category: literaturenote citekey: hosseinpourpredictingcrashfrequencymulti2018 title: "Predicting crash frequency for multi-vehicle collision types using multivariate Poisson-lognormal spatial model: A comparative analysis" authors: "Hosseinpour, Mehdi; Sahebi, Sina; Zamzuri, Zamira Hasanah; Yahaya, Ahmad Shukri; Ismail, Noriszura" year: 2018 date: 2018-00-00 2018 doi: 10.1016/j.aap.2018.05.003 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2018.05.003" zotero_key: 24EWPA25 zotero_storage: IQ2EWQUC collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Hosseinpour, Mehdi"

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Accident Analysis and Prevention

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

Predicting crash frequency for multi-vehicle collision types using multivariate Poisson-lognormal spatial model: A comparative analysis

Mehdi Hosseinpoura,⁎ , Sina Sahebib , Zamira Hasanah Zamzuric , Ahmad Shukri Yahayad , Noriszura Ismailc

^aDepartment of Civil Engineering, Central Tehran Branch, Islamic Azad University (IAUCTB), Tehran, Iran
^bDepartment of Civil Engineering, Sharif University of Technology, Tehran, Iran
School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600, Bangi, Malaysia
^d School of Civil Engineering, Universiti Sains Malaysia, 14300, Nibong Tebal, Malaysia

ARTICLE INFO

Keywords: Collision type Multivariate Poisson lognormal model Spatial correlation Two-stage model

ABSTRACT

According to crash configuration and pre-crash conditions, traffic crashes are classified into different collision types. Based on the literature, multi-vehicle crashes, such as head-on, rear-end, and angle crashes, are more frequent than single-vehicle crashes, and most often result in serious consequences. From a methodological point of view, the majority of prior studies focused on multivehicle collisions have employed univariate count models to estimate crash counts separately by collision type. However, univariate models fail to account for correlations which may exist between different collision types. Among others, multivariate Poisson lognormal (MVPLN) model with spatial correlation is a promising multivariate specification because it not only allows for unobserved heterogeneity (extra-Poisson variation) and dependencies between collision types, but also spatial correlation between adjacent sites. However, the MVPLN spatial model has rarely been applied in previous research for simultaneously modelling crash counts by collision type. Therefore, this study aims at utilizing a MVPLN spatial model to estimate crash counts for four different multi-vehicle collision types, including head-on, rear-end, angle, and sideswipe collisions. To investigate the performance of the MVPLN spatial model, a two-stage model and a univariate Poisson lognormal model (UNPLN) spatial model were also developed in this study. Detailed information on roadway characteristics, traffic volume, and crash history were collected on 407 homogeneous segments from Malaysian federal roads. The results indicate that the MVPLN spatial model outperforms the other comparing models in terms of goodness-of-fit measures. The results also show that the inclusion of spatial heterogeneity in the multivariate model significantly improves the model fit, as indicated by the Deviance Information Criterion (DIC). The correlation between crash types is high and positive, implying that the occurrence of a specific collision type is highly associated with the occurrence of other crash types on the same road segment. These results support the utilization of the MVPLN spatial model when predicting crash counts by collision manner. In terms of contributing factors, the results show that distinct crash types are attributed to different subsets of explanatory variables.

1. Introduction

Road traffic crashes in Malaysia have become a major socio-economic problem for road safety agencies and imposed a huge cost on the economy. For the year 2016, over 7000 people died in road accidents nationwide, a 6.7% increase compared to 2015 (IRTAD, 2017). Based on the Royal Malaysian Police (RMP) crash database, multi-vehicle crashes, including head-on, rear-end, angle, and sideswipe crashes, account for about 60% and 75% of total and fatal crashes, respectively. These figures indicate that multi-vehicle crashes are more frequent than single-vehicle crashes, and result in more serious consequences. Thus, it is highly justifiable to put more efforts into reducing the frequency and severity of multi-vehicle crashes. Safety researchers routinely apply crash prediction models (CPMs) to establish the relationship between crash occurrence and a set of explanatory variables (Xie et al., 2007; Geedipally and Lord, 2008; Li et al., 2008; Abdel-Aty and Haleem, 2011). In the context of crash frequency modelling, numerous studies have been conducted to relate traffic crashes to various risk indicating factors. From a methodological viewpoint, a number of published studies have modelled traffic crashes by collision type or injury severity.

E-mail addresses: h\_mehdi61@yahoo.com, meh.hosseinpour@iauctb.ac.ir (M. Hosseinpour).

^⁎ Corresponding author.

As documented by previous studies (Kim et al., 2006; Geedipally et al., 2010; Christoforou et al., 2011; Aguero-Valverde et al., 2016), modelling crash frequencies by collision type (i.e., crash-type model) can gain a better understanding of the effect of various factors on each crash type, leading to develop more effective countermeasures. In this regard, various modelling specifications have been developed in the literature. These models include univariate count models (e.g., the standard negative binomial (NB) model, Poisson-lognormal model, zero-altered model), two-stage approach, and multivariate count models (e.g., multivariate Poisson (MVP), multivariate Poisson-lognormal (MVPLN) model, and multivariate spatial models). However, much of the literature has applied univariate count models to separately model crash frequencies by collision manner, while two-stage models and multivariate modelling techniques have recently been applied. A limitation associated with univariate count models is that they cannot account for correlations, which may exist between different levels of classification (e.g., severity levels or collision types). The two stage method integrates the outputs of a crash-count model (e.g., NB model) and a discrete choice model (e.g., multinomial logit model), in which the former is adopted to estimate the total predicted crashes while the latter is employed to predict the probabilities of crash occurrence for each collision type and then to allocate the predicted crash counts to each collision type. The two-stage model is more appropriate for modelling the counts of crashes by collision type in cases where the crash data are characterized by small sample size and low sample mean (Geedipally et al., 2010; Wang et al., 2011; Aguero-Valverde et al., 2016). Regarding multivariate modelling specifications, an advantage of these models over their univariate counterparts is that they can accommodate possible correlations (i.e., presence of interdependencies) between different crash types, as they jointly model crash counts by collision manner. Among the multivariate specifications, the MVPLN model with spatial effects has been found to be more promising, since it not only accounts for unobserved heterogeneity (extra-Poisson variation) and possible interdependencies between crash types, but also spatial correlation between adjacent sites. The idea behind the inclusion of spatial correlation in the multivariate models is that neighbouring locations have similar geographical and environmental characteristics, and hence tend to be correlated instead of independent. Therefore, the presence of spatial correlations in crash data should be accounted for. Otherwise, neglecting to consider spatial dependence leads to underestimation of variability, which results in biased parameter estimates (Lee et al., 2015). Several researchers have recently proposed the MVPLN spatial model in order to address the issue of spatial correlation between neighbouring locations (Aguero-Valverde, 2013; Wang and Kockelman, 2013; Barua et al., 2014; Imprialou et al., 2016). However, despite advantages of the MVPLN spatial models, they were rarely applied in the safety literature to estimate the crash frequencies for different vehicle-to-vehicle collision types, which are considered as commonlyoccurring crash types (Aguero-Valverde et al., 2016). Therefore, this study aims to develop a MVPLN model with spatial effects for simultaneously estimating crash frequencies for four different multi-vehicle collision types, including head-on, rear-end, angle, and sideswipe collisions. The intent is to investigate the effects of various explanatory variables on the frequency of crashes across these four collision types while controlling for possible spatial correlations between adjacent roadways.

It is important to note that in the vast majority of the prior studies, the appropriateness of the multivariate modelling approaches were compared only to their univariate counterparts, UNPLN, or to other crash-type models, such as the MVP or conventional NB models. Nevertheless, it is also interesting here to find out how accurately the MVPLN spatial model can estimate the counts of different collision types when compared with other advanced models. To do this, a twostage model is also developed using the same data set. The reason for developing a two-stage model in this paper is that this model, despite its advantages, has rarely been applied in the literature (notable exceptions are works by Wang et al. (2011) and Geedipally et al. (2010) as discussed later). Therefore, this study intends to extend research by conducting a comparative analysis between the MVPLN spatial and the two-stage model to determine which model provides better statistical fit. Moreover, to evaluate the appropriateness of the MVPLN spatial model, it is also compared with separate univariate spatial Poissonlognormal (PLN) models and the MVPLN without spatial correlation. To accomplish the objectives of this study, 4-year (2007-2010) crash data were collected on 407 homogenous segments from Federal Roads in Malaysia. The organisation of this paper is as follows: Section 2 presents a review of previous research that has been focused on crash frequency modelling. Section 3 describes the methodology for estimating and comparing the proposed modelling techniques that are employed in this paper to model crash counts by type of collision. Section 4 presents the characteristics of the data used in this research. In Section 5, the results of model calibration and comparison are discussed followed by the discussion of the estimated parameters. The last section presents the key findings and conclusions of the study and offers specific recommendations and ideas for future research.

2. Literature review

There are a large body of studies in the literature that has estimated crash frequency either by collision type or injury severity. In this context, much of the earlier studies used conventional univariate count models, such as Poisson-gamma (i.e., NB), Poisson-lognormal, and zeroinflated models, to separately relate crash counts to a set of explanatory variables by collision type (Shankar et al., 1995; Greibe, 2003; Abdel-Aty et al., 2005; Qin et al., 2005; Kim et al., 2006; Ulfarsson et al., 2006; Jonsson et al., 2007, 2009; Venkataraman et al., 2013). However, a main concern when developing separate crash-type models is that they fail to correctly predict crash frequencies in cases where crash data are characterised by a small sample size and low sample mean (Lord and Mannering, 2010; Wang et al., 2011). By disaggregating total crashes into different collision types (e.g., head-on, rear-end, and rollover), the resulting subset will exhibit smaller sample sizes and lower sample means. This issue may cause less stable results and biased parameter estimates (Xie et al., 2007; Li et al., 2008; El-Basyouny and Sayed, 2009a; Bham et al., 2011; Aguero-Valverde et al., 2016). As documented by Lord (2006); Lord and Miranda-Moreno (2008), and Park and Lord (2009), this issue can negatively influence the estimation of the dispersion parameter of a Poisson-gamma model (i.e., negative binomial). They suggested that under such conditions, Poisson-lognormal models are preferred over Poisson-gamma models. More recently, other researchers introduced a more sophisticated methodology in which the total number of crashes is initially fitted by an aggregate count model. Then, the crash counts for each collision type are allocated using a discrete choice model. This approach is termed as the two-stage model (Wang et al., 2011). The literature has shown that the use of discrete choice models is more appropriate for modelling crash types when crash data for collision types exhibit the small sample size and low sample mean problem (Geedipally et al., 2010; Bham et al., 2011; Wang et al., 2011). The two-stage approach has been rarely applied in the safety literature. Notable exceptions are Geedipally et al. (2010) and Wang et al. (2011).

However, a critical drawback to the use of both univariate crashtype models and two-stage models is that they fail to accommodate the possible correlations that may exist across collision types. These correlations may be caused by shared latent information arising from common omitted factors or unobserved error terms. Ignoring such simultaneity may reduce the precision of parameter estimates and model efficiency (Aguero-Valverde and Jovanis, 2009; El-Basyouny and Sayed, 2009a; Lee et al., 2015). To cope with this problem, multivariate modelling approaches, which simultaneously model crash counts either by injury severity or collision type, have recently been proposed in the safety literature. The most popular and notable multivariate specifications that are applied in the literature include multivariate Poisson (MVP) model (Tsionas, 2001; Karlis, 2003; Ma and Kockelman, 2006), multivariate negative binomial regression model (MVNB) model (Zhao and Kockelman, 2018; Rodgers and Leland, 2005; Anastasopoulos et al., 2012), multivariate Poisson gamma mixture (MVPG) model (Mothafer et al., 2016), and multivariate Poisson-lognormal (MVPLN) model (Aguero-Valverde and Jovanis, 2009; Ye et al., 2009; El-Basyouny et al., 2014b; Wang et al., 2017).

A major restriction of the MVP model is that it fails to account for overdispersion (i.e., extra Poisson variation), which is a common issue in crash data. To address this problem, a multivariate negative binomial regression model (MVNB) was developed to analyse multivariate overdispersed crash counts. However, both MVP and MVNB models assume that the covariances between different categories are identical and positive, which does not accord with reality. To handle such drawbacks, several researchers have developed the MVPLN model (Chib and Winkelmann, 2001; Tunaru, 2002; Dong et al., 2014a; Li et al., 2015). The MVPLN approach is indeed a superior alternative to the MVP model as it not only accounts for extra Poisson variation but also accommodates a full correlation structure, which allows for both negative and positive correlations among response variables (Park and Lord, 2007; Ma et al., 2008; El-Basyouny and Sayed, 2009a). Nevertheless, a limitation associated with the MVPLN model is that it fails to account for possible spatial correlations which may exist between neighbouring sites. In fact, adjacent locations have similar characteristics in terms of geometric and environmental conditions, which may share unobserved effects. Neglecting to account for such similarities may result in biased parameter estimations and incorrect inferences. Traditionally, safety researchers incorporated spatial correlations into univariate crash prediction models. To do so, different types of spatial modelling specifications, such as spatial autoregressive (SAR) model, conditional autoregressive (CAR) model, and Bayesian hierarchical model, were developed and compared with non-spatial models (Aguero-Valverde and Jovanis, 2006; Quddus, 2008; El-Basyouny and Sayed, 2009b; Guo et al., 2010; Xie et al., 2013; Chen, 2015). In most cases, the spatial models outperformed their non-spatial counterparts.

To handle the issue of potential spatial correlations in a multivariate modelling approach, the MVPLN with spatial effects would be appropriate in most cases. The MVPLN spatial model is a promising approach that is capable of accommodating the possible correlations between adjacent roadway segments while accounting for dependencies between different crash classifications (e.g., crash types or injury severity levels). The MVPLN spatial models have recently been used in the safety literature (Song et al., 2006; Aguero-Valverde, 2013; Wang and Kockelman, 2013; Aguero-Valverde et al., 2016; Barua et al., 2016; Imprialou et al., 2016). For example, Barua et al. (2014) used a multivariate spatial model for severe (injury and fatal) and no-injury collisions using three years of collision data from the city of Richmond and the city of Vancouver. The results indicated that the multivariate model with both heterogeneous effects and spatial correlation resulted in the best fit when compared to its UNPLN spatial models. Lee et al. (2015) developed a multivariate Poisson-lognormal conditional autoregressive (MVPLN-CAR) approach for modelling crashes for motorized and nonmotorized modes of transportation (pedestrians and cyclists) at the macroscopic level. Islam et al. (2016) adopted both the univariate and multivariate full Bayesian (FB) spatial models in before–after safety analysis, and compared the results with those of non-spatial FB models. The results showed that the MVPLN-CAR model outperformed the other comparing models according to the deviance information criteria (DIC).

In summary, from the referenced studies, it can be concluded that despite the advantages of the MVPLN spatial model, it has rarely been applied in the literature, especially for modelling crash counts by collision type. A possible reason for this is that the MVPLN model with spatial correlation requires an extensive computational process, such as the time-consuming MCMC simulation-based method, to be formulated. The simulation time, which depends on the sample size and the model complexity, can take several days and the issue of time can still be a restricting factor in the model formulation. In addition, no built-in functions are available in most existing statistical software packages to fit these multivariate specifications, which lead safety researchers to disregard such sophisticated approaches.

3. Methodology

This section presents an overview of the modelling techniques that are used in this paper to estimate crash counts for four different multivehicle collision types, namely, head-on, rear-end, angle, and sideswipe crashes. The two approaches, including the MVPLN spatial and twostage models are briefly described in the following subsections.

3.1. MVPLN spatial model

This study develops a MVPLN spatial approach to simultaneously model crash frequency by collision type while taking into account the spatial dependency among adjacent roadways. The MVPLN spatial model is a generalisation of the MVPLN model where the latter fails to account for spatial correlations. The following describes the MVPLN spatial model applied in this study. Assume that N crashes fall into each of J collision types, and let Yik and μik be the observed and expected crashes of the collision type k on segment i, respectively. Yik is assumed to follow a Poisson distribution with mean μik. That is,

$$Y{ik} \sim \text{Poisson}(\mu{ik})$$ (1)

$$\log(\mu{ik}) = \beta{k0} + \sum{m=1}^{M} \beta{km} X{ikm} + \varepsilon{ik} + \varphi_{id(i)}$$ (2)

where βk⁰ is the constant term for crash type k (k = 1, 2, 3, 4); βkm is the coefficient of the mth covariate for crash type k; Xikm is the mth covariate for the ith segment and crash type k; mm M ( 1, 2, 3, , ) = … is the number of covariates; εik denotes unobserved heterogeneity random ^effect for the ⁱth segment and crash type ^k; φid i( ) represents the spatial random effect, which captures the spatial correlation between the roadway id i( ) and its neighbours. The function id i( ) defines roadway related to i th segment. The random effects accommodate the extra-Poisson heterogeneity among roadways. The MVPLN spatial can be reduced to the sole MVPLN model by excluding the spatial error term (φid i( )) from Eq. (2).

In order to account for the correlations between the unobserved heterogeneity, εik is assumed to follow multivariate normal distribution, as follows:

$$\varepsilon_{ik} \sim MN(0, \Sigma)$$ (3)

$$\varepsilon{ik} = \begin{pmatrix} \sigma{i1} \ \sigma{i2} \ \sigma{i4} \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \sigma{11} & \cdots & \sigma{14} \ \vdots & \ddots & \vdots \ \sigma{41} & \cdots & \sigma{44} \end{pmatrix}$$

$$\tag{4}$$

where Σ is the variance-covariance matrix.

The diagonal element σik of the covariance matrix represent the variance of εik, while the off-diagonal elements are the covariance of different collision types.

To control for the spatial correlation among the roadways, φid i( ) has a conditional autoregressive (CAR) distribution as follows:

$$\varphi{id(i)} | \varphi{id(j)}, id(i) \neq id(j), \tau_{id(i)}^2$$

$$\sim N \left( \frac{\sum{id(i) \sim id(j)} \varphi{id(j)} w{id(i)id(j)}}{\sum{id(i) \sim id(j)} w_{id(i)id(j)}} , \frac{\sigmas^2}{\sum{id(i) \sim id(j)} w_{id(i)id(j)}} \right)$$ (5)

where id i id j () ( ∼ ) specifies all the neighbours of roadway id(i); wid i id j () () denotes the weight between roadway id(i) and id(j); denotes; wid i id j () () = 1 if roadway id(i) and id(j) are adjacent, and wid i id j () () = 0 otherwise; σs ² is the variance for spatial correlation (Barua et al., 2014).

To estimate the MVPLN spatial model, one needs to specify prior distributions for *^β*0, βk, and ^Σ, which reflect prior knowledge about the parameters of interest (Barua et al., 2014). To do so, due to the absence of sufficient prior knowledge regarding the distributions of the parameters of interest, non-informative prior distributions need to be specified. For the regression parameters, a multivariate normal distribution with zero mean and large variance is used.

$$\beta \sim MVN(\beta0, R{\beta_0})$$ (6)

where β⁰ and Rβ⁰ are known non-informative hyperparameters.

For the term of Σ, a Wishart prior is used as the prior distribution of the inverse of the variance-covariance matrix for the heterogeneity ⁻ Σ ¹ (Aguero-Valverde, 2013):

$$\Sigma^{-1} \sim \text{Wishart}(\mathbf{R}, J)$$ (7)

where J is the degrees of freedom (here, J = 4); R is the scale matrix. To test the appropriateness of the MVPLN spatial model over its univariate counterpart, four independent UNPLN spatial models are also developed based on the Bayesian framework to estimate crash counts separately by collision type. Moreover, two additional MVPLN and UNPLN models without spatial effects are developed. The parameters of the MVPLN and UNPLN with and without spatial correlation developed herein were estimated using the Markov Chain Monte Carlo (MCMC) simulation under a Bayesian framework coded in WinBUGS.

3.2. Two-stage method

In the two-stage method, the process for predicting the number of crashes by collision type involves two stages. First, a crash-frequency model is adopted to predict the number of multi-vehicle crashes (in this paper, the aggregation of head-on, rear-end, angle, and sideswipe collisions) occurred on a certain road segment. Second, a multinomial-response model is employed to estimate the proportions of different collision types across individual crashes, given that a multi-vehicle crash has occurred. Then, the expected crash counts for a certain crash type can be obtained by multiplying the number of multi-vehicle crashes times the proportion of this crash type (Geedipally et al., 2010; Wang et al., 2011). Details on these two stages are discussed in the following subsections.

3.2.1. Modelling the total number of crashes

This study applies a Poisson lognormal (PLN) spatial model to estimate the frequency of multi-vehicle crashes as a whole, combining the crash counts of the four different vehicle-to-vehicle collision types. The PLN spatial model can be presented as follows:

$$Y{ik} \sim \text{Poisson}(\mu{ik})$$ (8)

$$\log(\mu{ik}) = \beta{k0} + \sum{m=1}^{M} \beta{km} X_{ikm} + \ln(Li) + \varepsilon{ik} + \varphi_{id(i)}$$ (9)

where all the parameters were already defined in the previous section.

For the PLN spatial model, the εik term, which account for segmentspecific heterogeneity, is assumed to be independent and to follow a prior normal distribution, ε Nσ ik ∼ (0, ) ^ε ² , where σε ² is within-segment (extra) variation. The spatial random effect term, φid i( ) , follows the univariate conditional autoregressive (CAR) structure:

$$\varphi{id(i)} | \varphi{id(j)}, id(i) \neq id(j), \tau_{id(j)}^2$$

$$\sim N \left( \frac{\sum{id(j)\sim id(j)} \varphi{id(j)} w{id(j)id(j)}}{\sum{id(i)\sim id(j)} w_{id(j)id(j)}}, \frac{\sigmas^2}{\sum{id(i)\sim id(j)} w_{id(j)id(j)}} \right)$$ (10)

where all the parameters were already defined in the previous section. As with the MVPLN spatial model, the parameters of the PLN spatial

model are estimated via the MCMC simulation under a Bayesian framework using the WinBUGS software.

3.2.2. Estimating the proportion for each collision type

This research adopts a mixed multinomial logit (MMNL) model to estimate the likelihood of occurrence of different multi-vehicle collision types. Compared with the conventional multinomial logit model (MNL), the MMNL model is a superior alternative and yields more precise results as the MMNL model relaxes the restrictive assumptions that are associated with a MNL model, such as the property of independence of irrelevant alternatives (IIA) and the property of independently and identically distributed (IID) errors. These assumptions are violated in most cases in which crash types may be correlated (i.e., sharing unobserved effects), which results in biased and incorrectly estimation (Chen and Chen, 2011; Wang et al., 2011). In addition, a limitation associated with this study is that information about driver characteristics, weather conditions, and vehicle factors were not provided in the current data set. Ignoring to account for such unreported variables may influence the occurrence of a collision type, and also cause potential unobserved heterogeneity. This issue may be problematic when a MNL model is used to predict the probability of crash occurrence, where the effects of all explanatory variables are assumed to be fixed and identical across crash observations (Pai, 2011). Instead, the MMNL model can appropriately account for the issue by considering the random effects of explanatory variables on the likelihood of collision type allowing a subset of explanatory variables to vary across crash observations (Haque and Chin, 2010; Kim et al., 2010a; Chen and Chen, 2011; Islam and Hernandez, 2013; Kim et al., 2013; Manner and Wünsch-Ziegler, 2013; Mosa and Esawey, 2013; Van Ristell et al., 2013; Behnood and Mannering, 2015; Wu et al., 2016; Seraneeprakarn et al., 2017). The MMNL model can be written as

$$P_{ln} = \int \frac{\exp(\betal X{ln})}{\sum_l \exp(\betal X{ln})} f(\beta_l | \varphi) d\beta$$ (11)

where Pin is the probability that a crash n is categorised as the collision type i, given that a vehicle-to-vehicle collision has occurred; Xin is a vector of explanatory variables; β is a vector of estimable coefficients; f ( ) β φi is the density function; and *φ*is a vector of parameters that is associated with the density function (mean and variance).

The MNL model is a special case of the MMNL model when β are fixed parameters across all crashes, which implies that the corresponding variables have a uniform effect over the observations. This study adopts the maximum simulation likelihood (MSL) method to estimate the MMNL model. The parameter estimates are based on 200 Halton draws, which have been reported to provide significantly more accurate results than purely random draws (Gkritza and Mannering, 2008; Milton et al., 2008; Anastasopoulos and Mannering, 2009; Haque and Chin, 2010; Chen and Chen, 2011; Wang et al., 2011). In the current research, the random coefficients are specified to be normally distributed, i.e., β ∼ N(b, σ), where b is the mean and σ is the standard deviation. A variable is considered to be random if the corresponding σ is statistically significant. All parameters of the MMNL model are estimated using a user-written command that is referred to as mixlogit in statistical software STATA (Hole, 2007).

Based on the estimated results of the PLN spatial and the MMNL models, the predicted crash counts for each of the four multi-vehicle collision types on a road segment can be determined by multiplying the total number of multi-vehicle crashes, which is estimated from the PLN spatial model, times the predicted probabilities of the collision types, which are obtained from the MMNL model. This can be calculated using the following equation:

$$\hat{N}{ij} = \hat{P}{ij}. \ \hat{N}_i \tag{12}$$

where *N*ˆ ij is the expected crash counts of collision type j on segment i; *N*ˆ i is the total number of crashes for segment i, as estimated by the PLN spatial model; and *P*ˆ ij is the probability of occurrence of the collision type j on segment i, which is obtained by aggregating the estimated probability of the collision type j across individual crashes that occur on segment i, which is calculated as follows (Wang et al., 2011):

$$\hat{P}{ij} = \frac{1}{N} \sum{n=1}^{N} \hat{P}_{ijn}$$ (13)

where *P*ˆ ijn is the predicted probability of the collision type j for the crash n that occurred on segment i, as determined by the MMNL model; and N is the observed number of crashes on segment i.

3.3. Model comparison

To assess the goodness-of-fit (GOF) of the three alternative methods developed in this study (i.e., MVPLN spatial model, UNPLN spatial model, and two-stage method), two different measures were used. These measures are the mean absolute deviance (MAD) and the mean squared prediction error (MSPE), as computed in Eqs. (14) and (15):

Mean Absolute Deviance (MAD) = $$\frac{1}{n} \sum_{i=1}^{n} |\hat{y}_i - y_i|$$ (14)

Mean Squared Prediction Error (MSPE) = $$\frac{1}{n} \sum_{i=1}^{n} (\hat{y_i} - y_i)^2$$ (15)

where n is the size of the sample; yi and *y*ˆ ⁱ are the observed number of crashes and predicted number of crashes, respectively, on segment i.

The interpretation of these criteria is as follows: the lower the values of MAD and MSPE, the better the model fit. In the case of the MVPLN (spatial) versus its univariate counterpart—the UNPLN (spatial) model—as these models are estimated using a Bayesian framework, an additional GOF measure, which is named the Deviance Information Criterion (DIC), is also used. This criterion, which is a Bayesian equivalent of the Akaike information criteria (AIC), is based on the posterior distribution of the deviance statistic. The DIC is computed as DIC D θ p D p = + =+ () 2 D D, where D θ( ) is the deviance of the posterior means, D is the posterior mean of the deviance, and pD is the ^effective number of parameters, which is given by ^p = −D Dθ( ) ^D . The DIC penalizes models with larger parameters; therefore, the model with the lower DIC is preferred. To decide whether there is a significant difference between the MVPLN model and the UNPLN model, if the difference in the DIC value is greater than 10, then the model with the lower DIC is considered to be significantly favoured over another. If the difference is between 5 and 10, then the model with the lower DIC is positively superior. A difference of less than 5 is considered to be weak evidence (Lan and Persaud, 2012; El-Basyouny et al., 2014b).

4. Study sites and data collection

To estimate crash frequencies by collision manner using the proposed models, detailed information on crash data, traffic conditions, and road geometric design should be collected. The relevant data in this study were collected from the Federal Roads in Peninsular Malaysia (known as West Malaysia). The federal roads comprise the main national road system in the country. However, they comprise nearly 40% of fatal crashes nationwide (IRTAD, 2017), which indicates that the

Table 1 Characteristics of the selected roadways.

Route	Code	From	To	State
Federal route 67	F67	Sungai Petani	Baling	Kedah
Federal route 76	F76	Baling	Pengkalan Hulu	Kedah
Federal route 2	F2	Taman Seri Mahkota	Karak	Pahang
Federal route 3	F3-1	Kampung Bukit Cendering	Paka	Terengganu
Federal route 3	F3-2	Permatang Sebayan	Tanjung Gemok	Pahang
Federal route 4	F4-1	Kuala Rui	Hentian Rehat Titiwangsa	Perak
Federal route 4	F4-2	Jeli	Tanah Merah	Kelantan

federal roads are ranked the highest among other road types (e.g., state, expressway, and municipal) in terms of crash casualties. The study sites were selected based on the availability of information related to roadspecific, traffic volume, and crash data. After reviewing several candidate roads, seven federal roads were finally selected as the study areas, comprising 498 km sections. Those roadways include F2, F3-1, F3-2, F4- 1, F4-2, F67, and F76 located in the states of Perak, Kedah, Kelantan, Pahang, and Terengganu. Table 1 presents characteristics of the study roadways.

For the study sections, detailed information on road geometric conditions (e.g., curvature, land use, shoulder width, number of lanes, etc.) and four-year (2007 to 2010) crash data (e.g., time and location of crashes, type of collision, and crash severity) were collected from the Malaysian Institute of Road Safety Research (MIROS), which were originally obtained from the Royal Malaysian Police (RMP) crash database. Data on traffic volume and composition, containing four-year (2007-2010) information on average daily light-vehicle traffic (LVT) and average daily heavy-vehicle traffic (HVT), were obtained from the Highway Planning Unit (HPU), which is operated under the Ministry of Works in Malaysia. By matching these three data sets, the study was provided with an integrated database that contains four years (2007 to 2010) of crashes, road-specific characteristics, and traffic conditions for each section. The study sites were then divided into homogeneous segments according to traffic flow, adjacent land use, shoulder width, number of lanes, and median type to ensure that each segment is uniform regarding to these attributes. Based on this splitting regime, a total of 407 homogeneous segments were produced with lengths ranged from 1 to 7 km and an average length of 1.2 km. During the 4-year period between 2007 and 2010, a total of 10,688 crashes occurred on the study segments, 6577 of which (approximately 60%) were multi-vehicle collisions (e.g., head-on, rear-end, angle, and sideswipe crashes) and 4111 crashes were single-vehicle collisions (e.g., pedestrian-involved, rollover, and run-off-the-road). The multi-vehicle collisions resulted in approximately 70% of the total fatal crashes in the selected sites. This indicates that these vehicle-to-vehicle collisions are more frequent than single-vehicle crashes with more severe consequences. To model crash counts by collision manner, multi-vehicle crashes were classified into four collision types, namely, head-on, rear-end, angle, and sideswipe crashes. The unit of crash frequency for these collision types is the number of crashes per segment for 4 years. Of the 6577 multi-vehicle crashes, 506 (8%) were head-on crashes, 2942 (45%) were rear-end crashes, 2063 (31%) involved angle crashes, and 1066 (16%) were sideswipe crashes. The most frequent and least frequent crash types were rear-end collisions and head-on collisions, respectively. Tables 2 and 3 contain descriptive statistics for continuous variables (both dependent and independent continuous variables) and categorical variables, respectively. As shown in Table 3, the vast majority of the segments under study are two-lane undivided roadways which are mostly

Table 2 Descriptive statistics of continuous variables for the 407 homogeneous segments.

Variable name	Description	Min.	Max.	Mean	Std. Dev.
Dependent variables
Head-on	Number of head-on crashes per segment	0	8	1.243	1.506
Rear-end	Number of rear-end crashes per segment	0	57	7.229	9.204
Angle	Number of angle crashes per segment	0	41	5.069	6.681
Sideswipe	Number of sideswipe crashes per segment	0	35	2.619	3.825
Multi-vehicle	Total number of multi-vehicle crashes, including head-on, rear-end, angle, and sideswipe collisions, per segment	1	121	16.16	18.565
Independent variables
Ln(LVT)	Natural logarithm of average daily light-vehicle traffic (e.g., motorcycles, passenger cars, light vans, SUVs)	7.934	10.221	8.985	0.618
Ln(HVT)	Natural logarithm of average daily heavy-vehicle traffic including bus, lorry, large van, truck, etc.	6.471	8.474	7.154	0.395
Segment length	The length of segment (km)	1	7	1.223	0.713
Speed limit	Actual posted speed limit (km/h)	50	90	81.327	11.669
PSW	Paved shoulder width (m)	0	2.4	1.242	0.595
UNPSW	Unpaved shoulder width (m)	0	2.4	1.174	0.713
Curvature	Horizontal curvature (1/km)	0.091	13.386	2.936	2.656
Access point	Number of minor intersections and driveways per km	0	9	1.12	1.535

located in rural areas.

5. Results and discussion

5.1. Results of two-stage model

As stated earlier, the two-stage model utilises the results of discrete choice models (the MNL and MMNL models in this paper) to allocate the total number of multi-vehicle crashes into different collision categories on a road segment. Table 4 presents the estimation results from the MNL and MMNL models, which are used for predicting the probability of a certain crash type among the four crash types. The head-on collision was chosen as the base case. Both models represent similar outputs in terms of the signs of the coefficients while the magnitudes of the coefficients are somewhat different. A likelihood ratio test (LRT) was performed between the MMNL (i.e., random parameter) model and the MNL (fixed parameter) model to test the appropriateness of the former over the latter. As shown in Table 4, the LRT is 103.39 with three degrees of freedom, which indicates that the MMNL model is substantially superior to its traditional MNL counterpart at any significance level. This result strongly rejects the null hypothesis that these two models are equal in terms of the LRT measure. The suitability of the MMNL model over the MNL model is primarily attributed to the fact that the MMNL model is capable of addressing the heterogeneity, which arises from observed and unobserved segment-specific attributes. As a result, allowing for random coefficients provides a better overall fit.

The results of the MMNL model show that the random coefficients associated with the model include paved shoulder width and number of access points. These variables are specified to vary across the crash observations as their standard deviations are statistically significant. As previously specified, random coefficients are considered to be normally distributed, i.e., β ∼ N(b, σ), where b is the mean and σ is the standard deviation. For example, paved shoulder width was random and normally distributed for both rear-end collisions and angle collisions. With respect to angle collisions, the coefficient has a mean of 0.215 and a standard deviation of 0.224. This can be interpreted such that for approximately 83% of multi-vehicle collisions, paved shoulder width has a positive effect on the likelihood of an angle collision, whereas it has a negative effect for the remaining 17% of the collision.

Variables with positive signs, such as Ln(LVT), the number of access points, and land use, imply that the likelihood of the occurrences of the corresponding response categories is higher than the base category, head-on collision. For example, Ln(LVT) was significant and positive for all three categories, which indicates that an increase in the number of light-vehicle traffic may increase the likelihood of rear-end, angle, and sideswipe collisions relative to head-on collisions. The coefficients of land use appeared to be statistically significant and positive in the rearend, angle, and sideswipe categories. This indicates that areas with high roadside activities are more likely to encounter these collisions rather than head-on crashes. The signs of horizontal curvature are significant and negative, except for the sideswipe collision, which is insignificant. Similar results can be derived for the effect of terrain type in which the likelihood of head-on and sideswipe collisions increases on roadway segments that are located in undulating terrain. Thanks to the MMNL estimation results, the expected number of crashes per segment for all study collision types is then estimated by multiplying the proportion of

Table 3 Descriptive statistics of categorical variables for the 407 homogeneous segments.

| Variable name | Description | Observation (proportion) in Sample | | | | |---------------------|-------------------------------------------------------------------------------------------------------------|------------------------------------|-------|-------|--| | | | 1 | 2 | 3 | | | Area type | Level of roadside development | 347 | 46 | 14 | | | | (1 for rural, 2 for semi-urban, 3 for urban) | (85%) | (11%) | (4%) | | | Land use | Level of activity along roadway | 242 | 99 | 66 | | | | 1 for no activity level (e.g., forested or agricultural areas) | (60%) | (24%) | (16%) | | | | 2 for low activity level (e.g., educational or industrial areas) | | | | | | | 3 for high activity level (e.g., residential or commercial areas) | | | | | | No. of travel lanes | Number of lanes for each travel direction | 357 | 50 | – | | | | (0 for one-lane, 1 for two or more lanes) | (88%) | (12%) | | | | Terrain Type | Indicator of vertical gradient along roadway | 306 | 101 | – | | | | (0 for flat terrain, 1 for undulating terrain) | (75%) | (25%) | | | | Side friction | Level of interaction between roadside activities (e.g., parking, bus stopping, trading) and through traffic | 337 | 70 | – | | | | (0 for low interaction, 1 for high interaction) | (83%) | (17%) | | | | Median type | Indicator of two opposing traffic flows are separated or not | 376 | 31 | – | | | | (0 for undivided, 1 for divided) | (92%) | (8%) | | |

Table 4 Parameter estimates for MNL and MMNL models.

| |-------------------------- | Rear-end collisions | Constant | Ln(LVT) | Horizontal curvature | Number of access points | Paved shoulder width | (Standard Deviation) | Land use: (1) No activity | Land use: (2) Low activity | Land use: (3) High activity | Terrain type | (1: undulating terrain, | Angle collisions | Constant | Ln(LVT) | Horizontal curvature | Number of access points | (Standard Deviation) | Paved shoulder width | (Standard Deviation) | Land use: (1) No activity | Land use: (2) Low activity | Land use: (3) High activity | Terrain type
(1: | Sideswipe collisions | Constant | Ln(LVT) | Horizontal curvature | Number of access points | Paved shoulder width | Land use: (1) No activity | Land use: (2) Low activity | Land use: (3) High activity | Terrain type
(1: | Summary statistics | Number of observations | Number of parameters | Log-likelihood at converge | 2
χ
LL MXL
LL | 2
χ(0.05,3) | 2
χ(0.01,3) | 2
χ(0.005,3) | MNL | | MMNL | | | | | --------------------------------|----------|------------|---------|-----------|------------|---------|--| | Coeff. | Std. Error | Z-value | Coeff. | Std. Error | Z-value | | | −3.167 | 0.859 | −3.69 | −3.020 | 0.922 | −3.28 | | | 0.502 | 0.095 | 5.29 | 0.489 | 0.102 | 4.80 | | | −0.073 | 0.022 | −3.39 | −0.073 | 0.023 | −3.12 | | | 0.046 | 0.033 | 1.37 | 0.052 | 0.037 | 1.43 | | | 0.262 | 0.086 | 3.04 | 0.251 | 0.093 | 2.71 | | | – | – | – | 0.251 | 0.035 | 7.17 | | | R | – | – | R | – | – | | | 0.289 | 0.136 | 2.13 | 0.265 | 0.146 | 1.81 | | | 0.366 | 0.163 | 2.25 | 0.350 | 0.177 | 1.98 | | | −0.378 | 0.121 | −3.11 | −0.491 | 0.133 | −3.69 | | 0: flat terrain) | | | | | | | | | | | | | | | | | −1.725 | 0.888 | −1.94 | −1.595 | | −1.66 | | | 0.298 | 0.098 | 3.05 | 0.279 | | 2.62 | | | −0.082 | 0.022 | −3.67 | −0.088 | | −3.6 | | | 0.063 | 0.034 | 1.85 | 0.094 | | 2.43 | | | – | – | – | 0.058 | | 2.32 | | | 0.251 | 0.089 | 2.82 | 0.215 | | 2.23 | | | – | – | – | 0.224 | | 5.37 | | | R | – | – | R | | – | | | 0.475 | 0.139 | 3.41 | 0.456 | | 3.04 | | | 0.479 | 0.167 | 2.86 | 0.540 | | 2.98 | | undulating terrain, 0: flat terrain) | −0.378 | 0.126 | −2.99 | −0.293 | | −2.13 | | | | | | | | | | | −3.580 | 0.959 | −3.73 | −3.530 | | −3.66 | | | 0.432 | 0.105 | 4.1 | 0.425 | | 4.02 | | | −0.030 | 0.024 | −1.24 | −0.025 | | −1.02 | | | 0.066 | 0.036 | 1.84 | 0.058 | | 1.63 | | | −0.105 | 0.137 | −0.77 | 0.079 | | 0.82 | | | R | – | – | R | | – | | | 0.201 | 0.154 | 1.3 | 0.223 | | 1.45 | | | 0.671 | 0.179 | 3.75 | 0.694 | | 3.85 | | undulating terrain, 0: flat terrain) | 0.082 | 0.096 | 0.86 | −0.100 | | −0.72 | | | | | | | | | | | 6577 | | | 6577 | | | | | 24 | | | 27 | | | | | −7859.01 | | | −7807.312 | | | | MNL
=
− 2[ ( )
(
)] | – | | | 103.396 | | | | | – | | | 7.815 | | | | | – | | | 11.345 | | | | | – | | | 12.838 | | | |

Note: "R" stands for reference category. Head-on crash type as the base case.

each collision type by the total expected number of multi-vehicle crashes, obtained from the PLN spatial model (the modelling results of the PLN spatial approach for the overall multi-vehicle crashes are omitted in this paper for brevity). The results of MAD and MSPE for the twostage model are presented and discussed in Section 5.3.

5.2. MVPLN spatial model

To obtain the posterior estimates of parameters for each model, the MCMC sampling procedure was run for 100,000 iterations with the initial 30,000 iterations were discarded as burn-in. The trace and autocorrelation plots were examined to ensure that the convergence of parameters was achieved. Table 5 provides the parameter estimates and their associated statistics for the UNPLN spatial model and the MVPLN spatial model (the parameter estimates of the UNPLN and MVPLN models without spatial effects are not shown due to space limitation). The results indicate that the UNPLN spatial and MVPLN spatial models present similar results in terms of the signs and magnitude of the estimated coefficients, although the UNPLN spatial model contains more significant variables than the MVPLN spatial model. For example, the natural logarithm of heavy-vehicle percentage (Ln(HVP)) and the number of travel lanes are, respectively, significant for the rear-end and sideswipe collisions in the UNPLN spatial models, but both of which are not significant in the MVPLN spatial model. Also, the category of semiurban for the variable area type is significant for the sideswipe collision in the UNPLN spatial model. The inclusion of more significant variables in the UNPLN spatial model is due to the fact that, unlike the MVPLN spatial model, the UNPLN spatial approach ignores interdependence between crash types, leading to the underestimation of the standard errors of the regression coefficients.

To test the appropriateness of the MVPLN spatial model over other models, the DIC was computed and presented in Table 5. The DIC statistics were found to be 6347, 6346, 6025, and 6137 for the UNPLN spatial, UNPLN, MVPLN spatial, and MVPLN models, respectively. Thus, the MVPLN spatial model is significantly preferred to the other models as the minimum difference in the DIC value between the MVPLN spatial and the MVPLN model is found to be 112, which is much greater than 10. In addition, the spatial variance is 2.849 (95% CI: 0.00057 and 29.77), which is statistically significant at the 95 credible interval. This confirms the existence of spatial dependence

Table 5 Parameter estimates of the UNPLN spatial and MVPLN spatial models under Bayesian framework.

Note: "R" stands for reference category. Dark-shaded cells indicate significant coefficients at 5% level.

among the crash data, which can be a surrogate for unobserved and relevant factors (El-Basyouny and Sayed, 2009b; Aguero-Valverde et al., 2016). The comparison between the multivariate and univariate models (MVPLN spatial vs. UNPLN spatial and MVPLN vs. UNPLN models) indicates that the inclusion of multivariate random effects considerably reduces the DICs. This finding is reasonable as the multivariate models can allow for the correlation between crash types, whereas the univariate models fail to handle the issue (Ma et al., 2008; El-Basyouny et al., 2014b; Li et al., 2015). Therefore, the MVPLN models result in more precise and robust estimates than its univariate counterparts. As a result, the MVPLN spatial model yields the best fit as indicated by its lower DIC value compared to the other models. These findings are consistent with the results of previous studies.

Table 6 presents the estimates of the covariance matrix and correlation matrix between crash types for the MVPLN spatial model. As indicated in the table, all crash types are highly and positively correlated with correlation coefficients ranging from 0.592 to 0.931. This finding implies that the occurrence of a specific collision type is highly associated with the occurrence of other crash types on the same road segment. The highest correlation was observed between ANGLE crashes and HO crashes (0.931) while the lowest correlation was observed between the RE and HO collisions (around 0.592), which is still high. Such high levels of correlation can be generated from unmeasured variables, such as weather conditions, driver attributes, and vehicle factors for which detailed information was not available in the current study (Ma et al., 2008; Ye et al., 2009). In these circumstances, applying the MVPLN spatial model is more appropriate as it can better deal with the issues of interdependence across crash types for a specific roadway and spatial correlation between adjacent roadways.

5.3. GOF comparisons among three modelling approaches

Two different measures of GOF, including MAD and MSPE, were calculated and presented in Table 7 to select the best fit model. Based on the values of MAD and MSPE, the MVPLN spatial and UNPLN spatial models fit the data significantly better than the two-stage model for all crash types while the MVPLN model outperforms the UNPLN spatial model. The superiority of the MVPLN spatial over its univariate counterpart—the UNPLN spatial—is also supported by the DIC statistic as its DIC value decreased significantly. In general, the MVPLN spatial

Table 6 Covariance-correlation matrices for the MVPLN spatial model (gray cells are correlation coefficients).

Note: p-values in parenthesis.

Table 7 Performance comparison of Two-stage, UNPLN spatial, and MVPLN spatial models.

Crash type	GOF Measures	Two-stage	UNPLN Spatial	MVPLN Spatial
HO	MAE	0.906	0.849	0.828
	MSPE	1.414	1.267	1.197
RE	MAE	2.000	1.890	1.871
	MSPE	10.453	7.246	7.243
ANGLE	MAE	1.835	1.558	1.536
	MSPE	8.745	5.082	4.997
Sideswipe	MAE	1.223	1.153	1.123
	MSPE	3.720	2.644	2.572

approach can be selected as the best fit model to simultaneously estimate crash counts by collision type. This is mainly due to the fact that the MVPLN spatial specification models can account for not only the dependence between crash types but also the spatial correlation between adjacent roadways.

5.4. Interpretation of MVPLN spatial results

This section presents the interpretation regarding the variables that are significantly associated with the MVPLN spatial model. To better discuss the effects of significant variables on the frequency of different crash types, the marginal effects (MEs) of all the explanatory variables were calculated.

The ME of the mth explanatory variable xm on dependent variable y can be calculated by taking the first derivative of the expected number of type specific crashes estimated by the MVPLN spatial model (Mothafer et al., 2016, 2018).

$$\frac{\partial E\left(y{ik} \mid x{ik}\right)}{\partial x{im}} = \beta{im} \exp(\betai^T x{ik}) \tag{16}$$

The ME of a specific explanatory variable may be interpreted as, at the mean values of the predictors in the model, a one unit increase in the value of the variable is correlated to an average increase (decrease) in the frequency crashes (Dong et al., 2014b). The ME values of all the contributing variables of the MVPLN spatial model are shown in Table 8.

Parameter estimates of the MVPLN spatial model given in Table 5 show that the effects of different explanatory variables vary across crash types. In other words, distinct crash types are affected by various explanatory variables in different ways. For example, horizontal curvature was found to be correlated with head-on crashes only, whereas the number of minor access points contributes significantly to all crash types, with a magnitude of 0.079 for sideswipe crashes to 0.197 for angle crashes. This finding was supported and confirmed by previous studies that focus on crash type modelling, which emphasized that different variables have unequal effects on different crash types (Greibe, 2003; McCartt et al., 2004; Abdel-Aty et al., 2005; Kim et al., 2006; Jonsson et al., 2007; Kim et al., 2007a; Pai and Saleh, 2008; Geedipally et al., 2010; Christoforou et al., 2011; Aguero-Valverde et al., 2016). Further explanation on each of those contributory variables is given in the following.

Table 8 Marginal effects of explanatory variables for the MVPLN spatial model.

Variables	Rear-end	Angle	Sideswipe	Head-on
Ln(LVT)	1.258	0.377	0.461	−0.252
Ln(HVT)	−0.357	−0.070	−0.035	0.306
Segment length	1.289	0.996	0.523	0.351
Speed limit	-0.059	-0.039	−0.014	-0.012
Horizontal curvature	−0.14	−0.090	0.017	0.048
Paved shoulder width	−0.385	−0.085	−0.303	−0.241
Unpaved shoulder width	−1.108	−0.312	−0.120	−0.116
Number of access points	0.566	0.555	0.165	0.109
Area type: (1) Rural	R	R	R	R
Area type: (2) Semi-urban	2.034	1.490	0.613	0.101
Area type: (3) Urban	4.151	0.509	1.458	−0.038
Number of travel lanes for each direction:	−0.009	−0.616	0.436	−0.068
(1: two or more lanes, 0 for one lane)
Land use: (1) No activity	R	R	R	R
Land use: (2) Low activity	0.417	0.897	0.193	−0.034
Land use: (3) High activity	0.377	0.998	0.685	−0.074
Terrain type (1: undulating terrain, 0: flat terrain)	−0.62	−0.095	0.251	0.343
Side friction (1: high interaction, 0: low interaction)	0.225	0.099	0.553	0.224

The number of access points is positively associated with all collision types. This result is intuitively reasonable and self-explanatory because as the number of driveways and minor intersections increases, the number of conflicting movements (e.g., merging and intersecting manoeuvres) increases, resulting in a high risk of vehicle-to-vehicle collisions. The ME values indicated that an increase in the number of minor intersections and driveways could result in an increase of all the four collision types. The natural logarithm of light vehicle traffic (Ln (LVT)) is among the variables that increase the risk of rear-end, angle, and sideswipe crashes. This finding is logical and consistent with previous studies (McCartt et al., 2004; Kim et al., 2007b; Ye et al., 2009; Geedipally et al., 2010; Abdel-Aty and Haleem, 2011; Bham et al., 2011; Das and Abdel-Aty, 2011; Meng and Weng, 2011; Aguero-Valverde et al., 2016). A higher volume of vehicle traffic increases the potential for greater conflicts among vehicles. Thus, it increases the risk of occurrence of multi-vehicle crashes (Milton and Mannering, 1998). In addition, as vehicle traffic increases the available gap between two vehicles decreases, which potentially increases the risk of a vehicle-tovehicle collision (Qin et al., 2005; Zhang et al., 2008; Christoforou et al., 2011). The ME values showed that one unit increase in the natural logarithm of light vehicle traffic would increase the average number of rear-end, angle, and sideswipe crashes by 1.258, 0.377, 0.461, respectively. The positive sign of segment length indicates that longer segments are prone to have more crashes (Lu et al., 2013). The marginal effects showed that a 1-km increase in segment length resulted in an average increase of 0.79 crashes for all the four collision types.

Area type was positively associated with all crash types except for head-on collisions. This finding implies that the risk of multi-vehicle crashes is higher in urban or semi-urban areas than rural areas, which is expected. In an urban setting, drivers are more likely to be distracted or faced with unexpected events; thus, they are likely to encounter multivehicle crashes (Bham et al., 2011). From the ME values, road segments located in urban areas were found to increase the frequency of rear-end and sideswipe crashes by 4.151 and 1.458, respectively. Speed limit is negatively associated with the frequency of rear-end and angle collisions. This finding may be attributed to the fact that lower speed limits are typically assigned to unsafe locations or roads with poor design elements, whereas safer and rural areas have higher posted speed limits. Consistent results were obtained in previous studies (Milton and Mannering, 1998; Golob and Recker, 2003; Greibe, 2003; Qin et al., 2005; Zhang and Ivan, 2005; Li et al., 2015). Based on the ME values, the number of rear-end and angle decreases as speed limit increases (ME M RE = − 0.059, EAngle = − 0.039). Land use contributes positively to the frequency of angle and sideswipe crashes. This indicates that segments with higher roadside activities (e.g., commercial or residential areas) are likely to experience more crashes compared with those with no roadside activity (e.g., forested or agricultural areas). This finding is also consistent with the finding from previous studies (Greibe, 2003; Kim et al., 2007b, b). The reason for positive association of land use on vehicle-to-vehicle crashes is due to the fact that road segments with high roadside activities attract more vehicular traffic, increasing the exposure to a vehicle-to-vehicle crash. Moreover, "land use" serves as a measure to other unsafe conditions, such as pedestrian movements and on-street parking, which could potentially increase the risk of angle and sideswipe collisions.

Paved shoulder width contributes significantly to head-on and sideswipe collision types, which is consistent with expectations. These collision types may occur when a vehicle crosses its lane either intentionally (e.g., manoeuvres relevant to turning, overtaking, lane changing, etc.) or unintentionally (e.g., losing control, driver distraction, drowsiness, inattention, etc.) and strikes a vehicle either in the opposing lane or adjacent lane. In these conditions, wider shoulder widths provide more room for the vehicle in the opposing lane or the adjacent lane to avoid a head-on or sideswipe collision (Gårder, 2006). This finding agrees with earlier studies (Milton and Mannering, 1998; Zhang and Ivan, 2005; Lee et al., 2006; Ma et al., 2008). According to the ME, at the mean values of the predictors in the model, the frequency of head-on and sideswipe crashes decreases by -0.303 and -0.241, respectively, for every one-unit increase in paved shoulder width. Unpaved shoulder width is negatively associated with rear-end crashes. In fact, larger shoulder widths provide more space to prevent vehicles from colliding with the leading vehicle in a traffic stream. The effect of horizontal curvature is only significant for head-on collisions, which indicates that head-on crashes are more likely to occur on horizontal curves than straight segments. This result is consistent with intuitive expectations as sharp curves reduce drivers' sight distance to see oncoming vehicles, which increase the risk of head-on crashes (Zhang and Ivan, 2005; Gårder, 2006; Ulfarsson et al., 2006). From the ME, a 1 ⁻ km ¹ sharper curve is associated with a 0.048 increase in the head-on crash frequency. Terrain type is significantly correlated with head-on crashes where the risk of head-on crashes is greater on undulating terrain relative to flat terrain. A similar finding was obtained in previous studies (Milton and Mannering, 1998; Zhang and Ivan, 2005; Ye et al., 2009). The ME value indicates that, at the mean values of the other predictors in the model, undulating terrain resulted in an average 0.343 increase in the number of head-on crashes. Sideswipe collisions increase with side friction. As the interactions between roadside activities (e.g., bus stops, loading zones, parking bays, etc.) and the through traffic increases, drivers are likely to be faced with unexpected events such as pedestrians or other vehicles engaged in roadside activities. Thus, they may be forced to suddenly change their positions or cross the centreline to avoid a possible collision, which may cause sideswipe crashes (Greibe, 2003). According to the ME value, at the mean values of the other predictors in the model, the number of sideswipe crashes increases by 0.553 where the interaction between roadside activities and the through traffic is high. The natural logarithm of heavy-vehicle traffic (Ln(HVT)), was found to have a positive impact on

Table 9 Summary of the explanatory variables associated with crash frequency for different crash types.

Variable	MVPLN Spatial model		Multi
	Rear-end	Angle	Sideswipe	Head-on	vehicle*
Ln(LVT)	+	+	+	N.S.*	+
Ln(HVT)	N.S.	N.S.	N.S.	+	+
Segment length	+	+	+	+	+
Speed limit	–	–	N.S.	N.S.	–
Horizontal curvature	N.S.	N.S.	N.S.	+	N.S.
Paved shoulder width	N.S.	N.S.	–	–	N.S.
Unpaved shoulder width	–	N.S.	N.S.	N.S.	–
Number of access points	+	+	+	+	+
Area type: (1) Rural	R*	R	R	R	R
Area type: (2) Semi-urban	+	+	N.S.	N.S.	+
Area type: (3) Urban	+	N.S.	+	N.S.	+
Land use: (1) No activity	R	R	R	R	R
Land use: (2) Low activity	N.S.	+	N.S.	N.S.	N.S.
Land use: (3) High activity	N.S.	N.S.	+	N.S.	N.S.
Terrain type (1: undulating terrain, 0: flat terrain)	N.S.	N.S.	N.S.	+	N.S.
Side friction (1: high interaction, 0: low interaction)	N.S.	N.S.	+	N.S.	N.S.

Note: N.S. denotes not statistically significant; R represents base level.

head-on crashes, which is also in line with previous studies (e.g., Abdelwahab and Abdel-Aty (2004)). The reason for this finding is that heavy vehicles travel at low speeds; thus, they increase the possibility of overtaking manoeuvres by their following vehicles. This conclusion is intuitively rational for this research where roughly 87% of the study segments are two-lane undivided roadways. In these circumstances, fast-moving vehicles have to cross the centreline to pass slow-moving vehicles and are likely to encounter a vehicle approaching from the opposite direction (Kim et al., 2007b). The ME value indicates that an increase in the natural logarithm of heavy vehicles corresponds to a 0.306 increase in the number of head-on crashes.

Table 9 presents the variables that were associated with each of the study collision types as well as the multi-vehicle crashes (the aggregation of head-on, rear-end, angle, and sideswipe collisions). As can be seen in the table, different crash types are attributed to different subsets of explanatory variables. By comparing the results of the MVPLN spatial model with those from the multi-vehicle model, it can be found that the multi-vehicle model fails to determine the real effects of explanatory variables on the frequency of each collision type. For example, terrain type and side friction were found to be significant determinants in headon and sideswipe crashes, respectively. However, such results cannot be seen in the multi-vehicle crash model. A similar finding was reached for the effect of paved shoulder width and horizontal curvature, which were found to be insignificant in the aggregate-crash model.

Based on the results of MVPLN spatial model, it is possible to develop cost-effective countermeasures to improve the overall safety of federal roadways. For example, the MVPLN spatial model shows that head-on crashes are more likely to occur on road segments with higher number of access points, narrower shoulder widths, higher number of heavy vehicles, sharper horizontal curves, and undulating terrain. Consequently, road safety plans should be implemented to reduce such dangerous conditions. For example, head-on crashes occurring at curved sections can be reduced by proposing a wide range of actions, such as straightening sharp curves, widening the pavement width, and improving superelevation.

^* Multi-vehicle is a UNPLN spatial model developed to estimate the frequency of multi-vehicle crashes as a whole, combining the frequencies of the four multi-vehicle crash types.

6. Conclusions and suggestions for future research

The objective of this research was to estimate crash counts for four different multi-vehicle collisions, including head-on, rear-end, angle, and sideswipe crashes. For this purpose, three modelling approaches were developed: (1) the MVPLN spatial specification, which simultaneously model crash counts by collision type while allowing for spatial correlation between adjacent roadways, (2) the UNPLN spatial model to separately estimate the number of crashes for each collision type, and (3) the two-stage method, which combines the outcomes of the MMNL model and the PLN spatial model of the total number of crashes. Detailed information on crash data, traffic conditions, and environmental attributes were collected on 407 homogeneous segments of Malaysian federal roads over 4-year (2007 to 2010) period. From the GOF measures (MSPE, MAD, and DIC), the MVPLN spatial model appeared to be considerably preferred over the two-stage model and the UNPLN spatial, as it yielded more accurate prediction results. The appropriateness of the MVPLN spatial model is mainly due to the fact that this model jointly estimates crash counts by collision manner through allowing for the interdependences between different collision types occurred on the same roadway while capturing the spatial correlation between neighbouring roadways. The results of the MVPLN spatial model proved the existence of the positive correlations between different pairs of crash types. These interdependencies may result from the latent factors that simultaneously influence the occurrence of crash types. Meanwhile, incorporating spatial correlations between neighbouring roadways significantly improved the model fit as specified by the DIC criterion.

This paper reported that the effects of various explanatory variables vary among different crash types. For example, based on the results of the MVPLN spatial model presented in Table 5, road segments with sharper horizontal curves tend to experience head-on crashes rather than other multi-vehicle crash types. The results also showed that terrain type is only associated with head-on crashes. Side friction has a significant and positive effect on sideswipe crashes but has an insignificant impact on other collision types. In general, light-vehicle traffic, number of access points, segment length, shoulder width, and area type significantly contribute to the majority of crash types, although the magnitude of their effects differs across the study crash types. These findings provide better insights into the real impact of various explanatory variables on the occurrence of different collision types and can help implement certain safety countermeasures for road segments that are identified as high risk to certain crash types.

A limitation associated with the present study is that the effects of other explanatory variables, such as driver-related factors, environmental conditions at the time of crash (e.g., weather and lighting conditions), and vehicle attributes, were not addressed in this study because detailed information about these variables, which are typically conducted at the individual crash level, was not available in the current data set. Although the issue was accounted for by the MMNL model herein, the inclusion of these crash-level data along with the aggregated segment-level data can potentially improve the prediction performance of the two-stage modelling approach. To recommend, further research with detailed data, especially data at the individual crash level, is warranted to confirm and extend the paper's findings.

There are very few studies in the literature that consider a set of covariates to be random in the multivariate modelling framework (Dong et al., 2014b; El-Basyouny et al., 2014a; Barua et al., 2016). The multivariate random-parameters spatial model is believed to produce more accurate results and to be superior to the multivariate fixedparameters spatial model because the former allows some or all covariates to vary across observations, thereby increasing the precision of parameter estimates and the model's prediction accuracy. Future research is recommended to examine the use of a multivariate randomparameters spatial technique for modelling crash counts by collision type. To generalise and confirm the main conclusions reached in this study, the research should be extended using a large data set from other national transportation infrastructures (e.g., expressways, state roads, or other federal roads) where roadway geometry, the environment, and traffic characteristics may differ from those collected from the case studies of this research. This may help reveal whether the primary conclusions of this paper are generalizable and valid for other locations.

Acknowledgments

The authors would like to thank the Malaysian Institute of Road Safety Research and the Highway Planning Unit for providing the data used in this study.

hosseinpourpredictingcrashfrequencymulti2018.md 117 KB

Історія Запис

status: converted

Contents lists available at ScienceDirect

Accident Analysis and Prevention

Predicting crash frequency for multi-vehicle collision types using multivariate Poisson-lognormal spatial model: A comparative analysis

ARTICLE INFO

ABSTRACT

1. Introduction

2. Literature review

3. Methodology

3.1. MVPLN spatial model

3.2. Two-stage method

3.2.1. Modelling the total number of crashes

3.2.2. Estimating the proportion for each collision type

3.3. Model comparison

4. Study sites and data collection

5. Results and discussion

5.1. Results of two-stage model

5.2. MVPLN spatial model

5.3. GOF comparisons among three modelling approaches

5.4. Interpretation of MVPLN spatial results

6. Conclusions and suggestions for future research

Acknowledgments

References

hosseinpourpredictingcrashfrequencymulti2018.md 117 KB Історія Запис

status: converted

Contents lists available at ScienceDirect

Accident Analysis and Prevention

Predicting crash frequency for multi-vehicle collision types using multivariate Poisson-lognormal spatial model: A comparative analysis

ARTICLE INFO

ABSTRACT

1. Introduction

2. Literature review

3. Methodology

3.1. MVPLN spatial model

3.2. Two-stage method

3.2.1. Modelling the total number of crashes

3.2.2. Estimating the proportion for each collision type

3.3. Model comparison

4. Study sites and data collection

5. Results and discussion

5.1. Results of two-stage model

5.2. MVPLN spatial model

5.3. GOF comparisons among three modelling approaches

5.4. Interpretation of MVPLN spatial results

6. Conclusions and suggestions for future research

Acknowledgments

References

hosseinpourpredictingcrashfrequencymulti2018.md 117 KB

Історія Запис