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Inference on Reported Vehicular Fatal Accidents in Nigeria Using a Bayesian Model

The study introduced a special case of the Poisson-Generalized Gamma empirical Bayes model to survey states in Nigeria with a higher risk of fatal accidents. Monte Carlo error and stationary dynamic trace plots were used to validate model convergence and accuracy of the posterior estimates. The main results included the disease mappings that revealed Ebonyi had the highest risk of road vehicular fatal accidents in Nigeria with a relative risk estimate of 1.4120 while Abuja had the lowest risk with a relative risk estimate 0.5711. In terms of geopolitical region, the risk of road vehicular fatal accident is highest in South-South region with a relative risk estimate of 1.1850 while North-Central had the lowest risk with a relative risk estimate of 0.7846. The study is to aid planned government programs to ameliorate vehicular road carnage in Nigeria. 


Introduction 1
The alarming rate of road vehicular accidents in Nigeria is becoming worrisome to individuals and government alike. According to the WHO (2018), accidents caused an estimated 1.35 million deaths worldwide. Further, as noted in the WHO (2018) report, road accident is one of the leading causes of death in the world. The risk of dying by road accident injury is highest in the African region with 26.6 traffic deaths per 100,000 people and lowest in the European region with 9.3 traffic deaths per 100,000 people. Nigeria statistics is succinctly put at 20.5 traffic deaths per 100,000 people and 615.4 traffic deaths per 100,000 motor vehicles, with a total fatality of 46,475 between 2013 and 2019 (NBS, 2019). These figures placed Nigeria among the highest in the rate of road accidents in the world and traffic deaths per inhabitants.
According to WHO (2018), the Nigeria figures are relatively very high compared with the United States' 12.4 traffic deaths per 100,000 populations, 14.2 traffic deaths per 100,000 motor vehicles, with the United Kingdom's 3.1 deaths per 100,000 people, 5.7 traffic deaths per 100,000 motor vehicles and with the China's 18.2 traffic deaths per 100,000 populations, 104.5 traffic deaths per 100,000 motor vehicles. As highlighted in NBS (2019), the vehicle population in Nigeria is put at 11,826,033, and the Nigeria's vehicle per population ratio is put at 0.06. The road is a primary means of commuting in Nigeria. According to NBS (2019), the current vehicular density in Nigeria is put at 60 vehicles every 1km, which poses a major challenge to road traffic. As discussed in   Vogelesang (1997). The study was motivated because of the growing number of fatal vehicular crashes on Nigerian roads. Also, the need to identify the hotspot states towards curbing vehicular crashes on Nigerian roads. The remainder of the paper is structured as follows: the special case of PGG EB model is presented in Section 2, Section 3 deals with the data application and results, discussions of results was done in Section 4 and Section 5 highlighted the Summary and conclusion.

Methodology
The proposed EB model is a special case of Poisson-Generalized Gamma model (PGG) introduced by Mbata et al. (2018). The model is built on Bayes' Theorem and has the form of Equation (1) (Gelman et al., 2004): Where, ∫ ( | ) ( |∅) is the unconditional marginal distribution whose inverse is the constant of proportionality ( ). The quantity is a normalizing constant to ensure that the posterior distribution (̃| , ∅) is a proper density. ∅ represents the hyperparameters of the prior distribution ( ( |∅)) usually estimated from the observed data ( ). It implies that: (i) (̃| , ∅) are the posterior distributions of the parameters given ( ) and hyperparameters (∅) in the model after observing the data. (ii) ( | ) is the likelihood function of the probability distribution with respect to , which reflects the relationship between the data and the parameter(s). (iii) ( |∅) is the prior distribution of the parameter given the hyperparameters (∅), which reflects the initial information on the parameter(s). Generally, a Bayesian model consists of a likelihood distribution and a prior distribution. The inference about the parameter of interest is based on the posterior distribution using MCMC sampling technique (Gelman et al., 2004).
Therefore, the PGG EB model is a conjugate Poisson-Generalized Gamma model where the Poisson distribution represents the observed data likelihood and the Generalized Gamma (GG) distribution is used as the prior distribution of the Poisson parameter. As discussed in Vogelesang (1997), Hauer (1995), the Poisson distribution has become a standard for analysing accident data. However, the choice of prior for appropriate modelling differs and depends on the nature of the study. Thus, under PGG conjugacy, as highlighted in Mbata et al. (2018), the posterior density distribution is obtained as Equation (2) The PGG relative risk estimator, variance (Var) and standard deviation (SD) are obtained as The relative risk estimator, variance (Var) and standard deviation (SD) are derived as To completely specify the posterior distribution model, the hyperparameter α of the prior distribution is estimated from the GG distribution using a method of moment estimation proposed by Huang and Hwang (2006). Given the pdf of GG distribution, when β = 1, as Thus, the rth Moment is expressed as The mean and variance are expressed in Equation (13) and Equation.
(15) as Therefore, square of the coefficient of variation (CV) is obtained as [9], opined that when = 1.0 gives a Gamma distribution and when = 2.0 approximately gives a Generalized Normal distribution. To optimize , value 0.5 is assumed. Therefore, when = 0.5 in Equation (16), we have; .
Simplifying by completing the squares, we have; According to Marshall (1991), and 2 are estimated as Monte Carlo error (MCE) and stationary dynamic trace plots are carried out to evaluate the accuracy and convergence of posterior estimates of the EB model. MCE estimates the difference between the mean of the sampled values and the true posterior mean value. As a rule of thumb, the simulation is run until the Monte Carlo error is less than about 5% of the sample standard deviation (Brooks and Gelman, 1998). For good details of Bayes theorem, see Gelman et al. (2004).

Results
The special case of PGG EB model is applied to reported road vehicular fatal accidents in Nigeria by states. The data were sourced from the National Bureau of Statistics (2013, 2014, 2015, 2017, 2018 and 2019), and the Nigerian states by geo-political zone aggregated data with the estimated expected counts are presented in Table 1. However, the 2016 data was excluded due to inconsistency in reporting with other years. The investigation is carried out using MCMC sampling technique by OpenBUGS statistical software and the program codes are found in the appendix. The posterior results of the EB model are presented in Table 2. The diagnostic dynamic trace plots are presented in Figure 1 for the six geo-political zones, respectively. The relative risk of fatal accident mapping is presented in Figures 2 and 3, respectively.    Table 1 with the corresponding total number of accidents and estimated expected fatal road accidents. The results from Table 2 indicate that the estimates of relative risk (RR) of road vehicular fatal accidents by state in Nigeria range from 0.5711 (FCT Abuja, the lowest) to 1.4120 (Ebonyi state, the highest). This implies that the risk of having a road vehicular fatal accident is highest in Ebonyi State and lowest in FCT Abuja. Meanwhile, RR ≥ 1 implies higher risk while RR < 1 implies lower risk. Therefore, the relative risk estimates of twenty-three (23) states (Ebonyi, Borno, Ogun, Rivers, Yobe, Osun, Bayelsa, Jigawa, Delta, Cross-River, Oyo, Akwa-Ibom, Sokoto, Kwara, Bauchi, Ondo, Imo, Taraba, Edo, Zamfara, Kano, Gombe and Anambra) indicate higher risk of road vehicular fatal accidents. While fourteen (14)  The results indicated that there is the accuracy of the posterior estimates of the EB model since MCE<5%SD respectively. Consequently, the convergence of MCMC sampling as the chains overlap each other is shown in the stationary dynamic trace plots presented in Figure 1. Suggesting that the posterior estimates of the special case of PGG EB model is highly reliable.

Conclusion
The analyses have shown that South-East, North-Central and North-West regions have a lower risk of road vehicular fatal accidents while South-South, South-West and North-East regions have a higher risk of road vehicular fatal accidents. Therefore, it can be inferred that accidents in the South-East, North-Central and North-West are not usually fatal, though most major highways in the South-East are in poor condition, unlike in the North-Central and North-West regions where most highways are relatively in good condition. In addition, it was found that Ebonyi state has the highest risk of road vehicular fatal accidents in Nigeria because more than 50% of the crashes are fatal as a result of speed violation and poor condition of vehicles, as viewed by Ohakwe et al. (2011).
The high risk of road vehicular fatal accidents in the South-South and South-West can be attributed to high vehicular traffic density and over-speeding in most major highways in the regions, while North-East region is as a result of the poor condition of the major highways. Also, it can be deduced that the risk of road vehicular fatal accident in Lagos state and FCT Abuja is lower compared to Kano and Rivers states at higher risk of road vehicular fatal accident. Though Lagos state has a high vehicular density, the high traffic congestion in Lagos state and FCT Abuja alike is highly likely to reduce fatal crashes.
Finally, based on the results obtained in terms of geopolitical region, the risk of road vehicular fatal accident is highest in the South-South region with a relative risk estimate of 1.1850 while North-Central had the lowest risk of road vehicular fatal accidents with a relative risk estimate of 0.7846. Generally, the major highways in Nigeria are highly vulnerable to fatal accidents due to the deplorable condition of the roads. This has been previously highlighted in Atubi (2010). The study is highly likely to aid planned government programs towards ameliorating and curbing vehicular road carnage in Nigeria. The study recommends comprehensive rehabilitation and reconstruction of major highways in Nigeria. The study has added to the body of literature the use of a special case of PGG model in analyzing and mapping accident data.