Statistical Analysis of Pattern on Monthly Reported Road Accidents in Nigeria

Road accidents in Nigeria have always been in the increase. Efforts made by Federal Road Safety Commission (FRSC) in tackling the menace have not yielded much result. This paper aims to find a suitable time series model to forecast the future characteristics of the road accident data on Oyo-Ibadan express road. The data used for this paper was monthly data collected for a period of Eleven years between 2004-2014. In achieving this, the additive model approach was adopted in the analysis. It includes the estimation of trend, seasonal variation and random variation using moving average method. Autoregressive Moving average model were also fitted to the data and the best order was choosing using Akaike Information Criteria (AIC). The order = c (1, 1, 2), seasonal = c (1, 1, 2)) gives the best description of the data with minimum (AIC). A forecast based on the model obtained was made by the use of m-step predictor. The time plot plotted shows that the graphs maintain a constant movement from 2004-2008 but increases abnormally in 2010 and later drop again maintaining appreciable downward movement as the year progresses. Judging from the result, accidents and deaths are higher during the festive period months because of the various festivities lined up during this period, which involve much more traveling than usual.


Introduction
Road traffic fatalities are forecast to increase over the next ten years from a current level of more than 1.3 million to more than 1.9 million by 2020. The Commission for Global Road Safety believes that the urgent priority is to halt this appalling and avoid able rise in road injury and then begin to achieve year on year reductions. The world could prevent 5 million deaths and 50 million serious injuries by 2020 by dramatically scaling up investment in road safety, at global, regional and national levels. Each year nearly 1.3 million people die as a result of a road traffic collision, more than 3000 deaths each day and more than half of these people are not travelling in a car. According to John Cohen in his book, causes and prevention of road accidents "says if we had the will, we should find ways, for we cannot assume to the problem of road safety are beyond the wit of man to solve once they are identified. We do not have the will because we are not sufficiently moved by disaster on the road". [1] Define accident as a chance occurrence, which produce unexpected and unpleasant consequences resulting from unforeseen and often a disastrous event.
According to the UN Secretary General, Ban Ki-Moon, lives will be saved through this decade of action. Following the declaration by the UN in 2011, the Federal Road Safety Commission (FRSC) in Nigeria set out to adopt and domesticate the UN action plan by developing a number of programmes suitable for every road user in the country. Despite integrated efforts towards reducing fatal road accidents, Nigeria still remains one of the worst hit countries. With a human population of about 167 million, a high level of vehicular population estimated at over 7.6 million, a total road length of about 194,000 kilometres (comprising 34,120 km of Federal, 30,500 km of State and 129,580 km of local roads), the country has suffered severe losses to fatal car accidents. Its population density varies in rural and urban areas at about 51. Undoubtedly, this immense pressure contributes to the high road traffic accidents in the country [2]. Nigeria is ranked secondhighest in the rate of road accidents among 193 countries of the world Thus, Nigeria's annual 8,000 to 10,000 traffic accident deaths between 1980 and 2003 were a major personal and traffic safety problems as well as a terrible waste of human resources for the country. In terms of the personal safety problem, Nigeria is a high risk region with an average of 32 traffic deaths per 1,000 people [3]. This is very high compared with the United State's 1.6 traffic deaths per 1,000 populations and with the United Kingdom's 1.4 deaths per 1,000 people [4]. In terms of traffic safety, there are on average 23 accidents per 1,000 vehicles in Nigeria (i.e. 230 per 10,000 vehicles) far in excess of the accident rate in the USA (2.7 accident per 10,000 vehicles) and the UK (3.2 accidents per 10,000 vehicles).
According to [5], between 1970 and 2001, Nigeria recorded a total of 726,383 road traffic accidents resulting in the death of 208,665 persons and 596,425 injuries. In that period, each succeeding year recorded more accidents, deaths and injuries. Indeed, the Nigeria accident pattern seems to suggest that the better the road, the higher the accident and fatality rate as well as the severity and non-survival indices because of driver noncompliance with speed limits [3,6,7]. Road accidents claim the largest toll of human life and tend to be the most serious problem all over the world [8]. Worldwide, the number of people killed in RTA each year is estimated at almost 1.2 million while the number of people injured could be as high as 50 million (WHO, 2004). Currently, motor vehicle accidents rank 9 th in order of disease burden and are projected to be ranked 3 rd in the year 2020. Nearly three-quarters of deaths resulting from motor vehicle crashes occur in developing countries [9] and this problem appears to be increasing rapidly in these countries [10]. Apart from humanitarian aspect of the problem, traffic accidents and injuries in these countries incur an annual loss of $65 billion to $100 billion annually. These costs include both loss of income and the burden placed on families to care for their injured relatives. The Americans bear 11% of the burden of road traffic injury mortality (WHO, 2002). The socioeconomic costs of RTA in Nigeria are immense and the direct cost of traffic casualties can perhaps, at best be understood in terms of the labour lost to the nation's economy [11]. It has been estimated that persons injured in accidents on Nigerian highways and streets no longer participate in the economic mainstream and this amounts to a loss of labour of millions of person's years to the nation [12].
It is also evident that Nigeria is worse than most other countries in terms of traffic accidents, in spite of her relatively good road network. As a 2004 World Bank report asserts "from the view -point of road development, Nigeria would no longer be regarded as a developing country [13]. But unlike in most countries where improved road development and vehicle ownership (as barometers of economic advancement) is accompanied by better traffic management, higher road safety awareness, and a relative decrease in the number of motor accidents, the opposite is true of Nigeria.
The problem of transportation and its safety is of great importance. An analysis of the traffic crashes data recorded over a seven-year period of 2000 -2006 shows that 98,494 cases of traffic crashes were recorded out of which 28,366 were fatal and resulted into 47,092 deaths [14]. This revealing statistics show that Nigeria is placed among the fore front nations (especially the third world nations) experiencing the highest rate of road tragedies in the world. Many researches had been conducted on road accident in different states in Nigeria: Kogi State, Katshina State, Ibadan [15,16]. Casualties in road accidents were studied by [17].
Armed robbers attempts and attacks are discovered to be one of the major causes of auto-crash on Nigeria roads [18]. A conducted survey revealed that auto-crash is a function of traffic volume as shown by [19]. The research showed that peak rainy season (June, July and September) which may be due to slippery road and poor visibility of the drivers, and the September, October, November and December in Lagos state Nigeria.
The peak accident that occurs in the dry season is always due to impatience [19,20]. [21] reported that road failure is another cause of road accidents in Nigeria. According to [22], the causes of road traffic accidents are multi-factorial. These factors can be divided broadly into driver factors, vehicle factors and roadway factors. Accidents can be caused by a combination of these factors.

Methodology
A predictable change or pattern in a time series that recurs or repeats over a one-year period can be said to be seasonal. Seasonality can be seen in many time series, and it's more common than you might think. For example, if you live in a climate with cold winters and warm summers, your home's heating costs probably rise in the winter and fall in the summer. You would reasonably expect the seasonality of your heating costs to recur every year. Similarly, a company that sells sunscreen and tanning products would see sales jump up in the summer, but drop in the winter. It is important to remember the effects of seasonality when analysing stocks from a fundamental point of view. The statistical technique used is the class Autoregressive Integrated moving average model (ARIMA). The statistical package used was R-Language.
A time series is a collection of observations of welldefined data items obtained through repeated measurements over time. For example, measuring the value of retail sales each month of the year would comprise a time series. This is because sales revenue is well defined, and consistently measured at equally spaced intervals. Data collected irregularly or only once are not time series. An observed time series can be decomposed into three components: the trend (long term direction), the seasonal (systematic, calendar related movements) and the irregular (unsystematic, short term fluctuations).

Unit Root Tests
Unit Root tests are usually performed on variables to determine if they stationary (i.e. zero mean and constant variance) and if otherwise, to determine their order of integration (i.e. number of times they are to be differenced to achieve stationarity). The time series characteristics of the variables using the Augmented Dickey-Fuller (ADF) and Phillips-Perron (P-P) tests were examined. Basically, the idea is to ascertain the order of integration of the variables as to whether they are stationary I (0) or non-stationary; and, therefore, the number of times each variable has to be differenced to arrive at stationarity. The standard DF test is carried out by estimating the following; (1) After subtracting from both sides of the equation: (2) Where = The simple Dickey-Fuller unit root test described above is valid only if the series is an AR (1) process. If the series is correlated at higher order lags, the assumption of white noise disturbances is violated. The Augmented Dickey-Fuller (ADF) test constructs a parametric correction for higherorder correlation by assuming that the y series follows an AR (P) process and adding P lagged difference terms of the dependent variable y to the right-hand side of the test regression: The usual practice is to include a number of lags sufficient to remove serial correlation in the residuals and for this; the Akaike Information Criterion is employed. Therefore, the ADF test given in equation (3) above is first used and then the Phillips Perron test described below.
Phillips and Perron propose a non-parametric alternative method of controlling for serial correlation when testing for a unit root. The P-P method estimates the non-augmented DF test equation (2), and modifies the t-ratio of the coefficient so that serial correlation does not affect the asymptotic distribution of the test statistic. The PP test is based on the statistic: Where is the estimate, and the t-ratio of , is the coefficient standard error, and s is the standard error of the test regression. In addition, is a consistent estimate of the error variance in equation (2) (calculated as (T -K) s 2 where k is the number of regressors). The remaining term, f 0 , is an estimator of the residual spectrum at frequency zero. Therefore, both equation (3) and (4) are used to test for the stationarity of the variables.

Stationary Time Series Models
White noise process, covariance stationary process, AR (p), MA (p) and ARIMA processes, stationarity conditions, diagnostic checks.
White noise process ARMA (p, q): Using the lag operator: The process is stable if for all numbers satisfying . Then we can write (5) If x is stable, it is covariance stationary: 1. or 0 -finite 2.

Partial Autocorrelation (PAC)
In AR (p) processes all x's are correlated even if they do not appear in the regression equation.
AR (1) ; ; We want to see the direct autocorrelation between and by controlling for all x's between the two. For this, construct the demeaned series and form regressions to get the PAC from the ACs. 1 st PAC: 2 nd PAC: .
Identification for an AR (p) process PACF for s>p: Hence AR (1): To evaluate it, use the relation : , substitute it to get:

Stability Condition for an AR (P) Process
The process is stable if for all z satisfying , or if the roots of the characteristic polynomial lie outside the unit circle. Then, we can write: .
Then we have the usual moment conditions: a. or 0 -finite b.

Invertibility requires
Then the AR representation would be: finite finite.
Represented in the above Fig. 1 is the time plot of all the variables used in this analysis. A critical look at the plots show that the series exhibited trend and seasonal effect. Since the plots cannot provide sufficient evidence to render the series not stationary, it is essential to use standard tests of stationarity.

Stationarity Test
The time series behaviour of each of the series is presented in Tables 2 below, using the ADF and PP tests at both levels.
H 0 = There is unit root H 1 = There is no unit root Since the absolute value of Augmented Dickey-Fuller (ADF) and Philip Perron (PP) test are 4.7613 and 112.99 which is greater than absolute critical value at 1%, 5% and 10% and also the p-value is less than all the critical value. Therefore, this leads to the rejection of the null hypothesis indicating that there is no unit root. This implies that the data is stationary.  Although the ADF and PP test shows that our data is weakly stationary but taking a critical look at the time plot above, we could notice a slight trend of non stationarity. To clear the element of doubt, we can remove this threat of non stationarity by differencing the series and then run the unit root test on the first difference.
This shows that the slight trend of non stationarity has disappeared which shows the real stationarity picture. The following is the first difference ADF and PP test to confirm the claim.
ADF and PP Test of the First Difference of the Data H 0 = There is unit root H 1 = There is no unit root This also shows that the absolute value of ADF (7.3095) and PP (144.07) are still greater than all the critical values of 1%, 5% and 10% and also p-value is less than all the critical values, with this, can be rejected and this implies that the data is stationary.

Model Fitting, Selection and Diagnostics
We considered both the deterministic and stochastic approach. Under the deterministic, least square trend was fitted for the variables and the estimated trend equation was recorded. The stochastic part include ACF and PACF inspection of the variables, fitting of the appropriate model as subjected by the ACF and PACF, selection of adequate model using Akaike information criterion (AIC) and model diagnostic.

Least Square Trend
The trend equations of the variables are shown below alongside the time plot. This helps us to visualize the movement of the series along the trend line. The trend equation will also help us to determine the growth rate of the series as well as the direction of the growth with changing in time. Eventually the following trend equation surfaced for the series.
Least square model (17) Trend model by least square. (18)   The graph above shows accident rate increasing by 1.9016 2 every month. Examining fig. 3, one could find out that ARMA model should be used for this but due to the seasonality that was discovered in the time plot fig. 1, we shall use the seasonal ARMA (i.e. SARIMA) model for the data. Here are the feasible tentative models with their Akaike's information criterion (AIC)

Diagnostics Checking
Checking whether the model capture the data well enough, we carry out the following diagnostics test for residual of the model.
From figure 4 below, it is observed that the residual plot suggested is rectangular scattered and irregular round a zero horizontal level with no trends whatsoever and also the point are within the tolerance line in the p-value, with all this, we can say the model capture the data well enough.

Conclusion and Recommendation
Having carried out the statistical analysis with the aid of time series, we obtained some results. In the time series analysis, we used the method of moving averages, autocorrelation and partial autocorrelation to analyse the series on the cases of road accidents along Ibadan-Oyo road, Oyo State. From the summary, results were observed that Christians celebrate Christmas in December and Easter in March or April. Also, there is Eid-el-kabir and Eid-el-_tri June-September, within these periods, people travel to celebrate with their loved ones. Accidents and deaths are higher during the festive periods and these EMBER months because of the various festivities lined up during this period, which involve much more traveling than usual. It is a period when commercial drivers make more money through overloading and excessive speeding, among other factors.

Recommendation
Based on the summary of the findings enumerated above on the number of road accidents, the number of cases will be reduced to the beeriest minimum by the government if the following recommendations are considered.
1. Creating awareness about Federal Road Safety: The government will have to increase efforts to promote awareness about the road safety issues and their social economic implications. The strategy to implementing this policy is by rising existing awareness, among stakeholders for planning and promoting road safety and their roles and responsibility. 2. Providing enabling legal, institutional and financial environment for road safety: Many government departments as well as various public and private agencies, share the responsibility of the various safety information data base: Detailed analysis of road accidents is essential if the causes of the accident are to be fully understood. At present time, the policy prepares a report for the accident that they are aware of. Accident report requires a precise location of the accident and condition at the time of the accident. 3. The government should improve data collection details at the scene of accident, improve the storage and accessibility of all data relevant to an accident such as vehicles involved, road, environment and drivers detail, etc.