Modelling the impact of intervention measures on total accident cases in Nigeria using Box-Jenkins methodology: A case study of federal road safety commission

Abstract

Road traffic accidents (RTA) have been a very big problem in many developing countries including Nigeria, causing many deaths and disabilities. The aim of this research is to model the effects of intervention measures adopted by the Nigerian government in curbing RTA. In the research road traffic accident data from 1960 to 2014 were analyzed using Box-Jenkins intervention methodology. The result of the modeling and analysis showed that the establishment of Federal Road Safety Commission in 1987, a Nigerian Government intervention measure, had an abrupt temporary impact on RTA in Nigeria (ω₀ = −2,423). The findings also showed that the total number of accident cases in Nigeria from 1961 to 1987 (26 years), the period before the intervention, was 657,280 while the total number of accident cases from 1988 to 2014 (26 years), the period after the intervention, was 430,721. This represents a 34.5% reduction in total accident cases after the intervention. In terms of accident density there was a 67.4% reduction in accident density during the post intervention period under consideration. It can be concluded that the establishment of the road safety agency has a positive impact on total cases of RTA in Nigeria by reducing it significantly, although RTA still continues to be a big problem in Nigeria. This model and analysis will assist road safety agencies to re-strategize in their policy implementation in order to further reduce RTA occurrence, the number of persons killed and injured in Nigeria.

Keywords:

• Box-Jenkins methodology
• intervention analysis
• modelling
• road traffic accidents

Public Interest Statement

Usually governments and regulatory agencies initiate certain policy interventions in order to achieve certain outcomes geared towards improving the living standards and for general societal wellbeing. Assessment of the outcomes of such policy interventions are usually done to evaluate the effectiveness of such policies in order to determine the need for modifications or improvement. Quantitative assessment of the effectiveness of such policies is invaluable to policy makers and regulators in improving the quality of social interventions.

In this work a quantitative assessment of the effectiveness of the establishment in Nigeria of a road safety agency on yearly total accident cases was successfully carried out using Box-Jenkins intervention analysis model.

The model effectively modeled the effect of the policy intervention on total road traffic accidents in Nigeria. The model would provide an invaluable quantitative tool to the road safety agency in improving the quality of its road safety interventions.

1. Introduction

Road Traffic Accidents (RTA), which has caused the deaths of millions of people, many injuries and permanent disabilities as well as numerous properties losses, has been a major problem in many parts of the world. According to World Health Organization WHO (2008), about 1.2 million persons died annually which approximate about 125 persons per hour. RTA is one of the leading causes of morbidity and mortality worldwide, especially in developing countries. The total annual costs of road crashes to developing countries are estimated to be about US$ 65 billion, which is more than the total annual revenue received in development assistance (Jacobs & Aeron-Thomas, 2000). According to Shbeeb and Awad (2016), road traffic injuries ranked as the ninth leading cause of death in 2004 and expected to become the fifth leading cause of death in 2030. Recent studies have shown an increase in the rate of RTA in many developing nations (Peden, 2004). This is attributed primarily to human, technical/Vehicular and environmental factors as well as urbanization, industrialization, population growth and increase in the number of motor vehicles on the roads. Nigeria, as a developing nation is equally bedeviled with this canker worm called RTA, hence the need for the Federal Government to introduce counter-measures in reducing Road Traffic Accident in Nigeria. These measures are what we shall refer to as interventions.

The literature is replete with various interventions measures taken to control RTA in various countries and regions of the world. Yang and Kim (2003) studied the effects of policy interventions prior to the 2002 world cup on RTA in Korea. They analyzed road accident data from 1970 to 2000, before the preparations for 2002 World Cup began, and after the preparation began in 2001. From their findings, through multiple policy interventions, partly in response to the 2002 FIFA World Cup preparation, about two thousand road traffic deaths and nine thousand traffic-related disabilities were averted in 2001 alone. They equally found that through multiple policy interventions, road traffic crashes in Korea were reduced in a relatively short period of time, along with their associated injuries and fatalities. They however opined that road traffic crashes still pose a major public health problem, threatening the quality of life of the Korean people. Khayesi (2012) studied why despite the intervention measures the risk of RTA in Kenya appear to be increasing when road safety measures exist. His findings showed that though the basic policy guidelines, institutional framework and measures to tackle RTA in Kenya exist, they are faced with a major problem of lack of commitment by key stakeholders. Other problems he found out include financial constraints, poor coordination, ineffective enforcement and limited community involvement and limited community involvement. The overall recommendation emanating from the analysis of his research was that there is need for a sustainable road safety strategy in Kenya that should go beyond the existing state of procrastination and symbolic actions. Sebego et al. (2014), examined the impact of the implementation of alcohol and road safety-related policies on crash rates, including overall crash rates, fatal crash rates, and single-vehicle nighttime fatal (SVNF) crash rates, in Botswana from 2004 to 2011. Their findings showed that the overall crash rate declined significantly in June 2009 and June 2010, such that the overall crash rate from June 2010 to December 2011 was 22% lower than the overall crash rate from January 2004 to May 2009. They equally found out that there were significant declines in average fatal crash and SVNF crash rates in early 2010. Their findings also showed that Botswana’s then recent crash rate reductions occurred during a time when aggressive policies and other activities (e.g. education, enforcement) were implemented to reduce alcohol consumption and improve road safety. They opined that while it is unclear which of the policies or activities contributed to these declines and to what extent, these reductions are likely the result of several, combined efforts. Pridemore, Chamlin, Kaylen, and Andreev (2013) studied the impact of a suite of 2006 Russian alcohol control policies on deaths due to traffic accidents in the country. They used autoregressive integrated moving average (ARIMA) interrupted time–series techniques to model the impact of the intervention on the outcome series from 2000 to 2010. Their findings show that the 2006 set of alcohol policies had no impact on female deaths due to traffic accidents. However, the intervention model revealed an immediate and sustained monthly decrease of 203 deaths due to transport accidents for males, which represents an 11% reduction relative to pre-intervention levels. They therefore concluded that the implementation of the suite of 2006 Russian alcohol control policies is partially responsible for saving more than 2400 male lives annually that would otherwise have been lost to traffic accidents. In Nigeria, the agency saddled with the responsibility for the implementations of road traffic accident reduction intervention policies is the Federal Road Safety Corps (FRSC). The impact of this agency had been studied in Edo and Oyo states of Nigeria by Igboanugo and Ekhuemelo (2007) and Charles-Owaba and Adebiyi (2001) respectively. From their analysis it was seen that there is a significant impact of intervention measures in RTA reductions in those areas since the inception of FRSC in 1988.

Concerted efforts had been made by several researchers over the years, as enumerated above, to identify the root cause of RTAs and their possible solutions. Also attempt had made to developed various methods of analyzing RTA through the use of univariate regression methods, descriptive statistics and charts in organizing and analyzing accidents data in selected roads and State in Nigeria and their possible solutions. These methods to an extent have helped in assessing RTAs situations in Nigeria. However these methods do not give holistic information and estimates of road traffic accident trend, and its related outcomes. Again, it does not predict accurate estimates of RTA cases, number of injuries and death in the near future. Also these methods do not give detailed evaluation of intervention measures implemented by road safety agencies in reducing RTA cases in Nigeria. Hence, the needful for a robust approach in identifying and predicting the extent of intervention policy on RTA in Nigeria.

The aim of this study therefore is to assess the impacts of the establishment of the road safety agency (FRSC) in Nigeria and to develop an intervention model for quantifying the impact of the agency on total accident cases in Nigeria.

2. Theoretical background

2.1. Intervention model

Generally intervention model function constitutes two main components namely; an intervention function and an ARIMA noise model (DeLurgio, 1998; Pridemore et al., 2013; Sebego et al., 2014).$$\text{Intervention ; model}=\text{intervention; function}+\text{ARIMA ; noise ; model}$$

Mathematically, the intervention model according to Box, Jenkins, and Reinsel (2008) is expressed as(1) $$Y_{(t)\text{Intervention}}=f\left(I_t\right)+N_t$$(1)

where $f(I_t)$ = the intervention function; N_t = ARIMA (p,d,q) $I_t=\left\{\begin{array}{c}{1}_{\text{when;intervention;occurs}}\hfill \\ 0_{\text{otherwise}}\hfill \end{array}\right.$.

The general intervention model can be re-expressed in the this form(2) $$Y_{(t)\text{Intervention}}=\frac{\left({\mathit{\omega }}_0-{\mathit{\omega }}_{1}B-{\mathit{\omega }}_2{B}^2-\cdots -{\mathit{\omega }}_s{B}^s\right)}{\left(1-{\mathit{\delta }}_{1}B-{\mathit{\delta }}_2{B}^2-\cdots -{\mathit{\delta }}_r{B}^r\right)}{X}_{t-b}+N_t$$(2)

where ω = intervention effect; δ = gradual increases or decreases term; B = backshift operator; Y_t and X_t represent either deviations from their means for stationary series; r = level of autoregression or power of B in the denominator; S + 1 = number of ω terms in the numerator; b = delay in the intervention impact as shown by X_t−b and the notation $N_t$ is ARIMA noise model which can be expressed in the form (Box et al., 2008): (2) $$N_{(t)}=\frac{\left(1-{\mathit{\theta }}_{1}B-{\mathit{\theta }}_2{B}^2-\cdots -{\mathit{\theta }}_q{B}^q\right)}{\left(1-{\mathit{\varphi }}_{1}B-{\mathit{\varphi }}_2{B}^2-\cdots -{\mathit{\varphi }}_p{B}^p\right)}{e}_t$$(2)

where $\mathrm{\varnothing }$ = autoregressive coefficient; e_t = White noise residuals/errors; θ = moving average coefficient; q = Order of moving average operator; p = order of autoregressive operator.

2.2. Common intervention types

Majorly intervention model can be classified into two types; Zero-Order Intervention Functions and First-Order Intervention Functions.

(1)

Zero-order intervention functions-consist of the following components (a) Abrupt, permanent impact-can be expressed as (4) $$Y_{(t)\text{Intervention}}={\mathit{\theta }}_0+{\mathit{\omega }}_0{I}_t$$(4) (b) Abrupt Temporary impact-can be expressed as (5) $$Y_{(t)\text{Intervention}}={\mathit{\theta }}_0+({\mathit{\omega }}_0-{\mathit{\omega }}_{1}B)I_t$$(5)

(2)

First-Order Intervention Functions-consist of the following components (a) Gradual, Permanent impact-can be expressed as (6) $$Y_{(t)\text{Intervention}}={\mathit{\theta }}_0+\frac{{\mathit{\omega }}_0}{(1-{\mathit{\delta }}_{1}B)}{I}_t$$(6) (b) Abrupt Temporary impact-can be expressed as (7) $$Y_{\left(t\right)\text{Intervention}}={\mathit{\theta }}_0+\frac{{\mathit{\omega }}_0(1-B^5)}{(1-{\mathit{\delta }}_{1}B)}{I}_t$$(7)

where I = 1 or 0 otherwise; θ₀ = N_t = normal central value of the total accident cases obtained from ARIMA analysis; ω₀ = initial intervention effect; −ω₁ = delay intervention effect; δ1 = gradual increases or decreases term of the first order intervention model.

3. Intervention modelling procedures

In order to model the impact of an intervention, DeLurgio (1998), stated that we need to model the time series without that impact. It implies that we model the pre-intervention series. Hence the following are steps for intervention modelling (DeLurgio, 1998; Pridemore et al., 2013; Sebego et al., 2014):

(1)	The date of intervention should be known
(2)	Theory suggesting the types of impact on time series should be determined
(3)	Sufficient observation to identify the ARIMA noise model before the intervention
(4)	Sufficient intervention and post-intervention observation.

3.1. ARIMA (Noise) modelling procedure

In time series analysis, an autoregressive model is represented in the following form (Box et al., 2008; Cryer & Chan, 2008; Pridemore et al., 2013; Sebego et al., 2014). 

(8) $$y_t={\mathrm{\varnothing }}_1{y}_{t-1}+{\mathrm{\varnothing }}_2{y}_{t-2}+\dots +{\mathrm{\varnothing }}_p{y}_{t-p}+e_t$$(8)

where y_t = time series variable; ∅ = autoregressive coefficient; e_t = white noise residuals/errors.

Similarly, a moving average model is represented in the following form (Box et al., 2008; Cryer & Chan, 2008).(9) $$y_t=e_t+{\mathit{\theta }}_1{e}_{t-1}+{\mathit{\theta }}_2{e}_{t-2}+\cdots +{\mathit{\theta }}_q{e}_{t-q}$$(9)

where θ = moving average coefficient.

ARIMA Model was pioneered by the land mark work of Box and Jenkins in 1970 (Nwobi-Okoye & Igboanugo, 2012, 2015); hence it is often referred to as Box–Jenkins method. This model integrates the autoregressive and moving average models with appropriate differencing to achieve stationarity. In order words, ARIMA extends the combination of AR and MA process to non stationary processes.

From Equations (8) and (9), Equation (10) is obtained:(10) $$y_t={\mathrm{\varnothing }}_1{y}_{t-1}+\dots +{\mathrm{\varnothing }}_p{y}_{t-p}+e_t+{\mathit{\theta }}_1{e}_{t-1}+\dots +{\mathit{\theta }}_q{e}_{t-q}$$(10)

Equation (9) could be represented in shortened form as:(11) $$\mathrm{\varnothing }(B)x_t=\mathit{\theta }(B)e_t$$(11)

If the output x_t is differenced d times to achieve stationarity, Equation (5) is obtained.(12) $${\mathrm{\nabla }}^d{y}_t={(1-B)}^d{y}_t$$(12)

where d = number of differencing.

In general by combining Equations (5) and (6), the model could be written as:(13) $$\mathrm{\varnothing }(B){(1-B)}^d{y}_t=\mathit{\theta }(B)e_t$$(13)

In order to realize the ARIMA model based on Equation (7), a plot of the 27-year pre-intervention accident data was done using SPSS software. After the plot, the data was investigated for stationarity, using the plots of the autocorrelation functions (ACF) and Partial autocorrelation functions (PACF). The accident cases series derived from the plots were found not to be stationary, hence differencing was used to achieve stationarity. Stochastic regularity was achieved after the first differencing. Following the achievement of stationarity of the data, a univariate model was fitted to y_t.

4. Results and discussion

4.1. Intervention modelling of road traffic accident cases in Nigeria

Generally as earlier noted, intervention model constitute two main components namely: an intervention function and an ARIMA noise model. The models shown in Sections 2 and 3 were used in this section to develop the intervention models for analyzing the impact of FRSC on total RTA in Nigeria. The details of the analysis are presented in here in Section 4.

4.2. Pre-intervention analysis and results

To model the impact of government policies (intervention), it is imperative to model the times series data of the total accident cases when there no intervention as described in Section 3.1. This required that we model the pre-intervention series which exist as noise N_t in the equation above (i.e. ARIMA model). Let the pre-intervention period for road traffic accident cases in Nigeria be 1960–1987. That is, before the establishment of FRSC. To obtain a parsimonious model various component of ARIMA has been analyzed below.

The plot in Figure 1 shows the actual versus predicted value of RTA in Nigeria from 1960 to 2019, while the plot in Figure 2 below shows the actual figures of RTA in Nigeria from 1960 to 1987 (pre-intervention period).

Figure 1. Predicted trends of total accident cases in Nigeria vs. years (1960–2019).

Figure 2. Plot of total accident cases in Nigeria vs. years (1960–1987) (pre-intervention period).

A close examination of ACFs and PACFs of the pre-intervention series shown in Figures 3 and 4 suggest an ARIMA (1,1,1)1 model. From Figure 1, impact of the intervention was noticeable in 1988 with an exponential declining profile. And it was observed that the trends were of Gradual, temporal impact of the first order intervention functions. To determine the best ARIMA noise model, Tables 1–3, indicated ARIMA (1,1,1)1 which gave the highest R², lowest BIC, and relatively low Ljung Box. From Box et al. (2008) and DeLurgio (1998), ARIMA (1,1,1)1 can be expressed, as shown in Equation (9), as:(14) $$Y_t={\mathit{\theta }}_0+{\mathrm{\varnothing }}_1{Y}_{t-1}-{\mathit{\theta }}_1{e}_{t-1}+e_t$$(14)

Figure 3. Plot of ACF vs. Lag of total accident cases (pre-intervention period).

Figure 4. Plot of PACF vs. Lag of total accident cases (pre-intervention period).

Table 1. Parameters estimation of ARIMA model for total accident cases (1960–1987)

Model	Number of values	Min value	Max value	Sum	Mean	Std. error	Std. deviation	Variance
Total cases	28	12,163	37,881.00	6.70 × 10^5	2.3943 × 10^4	1.6097 × 10^3	8,517.77	7.26 × 10^7

Table 2. Summary of diagnostics test evaluation of ARIMA models for pre-intervention period of total accident cases in Nigeria from 1960 to 1987

Statistical parameter	ARIMA (1,0,0)1	ARIMA (0,1,0)1	ARIMA (1,1,0)1	ARIMA (1,1,1)1	ARIMA (0,1,1)1	ARIMA (0,0,1)1	ARIMA (1,0,1)1
R^2	0.728	0.737	0.773	0.791	0.761	0.413	0.741
RMSE	4,510	4,338	4,096	4,008	4,214	6,653	4,505
MAPE	14.260	12.688	12.771	12.347	12.511	26.125	14.228
MaxAPE	63.492	36.293	31.172	30.150	33.204	69.050	61.153
MAE	3,243	3,101	3,039	2,997	2,968	5,386	3,249
MaxAE	14,340	13,750	11,810	11,420	12,580	15,690	13,550
BIC	17.066	16.872	16.880	16.858	16.936	17.844	17.183

Table 3. Summary of Ljung-Box diagnostics test of ARIMA models for pre-intervention period of total accident cases from 1960 to 1987

Model	Ljung-Box Q(18)
	Statistics	DF	Sig.
Total cases-ARIMA (1,0,0) model	37.216	17	0.003
Total cases-ARIMA (0,1,0) model	32.895	18	0.017
Total cases-ARIMA (1,1,0) model	14.450	17	0.635
Total cases-ARIMA (1,1,1) model	12.111	16	0.736
Total cases-ARIMA (0,1,1) model	19.459	17	0.303
Total cases-ARIMA (0,0,1) model	92.721	17	0.000
Total cases-ARIMA (1,0,1) model	28.276	16	0.029

where Y_t = total accident cases; θ₀, ∅₁, θ₁. are coefficients as defined previously above.

In terms of the observed series the model can be expressed as:

(15) $$Y_t-Y_{t-1}={\mathit{\theta }}_0+{\mathrm{\varnothing }}_1\left(Y_{t-1}-Y_{t-2}\right)-{\mathit{\theta }}_1{e}_{t-1}+e_t$$(15)

(16) $$Y_t={\mathit{\theta }}_0+{\mathrm{\varnothing }}_1(Y_{t-1}-Y_{t-2})+Y_{t-1}-{\mathit{\theta }}_1{e}_{t-1}+e_t$$(16)

Rearranging Equation (16), Equation (17) is obtained:(17) $$Y_t={\mathit{\theta }}_0+(1+{\mathrm{\varnothing }}_1)Y_{t-1}-{\mathrm{\varnothing }}_1{Y}_{t-2}-{\mathit{\theta }}_1{e}_{t-1}+e_t$$(17)

Therefore applying equation for ARIMA (1,1,1)1, (Box et al., 2008; DeLurgio,1998) and fitting total accident cases model by applying Equations (14)–(17) and Table 4, we have :(18) $${\mathit{\theta }}_0=473.312+(0.188)Y_{t-1}+0.812Y_{t-2}+0.54e_{t-1}+e_t$$(18)

Table 4. ARIMA (1,1,1)1 from 1960 to 1987

		Estimate	SE	t	Sig.
Constant		473.312	657.640	0.720	0.479
AR	Lag 1	−0.812	0.245	−3.315	0.003
Difference		1
MA	Lag 1	−0.540	0.361	−1.497	0.147

In forecasting form Equation (19) becomes:(19) $${\widehat{Y}}_t={\mathit{\theta }}_0=473.312+(0.188)Y_{t-1}+0.812Y_{t-2}+0.540e_{t-1}$$(19)

Figure 5 shows the actual vs. fitted value of ARIMA (1,1,1)1 for total accident cases (1960–1987) using Equation (20).

Figure 5. Actual vs. fitted value of ARIMA (1,1,1)1 for total accident cases (1960–1987).

4.3. Intervention analysis and results

We hereby present the intervention analysis modeling results used to access the impact of FRSC on total accident cases in Nigeria. Examining the graph of the yearly total accident cases in Nigeria from 1960 to 2014 shown in Figure 1, shows that the impact of the intervention in 1987 could be modeled as a second order intervention model with abrupt temporary impact.

Recall that:$${\mathit{\theta }}_0=N_t$$

From Equation (7) which represents a second order intervention model with abrupt temporary impact, we have:

$$Y_{(t)\text{Intervention}}={\mathit{\theta }}_0+\frac{{\mathit{\omega }}_0(1-B^5)}{(1-{\mathit{\delta }}_{1}B)}{I}_t$$

δ is a parameter of Gradual, temporal impact of intervention model employed, constrained in an interval of −1 ≤ δ ≥ 1. For model stability a central value of 0.5 was chosen.

Therefore, let δ = 0.5

Expanding(20) $$Y_{(t)\text{Intervention}}=0.5Y_{t-1}+0.5{\mathit{\theta }}_0+{\mathit{\omega }}_0\left(1-B^5\right)I_t$$(20) (21) $$Y_{(t)}=0.5Y_{t-1}+0.5[473.312+(0.188)Y_{t-1}+0.812Y_{t-2}+0.54e_{t-1}]-I_t+I_{t-5}$$(21)

Table 5 shows the result of the intervention model for road traffic accident in Nigeria from 1960 to 2014. Table 6 shows the summary of reported road traffic crashes trends in Nigeria from 1960 to 2014. It could be seen from Table 6 that the first gradual reduction in road traffic accident cases was period 27 and 28 which corresponds to 1987 and 1988.

Table 5. Results of intervention model for road traffic accident cases in Nigeria

Year	t	Actual value (Yt)	I	It−5	Intervention model for RTAs in Nigeria
1960	0	14,130	0	0
1961	1	15,963	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1962	2	16,317	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1963	3	19,835	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1964	4	15,927	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1965	5	16,904	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1966	6	13,550	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1967	7	13,000	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1968	8	12,163	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1969	9	12,988	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1970	10	16,656	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1971	11	17,745	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1972	12	23,187	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1973	13	24,181	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1974	14	28,893	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1975	15	23,611	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1976	16	37,881	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1977	17	35,716	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1978	18	36,111	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1979	19	29,271	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1980	20	32,138	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1981	21	33,777	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1982	22	37,094	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1983	23	32,109	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1984	24	28,892	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1985	25	29,968	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1986	26	25,188	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1987	27	28,215	0	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1988	28	25,792	1	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1989	29	23,987	1	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1990	30	21,934	1	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1991	31	22,546	1	0	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1992	32	22,864	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1993	33	21,459	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1994	34	18,204	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1995	35	17,030	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1996	36	16,442	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1997	37	17,488	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1998	38	16,138	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
1999	39	15,865	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2000	40	16,606	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2001	41	20,530	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2002	42	14,544	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2003	43	14,364	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2004	44	14,274	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2005	45	9,062	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2006	46	9,114	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2007	47	8,477	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2008	48	11,341	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2009	49	10,854	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2010	50	11,385	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2011	51	13,196	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2012	52	13,262	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2013	53	13,583	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5
2014	54	10,380	1	1	Yt = 0.5Yt−1 + 0.5[473.3 + (0.188)Yt−1 + 0.812Yt−2 + 0.54et−1] − ω0It + ω0It−5

Table 6. Road traffic accidents data in Nigeria (from 1960 to 2014)

Summary of reported road traffic crashes trends in Nigeria (1960–2014)
Year	Fatal cases	Serious cases	Minor cases	Total cases	Number killed	Number injured	Total casualty
1960	826	9,065	4,239	14,130	1,083	10,216	11,299
1961	193	9,982	5,788	15,963	1,313	10,614	11,927
1962	1,263	9,159	5,895	16,317	1,578	10,341	11,919
1963	967	6,918	11,950	19,835	1,532	7,771	9,303
1964	911	7,371	7,645	15,927	1,769	12,581	14,350
1965	1,029	7,762	8,113	16,904	1,918	12,024	13,942
1966	1,680	5,600	6,270	13,550	2,000	13,000	15,000
1967	1,560	5,200	6,240	13,000	2,400	10,000	12,400
1968	459	5,865	5,839	12,163	2,808	9,474	12,282
1969	1,559	5,199	6,230	12,988	2,347	8,804	11,151
1970	1,999	6,666	7,991	16,656	2,893	13,154	16,047
1971	1,129	8,098	8,518	17,745	3,206	14,592	17,798
1972	2,782	9,275	11,130	23,187	3,921	16,161	20,082
1973	2,981	9,275	11,925	24,181	4,537	18,154	22,691
1974	3,467	11,557	13,869	28,893	4,992	18,660	23,652
1975	2,834	9,446	11,331	23,611	5,552	20,132	25,684
1976	905	17,352	19,624	37,881	6,761	28,155	34,916
1977	4,242	14,140	17,334	35,716	8,000	30,023	38,023
1978	4,333	14,444	17,334	36,111	9,252	28,854	38,106
1979	3,513	11,708	14,050	29,271	8,022	21,203	29,225
1980	1,856	14,855	15,427	32,138	8,736	25,484	34,220
1981	4,053	13,510	16,214	33,777	10,202	26,337	36,539
1982	4,451	14,838	17,805	37,094	11,382	28,539	39,921
1983	3,853	12,844	15,412	32,109	10,462	26,866	37,328
1984	4,467	10,557	13,868	28,892	8,830	23,861	32,691
1985	3,597	11,991	14,380	29,968	9,221	23,853	33,074
1986	3,022	10,075	12,091	25,188	8,154	22,176	30,330
1987	3,385	11,286	13,544	28,215	7,912	22,747	30,659
1988	4,127	11,091	10,574	25,792	9,077	24,413	33,490
1989	3,838	10,314	9,835	23,987	8,714	23,687	32,401
1990	6,140	8,796	6,998	21,934	8,154	22,786	30,940
1991	6,719	8,982	6,845	22,546	9,525	24,508	34,033
1992	6,986	9,324	6,554	22,864	9,620	25,759	35,379
1993	6,735	8,443	6,281	21,459	9,454	24,146	33,600
1994	5,407	7,522	5,275	18,204	7,440	17,938	25,378
1995	4,701	7,276	5,053	17,030	6,647	14,561	21,208
1996	4,790	6,964	4,688	16,442	6,364	15,290	21,654
1997	4,800	7,701	4,987	17,488	6,500	10,786	17,286
1998	4,757	7,081	4,300	16,138	6,538	17,341	23,879
1999	4,621	6,888	4,356	15,865	6,795	17,728	24,523
2000	5,287	6,820	4,499	16,606	8,473	20,677	29,150
2001	6,966	8,185	5,379	20,530	9,946	23,249	33,195
2002	4,029	7,190	3,325	14,544	7,407	22,112	29,519
2003	3,910	7,882	2,572	14,364	6,452	18,116	24,568
2004	3,275	6,948	4,051	14,274	5,351	16,897	22,248
2005	2,299	4,143	2,620	9,062	4,519	15,779	20,298
2006	2,600	5,550	964	9,114	4,944	17,390	22,334
2007	2,162	4,812	1,503	8,477	4,673	17,794	22,467
2008	3,024	5,671	2,646	11,341	6,661	27,980	34,641
2009	2,460	6,024	2,370	10,854	5,693	27,270	32,963
2010	2,388	6,815	2,182	11,385	6,052	35,691	41,743
2011	2,840	8,357	1,999	13,196	6,054	41,165	47,219
2012	2,935	8,277	2,050	13,262	6,092	39,348	45,440
2013	3,294	8,589	1,700	13,583	6,544	40,057	46,601
2014	3,117	6,356	907	10,380	5,996	32,063	38,059

Source: Federal road safety corps: policy, research and statistics (2015).

Therefore, the intervention effect ω₀ is the difference in year 1987 and 1988, which can be expressed as:$${\mathit{\omega }}_0=25792-28215=-2423$$

(22) $$Y_{(t)}=0.5Y_{t-1}+0.5[473.312+(0.188)Y_{t-1}+0.812Y_{t-2}+0.54e_{t-1}]-2423I_t+2423I_{t-5}$$(22)

Figure 6 shows the plot of total accident cases and predicted accident cases without intervention. As shown in Figure 6, without intervention in 1988, the total accident cases recorded would have sky rocketed causing pains and hardship to the populace. Figure 7 shows the plot of total accident cases versus the predicted total accident cases with the intervention model. As Figure 7 shows, there is a gradual reduction in total accident cases from 1988 downwards, clearly showing the effects of the intervention.

Figure 6. Plot of total accident cases and predicted accident cases without intervention.

Figure 7. Plot of total accident cases and predicted accident cases with intervention.

Generally, the impact of intervention measure on RTA occurrence was modelled as shown in Equation (23), and Table 5 shows a summary of the expected intervention impacts. From the analysis, the impact of intervention measures on road traffic accident occurrence in Nigeria yields considerable results with gradual reduction of total accident cases (ω₀ = −2,423). For example the total number of accident cases from 1961 to 1987 (26 years), the period before the intervention, was 657,280 while the total number of accident cases from 1988 to 2014 (26 years), the period after the intervention, was 430,721. This represents a 34.5% reduction in total accident cases after the intervention. Considering the fact that the average yearly population of Nigeria from 1961 to 1987 was 64,159,442 while the average from 1988 to 2014 was 128,973,480, if accident density is defined as the number of accidents per person, the average accident density in percentage from 1961 to 1987 was 1.0244 and from 1988 to 2014 it was 0.3340. Hence, there was a 67.4% reduction in accident density during the post intervention period under consideration. If road traffic agencies in Nigeria would continue to build on the existing structures, there would be a dramatic reduction of RTA occurrence in Nigeria.

However from Figures 8 and 9 and Table 6, it could be observed that the number of persons killed seems to be on the increase while the number injured in RTA in Nigeria seems to decrease. Hence, it appears that the declining rate of RTA cases did not translate to reduction in the number of persons killed when compared to the pre-intervention period. But there is a reduction in the number of persons injured during the post-intervention period in Nigeria.

Figure 8. Plot of number of persons injured in Nigeria vs. years.

Figure 9. Plot of number of persons killed in Nigeria vs. years.

For instance the total number of fatal accident cases from 1961 to 1987 (26 years), the period before the intervention, was 66,490 while the total number of accident cases from 1988 to 2014 (26 years), the period after the intervention, was 114,207. This represents a 71.77% increase in serious accident cases after the intervention. Considering the fact that the average yearly population of Nigeria from 1961 to 1987 was 64,159,442 while the average from 1988 to 2014 was 128,973,480, if fatality density is defined as the number of accidents per person, the average fatality density in percentage from 1961 to 1987 was 0.1036 and from 1988 to 2014 it was 0.0885. Hence, there was a 14.55% reduction in fatality density during the post intervention period under consideration.

Furthermore, the total number of serious accident cases from 1961 to 1987 (26 years), the period before the intervention, was 274,973 while the total number of accident cases from 1988 to 2014 (26 years), the period after the intervention, was 201,001. This represents a 26.54% decrease in serious accident cases after the intervention. Considering the fact that the average yearly population of Nigeria from 1961 to 1987 was 64,159,442 while the average from 1988 to 2014 was 128,973,480, if seriousness density is defined as the number of serious accidents per person, the average seriousness density in percentage from 1961 to 1987 was 0.4286 and from 1988 to 2014 it was 0.1566. Hence, there was a 63.45% reduction in seriousness density during the post intervention period under consideration.

From the analysis above, it could be concluded that the increase in fatalities during the post intervention period could be due to population and economic growth which has led to increase in the number of roads and road users. Concerted effort should be made by road traffic agencies and stakeholders to explore fully the root-cause of high number of deaths and injuries still occurring on our roads despite the government interventions. As noted by Nwobi-Okoye (2011), most road traffic problems in Nigeria are due to the tyranny of small decisions by motorists, citizens and law enforcement agents. Nwobi-Okoye identified inadequate law enforcement by government regulatory agencies as being responsible for the road traffic problems in Nigeria. It appears that after the temporary impact on account of the initial government intervention, the government regulatory agencies local, state and federal government had not done enough to sustain the tempo and curb to the barest minimum defective behaviours that Nwobi-Okoye (2011) identified as the main causes of road traffic problems in Nigeria. These defective behaviours include but not limited to:

(1)	Driving against the traffic.
(2)	Following illegal routes.
(3)	Dropping/picking passengers at unauthorized points along the road.
(4)	Abandoning broken down vehicles on the road.
(5)	Buying goods from hawkers without clearing away from the road.
(6)	Illegal parking of vehicles along the main road.
(7)	Construction and use of illegal structures for businesses along the main road.
8)	Unrestricted use of private cars.
(9)	Illegal check points mounted by the police and other law enforcement agents etc.

ChikkaKrishna, Parida, and Jain (2015) noted that there is a significant need to improve the highway safety during roadway planning, design and operations in developing countries and Nigeria is not an exception. It is imperative that the authorities and policy makers in Nigeria adopt this approach in the development and management of its roads.

5. Conclusions

The intervention model developed in this work has successfully been used to assess the impact of Government policies on reducing RTA in Nigeria through the road safety agency known as FRSC. The conclusions from this study are as follows:

(1)	The establishment of the road safety agency has had an abrupt temporary impact on the total accident cases in Nigeria. Hence there was a significant reduction in the total accident cases during the post intervention period.
(2)	The number of serious accident cases declined significantly in the post intervention period.
(3)	Although there seemed to be an increase in fatalities during the post-intervention period, the fact that the fatality density decreased significantly during the post-intervention period means that the increase could be attributed to population growth (the populations has doubled in the post-intervention period) and economic growth (economy has more than doubled during the post-intervention period, in comparison with the pre-intervention period, to become the largest in Africa).
(4)	This quantitative model would be invaluable to the Nigerian government and the road safety agency in re-strategizing their policy implementation in order to reduce RTA occurrence, the number of persons killed and injured in Nigeria. To do this requires the increase in the absolute value of ω0 from the current value of 2,423 to a much higher value such as 5,000 (ω0 = −5,000) or higher.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

Benjamin Ufuoma Oreko

Benjamin Ufuoma Oreko is a lecturer in the Department of Mechanical Engineering, Federal University of Petroleum Resources, Effurun, Nigeria. His academic research interest include: Engineering materials, Nanotechnology, Operation Research, Mechanical Design and Renewable Energy.

Chukwuemeka Chidozie Nwobi-Okoye

Chukwuemeka Chidozie Nwobi-Okoye is lecturer in the Faculty of Engineering of Chukwuemeka Odumegwu Ojukwu University, Nigeria. His research interest include: Operations Research, CAD/CAM, Game Theory, Simulation and Artificial Intelligence.

Stanley Okiy

Stanley Okiy is a training officer in the department of welding engineering and offshore technology, Petroleum Training Institute, Effurun, Nigeria. He is involved in active academic research and engineering consultancy. His research interest include: Operations research and project management.

Anthony Clement Igboanugo

Anthony Clement Igboanugo is a Professor in the department of Production Engineering, University of Benin, Nigeria. He has 21 years industrial and university teaching experience and has to his credit over 80 scholarly publications. He is involved in active academic research and engineering consultancy for government and industries.