of synthesizing the heterogeneous data collected from the entities that have repeated observations. In other words, these models are not ideal for balancing the panel data (i.e., with repeated observations in different time periods). To this end, a SMOTE method for panel data (SMOTE-P) is proposed.

Let V = {VC, VN} denotes the variable set, where VCrepresents the set of categorical variables and V*N*is the set of continuous variables. For a sample x from the minority class, observations that have the same values for V*C*are selected as the k nearest neighbors (k is set at 3 in this study) based on the Euclidean distance. Given that a sample *xR*is chosen from the nearest neighbors, a new synthetic sample can be defined as:

$$q = x + u(x_R - x) \tag{9}$$

where u is randomly selected from a uniform distribution, u U(0, 1). In a conventional SMOTE model, the augmented minority class tends to have smaller variance. This implies the low variability of original data (Lusa, 2013). This is attributed to the restrictive selection of the nearest neighbors. However, such selection process ensures that the synthetic data q are located in the same feature space as the original data x. To mitigate the effect of restrictive selection, *xR*is chosen using a weighted approach. Let dx(xR) denote the Euclidean distance between the original data x and one of the nearest neighbors xR. The weight assigned to *xR*is

calculated as, 1 To account for the possible over-dispersion of the data, negative binomial regression model is also considered. However, our preliminary results indicate that the over-dispersion parameter is not statistically significant.

$$a_R = \frac{EXP(d_x(xR))}{\sum{r=1}^{k} EXP(d_x(x_r))}$$ (10)

Therefore, probability of a nearest neighbor being chosen should increase with its distance from the original sample. This could then increase the variability of synthetic data and enhance the prediction performance.

Furthermore, a hybrid approach that combines both oversampling and under-sampling is also adopted for comparison. The mixed resampling approach employed a two-stage process: (i) In the first stage, CBUS method is used to under-sample the majority class; (ii) In the second stage, SMOTE-P is used to oversample the minority class. A ratio of 1 to 4 (crash to non-crash cases) for the synthetic data was adopted (Abdel-Aty et al., 2004; Cai et al., 2020). It was because the improvement in statistical power was incremental when the ratio increased beyond 1 to 4 (Zheng et al., 2010; Roshandel et al., 2015). In this study, the ratio of 1 to 4 is also adopted when balancing the data using SMOTE-P, RUMC, CBUS, and mixed approaches.

2.4. Prediction performance of crash frequency model

In this study, prediction performances of the proposed crash frequency models are assessed using the metrics including Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE), which are given by Equations (9) and (10) respectively (see, Huo et al., 2020) as follows,

$$RMSE = \sqrt{\frac{1}{N}} \sum{i=1}^{N} (\mu{ii} - Y_{ii})^{2}$$ (9)

$$MAE = \frac{1}{N} \sum{i=1}^{N} |\mu{ii} - Y_{ii}|$$ (10)

where N denotes the number of observations, μit denotes the predicted crash frequency, and Yit is the observed crash frequency. Despite that RMSE and MAE both indicate the average derivations of predicted frequencies from the observed frequencies, RMSE is more sensitive to the outliers.

3. Data

Data used in this study were collected from 88 roadway segments in Hong Kong in a four-year period from 2014 to 2017. Such dataset entails the crash record, traffic counts and road profiles. The crash record is obtained from the Transport Information System (TIS) of Hong Kong Transport Department. TIS provides the information on crash time, location, vehicle class, crash circumstances, and injury characteristics of every crash that involves personal injury. The traffic count data is obtained from the Annual Traffic Census (ATC) database. ATC covers 88.5% of all trafficable roads in Hong Kong. In the ATC dataset, detailed traffic counts by vehicle type and time of the day of 88 road segments are available. For example, proportion of bus in the traffic in every hour during the 16-hour period from 7:00 a.m. to 11:00 p.m. can be obtained. Therefore, crashes involving buses in the period between 11:00 p.m. and 7:00 a.m. are not considered. The road segments considered are widely distributed across the territory of Hong Kong and are representative. For details, readers may refer to Chen et al. (2021). For the road profiles, information on road segment length, speed limit, presence of bus

Fig. 1. Annual average bus-involved crash counts of the 88 road segments in 2014-2017Note: For each road segment, total number of observations = 64 (i.e., 4 years × 16 h).

priority lane, bus stop density, and number of intersections are obtained from the Hong Kong Road Network Dataset.

To this end, a total of 531 bus-involved crashes occurred at the 88 road segments (in the period between 7 a.m. and 11 p.m. on weekdays) were identified using the geographical information system (GIS) approach (see Fig. 1). Then, the traffic and crash data were mapped to the sample with respect to corresponding time intervals (i.e., 7 a.m. – 8 a.m., 8 a.m. – 9 a.m., 9 a.m. – 10 a.m., …, and 10 p.m. – 11 p.m.). Such a data collection procedure yields a sample of panel data comprising 5632 observations (88 segments × 4 years × 16 time intervals). Fig. 2 shows the geographical distribution of zero-crash observations, and Table 1 presents the summary statistics of the sample respectively.

To evaluate the predictive power of estimated models, the dataset is further divided into two subsets in accordance with the holdout validation method (see Gooch et al., 2018; Tang et al., 2019). Specifically, 80% of the observations (N = 4480) are used as training data for model estimation, and the remaining 20% (N = 1152) are used for validation purpose. Of the 4480 observations in the training dataset, 4091 have zero-crash case and 389 have at least one crash. Ratio of crash to noncrash observations is about 1 to 11. To mitigate the problem of excessive zero observations, SMOTE-P and RUMC techniques are applied to generate new training datasets with the ratio of 1 to 4 for crash to noncrash cases.

4. Results

Table 2 presents the results of parameter estimation. For instance, REP (Model 1) and REZIP (Model 2) models are established using the original dataset that has excessive zero observations. On the other hand,

Table 1 Summary statistics of the sample.

Variable Mean S.D. Max. Min. Frequency
(%)
Number of bus-involved
crashes		 0.09 0.34 4 0
Morning peak (8am to 11am,
yes = 1)		 – – 1 0 1056
(18.75%)
Evening peak (5 pm to 8 pm,
yes = 1)		 – – 1 0 1056
(18.75%)
Presence of bus priority lane
(yes = 1)		 – – 1 0 896 (15.91%)
Speed limit (km/h) 64.80 13.07 100 30
Number of lanes 4.90 2.30 12 2
Bus stop density (per
kilometer)		 2.92 3.38 17 0
Hourly traffic flow (1000
vehicle)		 2.52 2.09 10.49 0.03
Average lane width (meter) 3.59 0.45 5.25 2.70
Intersection density (per
kilometer)		 1.94 3.14 13.00 0
Road segment length
(kilometer)		 3.00 3.48 19.08 0.18
Percentage of bus in traffic
stream (%)		 6.96 6.31 71.50 0

Note: Sample size = 5632 (i.e., 88 road segments × 4 years × 16 h)

REP models are established using the synthetic data based on RUMC (Model 3), CBUS (Model 4), Mixed (Model 5), and SMOTE-P (Model 6) approaches. The intent of this study is to document the application of a basic count data model with a re-balanced dataset for the identification of explanatory factors that affect the bus-involved crashes. As shown in

Fig. 2. Number of zero-crash observations of the 88 road segments in 2014–2017.

Table 2 Results of parameter estimation.

Variables Coeff. Model 1: REP model using original data#
95% CIs		 Coeff. Model 2: REZIP model using original data
95% CIs		 Coeff. Model 3: REP model using synthetic data (RUMC)
95% CIs
Constant − 1.16* (− 2.11, − 0.21) − 1.14* (− 2.14, − 0.13) − 0.99** (− 1.93, − 0.07)
Morning peak IS IS IS
Evening peak 0.50** (0.28, 0.72) 0.50** (0.26, 0.73) 0.34** (0.11, 0.56)
Hourly traffic flow 0.26** (0.21, 0.31) 0.21** (0.17, 0.25) 0.16** (0.12, 0.20)
Speed limit IS IS IS
Average lane width − 0.61** (− 0.86, − 0.36) − 0.63** (− 0.89, − 0.37) − 0.47** (− 0.72, − 0.22)
Intersection density IS IS IS
Bus stop density 0.07** (0.03, 0.11) 0.08** (0.04, 0.12) 0.07** (0.03, 0.11)
Road segment length 0.08** (0.05, 0.10) 0.08** (0.05, 0.11) 0.07** (0.05, 0.10)
Percentage of bus 0.05** (0.04, 0.06) 0.05** (0.04, 0.07) 0.05** (0.04, 0.06)
Presence of bus priority lane IS IS IS
Number of observations 4480 4480 1945
Ratio of non-zero to zero-crash cases 1:11 1:11 1:4
Unrestricted loglikelihood − 1411.64 − 1362.79 − 1027.89
Restricted loglikelihood − 1515.80 − 1515.80 − 1144.51
Likelihood ratio test 208.32** 306.02** 233.24**
Variables
Model 4: REP model using synthetic data
(CBUS)		 (Mixed) Model 5: REP model using synthetic data (SMOTE-P) Model 6: REP model using synthetic data
Coeff. 95% CIs Coeff. 95% CIs Coeff. 95% CIs
Constant − 0.82** (− 1.62, − 0.02) − 1.57** (− 2.05, − 1.09) − 2.37** (− 2.93, − 1.82)
Morning peak IS IS 0.13* (0.02, 0.24)
Evening peak 0.29** (0.15, 0.43) 0.33** (0.12, 0.54) 0.36** (0.26, 0.45)
Hourly traffic flow 0.19** (0.15, 0.23) 0.16** (0.13, 0.19) 0.14** (0.12, 0.16)
Speed limit IS IS IS
Average lane width − 0.41** (− 0.68, − 0.14) − 0.35** (− 0.52, − 0.18) − 0.26** (− 0.40, − 0.12)

Note:

Dispersion parameters are insignificant, therefore negative binomial models are not applicable.

Intersection density IS IS IS

Number of observations 1945 2746 5258 Ratio of non-zero to zero-crash cases 1:4 1:4 1:4 Unrestricted loglikelihood − 1289.34 − 2157.36 − 2813.94 Restricted loglikelihood − 1433.61 − 2489.15 − 3268.64 Likelihood ratio test 321.57** 698.02** 909.40**

Bus stop density 0.07** (0.03, 0.11) 0.09** (0.07, 0.11) 0.10** (0.08, 0.13) Road segment length 0.08** (0.04, 0.12) 0.09** (0.08, 0.10) 0.10** (0.09, 0.12) Percentage of bus 0.04** (0.03, 0.05) 0.03** (0.02, 0.04) 0.03** (0.02, 0.03) Presence of bus priority lane IS − 0.19* (− 0.36, − 0.02) − 0.16* (− 0.31, − 0.01)

Table 2, results of parameter estimation (Model 2, 3, 4, 5, and 6) indicate that factors including evening peak, hourly traffic flow, average lane width, road segment length, bus stop density, percentage of bus, and presence of bus priority lane can affect the number of bus-involved crashes, all at the 5% level. In particular, evening peak, hourly traffic flow, road segment length, bus stop density, and percentage of bus are positively associated with the bus-involved crashes. In contrast, average lane width is negatively associated with the bus-involved crashes. More importantly, one additional significant explanatory factor (i.e., morning peak) is identified in the REP model based on balanced data using Mixed resampling approach (Model 5). Also, two additional significant factors (i.e., morning peak and presence of bus priority lane) are identified using SMOTE-P approach (Model 6). Specifically, presence of bus priority lane is associated with the decrease in bus-involved crashes, while morning peak is positively associated with the bus-involved crashes.

Furthermore, with respect to the predictive performance, the estimated parameters in each crash frequency model are used to calculate the predicted bus crash frequency based on the validation dataset. Table 3 summarizes the results of RMSE and MAE estimations. Note that higher values of RMSE and MAE reflect greater difference between the observed and predicted numbers of bus-involved crashes. As shown in Table 3, Model 6 (REP model based on synthetic data using SMOTE-P approach) has the lowest prediction errors (MAE and RMSE reduced by 15.7% and 12.7% respectively) while Model 5 (REP model based on

Table 3 MAE and RMSE of the crash frequency models.

Model MAE RMSE
REZIP model using original data [A] 0.20 0.33
Applications of Resampling method
REP model using synthetic data (RUMC) [B] 0.25 0.38
Relative difference [(B-A)/A] +27.9% +14.8%
REP model using synthetic data (CBUS) [C] 0.23 0.35
Relative difference [(C-A)/A] +15.0% +6.1%
REP model using synthetic data (Mixed) [D] 0.18 0.31
Relative difference [(D-A)/A] − 10.0% − 6.1%
REP model using synthetic data (SMOTE-P) [E] 0.17 0.29
Relative difference [(E-A)/A] − 15.7% − 12.7%

synthetic using mixed approach) also has lower errors (MAE and RMSE reduced by 10.0% and 6.1% respectively), compared with Model 2 (REZIP model based on original data). Moreover, both Model 3 (REP model based on synthetic using RUMC approach) and Model 4 (REP model based on synthetic using CBUS approach) have higher errors (MAE increased by 27.9% and 15.0%, and RMSE increased by 14.8% and 6.1%, respectively), compared with conventional zero-inflated model using original data (Model 2). Overall, crash frequency model based on SMOTE-P approach is the best among the candidate models in terms of statistical fit, prediction error, and number of significant

**Statistical significance at the 1% level.

*Statistical significance at the 5% level.

IS: Statistically insignificant. # Model 1 was estimated for reference purpose, no discussion on the parameter estimation results of Model 1 would be provided.

factors. Furthermore, marginal effects of significant factors on businvolved crashes are estimated and presented in Table 4.

5. Discussion

5.1. Application of crash frequency model with re-balanced data

Previous studies indicated that excessive zeros (crash observations) could result in poor statistical fit, inaccurate parameter estimates, and therefore inappropriate policy implications (Huang and Chin, 2010; Gu et al., 2020; Yuan et al., 2019). In this study, four resampling approaches-RUMC, CBUS, SMOTE-P and Mixed are used to generate balanced zero and non-zero crash cases. In particular, SMOTE-P is applied to boost up the non-zero crash cases, RUMC and CBUS are applied to eliminate excessive zero observations, and mixed approach is a hybrid of CBUS and SMOTE-P, respectively. A ratio of 1–4 for crash to non-crash cases proposed by Abdel-Aty et al. (2004) is adopted in this study, as Zheng et al. (2010) and Roshandel et al. (2015) also indicated that results of parameter estimation are fairly robust and the improvement in statistical power is incremental when the ratio further increases.

Overall, with respect to likelihood ratio tests, the five candidate models (REZIP model using original data, and REP models using synthetic data) fit well with the observations, all at the 1% level of significance. With respect to the identification of explanatory factors, eight significant factors that affect the bus-involved crashes are identified in the REP model based on SMOTE-P approach (seven significant factors are identified in the model based on mixed resampling approach and six in the models based on under-sampling approaches). Despite that the same significant explanatory factors are identified in the REP model based on under-sampling approach (i.e., RUMC and CBUS) and REZIP model based on original data, former has higher prediction errors. This can be attributed to the information loss due to the massive removal of zero-crash cases (Cai et al., 2020; Yuan et al., 2019). Similarly, although the REP model based on mixed resampling approach can identify one additional significant factor and has lower prediction errors, compared with conventional REZIP model, it does not outperform that based on SMOTE-P approach because of the information loss. To this end, it is essential to retain the majority class (i.e., zero-crash cases) in the dataset, especially for disaggregate safety analysis at the microscopic scale (i.e., individual road entities and short time periods), despite that mixed resampling-based model may have lower prediction errors sometimes (i.e., diagnostic analysis of crash severity or collision type) (see Al-Radaideh and Daoud, 2018; Morris and Yang, 2021).

From the perspective of statistical approaches, despite that zeroinflated models have been the most popular method that accounts for the excessive zero observations in crash frequency analysis, many studies did not explicitly give the validity of underlying perfectly safe state a careful thought (Lord et al., 2005, 2007; Lord and Mannering, 2010). In particular, excessive zeros could be attributed to the small temporal and spatial scales (e.g., Pei et al., 2016). Aggregate analysis of crash may mitigate such problem but create another problem of ignoring the effects of time-varying variables (Behnood and Mannering, 2019; Mannering, 2018). To this end, instead of accumulating more crash

Table 4 Marginal effects of significant factors of REP model using synthetic data (SMOTE-P).

Variable ∂μ/∂x Z-stat
Morning peak 0.02 1.95
Evening peak 0.04 3.49
Hourly traffic flow 0.02 4.10
Average lane width − 0.03 − 1.96
Road segment length 0.01 4.00
Bus stop density 0.01 3.86
Percentage of bus 0.003 3.44
Presence of bus priority lane − 0.02 − 2.00

cases for efficient statistical inference, applying resampling approach to generate crash cases (e.g., SMOTE-P in this study) could be a more promising approach. Previous studies employed SMOTE to balance the real-time traffic data for case-control analysis of crash occurrence and found that model using synthetic data based on SMOTE approach outperformed that using original data (Basso et al., 2018; Schlogl ¨ et al., 2019; Yuan et al., 2019). Consistently, results of current study indicate that crash frequency model using synthetic data based on SMOTE-P approach provides better a statistical fit and lower prediction errors. More importantly, more explanatory factors that affect the bus crash frequency can be revealed in the model using SMOTE-P approach.

5.2. Effects of bus exposure and operation

Our results indicate that an increase in the proportion of bus is associated with an increase in the number of bus-involved crashes. Indeed, it could be attributed to the increase in bus exposure to the mixed traffic stream (Chen et al., 2021; Dinu and Veeraragavan, 2011). Other exposure variables of buses such as bus kilometers travelled (Cheung et al., 2008), service frequency (Goh et al., 2014), and route length (Gu et al., 2020) are all found positively associated with bus crashes. Our results also indicate that an increase in the bus stop density is associated with an increase in the bus crash frequency. This finding is in line with that of Cheung et al. (2008), Chimba et al. (2010) and Goh et al. (2014), suggesting that higher frequency of "stop and start" acts for the buses at the bus stops can lead to more conflicts between buses and other vehicles on the roads. In addition, conflicts are expected to increase when the bus stops are located closer to one another.

Moreover, as reviewed in Goh et al. (2014), bus priority strategies mainly include space allocation (e.g., providing bus-only lane), time allocation (e.g., extending the green phase for buses at the signalized intersections) or a combination of both. There is a body of literature that provide evidence on the safety benefits of the implementation of bus priority. For example, introducing the bus lanes in Melbourne helped reduce the numbers of overall road crashes and bus crashes by 14% (Goh et al., 2013) and 54% (Goh et al., 2014) respectively. Consistently, our results suggest that the presence of bus priority lane is associated with a decrease in bus crashes, owing to the physical separation (or, reduced interactions) between buses and other vehicles (De Ceunynck et al., 2017). However, an earlier study in Hong Kong revealed a mixed safety effect of bus lanes, suggesting that the bus lanes help reduce the public bus crashes but potentially increase other vehicle crashes (Tse et al., 2014). Indeed, conflicts may arise from the increased vehicle interactions on the other traffic lanes due to the reduced road space. It is worth further investigating the safety effects of bus lanes on non-bus traffic in a future study for the development of safer bus lane operation strategies.

5.3. Effects of road environment features

The road environment features include its geometric characteristics, traffic flow and traffic control. Our results suggest that the average lane width is negatively associated with bus-involved crashes. This finding is consistent with that of previous studies, suggesting that wider traffic lanes (to provide extra space as a safety buffer) contributes to the reduction in bus crashes (Chimba et al., 2010; Dai et al., 2020). Our results also suggest that road segment length and hourly traffic flow are positively associated with the number of bus-involved crashes, which is in line with the previous findings (Cheung et al., 2008; Goh et al., 2013), indicating an increase in the overall crash exposure (Chen et al., 2020b, 2021; Pei et al., 2012; Qin et al., 2004).

Interestingly, although findings in earlier studies suggested that the occurrence and severity of bus crashes tend to increase at the intersections (af Wåhlberg, 2004; af Wåhlberg, 2018; Kaplan and Prato 2012), our results did not reveal a significant relationship between intersection density and bus-involved crashes. This could be attributed to the higher sense of social responsibility and risk compensation behavior of bus drivers (Chen et al., 2020a; Mannering and Bhat, 2014; Paleti et al., 2010). Bus drivers tend to drive more cautiously as it usually takes longer time to pass through the intersection considering the weights and dimensions of double-deckers. To further improve bus safety at intersections, government agencies may consider introducing adaptive signal control at the intersections, giving priority to the public buses (Goh et al., 2014).

5.4. Temporal effects

With respect to time of the day, numbers of bus-involved crashes during the morning and evening peak periods were found to be significantly higher than that of other time periods at the 1% level. This is not surprising since the bus services are much more frequent during the morning peak hours. More importantly, bus drivers are prone to very high stress to adhere to the service schedule (Hlotova et al., 2014). This could in turn induce more driving errors and violation offenses, and therefore have adverse impact on overall road safety (Rowden et al., 2011). Also, after driving for long distances and hours, bus drivers tend to have higher crash risk due to possible distractions and driving under the influence of fatigue (Boufous and Williamson, 2006; Chen et al., 2019; Gu et al., 2020; Huting et al., 2016). Maynard et al. (2020) pointed out that bus drivers are more vulnerable to fatigue since they often have to work overtime, insufficient sleep, and high work pressure. On the other hand, the evening peak period from 5 pm to 8 pm covers the most common off-work hours in Hong Kong. It has been found that drivers tend to drive less cautiously during their off-work hours. This may therefore increase the possible conflicts between bus and other vehicle class (Chin and Huang, 2009; Chen et al., 2021). Indeed, Tamakloe et al. (2020) revealed that the probability of multiple-vehicle crashes involving a bus tends to be higher in the late afternoon. Similar to previous studies, results of our study also lend credence to the notion that disaggregate analysis of road crashes is important, and temporal characteristics such as year, season, day of the week, and time of the day should be incorporated in the model formulations (Behnood and Mannering, 2019; Mannering, 2018; Tamakloe et al., 2020; Pei et al., 2016). In the future study, it is worth examining the temporal stability of parameters that affect the crash frequency.

6. Conclusion

To address the deficiency of the conventional SMOTE model, this study introduces a modified data generation approach, namely SMOTE-P, to deal with the heterogenous panel data that have excessive zero observations for the disaggregate analysis of bus-involved crashes. A ratio of 1 to 4 for crash to non-crash cases is used for the data synthesis. Then, REP models are applied to model the synthetic data. Prediction performances of candidate models are assessed using the metrics including RMSE and MAE. Results indicate that the proposed crash frequency model based on synthetic data using SMOTE-P outperforms other counterpart, in terms of statistical fit, prediction error, and number of explanatory factors revealed. For example, morning peak, evening peak, hourly traffic flow, segment length, bus stop density, and percentage of bus are positively associated with the number of bus-involved crashes, while average lane width and presence of bus priority lane are negatively associated with bus-involved crashes.

This study also sheds light on statistical inference for crash analysis using imbalanced data. Indeed, excessive zero crash observations are not necessarily derived from the dual-state process, which is an important theoretical ground of conventional zero-inflated models. To this end, this study bridges the gap in the literature on the use of resampling approaches to deal with imbalanced datasets for disaggregate crash analysis. In addition, findings of this study provide an updated and reliable interpretation of the explanatory factors that affect the bus safety. This is crucial for the development of effective safety

interventions and bus management strategies. For example, the implementation of bus-only lane is generally considered effective in reducing the bus-involved crashes, which is consistent with the result of this study. It is also suggested that the priority strategies, in terms of space and/or time, for the buses should deserve further investigation. To sum up, the contribution of this study is two-fold. First, it provides a new direction to deal with excessive zeros for disaggregate crash analysis by the oversampling of panel data. Second, it identifies the explanatory factors to bus-involved crashes for the improvements of bus operation and management strategies.

Yet, this study is limited to the 88 major roads in Hong Kong that have continuous and detailed traffic count data on weekdays. Horizontal and vertical profiles are not included in the current study due to limited availability of required data. Moreover, the safety effects of discontinuous and continuous bus lanes should be investigated in an extended study when the information on geometric design and operation rule of the bus-only lanes is available. This study used the bus crash data in Hong Kong during the period from 2014 to 2017 for the case study, with an approximately 1:11 ratio of crash to non-crash cases. Although most studies using resampling algorithms to balance the data have adopted a benchmarking ratio of 1:4, sensitivity analysis of the crash to non-crash ratio can be conducted in an extended study to further enhance the robustness of data generation and crash frequency model.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The work described in this paper was supported by the grants from the Research Grants Council of Hong Kong (Project No. 25203717), and the Hong Kong Polytechnic University (H-ZJMQ).

References