Modeling road traffic fatality cases in Ghana.

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Modeling Road Traffic Fatality Cases in Ghana.
Adu-Poku, K.A. 1*, Avuglah, R.K. 2, and Harris, E.3
1Department of Mathematics, Kwame Nkrumah University of Science and Technology Kumasi, Ghana
2Department of Mathematics, Kwame Nkrumah University of Science and Technology Kumasi, Ghana
3Department of Mathematics, Kwame Nkrumah University of Science and Technology Kumasi, Ghana
*Corresponding Author’s Email: adupoku2009@yahoo.com
Abstract
Road traffic fatalities are of major concern in Ghana. According to the annual report of National Road Safety
Commission (NRSC), 2,047 road traffic fatality cases were recorded in the year 2007 which is an increase of
29.5% over that of the year 2000 (NRSC, 2007 Report). Relative to the year 2001(NRSC, 2001) the 2010 figures
for fatal crashes and fatalities also recorded corresponding increases of 34.1% and 19.6% respectively. Using
annual road fatality data for the period 1991-2011 from the National Road Safety Commission and Building and
Road Research Institute (BRRI) in Ghana, the Autoregressive Integrated Moving Average (ARIMA) model has
been utilized to study the trend and pattern of road fatalities in Ghana. The results showed that road traffic
fatalities in the country increased over the period of study. Moreover, a forecasting approach indicates that road
traffic fatalities would continue to increase over the next five years.
Keywords: Accident, Autoregressive (AR), Moving average (MA) and ARIMA.
1.0 Introduction
It is estimated that exactly 3,000 people die as a result of road traffic accidents all over the world on a given day.
(Penden et al., 2004). Over 90% of the world fatalities on roads occur in low-income and middle-income
countries (WHO, 2009). However, middle income countries have the highest annual fatality rates at 20.1 per
100,000 while high income countries have the lowest fatality rate at 8.7 per 100,000 (WHO, 2013). In 1999, it
was estimated that between 750,000 and 880,000 people died from road crashes and that the majority of these
deaths occurred in developing and transitional nations. (G Jacobs et al., 2000). Statistical projections have shown
that the highest fatality rates (death per 10,000 motor vehicles) worldwide occur in African countries particularly
Ethiopia, Uganda and Malawi. (G Jacobs et al., 2000).
According to Traffic Statistics, road traffic fatalities in the member states of the European Union annually claim
about 43,000 lives and leave more than 1.8 million people injured representing an estimated cost of 160 billion
euros (European Road Safety Observatory, 2008). In an effort of halving the number of road traffic fatalities by
2010, the European Commission published a white paper on transport policy on 12th September 2001
(European Commission , 2001). “Time to decide” was the subtitle of the white paper and this subtitle was
chosen because the commission had the view that the time had come as far as road safety was concerned
(European commission, 2001 memo).
Pedestrian death remains the leading effect of fatalities among urban road users in Ghana (James Damsere et al.,
2010). At least 1,800 deaths are recorded every year and 6 people die every day as a result of road traffic crashes
in Ghana (NRSC, 2013). Despite increased Road Safety Campaigns, the rate at which accidents occur on our
roads is very alarming. Annual report from the National Road Safety Commission indicates that the total number
of road traffic crashes in 2007 was 12,038 representing an increase of 3.2% over the 11,668 recorded in 2006
and 5.9% over the 2005 figure of 11,328 (NRSC, 2007 report). Similarly, road traffic fatalities have increased
accordingly over the years. Relative to the year 2001, the 2010 fatality figure of 1,986 deaths represented an
increase of 19.6% indicating an upward trend (BRRI, NRSC, 2010 report).
Road traffic fatalities occur as a result of several factors such as over speeding, poor vehicle maintenance, human
errors and pedestrian activity (Aworemi et al., 2010, Shinar 1978, Afuukar et al., 2003). Studies have indicated
that road side activities such as jaywalking, nighttime walking and street walking are also associated with
fatality cases in Ghana (James Damsere et al., 2010).

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Several researchers have used different statistical models in analyzing road traffic fatalities. For instance, Chang
and Mannering (1998) used nested logit model to analyze road fatalities and concluded that fatality is worsened
if the accident has a truck involved. Multinomial logistic regression model was used by (Bedard et al., 2002) to
analyze road fatalities and the results showed that female drivers with blood alcohol concentration greater than
0.3 were associated with high fatality rates. Similarly, the logistic regression model was also used by (Ballesteros
et al., 2004, Plurad et al., 2006, Macleod et al., 2010 and Mouden et al., 2009) to analyze road fatalities.
Correspondingly, the studies of accident variables including injuries and fatalities have extensively been
considered within the framework of time series modeling. Using ARIMA time series model, (Yuan Yuan pack et
al., 2013) indicated that China’s accident cases would increase from 107,579 to 401,536 over a period of one
year. (Rohayu Sarani et al., 2012) used ARIMA model to forecast Malaysia road fatalities for the period 2015-
2020 and the results showed that the predicted fatalities for the year 2015 will be 8,760 and 10,716 in 2020.
Similarly, time series analysis has also shown that road traffic fatalities are on the decrease with the exception of
Lagos Island local area (Atubi et al., 2013). The aim of the present study is to study the trends, patterns and
forecasts of road traffic fatalities in Ghana. The data set used is mainly road traffic fatalities for the period 1991-
2011 and was obtained from the National Road Safety Commission and Building and Road Research Institute in
Ghana.
2.0 Time Series
Time series is a sequence of observations ordered by a time parameter. Mostly these observations are collected at
equally spaced discrete time intervals.
2.1 Components of time series
 Trend: When there is a long term increase or decrease in a data.
 Seasonal: The periodic fluctuations in a time series within a certain time period.
 Cyclical: Long departure from the trend due to factors other than seasonality.
 Irregular: The movement or component left after accounting for trend, seasonal and cyclical movement
in a time series.
2.2 Stationary process
A stochastic process {
t
X
} is said to be strictly stationary if the joint distribution of
1 2 3
, , ,.....
t t t tn
X X X X
is the same as the joint distribution of
1 2 3
, , ,.....
t k t k t k tn k
X X X X
   
for all
1 2 3
, , ,.... n
t t t t
.
Mathematically,

( ) ( )
t t k
E X E X 


( ) ( )
t t k
Var X Var X 

for all t and k.

( , ) ( , )
t s t k s k
Cov X X Cov X X


for all t, s and k.
2.3 Differencing
Consider a time series
t
Y
, first order differencing is defined as
11t t t
y y y 

We can use backshift operator to express differencing as follows
11(1 )
t t t t t t
y y y y By B y

     
and
second order differencing is also expressed as
11 12
2
t t t t
y y y y

  
. In terms of backshift operator, the
equation becomes
11 2 2
12
2 2 (1 2 )
t t t t t t t t
y y y y y By B y B B y

        
For example, 
 
 or   and  .

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ARIMA (1,0,0) can be expressed in terms of backshift operator as      or
    but   put into the equation, we get    or     
Similarly, ARIMA (2,0,0) can be written as     
 
                  
 .
2.4 Integrated ARMA (or ARIMA models)
Autoregressive Integrated Moving Average (ARIMA) model is a generalization of an Autoregressive Moving
Average (ARMA) model. The model is generally referred to as an ARIMA (p, d, q) model where p, d and q are
integers greater than or equal to zero. The first parameter (p) refers to the number of autoregressive lags, the
second parameter (d) refers to the order of integration and the third parameter (q) gives the number of moving
average lags.
• Suppose a series {
t
Y
} is non-stationary, if the dth difference of the series is a stationary ARMA(p ,
q) process, then {
t
Y
} is said to be an ARIMA(p, d, q) process. The general Autoregressive integrated
moving average (ARIMA) is of the form:
• 

        


  
  

2.5 Box Jenkins Method
This method proposes three stages namely; identification, estimation and diagnostics checking.
The first stage in building the model is the identification of the appropriate ARIMA models through the study of
the autocorrelation and partial autocorrelation function. The sample autocorrelation plot and sample partial
autocorrelation plot are compared to the theoretical behavior of these plots when the order is known.
Other tools for model identification are:
 Akaike Information Criterion (AIC)
 Akaike Information Criterion Corrected (AICC)
 Bayesian Information Criterion. (BIC).
The BIC imposes a greater penalty for the number of estimated model parameters than AIC. The use of
minimum BIC for model selection results in a chosen model whose number of parameters is less than that
chosen under AIC.
The next stage is to estimate the parameters of the ARIMA model chosen. The nonlinear least squares and
maximum likelihood method is normally used.
The third stage is the diagnostics checking of the model. The Q statistics is used for the model adequacy check.
The fourth stage is where the analyst uses the model chosen to forecast and the process ends.
 
11
1 1 1 1
1 1 2 2 1 1
1 1 2 2 1 1 2 2
1 1 2
........... (1,0,0)
........... (1,0,1)
........... (2,0,1)
........... (2,0,2)
t t t
t t t t
t t t t t
t t t t t t
t t t t
y y ARIMA
y y ARIMA
y y y ARIMA
y y y ARIMA
y y y y

   
    
      



  
   
  

  
   
    
   ............. (1,1,0)
tARIMA



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3.0 Analysis and results
3.1 Preliminary Analysis
Fig 1: Time plot of fatality cases in Ghana from 1991 to 2011.
Fatalities in Ghana decreased from 1991 to 1994 and slightly increased from 1995 to 1996. It then declined from
1997 to 1998 but rose sharply from 1999 to 2004. An irregular pattern was then observed from 2005 to 2011. In
general, the trend of fatality cases in Ghana was on a rise but not consistent. The fatality time plot in fig 1 was
not stationary due to the trend component.
3.2 First differencing of fatality cases
Fig 2: First differencing of fatality cases in Ghana.
First differencing was performed to remove trend component in the original data
Fig 3: ACF plots of the differenced fatality cases in Ghana.

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From the correlogram, the sample ACF tails off to zero after the first significant lag. That is lag 1 exceeded its
significant bound and there after dies off exponentially representing a moving average of 1 (MA 1).
Fig 4: PACF plots of the differenced fatality cases
The partial Autocorrelation function tails off to zero after the first two significant lags. Comparing with the error
limits, we could see that lag 2 was significant, meaning an Autoregressive model of 2 (AR 2). From the
foregoing analysis, the following ARIMA models are therefore possible for the data series.
ARIMA(2,1,1)
ARIMA(2,1,0)
ARIMA(0,1,1)
PARAMETER ESTIMATION
TABLE I : ARIMA (2,1,1) MODEL
COEFFICIENT
ESTIMATE
STD. ERROR
T-VALUE
AIC
BIC
AR 1
1.0545
0.3405
3.09
11.37
10.57
AR 2
0.5820
0.2292
2.54
MA 1
0.2749
0.4242
0.64
The t-value for MA 1 is not statistically significant while that of AR 1 and AR 2 are statistically significant
since the t-value is greater than 2 in absolute terms. (Table I).
TABLE II : ARIMA (2,1,0) MODEL
COEFFICIENT
ESTIMATE
STD. ERROR
T-VALUE
AIC
BIC
AR 1
0.8483
0.1942
4.37
11.29
10.45
AR 2
0.4604
0.1980
2.33
From table II, the parameter based on the t-value test is statistically significant.
TABLE III: ARIMA (0,1,1) MODEL
COEFFICIENT
ESTIMATE
STD. ERROR
T-VALUE
AIC
BIC
MA 1
1.000
0.2025
4.94
11.20
10.30
The parameter based on the t-value test is statistically significant in table III.

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3.3 Model Diagnostics
Fig 5: Diagnostics of the residuals from ARIMA (0, 1, 1)
From figure 5, it is evident that there was no obvious trend and pattern for the standardized residuals plot. The
ACF of the residual plot shows that none of the lags exceeded its significant bound. Most of the residuals do not
deviate significantly from the line of best fit and its distribution looks approximately linear. Lastly, the Ljung-
Box statistics plot was not significant at any positive lag. In conclusion, the model was adequate and fits well. It
was also observed that ARIMA (2,1,1) and (2,1,0) models exhibited similar diagnostic characteristics as ARIMA
(0,1,1).
3.4 Selection of best model for forecasting fatality cases in Ghana.
The standardized residual plots of all the models were independently and identically distributed with a mean of 0
and some few outliers. There was no evidence of significance in the autocorrelation functions of the residuals of
all the models and the residuals appear to be normally distributed in all the models. The Ljung – Box statistics
was not significant at any positive lag for all the models. The AR and MA parameters in ARIMA (2,1,1) were
not significant at 5% level of significance which could have a negative effect on the forecast if used for
prediction. The parameters in ARIMA (2,1,0) and (0,1,1) were however significant.
The AIC, AICC, and BIC were good for all the models but they favored ARIMA (0, 1, 1) model since it had the
minimum AIC value.
From the discussion above, it is evident that ARIMA (0, 1, 1) model was the best model for forecasting fatality
cases in Ghana.
The fitted ARIMA (0, 1, 1) model for forecasting fatality cases from 1991-2011 was given by
 
11
71.8104
tt t
y
t
y



   

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TABLE IV: FORECASTING FATALITY CASES IN GHANA.
YEAR
FORECASTED
FATALITY
CASES
2012
2347.057
2013
2421.868
2014
2496.678
2015
2571.488
2016
2646.299
Fig 6 : Graph of the fatality cases, its forecasts and confidence intervals
The Figure gives the visual representation of the original fatality cases data (black line), its forecasts (red line)
and confidence interval (blue short dashes lines).
From the prediction values and the graph above, it can be observed that, fatality cases in Ghana would continue
to increase in the next 5 years.
4.0 Conclusions
Time series analysis of data from the years 1991 – 2011 showed that patterns of Road Traffic fatality cases were
increasing in Ghana.
ARIMA (0, 1, 1) was identified to be a suitable model for forecasting into the future of the fatality cases in
Ghana.
The study also revealed that road traffic fatality cases in Ghana would continue to increase over the next 5 years.
The findings of this study draw attention to the importance of implementing key road safety measures in order to
change the increasing pattern of road fatalities in the country. Future studies are therefore needed to understand
the mechanisms underlying road fatalities in Ghana.
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