Count Models Analysis of Factors Associated with Road Accidents in Nigeria

The injury prevention of the SDG that is related to road safety appears in SDG 3.6 and SDG 11.2 under “good health and well-being” and sustainable cities and communities,” respectively. This stress makes the environment safe for people. To achieve global sustainable growth, injury has to be drastically reduced. Ultrasonics Sonochemistry (2021) [1], the study by Ma et al. [2] highlighted thirteen more relevant areas of SDGs that relate to injury prevention beyond SDGs 3.6 and 11.2.

When a road vehicle collides with another vehicle, a person, an animal, or a topographical or architectural obstruction, it is called a road traffic accident (RTA). Injury, property destruction, and death are all possible outcomes. RTA kills 1.2 million people per year and injures four times as many [3]. A road traffic accident is defined in this study as a collision on the road between two or more objects, one of which must be a moving vehicle of any sort [4]. Road traffic accidents (RTAs) are on the rise in developing countries, and they are now one of the main causes of death.

Accidents and fatalities on the road are the results of a complex interaction of factors. In Nigeria, road users range from pedestrians, rickshaws, bicycles, hand carts, and tractor trolleys to various types of two/three-wheelers, cars, trucks, buses, and multi-axle commercial vehicles, among other things. Because of changes in people's standards of life and urbanization, the automobile population has been steadily expanding. The need and urgency for a well-thought-out policy on the subject of road safety have been highlighted by growth in vehicle population combined with limited road space used by a wide variety of vehicles.

Accidents are becoming more common in Nigeria as the country's population of vehicles grows. Road traffic accidents are human tragedies that cause a significant deal of suffering. They impose a significant socio-economic cost in terms of untimely deaths, injuries, and lost revenue opportunities. Road traffic accidents can have devastating implications, affecting not only people, their health, and well-being, but also the economy. As a result, road safety has become a national concern. Road safety is a multi-faceted and multi-sectorial problem. This includes things like improving and supervising road infrastructure, providing safer vehicles, enforcing rules and laws, mobility planning, providing health and hospital services, child safety, and urban land use planning, among other things. In other words, its scope includes engineering elements of both automobiles and roads on the one hand, as well as health and medical services for trauma cases in the aftermath of a crash on the other.

The biggest issue that Nigerian roads are facing is deterioration owing to a lack of road quality construction by the engineering business in charge, or simply a lack of road maintenance [5]. Accident rates have resulted in many lives being lost over many years, and the trend of death continues to rise. Until now, Nigeria's major roadways have remained in a state of terror, with people fearful of traveling on them for their daily needs or any other reason. According to a report by Saliu et al. [6], numerous locations across the country had extremely inhumane road conditions that had been neglected for many years, and the governments of the various states had never considered constructing them.

According to statistics, around 24% of national highways in Nigeria are in inconvenient shapes, and this trend has continued to rise to 60% since 1999 [7]. According to the report [7], it will cost nearly N350 billion to bring Nigerian roads up to standard or in good repair over the next 9 to 10 years, and upkeep of Nigerian roads will cost N25 billion each year on average, with a cost of N33 billion for road rehabilitation. Ignoring these roads will inevitably result in traffic accidents, resulting in a high number of fatalities and injuries, and, as a result, the country's public and profit-making development would be hampered. The major goal of this research study is to examine the factors that cause road accidents in Nigeria using the quantitative tool of higher extensions of the Poisson regression model because the current state of all Nigerian roads is in poor condition and reports of accidents have been recorded across the federation.

Vehicle-related, environment-dependent, and driver dependent (i.e., pedestrian dependent for hazard between a motor and a pedestrian) are the three categories of risk factors discovered in road accidents.

According to references [8-10], the majority of road crashes are caused by driver factors, which account for 60 percent to 90 percent of all road accidents. 'The constant rising toll of catastrophes or fatalities and wounds or injuries from road crashes in low-income countries is a significant item or an essential success factor in the spontaneous growth in the value of the motor, including vehicles, cars, and trucks [11-13].

Corruption is also a problem in some nations, like Nigeria, where an unskilled individual or driver would be unable to get through driving school and obtain a driver's license. With bribery, the driver will seek alternate means of obtaining licenses through corruption by meeting with license issuance authorities, and as a result, there is a greater likelihood of a death occurring while he or she is on the road. The issue will be raised between the driver and the police, blame between the public and the driver, or the public will blame the police and the drivers, and the police and drivers will blame the public in either direction.

Many authors' efforts and contributions on the factors that cause road accidents are well appreciated and will be remembered for a long time. In various areas of the world [14-16], and also in Nigeria [17-20] different studies have been carried out on factors associated with road accidents. Most of these studies focused to identify determinants of factors associated with road accidents and also used binary logistic regression and Cox proportional regression models. Moreover, different approaches are commonly used to model the number of factors associated with road accidents in different studies. These models might be more recommended to measure the incidence of factors associated with road accidents and associated factors. There is a need for data on the causes that cause road accidents, and additional study is needed to update policy design and programme implementation for the best health intervention.

All of the aforementioned authors, on the other hand, used a variety of statistical methodologies to address the elements that could lead to road accidents, resulting in a variety of statistical outcomes. However, most of the authors' work is focused on regression models, which are unable to capture overdispersion; so, the purpose of this research study is to investigate overdispersion as a gap to be filled.

Firstly, descriptive statistics of all stimulus variables and the controlled variable are found to enable us to have a glimpse of these variables, then followed by the test of collinearity of all stimulus variables. The collinearity test was carried out using the variance inflation factor (VIF), mathematically, VIF is written as:

$V I F=G V I F^{\frac{1}{2 d f}}$

where, GVIF $=$ is the generalized variance inflator factor and is the degree of freedom. But and are reported as well. It is important to note that a variance inflation factor (VIF) value greater than 5 indicates the presence of a correlation between a given independent variable and other independent variables in the model. Once the test of collinearity is done, we proceed to perform a backward stepwise vector generalized linear regression model to enable us to know the variables to use in the actual or final model building or simply the model selection using Alkaike information criteria to select the best model, though VIF will identify that we just to do perform this as well. Stepwise regression is the step-by-step repeated process of building of a regression model that entails the selection of stimulus variables to be worn in a final model. This entails deleting or adding control variables in cycles and testing for statistical significance after each repetition. The mathematical methodology of backward regression model is described below and estimators $\alpha, b_1-b_{18}$ are estimated using maximum likelihood estimation method:

$Y_t=\alpha+b_1 X_1+b_2 X_2+b_3 X_3+b_4 X_4+b_5 X_5+b_6 X_6+b_7 X_7+b_8 X_8+b_9 X_9+b_{10} X_{10}+b_{11} X_{11}+b_{12} X_{12}+b_{13} X_{13}+b_{14} X_{14}+b_{15} X_{15}+b_{16} X_{16}+b_{17} X_{17}+b_{18} X_{18}+\epsilon_t$                      (1)

where, $\epsilon_t=$ stepwise regression error or stepwise regression residual and $Y_t=$ number of road accidents and finally $\alpha=$ stepwise regression constant.

Eq. (1) is built and each explanatory variable is deleted at a time as the regression advances and this is based on the following criteria; at each iteration, the variable with the lowest “F to remove” statistic is deleted from the model. Once the explanatory variables are all identified, then we move ahead to build our proposed model, which is a Zero truncated Poisson regression model to account for zero truncation. From the concept of Poisson regression model which models count of data and is described as:

$p\left(y_i\right)=\frac{e^{-\mu_i} \mu_i^{y_i}}{y_{i} !}$

$i=1,2,3,4,5, \ldots \ldots$          (2)

where, $\mu_i=$ long-run expected accidents count occurring at $i^{t h}$ road at a particular time. And also $p\left(y_i\right)$= is the probability of y accidents occurring at the i road during a particular period. Now in the Poisson regression model, the long-run expected accident count is assumed to be a function of control predictors described mathematically as:

$\mu=\exp \left(\vartheta_0+\vartheta_1 X_1+\vartheta_2 X_2+\vartheta_3 X_3+\vartheta_4 X_4\right. \left.+\cdots \vartheta_n X_n\right)$             (3)

where, $X_1, X_2, X_3, \ldots, X_n$ are the predictor variables and their respective coefficients are $\vartheta_1, \vartheta_2, \vartheta_3, \ldots, \vartheta_n$ which all stands for statistical significance of each predictor variable, and they are estimated using the maximum likelihood or the through the method of moment. By taking the logarithm of Eq. (3), thus we have:

$\log \left(\mu_i\right)=\log \left(\exp \left(\vartheta_0+\vartheta_1 X_1+\vartheta_2 X_2+\vartheta_3 X_3+\vartheta_4 X_4\right.\right. \left.\left.+\cdots \vartheta_n X_n\right)\right)$

$\log \left(\mu_i\right)=\left(\vartheta_0+\vartheta_1 X_1+\vartheta_2 X_2+\vartheta_3 X_3+\vartheta_4 X_4+\cdots \vartheta_n X_n\right)+\epsilon_t$

Now recall that

$\mu_i=E\left(Y_i\right)$;

$\rightarrow \log \left(E\left(Y_i\right)\right)=\left(\vartheta_0+\vartheta_1 X_1+\vartheta_2 X_2+\vartheta_3 X_3+\vartheta_4 X_4+\cdots \vartheta_n X_n\right)+\epsilon_t$

$\log \left(Y_i\right)=\left(\vartheta_0+\vartheta_1 X_1+\vartheta_2 X_2+\vartheta_3 X_3+\vartheta_4 X_4+\cdots+\vartheta_n X_n\right)+\epsilon_t$          (4)

From Eq. (4) is the Poisson regression model that model the dependent variable of counts (which can include zero counts i.e. no zero truncation) as a function of many explanatory variables, But this research is now because the dependent variable is disregarding zero i.e. zero is been truncated, hence if $Y_1=0$, its $P_r\left(Y_1=0\right)=e^{-\mu_1}$ and this is obtained from Eq. (2), then its conditional probability mass function is defined as

$P_r\left(Y_1=y_1 \mid Y_1>0\right)=\frac{P_r\left(Y_1=y_1\right)}{P_r\left(Y_1>0\right)}=\frac{P_r\left(Y_1=y_1\right)}{1-P_r\left(Y_1=0\right)}$             (5)

Now using Eq. (2), the zero truncated probability mass function for $\left(Y_1 \mid Y_1>0\right)=$

$f_1^*\left(y_1\right)=P_r\left(Y_1=y_1 \mid Y_1>0\right)$

$=\frac{e^{-\mu_1} \times \mu_1^{y_1}}{y_{1} !} \times \frac{1}{\left(1-e^{-\mu_1}\right)}=\frac{\mu_1^{y_1}}{y_{1} !\left(e^{-\mu_1}-1\right)}$           (6)

Then taking the exponential form of the mass function in Eq. (6), thus we have:

$f_1^*\left(y_1\right)=\exp \left[y_1 \ln \left(\mu_1\right)-\ln \left(y_{1} !\right)-\ln \left(e^{\mu_1}-1\right)\right]$                 (7)

Similarly, the zero truncated conditional distribution of $\left(Y_2 \mid y_1, Y_2>0\right)$ is

$P_r\left(Y_2=y_2 \mid Y_2>0\right)=\frac{P_r\left(\left(Y_2=y_2 \mid y_1\right)\right)}{P_r\left(Y_2>0 / y_1\right)}=\frac{P_r\left(\left(Y_2=y_2 \mid y_1\right)\right)}{1-P_r\left(Y_2=0 / y_1\right)}$            (8)

The zero truncated condition Poisson distribution of Eq. (8) is given as

$f_2{ }^*\left(y_2\right)=f_2{ }^*\left(\left(Y_2=y_2 \mid y_1, Y_2>0\right)\right)$

$=\frac{e^{-\mu_2 y_1} \times\left(\mu_2 y_1\right)^{y_2}}{y_{2} !} \times \frac{1}{\left(1-e^{-\mu_2 y_1}\right)}=\frac{\left(\mu_2 y_1\right)^{y_2}}{y_{2} !\left(e^{-\mu_2 y_1}\right)}$              (9)

Now taking the exponential form of the mass function in Eq. (10), thus we have:

$f_2^*\left(y_2\right)=f_2^*\left(\left(Y_2=y_2 \mid y_1, Y_2>0\right)\right)$

$=\exp \left[\left(y_2 \ln \left(\mu_2\right)+y_2 \ln \left(y_1\right)-\ln \left(y_{2} !\right)-\ln \left(e^{\mu_2 y_1}-1\right)\right)\right]$              (10)

Both Eq. (6) and Eq. (9) are two zero truncated conditional distributions, finding their joint distribution will lead to another model called zero truncated bivariate Poisson model and which is not our proposed model. Eq. (7) is used to estimate our long-run expected frequency $\mu_1$ from the conditional distribution.

Hence, from Eq. (7) we say:

Let $L=f_1^*\left(y_1\right)$, then

$L=\exp \left[y_1 \ln \left(\mu_1\right)-\ln \left(y_{1} !\right)-\ln \left(e^{\mu_1}-1\right)\right]$            (11)

Now $\ln \left(\mu_1\right)=X^{\prime} \varphi_i$, where $X^{\prime}=\sum_{i=1}^n X_i$ now from Eq. (11), we have that:

$L=\exp \left[y_1 \ln \left(\sum_{i=1}^{18} \varphi_i X_i\right)-\ln \left(y_{1} !\right)-\ln \left(e^{\sum_{i=1}^{18} \varphi_i X_i}-1\right)\right]$           (12)

Now differentiating Eq. (12) with respect to the parameters $\varphi_i$ and then equate to 0 to estimate the parameters $\varphi_i$. Then this $\varphi_i$ are now substituted into this equation:

$\hat{Y}_t=\varphi_0+\varphi_1 X_1+\varphi_2 X_2+\varphi_3 X_3+\varphi_4 X_4+\varphi_5 X_5+\varphi_6 X_6+\varphi_7 X_7+\varphi_8 X_8+\varphi_9 X_9+\varphi_{10} X_{10}+\varphi_{11} X_{11}+\varphi_{12} X_{12}+\varphi_{13} X_{13}+\varphi_{14} X_{14}+\varphi_{15} X_{15}+\varphi_{16} X_{16}+\varphi_{17} X_{17}+\varphi_{18} X_{18}+\epsilon_t$                 (13)

where, $\hat{Y}_t=$log expected number of road accidents and $X_i$ are the stimulus variables and which are reduced by a variance inflation factor. Once Eq. (13) has been estimated, we determined whether it is good or appropriate to describe such explanatory variables, and this is done by testing for model fitness, this test explains whether all covariate variables will explain the response variable (number of road accidents).

The model fitness is checked by applying the residual deviance which measures the change in the deviance between the given model and the observed model i.e., the model that fits the data. Residual deviance has a degree of freedom n-p, where n= number of observations and p= number of parameters in the model. In this research, the number of observations we had is 36 observations plus 19 parameters, constant inclusive. Now, if the residual deviance is greater than its corresponding degrees of freedom then the model does not fit the data i.e., the observed values are not close to the predicted means and causing both of them to be large or have a great discrepancy.

In a nutshell, the residual deviance should be relatively small. For a fitted zero truncated Poisson regression model, the deviance D is given by:

$D=2 \sum_{i=1}^N\left[y_i \log \left(\frac{y_i}{\mu_i}\right)-\left(y_i-\mu_i\right)\right]$          (14)

where, $y_i$ observed number of road accidents that occurred at a particular period for any state and $\mu_i$ predicted number of road accidents that occurred at a particular period for any state.

Once the model of the goodness of fit test is done, we determine whether the model in Eq. (13) is over dispersed i.e., we test whether we need to estimate the overdispersion parameter. The mathematical illustration of how to test for overdispersion of the model for this research study is illustrated below:

$\operatorname{disp}_{\text {change }}=2 \times\left(\begin{array}{c}\log \operatorname{like}\left(m_1\right)- \\ \operatorname{loglike}\left(m_2\right)\end{array}\right)$

where, $m_1$ is the log-likelihood obtained when Eq. (13) is modeled with positive negative binomial distribution and $m_2$ is the log-likelihood obtained when Eq. (13) is modeled with positive Poisson regression.

Then dispersion $_{\text {change }}$ is tested using chi-squared under the degree of freedom of the number of dispersed parameters.

The hypothesis is:

$H_0$: No need for dispersion parameter

$H_1$: There is a need for dispersion parameter

The decision rule: if the observed p-value is lower than the exact p-value then $H_0$ is rejected, hence otherwise.

The essence of checking for overdispersion is that if found that the model is over dispersed, it has a great negative effect in estimating the standard error of estimates of the parameters. Once there is a presence of overdispersion, the proposed model is being corrected with a zero truncated negative Poisson regression model.

The main objective of the study was to develop a statistical method of predicting factors that lead to road accidents in Nigeria. A cross-sectional study design was adopted and secondary data from the National Bureau of Statistics (2020) was used within a sample period of the 1st quarter of 2006 to the 2nd quarter of 2020. The statistical method adopted was the extension of the Poisson regression model such as the zero truncated Poisson regression model to truncate zeros that occurred in the points of the dependent variable (number of road accidents) and a further Poisson regression model was proposed to correct for overdispersion and which is the Zero truncated negative Poisson regression model (ZTNPRM). From the ZTNPRM, the model indicates that human errors contribute to a large proportion of 41.4% (3444/8319) in leading to road accidents. Such human factors are the greatest cause of road accidents and such factors are; the use of phones while driving (2.224,0.0262), dangerous overtaking (1.141,0.02538), wrongfully overtaking (6.001,1.96e-09), dangerous driving (6.886,5.74e-12), road obstruction violation (1.622,0.01048), route violation (2.401,0.0164) and sign light indicator violations (3.253,0.0011) as these variables were statistically significant and positively related in predicting the occurrence of road accidents in Nigeria. Other vehicular factors were also positive and statistically significant and such variables were Tyre burst (7.252,4.09e-13), brake failures (4.703,2.56e-06).

The goodness of fit test was carried out on the proposed model to fit count data used and the result showed that there is a presence of overdispersion, the estimated residual deviance was lower than the corresponding degree of freedoms, hence indicating that the model used deeply explains the structure of the contributing factors in question, and similarly, overdispersion was present and statistically significant, hence the model of Zero truncated Poisson regression model was improved or corrected by Zero truncated negative Poisson regression model. Finally, the study recommends that Nigerian car users strictly follow all rules and regulations associated with safe driving, such as not drinking, not breaking speed limits, not overtaking unnecessarily or incorrectly when such is not needed, etc. All the factors identified by this model that cause road accidents predicted low road accidents as a result. Before starting a trip, vehicle factors can be improved by testing the tire strength, and all brake systems should be fully functional and in good condition.