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Computational Statistics and Data Analysis 53 (2009) 3412–3425
Contents lists available at ScienceDirect
Computational Statistics and Data Analysis
journal homepage: www.elsevier.com/locate/csda
Statistical computation and analyses for attribute events
Yafen Liu a , Zhen He a , Lianjie Shu b , Zhang Wu c,∗
a
School of Management, Tianjin University, 300072, China
b
Faculty of Business Administration, University of Macau, 999078, Macau
c
School of Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore
article
info
Article history:
Received 8 June 2008
Received in revised form 11 February 2009
Accepted 16 February 2009
Available online 23 February 2009
a b s t r a c t
This article studies the monitoring of the attribute events based on statistical computation
and analyses. The size of an attribute event is an integer rather than a continuous variable.
For example, the detection of a product lot containing defectives is an attribute event, the
size of which is the number of defectives found in this lot. While many control charts
have been developed for monitoring the time interval (T ) between the occurrences of an
event, many other attribute charts may be employed to examine the size (C ) of the event.
However, these two types of control charts have been investigated and applied separately
with limited syntheses in Statistical Process Control (SPC). This article presents a single SPC
chart (called the rate chart for attribute, or rate chart in short) for simultaneously monitoring
the time interval T and size C of an attribute event based on the ratio between C and T . Our
studies show that the new chart is more effective for detecting the out-of-control status
of the attribute event compared with an individual t chart or an individual c chart, as well
as a combined t & c chart. More profound is that the rate chart performs more uniformly
than other charts for detecting both T shift and C shift, as well as the joint shift in T and
C . The rate chart has demonstrated its potential for both manufacturing systems and nonmanufacturing sectors (e.g., supply chain management, office administration and health
care industry), especially for the latter.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
In general, there are two types of events: a negative one and a positive one. A negative event may be the rejection of a lot
of a product, a failure of a system, or a traffic accident; whilst a positive event may refer to a purchase order of product lots,
the success of a program, the arrival of a tourist group. For example, in a manufacturing production line, a critical sensing
device may go wrong (a negative event) at a random time. Each malfunction of this device will result in several (C ) defective
products. The time between two malfunctions is a random variable T . It is different from the constant sampling interval h
used by many control charts. The size C of an event is the number of items that are incurred by an event occurrence and
also decide the characteristic of the event. It may refer to the number of defects in a rejected lot, the death toll in a traffic
accident, the number of lots sold in a product order, and so on. An SPC system can continuously analyze and monitor the
collected data of an event which involves the time interval T and the size C , and accordingly decides whether the situation
is under control or out of control. Both T and C are random variables, but T is a continuous variable, while C is an integer
one. In this article, T and C are assumed to be independent of each other.
An upward shift (i.e., a decrease of T and/or an increase of C ) or an increase of the ratio C /T (called event ratio and denoted
by R) indicates a move in a loss direction for a negative event. It means a higher rate in damage, cost, or loss incurred by the
∗
Corresponding author. Tel.: +65 67904445; fax: +65 67911859.
E-mail address: mzwu@ntu.edu.sg (Z. Wu).
0167-9473/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.csda.2009.02.010
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Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425
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occurrences of the event. However, an upward shift represents a move in a gain direction for a positive event, suggesting that
the gain or success has been achieved more often and in a larger scale. On the contrary, a downward shift (i.e., an increase
of T and/or a decrease of C ) or a decrease of the event ratio R indicates a gain movement for a negative event, showing that
the severity of the problem is alleviated or improved. But a downward shift represents a loss movement for a positive event,
meaning that the gain and profit is made less frequently and in a smaller amount. In SPC practices, one is more interested
in detecting the upward R shifts for a negative event and the downward R shifts for a positive event, because these shifts
represent a transit to a worse status, and the users should be alarmed as soon as possible.
Many control charts have been developed to monitor the time interval T of an event, including the TBE (Time Between
Events) or exponential chart (Chan et al., 2000; Xie et al., 2002; Zhang et al., 2008), the Gamma chart (Wu et al., 2001; Zhang
et al., 2007), the exponential CUSUM chart (Lucas, 1985; Vardeman and Ray, 1985; Gan, 1992; Borror et al., 2003; Liu et al.,
2006), and the exponential EWMA chart (Gan, 1998). As a matter of fact, the effect (e.g., the amount of damage or loss) of
an event in a certain time period is, however, not only dependent on T (or the frequency of occurrence), but also on the
size C . For example, the loss of market competitiveness of an aged product is indicated not only by the decreased number
of purchase orders but also by the decreased number of lots to be delivered in each order. Similarly, a task team for bird flu
prevention and elimination must be enhanced not only when the outbreak of this infection becomes very frequent but also
when the number C of poultry infected in each outbreak gets very high. Many control charts for attributes with different
advanced features have been proposed recently by the researchers (Shepherd et al., 2007; Wu and Jiao, 2008; Chiu and Kuo,
2008; Chakraborti and Human, 2008; Höhle and Paul, 2008; Chiu and Sheu, 2008; Sim and Lim, 2008). The individual c chart
used widely in SPC may be adopted to monitor the size C of an attribute event. Whenever the event occurs, the sample value
of C will be checked against the control limits of the c chart, and the process status is decided.
In SPC literature, little effort has been made to explore the algorithms for monitoring T and C concurrently. A combined
t & x scheme consisting of a t chart and an x chart was recently investigated for examining both the time interval T and
magnitude X of an event, in which X is a continuous variable (Wu et al., in press). It is similar to an x̄ & R (or x̄ & S) chart
combination for monitoring the mean and variability of a variable (Khoo and Yap, 2005). In reality, the size of an event is an
integer or count data in many SPC applications (Zhang and Lin, 2008; Rigby et al., 2008), especially in non-manufacturing
sectors (e.g., health-care industry (Woodall, 2006), supply chain systems, language data process (Yu et al., 2003)). In these
applications, a synthesized scheme is needed for monitoring the attribute events.
This article investigates a single rate chart that monitors both T and C by checking a single statistic R, or the ratio between
C and T . Two types of rate charts will be developed, one for detecting increasing R shifts of negative events, another for
detecting decreasing R shifts of positive events.
The single rate chart makes use of information about both T and C , and thus has higher detection effectiveness than an
individual t chart or an individual c chart, or even a joint t & c chart. In this research, the effectiveness of a control chart is
measured by the steady-state Average Time to Signal (ATS) which measures the average time required to signal a process
shift once it occurs.
2. Data analysis for an attribute event
A rate chart is operated by checking the sample value of the event ratio R (= C /T ). It has only one charting parameter:
the upper control limit UCLr for detecting the upward shift (or an increase of the event ratio R) or the lower control limit
LCLr for detecting the downward shift (or a decrease of R). The procedure to operate a rate chart and to analyze the sample
data for detecting upward R shifts is outlined below:
(1) When an occurrence of the event is detected, find the time interval T between the current and the last occurrences and
meanwhile count the size C for this instance.
(2) Calculate the event ratio R.
C
.
(1)
T
(3) If R goes above UCLr , the process is thought to be out of control due to an increase in event ratio R.
(4) If a R shift is detected, go to step (5). Otherwise, the process is considered to be in control, and go back to step (1) to wait
for the next occurrence of the event.
(5) Stop the process immediately, and subsequently investigate and remove the assignable cause. Afterwards, resume the
process and start with step (1) again.
R=
The implementation of a rate chart detecting the downward R shifts (increasing T and/or decreasing C ) is quite similar
except step (3) being modified as follows:
(3) If R falls below LCLr , the process is considered as out of control due to a decrease in R.
3. Statistical computation of the parameters of a rate chart
To design a rate chart, the following three specifications need to be determined:
(1) The allowed minimum value, τ , of the in-control Average Time to Signal (ATS 0 ). The value of τ is decided with respect to
the tolerable false alarm rate. The resultant (or actual) ATS 0 value must be no smaller than τ . Moreover, when comparing
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the effectiveness of control charts, all charts are required to produce same or close ATS 0 values, so that all of them result
in the similar false alarm rate and the comparison is fair (Montgomery, 1994; Reynolds and Stoumbos, 2004). Usually,
this requirement can be met by adjusting the control limits of the charts.
(2) The in-control values of the parameters associated with the probability distribution of T . Most of the reported works
(Xie et al., 2002) assume that T follows an exponential distribution. Therefore, the only distribution parameter, λ, is the
reciprocal of the mean value of T . The in-control value λ0 can be estimated from the in-control sample mean of T that is
usually obtained by analyzing historical data. The probability density function and the cumulative probability function
of an exponential distribution are given by,
fT (T ) = λe−λT ,
(2)
FT (t ) = Pr(T ≤ t ) = 1 − e
−λt
(3)
.
(3) The in-control values of the parameters of the probability distribution of C . The random number C is counted on a per
event occurrence basis and usually follows a Poisson distribution. In fact, any random phenomenon that occurs on a per
unit (or per unit area, per unit volume, per unit time, etc.) basis is often well approximated by the Poisson distribution
(Montgomery, 2005). This distribution has only one parameter, g, which is the mean value of C . The in-control value
g0 can be estimated by analyzing the average of the observed C values from m samples when the process is in control
(Montgomery, 2005).
g0 ≈
m
P
i=1
ci
.
m
The probability mass function and the cumulative probability function of a Poisson distribution are given by,
e−g g C
p(C ) =
C!
(4)
(5)
,
FC (c ) = Pr(C ≤ c ) =
c
X
e−g g C
C =0
C!
(6)
.
When a process is in control, ATS 0 is determined by the following formula (Wu and Spedding, 1999):
1
ATS0 =
λ0
· ARL0 =
1
λ0
·
1
α
=
1
λ0 α
(7)
,
where ARL0 is the in-control Average Run Length, and α is the type I error of the rate chart. If ATS 0 is made equal to τ , we
obtain (from Eq. (7)),
α=
1
λ0 · τ
(8)
.
The cumulative probability function of the statistic R can be derived as follows:
FR (r ) = Pr(R ≤ r ) = Pr
=
∞
X
=
∞
X
C =0
C =0
Pr T ≥
exp −
y
r
λC
r
C
T
≤r
= Pr T ≥
y = C p(C ) =
−g
gC
≈
C!
∞
X
Uc
X
C =0
C =0
C
r
exp −
exp −
λC
r
λC
r
−g
e−g g C
C!
gC
C!
,
(9)
where p(C ) is the probability mass function of C given by Eq. (5). An upper limit Uc is used in Eq. (9) for the numerical
computation. It ensures that the error
a specified level. In this article, Uc is determined as the smallest integer
is smaller than
that makes the expression of exp −
λ(Uc +1)
r
−g
g Uc +1
(Uc +1)!
equal to or smaller than 0.0000001.
The design of a rate chart is simply to determine the control limit LCLr or UCLr . We can first calculate α corresponding to
the specifications λ0 and τ (Eq. (8)). Then, LCLr for detecting downward R shifts is calculated by
LCLr = FR−1 (α),
(10)
UCLr = FR−1 (1 − α).
(11)
where FR−1 () is the inverse function of the cumulative probability function FR () (Eq. (9)). For detecting upward R shifts, UCLr
is determined by
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The above design algorithm for the rate chart has been implemented in a computer program (attr_rchart.c) in C language.
The computer program is available from the authors upon request. The users only need to key in the specifications τ , λ0 and
g0 , and the computer program will subsequently complete the design of a rate chart almost in no time in a personal computer.
4. Analyses of statistical characteristics
The effectiveness of four control charts (the individual t chart and individual c chart, the t & c chart, and the rate chart) is
compared in this section. The control limits of the individual t chart, c chart and t & c chart can be easily determined based
on type I error α (see Eq. (8)).
Individual t chart
ln(1 − α)
, (detecting upward R shifts)
λ0
ln(α)
. (detecting downward R shifts)
UCL = −
λ0
LCL = −
(12)
Individual c chart
Its control limits can be obtained by rounding off the results from the following equation so that ATS 0 ≥ τ (or the
resultant false alarm rate being no higher than the specified level).
UCL = FC−1 (1 − α),
−1
(detecting upward R shifts)
(detecting downward R shifts)
LCL = FC (α).
(13)
−1
where FC () is the inverse function of the cumulative probability function of the Poisson distribution (Eq. (6)).
It is noted that the sampling interval for a conventional c chart is usually fixed. But when the c chart is adopted to monitor
the size of an attribute event, the sampling interval must be equal to the random time interval T between two occurrences
(i.e., taking a sample only when the event happens). Otherwise, the conventional control chart is almost incapable of
detecting the changes of the size C of an event (let alone the detection effectiveness), because a sample taken at a fixed
interval is unlikely to catch the occurrence of an event which happens randomly.
t & c chart
A t & c chart runs a t chart and a c chart simultaneously. If a t & c chart is to detect upward R shifts, it uses a t chart with
a lower control limit LCLt and a c chart with an upper control limit UCLc . The t & c chart will produce an out of control signal
when the sample value of T of an occurrence falls below LCLt and/or the sample value of C goes above UCLc . In contrast, a
t & c chart for detecting downward R shifts uses a t chart with an upper control limit UCLt and a c chart with a lower control
limit LCLc . It will produce an out of control signal when a sample value of T goes above UCLt and/or a sample value of C falls
below LCLc .
When a t & c chart for detecting upward R shifts is to be designed, the control limit UCLc of the c chart can be determined
by
UCLc = FC−1 (1 − αc ),
(14)
and LCLt of the t chart is determined by
LCLt = −
ln(1 − αt )
λ0
,
(15)
where FC−1 () is the inverse function of the cumulative probability function FC () for a Poisson distribution (Eq. (6)), and αt and
αc are the type I errors allocated to the t chart and c chart, respectively. In this study, we make
αt = αc = 1 −
√
1−α
(16)
so that the total type I error of the t & c combination is equal to the overall type I error α (see Eq. (8)).
On the other hand, if a t & c chart is designed to detect the downward R shifts, the control limit LCLc of the c chart will
be obtained by
LCLc = FC−1 (αc ),
(17)
and the control limit UCLt of the t chart is determined as follows:
UCLt = −
ln(αt )
λ0
.
(18)
We first study the chart performance for detecting an increasing event ratio R, and then for detecting a decreasing R.
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Table 1
ATS values of four control charts for increasing R shift.
δλ
Chart
δg
1.00
1.10
1.20
1.30
1.41
1.51
10.00
t
c
t&c
Rate
114.5
1229.7
148.1
115.6
114.5
665.7
142.8
106.4
114.5
391.7
134.6
98.7
114.5
247.1
123.6
92.2
114.5
165.2
110.6
86.6
114.5
116.2
96.6
81.8
8.20
t
c
t&c
Rate
166.3
1499.6
214.6
167.6
166.3
811.9
205.2
153.9
166.3
477.7
191.2
142.4
166.3
301.3
173.0
132.8
166.3
201.5
151.9
124.5
166.3
141.7
130.2
117.3
6.40
t
c
t&c
Rate
266.4
1921.4
341.6
268.0
266.4
1040.2
322.8
245.4
266.4
612.1
295.6
226.7
266.4
386.1
261.6
210.9
266.4
258.2
224.1
197.3
266.4
181.5
187.3
185.6
4.60
t
c
t&c
Rate
502.8
2673.2
634.8
504.9
502.8
1447.2
588.0
461.4
502.8
851.6
523.5
425.2
502.8
537.1
447.5
394.6
502.8
359.2
369.7
368.4
502.8
252.5
298.5
345.8
2.80
t
c
t&c
Rate
1322.6
4391.7
1600.4
1325.4
1322.6
2377.6
1422.3
1208.1
1322.6
1399.1
1199.3
1110.7
1322.6
882.4
965.1
1028.4
1322.6
590.1
752.2
957.9
1322.6
414.8
577.5
897.0
1.00
t
c
t&c
Rate
10000.0
12296.7
10000.0
10000.0
10000.0
6657.3
7760.8
9082.7
10000.0
3917.5
5647.6
8320.2
10000.0
2470.8
3973.8
7676.4
10000.0
1652.3
2779.7
7125.7
10000.0
1161.5
1968.9
6649.0
Study 1: Detection of increasing event ratio R
The comparison is first conducted with the following general condition:
τ = 10 000,
λ0 = 0.01,
g0 = 4.
(19)
The specification of (λ0 = 0.01) means that, on average, the event takes place once for every 100 time units when the
process is in control, and (τ = 10 000) implies that a false alarm is produced for every 100 occurrences of the event. Based
on the above specifications, the four charts are designed and the charting parameters are listed below:
Individual t chart:
Individual c chart:
t & c chart :
Rate chart :
LCL = 1.0050
UCL = 9
LCLt = 0.7207,
UCLr = 3.9750.
UCLc = 10
Since the mean value of C is equal to the parameter g, the Poisson distribution of C will move to the right when g becomes
larger than g0 , and the C value as well as the event ratio R will get greater in general. Likewise, if the parameter λ increases,
R becomes larger because of the decrease of T . We can set
λ = δλ λ0 ,
g = δg g0 ,
(20)
where δg represents the C shift in terms of g0 , and δλ indicates the change of T in terms of λ0 . A larger δg and/or δλ lead to a
greater value of the event ratio R, and vice versa. The process is in control when (δg = δλ = 1). Then a domain of increasing
R shift can be defined as follows:
1 ≤ δλ ≤ 10, 1 ≤ δg ≤ 1.51.
(21)
The maximum value δλ,max (= 10) of δλ and the maximum value δg ,max (= 1.51) of δg are determined so that the power
of the individual t chart for detecting the pure decreasing T shift at (δλ = δλ,max ) is equal to the power of the individual c
chart for detecting the pure increasing C shift at (δg = δg ,max ). The purpose is to leverage the sizes of the T shift and C shift
in the performance study.
Table 1 tabulates the ATS values of the four charts in the above shift domain (21). The formulae for calculating the ATS
of the rate chart are given in the Appendix. There are a total of 35 out-of-control cells (combinations of the discrete values
of δg and δλ ) and one in-control cell (δg = δλ = 1) at the bottom-left corner. Each cell contains the ATS values of the four
charts for a particular combination of the values of δg and δλ . It is interesting to observe the followings:
(1) Firstly, the t chart, t & c chart and rate chart generate an identical ATS 0 value equal to τ when the process is in control.
However, the c chart generates an ATS 0 value larger than τ due to the discrete nature of the size of C . Any attempt to
reduce the ATS 0 of the c chart (e.g., reduce the UCL of the c chart by one) will make ATS 0 smaller than τ and therefore is
not allowed.
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(2) For detecting the pure C shift (see the out-of-control cells at the bottom row, corresponding to δg > 1 and δλ = 1), the
individual c chart suggests itself to be most effective, followed by the t & c chart, and then the rate chart. The individual
t chart is completely insensitive.
(3) On the other hand, for detecting the pure T shift (see the out-of-control cells at the left-hand side column, where δg = 1
and δλ > 1), the individual t chart and the rate chart are most, and almost equally, effective. The other two charts are
also capable of detecting the pure T shift, but the power, especially the power of the individual c chart, is very low. The
ATS value of the c chart becomes shorter along with the increase of λ, because the sampling interval of the c chart is
equal to T .
(4) For detecting the joint (or simultaneous) shifts in T and C (see the out-of-control cells in which δg > 1 and δλ > 1), the
rate chart outperforms other three charts in most of the cases. It is only slightly inferior to the individual c chart in a few
cells where the C shift prevails.
In order to have a more general and quantitative analysis of the relative effectiveness of the charts, the average of the
ratios (AR) of ATS values across a shift region of interest is calculated for each chart.
AR =
m
P
i=1
ATS (δg ,i ,δλ,i )
ATSr (δg ,i ,δλ,i )
(22)
,
m
where δg ,i and δλ,i denote the discrete values of δg and δλ in the ith cell and m is the number of out-of-control cells in the shift
region. ATS (δg ,i , δλ,i ) is the value of ATS produced by a particular chart at (δg ,i , δλ,i ) and ATS r (δg ,i , δλ,i ) is the value generated
by the rate chart in the same cell. Obviously, if the AR value of a chart is larger than one, this chart is generally less effective
than the rate chart in the shift region, and vice versa. The AR values are studied in the following four regions:
Region 1 (δg > 1 and δλ = 1): ARc is calculated in this region to compare the ATS performance against the pure C shifts,
with m = 5 (see Table 1).
Region 2 (δg = 1 and δλ > 1): ARt is calculated to compare the ATS performance against the pure T shifts, with m = 5.
Region 3 (δg > 1 and δλ > 1): ARc +t is calculated to compare the ATS performance against the concurrent T and C shifts,
with m = 25.
Region 4 ARoverall is derived by merging regions 1, 2 and 3, and is used to compare the overall ATS performance over all the
out-of-control cells in Table 1, with m = 35.
The AR values for the individual t chart, individual c chart and t & c chart are calculated using the data in Table 1 and
the results are listed in RUN 0 in Table 2. These AR values reveal that the rate chart outperforms the two individual charts
and the t & c chart in region 3 (where δg > 1 and δλ > 1) and region 4 (the overall region). Specifically, the ARoverall values
indicate that the rate chart is, on average, more effective than the individual t chart, individual c chart and t & c chart by
23.0%, 161.9% and 8.5%, respectively. Although the individual t chart is effective in the pure T shift region and the individual
c chart and t & c chart are very sensitive to the pure C shift, each of them is either completely insensitive or very ineffective in
some other circumstances. An apparent advantage of the rate chart is its uniform performance over the whole shift domain
(sensitive to both T and C shifts).
Next, a 23 factorial experiment (Montgomery, 2005) is carried out to further study the relative effectiveness of the charts
under different conditions. The three specifications, τ , λ0 and g0 , are used as the input factors and each of them varies at
two levels, resulting in eight runs. The low and high levels for each factor are decided below:
τ
g0
λ0
5000
2
0.003
20 000
8
0.03.
(23)
The eight resultant combinations of the values of τ , λ0 and g0 are listed at the left hand side of Table 2.
The results of all eight runs are shown in Table 2. Similar to RUN 0, the rate chart is, on average, the most effective chart.
The two individual charts excel only in a small region. The t & c chart has an ARoverall value slightly smaller than one in RUN
1 and RUN 2, but it is considerably less effective than the rate chart in other RUNs.
Finally, a grand average AR that indicates the average of AR values encompassing all the eight runs is calculated. The results
are listed at the bottom of Table 2. The values of ARoverall indicate that, from a most comprehensive viewpoint (covering all
different values of τ , λ0 , g0 , δλ and δg ), the rate chart is more effective than the individual t chart, individual c chart and t & c
chart by 23.0%, 174.6% and 14.7%, respectively. More importantly, the rate chart demonstrates a unique strength in achieving
a balanced effectiveness for detecting both T shift and C shift, as well as the joint shift. Its performance is satisfactory across
the entire shift domain. Conversely, the individual t chart is only effective for detecting a T shift yet fails to sense the C shift.
Similarly, the individual c chart is only powerful for detecting a C shift but intolerably ineffective for a T shift.
Study 2: Detection of decreasing event ratio R
In this case, the four charts are first compared under the following condition:
τ = 10 000,
λ0 = 0.01,
g0 = 13.
(24)
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Table 2
AR values of the control charts for increasing R shift.
RUN
τ
λ0
g0
Chart
ARc
ARc +t
ARoverall
0
10000
0.010
4
t
c
t&c
1.303
0.386
0.547
0.994
7.071
1.260
1.262
2.176
1.158
1.230
2.619
1.085
1
5000
0.003
2
t
c
t&c
1.711
0.469
0.639
0.953
4.586
1.080
1.311
1.413
1.042
1.317
1.731
0.990
2
5000
0.003
8
t
c
t&c
1.330
0.361
0.443
0.988
3.939
1.363
1.174
1.239
0.974
1.170
1.499
0.954
3
5000
0.030
2
t
c
t&c
1.396
0.460
0.401
0.992
8.707
2.151
1.353
2.672
1.429
1.308
3.218
1.385
4
5000
0.030
8
t
c
t&c
1.198
0.515
0.409
0.998
10.696
1.772
1.182
3.044
1.229
1.158
3.776
1.190
5
20000
0.003
2
t
c
t&c
1.472
0.357
0.595
0.981
5.367
1.213
1.360
1.807
1.205
1.322
2.109
1.119
6
20000
0.003
8
t
c
t&c
1.236
0.563
0.430
0.995
10.976
1.554
1.193
2.979
1.142
1.171
3.776
1.099
7
20000
0.030
2
t
c
t&c
1.311
0.470
0.402
0.998
9.532
2.145
1.302
2.937
1.439
1.260
3.527
1.392
8
20000
0.030
8
t
c
t&c
1.159
0.328
0.481
0.999
6.570
1.456
1.156
2.056
1.170
1.134
2.454
1.112
ARc +t
ARoverall
1.255
2.258
1.199
1.230
2.746
1.147
ARc
t
c
t&c
1.346
0.434
0.483
ARt
ARt
0.989
7.494
1.555
Accordingly, the parameters of control charts can be designed as listed below:
Individual t chart:
Individual c chart:
t & c chart:
Rate chart:
UCL = 460.5170
LCL = 5
UCLt = 506.9858,
LCLr = 0.0229.
LCLc = 5
The following domain for the decreasing R shifts is used (see Table 3):
0.10 ≤ δλ ≤ 1, 0.37 ≤ δg ≤ 1.
(25)
The in-control cell is now located at the top-right corner in Table 3. The minimum value (δλ,min = 0.10) of δλ and the
minimum value (δg ,min = 0.37) of δg are determined so that the sizes of the T shift and C shift are nearly leveraged. Table 3
tabulates the ATS values of the four charts within the shift domain. It can be found that:
(1) The t chart, the t & c chart and the rate chart generate an identical ATS 0 value equal to τ when δλ = δg = 1 at the
top-right corner. However, the c chart again generates an ATS 0 value larger than τ due to the discrete nature of the size
C.
(2) For detecting the pure decreasing C shift (see the out-of-control cells at the top row where δg < 1 and δλ = 1), the t & c
chart is most effective, followed by the individual c chart and the rate chart. The individual t chart is again insensitive
at all.
(3) For detecting the pure increasing T shift (see the out-of-control cells at the right-hand-side column where δg = 1 and
δλ < 1), the individual t chart is most sensitive, followed by the t & c chart and the rate chart. However, the out-ofcontrol ATS values generated by the individual c chart are even larger than the in-control ATS 0 . It is because that the
sampling intervals become larger when T is increasing.
(4) For detecting the joint shift in T and C (see the out-of-control cells in which δg < 1 and δλ < 1), the rate chart
outperforms the other three charts in general.
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Table 3
ATS values of four control charts for decreasing R shift.
Chart
δλ
δg
0.37
0.50
0.62
0.75
0.87
1.00
1.00
t
c
t&c
Rate
10000.0
211.9
210.4
552.3
10000.0
438.1
429.0
985.7
10000.0
1059.9
999.6
1759.3
10000.0
2873.5
2447.0
3139.8
10000.0
8482.1
5555.9
5603.5
10000.0
26736.6
10000.0
10000.7
0.82
t
c
t&c
Rate
5010.3
258.4
248.3
396.8
5010.3
534.3
490.7
693.5
5010.3
1292.6
1080.3
1204.8
5010.3
3504.2
2342.9
2065.3
5010.3
10344.0
4285.8
3495.3
5010.3
32605.7
6040.2
5855.9
0.64
t
c
t&c
Rate
2752.4
331.1
307.6
392.9
2752.4
684.6
577.7
614.9
2752.4
1656.1
1141.7
963.5
2752.4
4489.8
2036.4
1497.9
2752.4
13253.3
2927.4
2306.7
2752.4
41776.0
3457.5
3522.7
0.46
t
c
t&c
Rate
1638.4
460.7
398.6
419.2
1638.4
952.4
674.2
581.2
1638.4
2304.2
1109.1
811.1
1638.4
6246.7
1565.3
1129.7
1638.4
18439.3
1863.6
1565.6
1638.4
58123.1
1997.1
2157.8
0.28
t
c
t&c
Rate
1162.9
756.9
566.8
527.7
1162.9
1564.7
800.0
642.7
1162.9
3785.4
1045.2
789.0
1162.9
10262.4
1213.9
970.8
1162.9
30293.2
1295.3
1193.7
1162.9
95488.0
1326.6
1465.2
0.10
t
c
t&c
Rate
1475.5
2119.3
1221.6
1142.9
1475.5
4381.2
1369.1
1222.1
1475.5
10599.2
1469.2
1311.8
1475.5
28734.6
1519.2
1410.7
1475.5
84820.9
1539.6
1518.4
1475.5
267366.4
1546.9
1635.0
A similar 23 experiment is also carried out in which the low and high levels of the specifications are determined, as below:
τ
g0
λ0
5000
10
0.003
20 000
17
0.03.
(26)
For this 23 experiment, the AR values are calculated in the following four regions and the results are shown in Table 4:
Region 1:
Region 2:
Region 3:
Region 4:
ARc for pure decreasing shift in C (δg < 1 and δλ = 1)
ARt for pure increasing shift in T (δg = 1 and δλ < 1)
ARc +t for joint shift in T and C (δg < 1 and δλ < 1)
ARoverall for the rally of all above.
As shown in Table 4, the AR values indicate that the rate chart outperforms the t & c chart slightly from an overall
viewpoint. Usually, the rate chart is more effective for detecting joint shifts in T and C and/or when g0 is smaller; and the
t & c chart is more sensitive to pure C or pure T shifts and/or when g0 is larger. Both the rate chart and t & c chart outperform
the individual t chart and individual c chart to a degree much more significant than the cases for detecting upward R shifts.
The individual t chart is too ineffective for detecting any decrease in C , whereas the individual c chart is extremely dull
for detecting any increase in T . The values of the grand ARoverall at the bottom of Table 4 manifest that the rate chart is, on
average, more effective than the individual t chart and individual c chart by 336.5% and 1410.2%, respectively.
5. Examples
Example 1: Detection of increasing event ratio R
A municipal department is developing a SPC tool to monitor the occurrence of the disastrous accidents. The SPC system
can help decide when an immediate and reinforced action should be taken. In this case, T is the time interval between two
consecutive accidents and C is the death toll in an occurrence. The department has the records of ti and ci for 25 accidents
(Table 5) for a time period when the situation was thought normal (in an in-control status). It is found that the data of T can
be fitted very well to an exponential distribution and the data of C to a Poisson distribution. The in-control parameters of
the distributions can be estimated from the records:
c1 + c2 + · · · + c25
= 4.0400,
g0 ≈ C̄ =
25
1
1
= 0.0720.
λ0 ≈ =
(t1 + t2 + · · · + t25 )/25
T̄
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Table 4
AR values of the control charts for decreasing R shift.
RUN
τ
λ0
g0
Chart
0
10000
0.010
13
t
c
t&c
7.781
0.772
0.631
0.818
54.611
0.958
2.791
7.073
1.102
3.222
12.964
1.014
1
5000
0.003
10
t
c
t&c
2.860
0.892
0.707
0.903
13.088
1.056
1.635
3.292
1.059
1.706
4.349
1.009
2
5000
0.003
17
t
c
t&c
2.317
0.514
0.596
0.945
8.107
1.082
1.509
2.107
0.992
1.544
2.737
0.949
3
5000
0.030
10
t
c
t&c
14.590
0.882
0.756
0.715
60.185
0.859
3.893
7.839
1.181
4.967
14.323
1.074
4
5000
0.030
17
t
c
t&c
7.908
0.356
0.561
0.842
35.059
0.958
2.822
4.332
1.070
3.265
8.154
0.982
5
20000
0.003
10
t
c
t&c
7.106
0.656
0.599
0.803
22.064
1.088
2.638
3.971
1.100
3.014
6.082
1.027
6
20000
0.003
17
7
20000
0.030
10
t
c
t&c
t
c
t&c
4.646
0.409
0.567
49.536
1.250
1.047
0.890
19.734
1.012
0.552
207.176
0.629
2.157
3.167
1.029
7.885
19.368
1.351
2.332
5.140
0.961
12.787
43.609
1.204
8
20000
0.030
17
t
c
t&c
20.724
0.545
0.477
0.753
196.434
0.914
4.732
14.587
1.103
6.448
38.559
0.987
ARc
ARt
ARc +t
ARoverall
t
c
t&c
ARc
13.052
0.697
0.660
ARt
0.802
68.495
0.951
ARc +t
3.340
7.304
1.110
ARoverall
4.365
15.102
1.023
The value of λ0 indicates that, on average, there is an accident for every 13.9 days. Other specifications are decided as
τ = 365 days. The main concern of the department is to alarm any rise of event ratio R, that is, the increase in the frequency
of the accidents and the increase of death toll in each occurrence.
Using the computer program attr_rchart.c, the four charts aforementioned can be easily designed and the charting
parameters are listed below:
Individual t chart:
Individual c chart:
t & cchart:
Rate chart:
LCL = 0.5389
UCL = 8
LCLt = 0.2217,
UCLr = 7.4600.
UCLc = 8
The ATS values of the four charts are given in Table 6, where the in-control cell is located at the bottom-left corner. The
performance comparison of the four charts is similar to that summarized in Study 1 in the section of analyses of statistical
characteristics. The rate chart is generally more powerful than other charts except for prevailing C shifts. The values of the
average ratio are ARc = 1.410, ARt = 0.983, ARc +t = 1.257, ARoverall = 1.240 for the t chart; ARc = 0.509, ARt = 7.603,
ARc +t = 2.119, ARoverall = 2.672 for the c chart; and ARc = 0.428, ARt = 1.608, ARc +t = 1.119, ARoverall = 1.090 for the
t & c chart. The AR overall values indicate that the rate chart is more effective than the individual t chart, individual c chart
and t & c chart by 24.0%, 167.2% and 9.0%, respectively.
Example 2: Detection of decreasing event ratio R
An electronic company is planning to develop a SPC chart to monitor the selling of one of its main products, an integrated
circuit chip. In this case, a positive event arises, referring to the purchase order made by customers. Here, T is the time interval
(in working hours) between two consecutive orders and C is the number of lots purchased in an order. The company has
the records of ti and ci for 50 orders (Table 7) received during a time period when the selling of the product was thought
normal. The in-control parameters of the exponential distribution for T and the Poisson distribution for C can be estimated
from the records:
c1 + c2 + · · · + c50
g0 ≈ C̄ =
= 11.4600,
50
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Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425
Table 5
Sample data of T and C in example one.
No.
ti (day)
ci (death toll)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
8.0
6.7
13.3
9.5
0.1
3.4
16.5
5.5
20.4
38.8
2.2
47.7
19.3
7.0
10.7
0.9
12.1
3.9
32.4
38.2
3.3
14.1
15.1
9.6
8.5
2
1
5
6
5
7
5
5
2
4
6
5
3
2
1
2
5
6
6
4
1
2
5
6
5
λ0 ≈
1
T̄
=
1
(t1 + t2 + · · · + t50 )/50
= 0.0561.
The value of λ0 indicates that, on average, there is one purchase order for every 17.8 h or 2.23 days (eight hours per day).
Other specifications are decided as τ = 1440 h (or six months). The main concern of the company is to signal any decline
of sales, that is, the decrease in the frequency of the purchase orders and the reduction in the number of lots of each order.
The four charts are designed and the charting parameters are listed below:
Individual t chart:
Individual c chart:
t & c chart:
Rate chart:
UCL = 78.2572
LCL = 5
UCLt = 84.0564,
LCLr = 0.1161.
LCLc = 4
The ATS values of the four charts are given in Table 8, where the in-control cell is located at the top-right corner. For this
case, the rate chart is more powerful for detecting most of the shifts, except for prevailing T shifts or prevailing C shifts. The
ARoverall values indicate that the rate chart is more effective than the individual t chart, individual c chart and t & c chart by
216.0%, 420.8% and 7.2%, respectively.
6. Conclusions
This article discusses the computation of the parameters of a single rate chart for monitoring the time interval T and
size C of an attribute event, and also the analyses of the statistical characteristics of this chart. Unlike the t & c combination,
the rate chart manipulates only a single statistic R (event rate). But, it is also able to make use of information related to
both T and C . The statistical performance of the rate chart significantly excels the individual t chart and individual c chart,
especially for detecting decreasing R shifts. It also outperforms the combined t & c chart from a holistic viewpoint.
In addition to high detection effectiveness, the merit of the rate chart lies in its uniform statistical performance (sensitive
to both T and C shifts). This feature is important as T and C shifts happen simultaneously in most attribute events. Reynolds
and Stoumbos (2004) pointed out that, in most of the cases, it is very difficult to predict the types and magnitudes of process
shifts. Therefore a control chart should have an excellent overall performance for different shifts.
Another potential merit of the rate chart is that control charts with even better statistical properties may be further
developed based on this single rate chart. For example, similar to the CUSUM scheme for monitoring the process mean and
variance used in variable SPC (Reynolds and Stoumbos, 2004; Zhao et al., 2005; Shu and Jiang, 2006), a CUSUM type rate
chart for simultaneously monitoring T and C may be an avenue for further research.
In this article, it is assumed that T and C follow the exponential distribution and Poisson distribution, respectively, and
they are mutually independent. It would be interesting to carry out further statistical analyses on the rate chart for the cases
in which T and/or C follow other probability distributions, or T and C are interdependent. For interdependent T and C , a
joint probability distribution has to be used.
Author's personal copy
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Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425
Table 6
ATS values of four control charts in example one.
δλ
Chart
δg
1.00
1.14
1.28
1.56
1.70
10.00
t
c
t&c
rate
5.7
61.5
9.5
5.8
5.7
30.5
8.5
5.4
5.7
17.3
7.3
5.0
1.42
5.7
10.9
6.1
4.7
5.7
7.4
5.1
4.5
5.7
5.4
4.2
4.3
8.20
t
c
t&c
rate
7.9
75.0
13.3
8.1
7.9
37.2
11.6
7.4
7.9
21.1
9.8
6.8
7.9
13.3
8.0
6.4
7.9
9.1
6.5
6.1
7.9
6.6
5.3
5.8
6.40
t
c
t&c
rate
12.0
96.1
20.2
12.2
12.0
47.7
17.2
11.1
12.0
27.1
13.9
10.2
12.0
17.0
11.1
9.6
12.0
11.6
8.8
9.0
12.0
8.5
7.1
8.5
4.60
t
c
t&c
rate
21.4
133.7
35.2
21.7
21.4
66.4
28.5
19.6
21.4
37.6
22.1
17.9
21.4
23.7
16.9
16.6
21.4
16.2
13.1
15.5
21.4
11.8
10.3
14.6
2.80
t
c
t&c
rate
53.1
219.7
79.2
53.4
53.1
109.1
59.2
47.7
53.1
61.8
42.8
43.3
53.1
38.9
31.1
39.7
53.1
26.6
23.1
36.8
53.1
19.4
17.8
34.4
1.00
t
c
t&c
rate
365.0
615.2
365.0
365.0
365.0
305.4
229.2
320.8
365.0
173.2
146.6
286.4
365.0
108.9
98.3
258.8
365.0
74.4
69.6
236.1
365.0
54.4
52.0
217.2
Table 7
Sample data of T and C in example two.
No.
ti (h)
ci (No. of lots)
No.
ti (h)
ci (No. of lots)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
9.6
8.0
16.0
11.4
0.1
4.1
19.7
6.6
24.4
46.5
2.7
57.3
23.2
8.4
12.8
1.1
14.6
4.7
38.8
45.9
3.9
17.0
18.1
11.6
10.2
8
6
13
15
13
16
13
12
8
11
15
14
9
8
6
8
13
14
15
10
5
7
13
14
12
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
7.4
20.3
32.1
72.9
29.5
8.8
12.2
4.1
12.5
2.7
2.7
11.0
2.2
11.0
2.0
1.2
6.5
29.4
9.3
46.3
1.3
88.8
11.1
22.2
26.4
12
8
13
16
13
14
19
17
7
6
11
10
10
9
20
12
7
14
11
17
9
12
12
6
10
It is noted that the rate chart is insensitive to the simultaneous changes of T and C in the same direction (e.g., both T
and C increase at the same time). However, most of the users may not be interested in signalling such changes, as the event
rate (in terms of the loss per unit time for a negative event or the gain per unit time for a positive event) will be untouched.
However, in some applications, a user may want to signal any change of T regardless of C , then he should use a t chart.
Similarly, if a user is interested in any change of C regardless of T , he may select a c chart.
Author's personal copy
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Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425
Table 8
ATS values of four control charts in example two.
Chart
δλ
δg
0.35
0.48
0.61
0.74
0.87
1.00
1.00
t
c
t&c
Rate
1440.0
28.2
40.3
82.0
1440.0
49.4
84.4
145.5
1440.0
101.6
195.2
258.1
1440.0
235.3
452.1
457.7
1440.0
595.4
916.2
811.9
1440.0
1613.0
1440.0
1440.0
0.82
t
c
t&c
Rate
742.4
34.4
47.0
60.2
742.4
60.2
94.2
104.3
742.4
123.9
201.9
180.9
742.4
287.0
403.6
309.8
742.4
726.1
657.4
524.0
742.4
1967.1
846.2
876.8
0.64
t
c
t&c
Rate
423.6
44.1
57.6
61.9
423.6
77.2
108.1
96.2
423.6
158.8
203.5
150.4
423.6
367.7
332.0
233.9
423.6
930.3
439.3
360.4
423.6
2520.4
496.3
550.7
0.46
t
c
t&c
Rate
262.2
61.3
73.4
68.6
262.2
107.4
122.0
94.5
262.2
220.9
189.3
131.7
262.2
511.5
250.1
183.5
262.2
1294.3
285.7
254.6
262.2
3506.6
300.9
351.4
0.28
t
c
t&c
Rate
194.0
100.7
102.3
89.5
194.0
176.4
140.6
108.6
194.0
362.9
176.3
133.2
194.0
840.4
198.5
164.0
194.0
2126.4
208.7
201.8
194.0
5760.8
212.6
248.0
0.10
t
c
t&c
Rate
257.0
281.9
217.4
200.4
257.0
493.9
240.5
214.1
257.0
1016.2
254.9
229.7
257.0
2353.1
261.8
247.1
257.0
5953.9
264.6
266.0
257.0
16130.3
265.6
286.6
Acknowledgments
This research work is supported by Natural Science Foundation of China (No. 70572044).
Appendix. Computation of ATS of the rate chart
Firstly, let ATS z be the zero-state ATS of a rate chart. Under the zero-state mode, the process shift is assumed to take
place from the beginning or immediately after an occurrence of the attribute event. ATS z is determined as follows (Wu and
Spedding, 2000):
ATSz =
1
λ · Pz
,
(27)
where Pz is the power of the rate chart under zero-state mode.
Computation of ATS of the rate chart for detecting upward R shifts
For detecting upward R shifts,
Pz = 1 − FR (UCLr ).
(28)
The cumulative probability function FR () is calculated by Eq. (9).
Next, suppose that an R shift occurs at a time moment Ts between the (k − 1)th and the kth occurrences of the event.
The time interval between the (k − 1)th occurrence and Ts is denoted as T0 and the time interval between Ts and the kth
occurrence is indicated by T1 . The time interval T between the (k − 1)th and the kth occurrences is given as,
T = T0 + T1 .
The cumulative probability function of T in the above equation is calculated by
FT (t ) = FT0 +T1 (t ) = Pr(T0 + T1 < t )
t
=
Z
t
=
Z
Pr(T0 + y < t |y = T1 ) · fT1 (T1 )dT1
0
0
Pr(T0 < t − y|y = T1 ) · fT1 (T1 )dT1
(29)
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Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425
=
Z
t
1 − e−λ0 (t −T1 ) · λ · e−λT1 dT1
0
e−λ0 t (1 − e−(λ−λ0 )t )
1 − e−λt
λ
,
−
=
λ
λ
−
λ
0
1 − e−λ0 t − λ0 te−λ0 t , if λ = λ0 .
if λ 6= λ0 ,
(30)
It is noted that both T0 and T1 are exponentially distributed random variables with the in-control parameter λ0 and
out-of-control parameter λ, respectively.
Next, let Pk be the probability that the R shift is signalled at the kth occurrence of the event (or the first occurrence after
the R shift). Then, (1 − Pk ) is the complementary probability that the R shift is not detected by the kth occurrence, but by
a following one. For the first case, ATS is equal to 1/λ; and for the complementary case, ATS equals (1/λ + ATS z ). It is noted
that, after the R shift occurs, λ always takes the out-of-control value and therefore the zero-state mode always come to play
after the kth occurrence of the event. Now, the steady-state ATS can be calculated by (Wu and Spedding, 1999),
ATS = Pk ·
1
+ (1 − Pk )
λ
1
λ
(31)
+ ATSz .
Finally, the computation of Pk needs be elaborated. It is the probability that the sample value of C /T (or C /(T0 + T1 )) in
the kth occurrence falls beyond the control limit. For detecting upward R shifts,
Pk = Pr
= Pr
=
∞
X
C =0
C
> UCLr
T
C
T0 + T1
FT0 +T1
> UCLr
C
UCLr
= Pr T0 + T1 <
· p(C ) ≈
Uc
X
C =0
FT0 +T1
C
UCLr
C
UCLr
· p(C ),
(32)
where FT0 +T1 () is calculated by Eq. (30) and p(C ) by Eq. (5).
The procedure for calculating the steady-state ATS of the rate chart detecting upward R shifts is summarized below:
(1) Calculate the power Pz using Eq. (28).
(2) Calculate the zero-state ATS z using Eq. (27).
(3) Calculate the probability Pk using Eq. (32).
(4) Finally, compute the ATS using Eq. (31).
Computation of ATS of the rate chart for detecting downward R shifts
The above four steps can also be used to calculate the steady-state ATS of the rate chart for detecting downward R shifts
subject to some modifications for steps (1) and (3). In step (1), the power Pz will be calculated as follows:
(33)
Pz = FR (LCLr ).
In step (3), the probability Pk is determined by
Pk = Pr
= Pr
C
< LCLr
T
C
T0 + T1
< LCLr
=
∞
X
≈
Uc
X
C =0
C =0
1 − FT0 +T1
1 − FT0 +T1
C
= Pr T0 + T1 >
C
LCLr
C
LCLr
LCLr
· p(C )
· p(C )
All the formulae derived above have been verified by simulation results.
(34)
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Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425
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