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Road crash prediction models are very useful tools in highway safety, given their potential for determining both the crash frequency occurrence and the degree severity of crashes. Crash frequency refers to the prediction of the number of crashes that would occur on a specific road segment or intersection in a time period, while crash severity models generally explore the relationship between crash severity injury and the contributing factors such as driver behavior, vehicle characteristics, roadway geometry, and road-environment conditions. Effective interventions to reduce crash toll include design of safer infrastructure and incorporation of road safety features into land-use and transportation planning; improvement of vehicle safety features; improvement of post-crash care for victims of road crashes; and improvement of driver behavior, such as setting and enforcing laws relating to key risk factors, and raising public awareness. Despite the great efforts that transportation agencies put into preventive measures, the annual number of traffic crashes has not yet significantly decreased. For in-stance, 35,092 traffic fatalities were recorded in the US in 2015, an increase of 7.2% as compared to the previous year. With such a trend, this paper presents an overview of road crash prediction models used by transportation agencies and researchers to gain a better understanding of the techniques used in predicting road accidents and the risk factors that contribute to crash occurrence.

Road traffic accidents are the world’s leading cause of death for individuals between the ages of one and twenty-nine [

Traffic accidents prediction models are very useful tools in highway safety, given their potential for determining both the frequency of accident occurrence and the contributing factors that could then be addressed by transportation policies. Vehicular crash data can be used to model both the frequency of crash occurrence and the degree of crash severity. Crash frequency refers to the prediction of the number of crashes that would occur on a specific road segment or intersection in a time period [

A traffic accident may have many contributing factors, such as those related to driver behavior, road geometry, traffic volumes, vehicle, and environment. The influence of such variables on crash occurrence could significantly vary on a case-by-case basis, but in general, both behavioral factors related to the driver’s errors, and non-behavioral factors related to road geometry, traffic flow conditions, vehicle, and environment are thought to significantly affect traffic crashes [

1) Driver behavior: alcohol and drug use, reckless operation of vehicle, failure to properly use occupant protection devices, the use of cell phones or texting, and fatigue.

2) Vehicle factors: vehicle type, and the engineering and the safety design standards for vehicle performance. For example, the design of windshield glass and the location and durability of gas tanks can increase safety. Passenger protection systems in vehicles (i.e. air bags, safety belts), if used, can eliminate injuries or reduce their severity.

3) Roadway characteristics: road geometries and road side conditions, such as well-designed curves and grades, wide lanes, adequate sight distance, clearly visible striping, flared guardrails, good quality shoulders, roadsides free of obsta- cles, well-located crash attenuation devices, and well-planned use of traffic signals.

4) Traffic volumes: average annual daily traffic (AADT) or the vehicle miles travelled (VMT). AADT is the average number of vehicles passing a point along a particular road section each day. Thus, AADT represents the vehicle flow over a road section on an average day of the year. VMT refers to the distance travelled by vehicles on roads. It is often used as an indicator of traffic demand and is commonly applied to evaluate mobility patterns and travel trends.

5) Environmental factors: weather conditions, and light conditions.

6) Time factors: the season of the year, the month of the year, weekdays, and the hour of crash occurrence.

The highest cost of traffic crashes is in the loss of human lives; however, society also bears the consequences of many costs associated with motor vehicle crashes. Highway crashes currently cost the USA about $1078.0 billion a year, approximately 5.0 percent higher than 2000. Total costs include both economic costs and societal harm [

Early crash analysis models were generally based on simple multiple linear regression methods assuming normally distributed errors. However, researchers soon discovered that crash occurrence could be better fitted with a Poisson distribution. Hence, a Poisson regression model based upon a generalized linear framework was soon adopted over conventional multiple linear regression techniques. Several such Poisson regression approaches for exploring the relationship between the risk factors and crash frequency have been proposed [

There are different statistical approaches for modeling traffic crashes. The following approaches present some of the mostly used methods.

Early models of traffic accident models were based on the simple multiple linear regression approach assuming normally distributed errors. The general form of the linear crash prediction model can be expressed as follows:

where,

Y: the dependent variable (i.e. crash frequency),

θ: the crash dataset,

Dist(θ): the model distribution,

X: a vector representing different independent variables (i.e. risk factors),

β: a vector of regression coefficients,

f(.): link function that relates X and Y together,

ε: the disturbance or error terms of the model.

Although multiple linear regression models have been widely applied, it has been found that crash occurrence can often be better fitted with a Poisson distribution. One frequent pitfall is to model crash data as continuous data by applying an ordinary least square regression [

where,

P (n_{i}): the probability of n crashes occurring on a highway segment i,

n_{i}: the number of observations per time period (such as a year),

λ_{i}: the expected crash frequency on road segment i per time period (i.e. the mean of distribution) which can be estimated as follows:

where

X_{i}: a vector of the independent variables (i.e. risk factors),

β: a vector of the estimates (coefficients) of the independent variables X_{i}.

This model is estimable by standard maximum likelihood methods, with the log likelihood (LL) function given as:

One assumption of Poisson Models is that the mean and the variance are equal, an assumption that is sometimes violated [

In order to overcoming the problem of over-dispersion, the Negative Binomial (NB) distribution (also called the Poisson-Gamma) has been investigated as an alternative to the Poisson distribution given that it relaxes the condition of mean equals to variance, and hence can take into account over-dispersion in the crash data counts [

The NB uses a Gamma probability distribution and can relax the assumption of the mean equals the variance and, hence, the NB can accommodate over-dis- persion that may exist in the crash data counts [

where EXP (ε_{i}) is a gamma-distributed error with mean equals one and variance equals α. The addition of this term allows the variance VAR (n_{i}) to differ from the mean E (n_{i}) as shown in Eq. 6:

This error term is called the over-dispersion parameter, and both α and β can be estimated from the maximum likelihood function. When α is zero, the model becomes Poisson regression, and if α is found to be significantly different from zero, then the NB regression can be used instead of the Poisson regression model to handle the over-dispersion in crash data. However, the NB model also has some limitations such as its inability to handle the case of under-dis- persion of the data count, when the mean of the crash counts is higher than the variance [

To address the limitations of the NB models, the Poisson-lognormal model was introduced, in which the error term is Poisson-lognormal rather than gamma- distributed so as to better handle under-dispersed data counts [_{i}) term used in the model is lognormal-rather than gamma-distributed. The Poisson-lognormal model provides more flexibility than the negative binomial model, but it does have some limitations, such as, its complex estimation of parameters due to the fact that the Poisson-lognormal distribution does not have a closed form [

Another widely used crash frequency modeling approach is the zero-inflated Poisson and zero-inflated negative binomial models, which have been introduced primarily to deal with the over-dispersion problem caused by excessive zeroes (i.e. locations where no crashes can be observed) in traffic data counts. The zero-altered procedure allows modeling the crash frequencies in two states, namely; the zero-crash state, and the non-zero crash state (where crash frequencies follow Poisson or negative binomial distribution), and the probability of a section being in zero or non-zero states can be found by a binary logit or probit model. In crash data, large numbers of zero observations are commonly present largely due to under reporting of minor crashes at these sites, the presence of dangerous crash sites (i.e. non-zero crash sites) in close proximity to the neighboring zero crash sites rendering the zero-crash sites to the safe mode, and given that some of zero crash sites may be free from only certain type of crashes, not all types of crashes [_{i} for the zero crash state, and (1-p_{i}) for the non-zero crash state, and the overall probability of crashes is the sum of the probabilities from each state. The probability of crash frequency in the zero state can be modeled as:

where R_{i}(0) is the probability of zero crashes that occurs in the zero state. The probability of crash frequency in the non-zero state can be modeled as:

where R_{i} (n_{i}) is the probability of non-zero crashes in the non-zero state. Maximum likelihood estimates can be used to estimate the parameters of both ZIP and ZINB regression models and confidence intervals are constructed by likelihood ratio tests. In zero-inflated models, the two state process is assumed to follow a logit (logistic) or probit (normal) probability process [

The Conway-Maxwell Poisson model has been recently investigated with respect to highway safety issues, but its application in crash frequency modeling has been rather limited [

Random-parameters models have also been investigated to take the effect of the unobserved heterogeneity from one roadway site to another [

The motivation for random-parameter models is to account for unobserved heterogeneity across observations. Random-parameter models can be derived by assuming that the estimated parameters vary across observations according to some distribution. Estimated parameters can be modeled as [

where

β_{n}: a vector of estimated parameters of the n observations,

ω_{n}: a randomly distributed term.

With this equation, the Poisson, and the Negative Binomial parameters become:

Given that a linear function may not sufficiently explain the relationship between the dependent variables and the associated independent variables in crash modeling, non-linear approximators such as fuzzy logic and neural networks have also been explored. Artificial Neural Networks (ANNs) are a class of computational intelligence tools that can be used for prediction and classification problems. ANNs can model very complex non-linear functions to high accuracy levels using a process of learning that is similar to the learning procedure of the cognitive system in the human brain. The network body is composed of input layers, hidden layers, and output layers. These models can be trained to appro- ximate any nonlinear function to a required degree of accuracy using a learning algorithm (such as back propagation) that would give the desired output, in a supervised learning process. ANNs have some advantages over the statistical models. For instance, regression models need a pre-defined relationship or functional form between the dependent variable (crash frequency) and the independent explanatory variables that can be estimated by some statistical approaches, whereas the ANNs do not require the establishment of these functional forms, and can be easily applied in the analysis. On the other hand, the ANNs differ from the statistical models in that they behave as black-boxes and do not provide interpretation for the parameter estimates [

Logit and Probit models can be applied to study crash severity modeling. The data used in modeling crash severity is often attributed with many details relating to the crash occurrence (i.e. such as the number of vehicles involved, age of victims, weather conditions, types of vehicles involved, and crash type) which can be integrated in statistical models. Since the dependent variable (i.e. crash severity) usually has two or more outcome categories (i.e. fatal, injury, property-damage-only), logit and probit models are often used to model the severity of crash data. Discriminant analysis could also be used to model crash severity, but given its rigid assumptions, logit and probit models have been viewed as preferable [_{i}= 1)):

where,

X_{i}: a vector of explanatory variables (i.e. risk factors),

β: a vector of regression coefficients.

As ^{−1}) that can be expressed as:

There are many types of the multinomial models that can be used in modeling crash severity, such as, the multinomial logistic regression (MNL), the nested logistic regression, the mixed logistic regression, and the multinomial probit models. For example, The MNL tries to find the best fitted model to describe the relationship between the polytomous dependent variable with more than two categories and a set of independent variables. The logistic regression model is a non-linear transformation of the linear regression model, as it consists of an S-shaped distribution function [

where,

p: the probability of presence of an outcome of interest,

X_{k}: the vector of k independent variables,

b_{0}: the regression coefficient on the constant term (intercept),

b_{k}: the vector of regression coefficients on the independent variables X_{k},

The odd ratio is the probability of the event divided by the probability of the nonevent, and is defined as follows [

When p = 0, then odd (p) = 0, when p = 0.5, then odd (p) = 1.0, and when p = 1.0, then odd (p) =

The transformation from odds to log of odds is the log transformation, and this is a monotonic transformation. That is, the greater the odds, the greater the log of odds and vice versa. Logit (p) can be back-transformed to p by the following formula:

The transformation from probability to odds is a monotonic transformation as well, meaning the odds increase as the probability increases or vice versa. Probability ranges from 0.0 and 1.0. Odds range from 0.0 and positive infinity [

Traffic crash prediction models are very useful tools in road safety programs used by transportation agencies, police, health departments, education institutions that oversee road safety, vehicles, and the driver’s education. They can be used to predict both the frequency of crash occurrence and the contributing factors that could then be addressed by transportation policies. According to the world health organization (WHO), road crashes are ranked as the ninth most serious cause of death in the world, and present the world's leading cause of death for individuals between the ages of one and twenty-nine. Each year, traffic accidents are responsible for killing about 1.25 million people and injuring approximately 50 million more. Following current trends, about two million people could be expected to be killed in motor vehicle crashes each year by 2030. The World Bank estimates that road traffic injuries cost 2.0 percent to 3.0 percent of the Gross National Product of developing countries. Given a such trend, this paper presented different types of traffic crash prediction models to gain a better understanding of the techniques used to predict road accidents and their contributing risk factors. A wide range of statistical approaches were presented including, Poisson regression, Negative Binomial regression, Zero-Inflated models, logit and probit models, and machine learning methods.

Abdulhafedh, A. (2017) Road Crash Prediction Models: Different Statistical Modeling Approaches. Journal of Transportation Technologies, 7, 190-205. https://doi.org/10.4236/jtts.2017.72014