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

1

m

i

i

Tm

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:

1

max 0,

iii

CXkC

(2.3)

For monitoring upward movement and for monitoring

downward movement, the statistic is

**

1

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

10

10

mm

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

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