Design of Cusum Scheme for Monitoring Road Accident Fatalities

doi:10.4236/ojs.2012.22026

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Open Journal of Statistics, 2012, 2, 213-218

http://dx.doi.org/10.4236/ojs.2012.22026 Published Online April 2012 (http://www.SciRP.org/journal/ojs) 213

Design of Cusum Scheme for M onitori ng Road

Accident Fatalities

Kayode Samuel Adekeye1*, Omololu Stephen Aluko2

1Department of Mathematical Sciences, Redeemer’s University, Redemption City, Nigeria

2Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

Email: *[email protected]

Received December 9, 2011; revised January 12, 2012; accepted January 25, 2012

ABSTRACT

In recent years, road accident fatalities in Nigeria have continued to be on the increase. Thus, urgent attention is needed

to reduce or eliminate road acciden ts fatalities. To achieve this goal, the cumulative sum (Cusum) control chart scheme

was designed for monitoring the road accident fatalities using the recorded occurrence of road accident fatalities in a

state in the western part of Nigeria. The designed Cusum detects the period of the years when the highest occurrence of

road accident fatalities occurred. These periods were observed to be festive periods such as Christmas, Easter, Eid-el-

Kabir, and Eid-el-Moluod. Therefore, the festive periods of the year should be used as benchmark by road managers as

periods where more attention or precaution measure should be put in place on the roads to drastically reduce or elimi-

nate high occurrence of road accident fatalities. The designed Cusum control chart can be adapted for other states in the

country and also for the larger society for detecting the periods when the rate of death as a result of road accidents was

prevalent.

Keywords: Cusum; Fatalities; Monitoring; Prevalent; Ro ad Accident

1. Introduction

The problem of deaths and injury as a result of road ac-

cidents is now acknowledge to be a global phenomenon

with authorities in virtually all countries of the world

concerned about the growth in the number of people

killed and seriously injured on their roads. Fatality means,

a death that is caused in an accident or war or the fact

that a particular disease or accident will result in death”.

In recent years there have been two major studies of

causes of death worldwide which have been published in

the “Global Burden of Disease” (2008). This publication

show that road accidents as a cause of death or disability

were by no means insignificant as it is in the ninth place

out of a total of over 100 separately identified causes of

death [1].

In recent years, road accident fatalities in Nigeria have

continued to be on the increase. This study is motivated

by recent interest of Nigerian government to minimize

the road fatal accidents by 50% in year 2015. Thus, ur-

gent attention is needed to be given to our road transpor-

tation system in order to reduce or eliminate road acci-

dents fatalities and save lives.

In this paper, our focus is to develop a Cusum control

chart that will be able to assist the government to achiev e

the above stated goal. The cumulative sum (Cusum) con-

trol chart scheme was designed for monitoring the road

accident fatalities using the recorded occurrence of road

accident fatalities in a state in the western part of Nigeria.

The designed Cusum control chart can be adapted for

other states in the country and also for the larger society

for detecting the time point (month) of the year when the

rate of death as a result of road accidents was prevalent

(high occurrence) and the cumulative trend for road ac-

cident fatalities so that appropriate actions would be

taken to forestall re-occurrence. The designed scheme is

expected to serve as a useful tool for road managers in

their effort to reduce road accident fatalities.

2. The Cumulative Sum (Cusum) Control

Chart

A Cumulative Sum (Cusum) control chart is a graphical

representation of the trend in the outcome of a series of

consecutive procedures performed over time. The Cusum

control charts attempt to device a simple procedure that

will make use of all available information. The unique

and important feature of Cusum is its ability to indicate

the trend of deteriorating performance early, prompting

preventive and corrective measures. A major advantage

of this chart over the ordinary Shewart control chart is

*Corresponding a uthor.

K. S. ADEKEYE ET AL.

214

that it is very effective in detecting relative small shifts.

They are more meaningful graphically, as process shifts

are often easy to detect and points of change can be eas-

ily located [2]. The Cusum control chart is a technique

for identifying whether a real problem has arisen and

provides a means of estimating the time at which the

problem arose. Such information assists in the identifica-

tion of management changes which might have caused a

problem or, alternatively, an improvement, in the attrib-

ute of interest. In effect, the Cusum chart functions as a

significance test in attempting to distinguish real effects

from sampling variance. It was introduced by Page [3],

which provided integral equations for approximately the

Average Run Length (ARL). Many authors like Lucas

[4], Ott et al. [5] and Montgomery [6] have worked on

the Cusum control chart. Lucas [4] gave a detailed pro-

cedure for designing a counted data Cusum chart based

on a poisson counts and implemented it on the occur-

rence of industrial accidents. Osanaiye and Talabi [7]

considered a versatile dimension on the application of

Cusum chart in a non-manufacturing sector and imple-

mented it on the cases of diabetes patients. Adekeye [8]

applied the concept of cusu m control chart to monitor the

crime rate in Nigeria. Nix, Rowland and Kemp [9] used

Cusum control chart to monitor rare congenital malfor-

mations, while Lin and Adams [10] applied Cusum to

monitor surgical performance.

Let 12 m

,,,

XX be a sequence of observed inde-

pendent valu es from a process. Without loss of generality,

the in-control process mean is assumed to be zero. The

principal feature of the Cusum control chart is that the

values of the random variable i

, are

compared with a pre-determined target or reference value

T. The cumulative sum of the deviation of the variable X

from T is

1,2, 3,,im



1ii i

SXTS



 (2.1)

It should be noted that when T is unknown, then an es-

timate of T given by





1im

(2.2)

is often used. In the literatures, S0 = 0 is the ideal value

used. To determine the trend or process shift, the values

of Si, are plotted on a chart or presented in a

table to detect an upward shift or a downward shift in

process quality (one-sided Cusum) or in both directions

(Two-sided Cusum).

3. Designing of Cusum Scheme for Accident

Fatality

The Cusum chart is often used to detect an upward or

downward shift in the process quality (one-sided Cusum

chart) or shift in both directions (Two-sided Cusum

Chart). To monitor a positive shift from the target value,

the Cusum statistic is given as:





max 0,

iii

CXkC







 (2.3)





For monitoring upward movement and for monitoring

downward movement, the statistic is





min 0,

iii

CXkC













Ch

Ch

(2.4)

The process is taken to be out-of-control if i

for an upward shift or i for a downward shift.

The procedures for determination of the parameters of

the Cusum chart (K and h) as discussed in the literatures

(see [3,4]) are presented below.

For a counted data Cusum, the parameters to be de-

termined are the reference value (k) and the decision in-

terval (h) The value k can be described as the reference

value for the process; which is usually chosen between

the acceptable process mean value (0



) and the mean

level that the Cusum scheme is inherited to detect

quickly (1



), otherwise known as the rejectable mean

value. The values of 0



and 1



are mean numbers of

counts per sampling interval. The reference value k for

the counted Cusum should be chosen close to

knm nm







(2.5)

The decision interval, h, is determined by specifying

the in-Control (IC) and out-of-Control (OC) average run

length (ARL). The IC-ARL is the average number of

consecutive procedures required for a Cusum chart to

cross a decision interval or signal during the period when

the process is performing at an acceptable level. This is

analogous to Type I error or false-positive error in hy-

pothesis testing. On the other hand, the OC-ARL is the

average number of procedures performed before the

Cusum chart signals, during the period when the process

is performing at an unacceptable level. It is a measure of

sensitivity and is analogous to power, Type II error or

false-negative error in hypothesis testing.

A design with short IC-ARL (large type I error) is

prone to false alarm while a design with short OC-ARL

(high power) will quickly detect poor performance. Ide-

ally, Cusum monitoring requires long IC-ARL (small

type I error) and short OC-ARL (high power) before the

chart signals an actual deterioration in performance. Un-

fortunately, thus ideal could not be reached, as a desira-

bly long IC-ARL (small type I error) will lead to unac-

ceptably long OC-ARL (low power). On the other hand,

the desired short OC-ARL (high power) will lead to more

frequent false alarm of short IC-ARL (large type I error).

Hence, a trade-off is made between them. For normally

K. S. ADEKEYE ET AL.

215

distributed random variables, Ewan & Kemp [11] deter-

mined the ARL at 0



(call it L0) and the ARL at 1



(call it L1). These ARLs are functions of ,





and n



. Thus, when the reference

value, k is obtained using Equation (2.5) the correspond-

ing decision interval, h is obtained using the nomograph

table (see [5]) or the tables of C1 and C2 schemes for

Poisson variable as presented in British Standard 5703,

part 4 [12].

The data used in this work was obtained from the De-

partment of Policy, Research and Statistics of the Federal

Road Safety Corps (FRSC), Ondo State Command, Akure,

Nigeria for the period January 2001 to December 2010.

Thus it covers the period of 120 months. The data is on

road accident fatalities that occurred within the State for

the period under consideration as recorded on the spot of

the road accidents.

The global expectation on road accident fatalities is to

reduce or minimize the occurrence of death from road

accidents. Therefore, a downward trend in the CUSUM

path will reflect the expectation of the globe. However, a

plot of the Cusum value of the recorded road accident

fatalities reflected an upward movement as shown in

Figure 1. Hence, we design a Cusum chart for an upward

shift. This will enable us to monitor the upward trend and

locate the point at which an unbearable number of oc-

currences are likely to happen.

From the data for the 120 months, the overall mean per

month is 9X8.2 and the standard deviation





Thus, in other to detect changes in the mean level of road

fatalities during this period, the acceptable mean level is

chosen nearer to the current mean level, i.e. 09





and

standard deviation 8.2



. Suppose the authority de-

mands a shift of 1.5σ from 0



to be the rejectable level,

then a shift of 1.5σ in the positive direction yields

121



. Using Equation (2.4), the reference value k is

computed. Thus,

 

21 912

21 9

knn









hTo obtain the decision interval, , the corresponding

value of h fro m table C2 of BS 5703, Part 4, when k = 12,

is equal to 11. Thus, the parameters of the designed

Cusum are k = 12 and h = 11. Therefore, an out-of-con-

trol signal will be indicated when





1iii

max 0,1211CXC



 . The plot of the Cusum

values with the obtained decision interval, h is plotted in

Figure 1.

From Figure 1, it is observed from the Cusum chart

that the Cusum values for the first 71 months are less

than the decision interval (h). From the 72nd to the 89th

months, an upward movement was observed. Thereafter

from the 90th to the 108th months a slight downward shift

in the Cusum values were observed. The last segment of

the Cusum control chart reflect an upward movement. It

should be noted that from the 72nd month to the 120th

month, the Cusum values were above the decision inter-

val. Thus, the process of the monthly road accident fa-

talities can be adjudged to be out-of-control. An impor-

tant feature of the Cusum Control chart is reflected in

Figure 1, the point of change is clearly visible.

In order to know the exact time of the year when the

road accident fatality is more prominent, the data were

broken down into yearly basis and the corresponding

Cusum statistic for each year are implemented to detect

increase in road accident occurrence as shown in Tables

1-10.

4. Discussion of Results

The design Cusum with reference value k = 12 and deci-

sion interval, h = 11 for the period of one hundred and

twenty (120) months as plotted in Figure 1 reflect that

the period under consideration can be categorized into

three segments. The first segment is the period 2001 to

November, 2006 where the road accident fatalities were

less than the decision interval. This implies that, though

Figure 1. Cusum control chart for road accident fatalities.

K. S. ADEKEYE ET AL.

216

Table 1. Cusum tabulation for road accident fatality for

year 2001.



k

i

Xk i

Jan 5 –7 –7 0

Feb 0 –12 –19 0

Mar 6 –6 –25 0

Apr 8 –4 –29 0

May 4 –8 –37 0

Jun 0 –12 –49 0

Jul 3 –9 –58 0

Aug 7 –5 –63 0

Sept 2 –10 –73 0

Oct 3 –9 –82 0

Nov 4 –8 –90 0

Dec 9 –3 –93 0

Table 2. Cusum tabulation for road accident fatality for

year 2002.



k

i

C i

Xk i

Jan 3 –9 –9 0

Feb 2 –10 –19 0

Mar 4 –8 –27 0

Apr 6 –6 –33 0

May 0 –12 –45 0

Jun 7 –5 –50 0

Jul 3 –9 –59 0

Aug 3 –9 –68 0

Sept 5 –7 –75 0

Oct 2 –10 –85 0

Nov 8 –4 –89 0

Dec 6 –6 –95 0

Table 3. Cusum tabulation for road accident fatality for

year 2003.

Table 4. Cusum tabulation for road accident fatality for

year 2004.





Xk

i

Jan 11 –1 –1 0

Feb 16 4 3 4

Mar 0 –12 –9 0

Apr 13 1 –10 1

May 2 –10 –20 0

Jun 9 –3 –23 0

Jul 8 –4 –27 0

Aug 5 –7 –34 0

Sept 4 –8 –42 0

Oct 9 –3 –45 0

Nov 7 –5 –50 0

Dec 4 –8 –58 0

Table 5. Cusum tabulation for road accident fatality for

year 2005.





Xk

i

Jan 6 –6 –6 0

Feb 2 –10 –16 0

Mar 8 –4 –20 0

Apr 4 –8 –28 0

May 7 –5 –33 0

Jun 3 –9 –42 0

Jul 5 –7 –49 0

Aug 0 –12 –61 0

Sept 6 –6 –67 0

Oct 3 –9 –76 0

Nov 9 –3 –79 0

Dec 11 –1 –80 0

Table 6. Cusum tabulation for road accident fatality for

year 2006.



k

i

C i

Xk i

Jan 7 –5 –5 0

Feb 0 –12 –17 0

Mar 0 –12 –29 0

Apr 3 –9 –38 0

May 6 –6 –44 0

Jun 4 –8 –52 0

Jul 5 –7 –59 0

Aug 6 –6 –65 0

Sept 0 –12 –77 0

Oct 7 –5 –82 0

Nov 3 –9 –91 0

Dec 10 –2 –93 0





Xk

i

Jan 4 –8 –8 0

Feb 0 –12 –20 0

Mar 2 –10 –30 0

Apr 2 –10 –40 0

May 11 –1 –41 0

Jun 6 –6 –47 0

Jul 2 –10 –57 0

Aug 0 –12 –69 0

Sept 8 –4 –73 0

Oct 10 –2 –75 0

Nov 0 –12 –87 0

Dec 26 14 –73 14*

K. S. ADEKEYE ET AL. 217

Table 7. Cusum Tabulation for road accident fatality for

year 2007.

XXk



Xk

i

C i i

Jan 4 –8 –8 0

Feb 18 6 –2 0

Mar 14 2 0 0

Apr 24 12 12 12*

May 16 4 16 16*

Jun 9 –3 13 13*

Jul 26 14 27 27*

Aug 6 –6 21 21*

Sept 17 5 26 26*

Oct 10 –2 24 24*

Nov 14 2 26 26*

Dec 18 6 32 32*

Table 8. Cusum tabulation for road accident fatality for

year 2008.

XXk



Xk

i

i i

Jan 13 1 1 1

Feb 15 3 4 4

Mar 15 3 7 7

Apr 20 8 15 15*

May 12 0 15 15*

Jun 4 –8 7 7

Jul 13 1 8 8

Aug 5 –7 1 1

Sept 1 –11 –10 0

Oct 14 2 –8 0

Nov 9 –3 –11 0

Dec 12 0 –11 0

fatalities exist within the said period but the rate is not

alarming. In the second and third segment, the Cusum

values were outside the decision interv al. This implies an

upward shift in the process. The cumulative sum of the

fatalities increases from December, 2006 to May, 2008,

thereafter a downward shift was observed from June

2008 to December, 2009. It should be noted that the

downward shift is not an improvement or reduction per

se, since the values at this period were still far above the

decision interval. The Cusum tabulation in Table 1 through

Table 10 shows that the road accident fatalities during

the period January 2001 to November 2007 (see Tables

1-6) were close and no alarming rate of occurrence was

observed. For the periods where alarming rate was dis-

played, it is very easy to detect the period of the year

when the highest occurrence of road accident fatalities

Table 9. Cusum tabulation for road accident fatality for

year 2009.





Xk

i

Jan 24 12 12 12*

Feb 26 14 26 26*

Mar 24 12 38 38*

Apr 28 16 54 54*

May 20 8 62 62*

Jun 19 7 69 69*

Jul 43 31 100 100*

Aug 10 –2 98 98*

Sept 5 –7 91 91*

Oct 33 21 112 112*

Nov 17 5 117 117*

Dec 0 –12 105 105*

Table 10. Cusum Tabulation for road accident fatality for

year 2010.





Xk

i

Jan 24 12 12 12*

Feb 26 14 26 26*

Mar 24 12 38 38*

Apr 28 16 54 54*

May 20 8 62 62*

Jun 19 7 69 69*

Jul 43 31 100 100*

Aug 10 –2 98 98*

Sept 5 –7 91 91*

Oct 33 21 112 112*

Nov 17 5 117 117*

Dec 0 –12 105 105*

occurred. From Table 6, December 2006 was the only

month that has a Cusum value that is more than the deci-

sion interval. For year 2007 (Table 7), December also

has the highest Cusum value. In year 2008 (see Table 8),

the highest Cusum values was in April and May. From

Tables 9 and 10 the highest occurrence is in the month of

November. It should be noted that the months where the

highest occurrence of road accident fatalities were ob-

served for the years are all festive periods such as

Christmas, Easter, Eid-el-Kabir, and Eid-el-Moluod.

5. Conclusion

The Cusum chart plotted for road accident fatality data is

capable of detecting small shift from the mean level.

More so, the points of change in the process are clearly

identified and points at which these occurred are easily

K. S. ADEKEYE ET AL.

218

located on the CUSUM in Figure 1. Thus, CUSUM chart

provides visual aids diagnosis and identifies the situa-

tions that require quick attentio n. The months where high

occurrence of road accident fatalities are identified should

be used as benchmark by road manager especially Fed-

eral Road safety Corps (FRSC) as periods where more

attention or precaution measure should be put in place on

the roads to drastically reduce or eliminate high occur-

rence of road accident fatalities. The design Cusum con-

trol chart can be adapted for other states in the country,

the entire country and for other countries of the world .

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