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CUSUM and Shewhart Charts for Poisson Distributed Counts Program Page
CUSUM
Poisson
Notation and Data
Example Shewhart
Example CUSUM
References
The CUSUM chart is used to detect small and persistent change, and is based on the cumulative sum of differences
between sampling measurements and the mean, (thus CUSUM). This assumes that, in the "in control" state the CUSUM would
hover around the expected mean level, as deviation around the mean would cancel each other out. In the "out of control"
state, there will be a bias away from the expected mean, and the CUSUM will drift away from the expected mean level.
The CUSUM programs on this site follow the approach outlined in the text book by Hawkins and Olwell (see references),
summarized as follows.
 The user defines the "in control". The central tendency and variance is defined, according to the nature of the data
In the normally distributed measurements, these are the mean and the Standard Deviation.
 Using specific algorithms, the level of departure (h) from the central tendency to decide that the "out of control"
alarm should be triggered is calculated. This level is abbreviated as h
 h is calculated, depending on the amount of departure (k) the system is designed to detect, and the sensitivity
of the detection, in terms of the averaged run length (ARL). The ARL is the expected number of observations
between false alarms when the situation is "in control". Conceptually this represents the probability of Type I
Error (α). An average run length of 20 is equivalent to α=0.05, ARL of 100, α=0.01.
 Once the chart and the ARL are defined, sampling takes place at regular intervals. The departure from the expected
is corrected by k, then added to CUSUM. If the CUSUM regressed to 0 or beyond, as it often does when the situation
is "in control", the CUSUM recommences at 0.
 In most cases therefore, two CUSUMs can be plotted, one for excessive increase in value, and one for excessive
decrease in value (two tails). In most quality control situation however, only one of the tails is of interest.
 In the programs of this site, 3 levels of h are offered in default, for ARLs of 20 (α=0.05) for yellow alert,
50 (α=0.02) for orange alert, and 100 (α=0.01) for red alert. The idea is that a yellow alert should trigger
a heightened expectancy, orange alert triggers an investigation, and red alert triggers immediate response. However,
these are merely recommendations, and users should define their own levels of sensitivity.
CUSUM for Poisson Distributed Counts
Poisson distribution concerns events within a defined environment, such as the number of cells in a volume of fluid, or number of
tadpoles in a pond.
The most common environment however is time, so most counts are in terms of a unit of time. In health care, commonly
used counts are number of complaints received by a hospital in a month, the number of falls in an age care facility per month,
number of adverse incidents in an Intensive Care Unit a week, and so on.
A disadvantage of using CUSUM to monito counts is that the same base line has to be maintained for comparison. So evaluations
can only take place at set intervals. The interval must be long enough for some events to occur, and a sudden change in
counts when the situation goes out of control cannot be detected until at least the next evaluation.
For this reason, the newer method in the CUSUM for Exponential Distributed Data Explained Page
is use, where the interval between events, the inverse of count, is increasingly used, as evaluation can take place after
each event.
The CUSUM for counts, this page and the CUSUM and Shewhart Charts for Poisson Distributed Counts Program Page,
are therefore presented for
legacy reasons. The method exists, had been useful in the past, and some users may wish to use it.
Shewart Chart for Counts
The original Shewhart chart for means and Standard Deviations can be modified for use with Poisson distributed counting of events.
Using the Poisson Test Program Page
the probability of departure from the bench mark
mean count (λ) can be calculated. The departure level with a probability of <0.02275
(the probability of being 2 standard deviations away from the mean in measurements) and of <0.00135
(the probability of being 3 standard deviations away from the mean in measurements)
are used, so that the same logic of triggering an alarm when a major change
occurs.
Of all the parameters in the program, only the count when processes are in control
(bench mark) is needed for Shewhart chart.
CUSUM Chart for Counts
The parameters required are as follows
The count when in control is the averaged count expected when the processes
are in control, in other words, the bench mark. This can be expressed as a
decimal pointed number if necessary. As the Poisson distribution are best at
handling low counts, the system works best when counts of below 20 are handled.
The departure when out of control is the change in the count in either or
both direction when the process is out of control. If the count in control is
2.5, and departure is 1.5, then the system is designed to trigger an alarm if
the count becomes more than 4 (2.5+1.5 = 4) or less than 1 (2.51.5 = 1). Please
note that the lower end of the out of control count is truncated at 1, as
the statistics cannot deal with a count of 0. An in control count of 2 with
a departure of 2 will trigger an alarm when the count exceeds 4 (2+2), or when
the count is less than 1 (22=0 but truncated at 1).
The average run length (arl) is the same as that from the CUSUM for Means Explained Page,
but the scale is the departure from the incontrol average count (λ), CUSUM will hover around 0 when
the situation is control, and deviates above 0 when the out of control count exceeds that when in control, and
deviates below zero when the out of control count drops below that when in control.
In this example, we will use the default example data in the
We aim to monitor complaints from patients in a hospital. On average we get
5 complaints every 2 month, and as we wish to monitor monthly, we set the bench
mark to 2.5 complains per month.
Shehart Chart
We can set out two alarm borders. The yellow alarm, at a count of 7 complaints per month,
has a probability of 0.02275 (2.3%) if the bench mark is maintained. This is the same
probability of a normally distributed measurement being 2 standard errors from the mean.
The orange alert, at a count of 9 per month, has a probability of 0.00135 (0.14%)
if the bench mark is maintained. This is the same probability of a normally
distributed measurement being 3 standard errors from the mean.
We would trigger an alarm and begin investigations if complaints exceed 7 per month in
3 consecutive months or exceed 9 per month in 2 consecutive months.
As can be seen from the chart on the left, as the complaint rate increased only from 2.5
(5 per 2 months) to 2.9 (3 per 2 months), the Shewhart chart is not sensitive enough to
detect this level of change.
We use the same parameters and data as that in the Shewhart chart example.
We run a hospital and on average we expect 5 complaints every
2 months (2.5 per month). We want our CUSUM program to trigger an alarm should
the monthly complaint number increases to 4 or more on average. The departure is therefore
set at 1.5 (2.5+1.5=4).
We set the yellow alert at arl=20, the orange alert at arl=50, and red alert at arl=100.
The Program iterates the decision line until the required arl is reached or exceeded,
these are displayed for checking purposes.
Dir  Yellow  Orange  Red 
Up  4  6  8 
Down  4  5  6 
The first display contains the decision lines, shown to the left. A yellow alert
will be trigger when the CUSUM is above 4, the orange alert at 6, and red
alert at 8.
Decision lines for CUSUM below 0 are also shown, yellow alert at 4, orange
at 5, and red at 6. However, these are only important if we are concerned
with both increase and decrease in counts, and they can be ignored in this
particular exercise as we are only interested in detecting an increase in complaints.
count  arlY+  arlY  arlY±  arlO+  arlO  arlO±  arlR+  arlR  arlR± 
1  8547  5  5  368978  6  6  16880438  7  7 
2  85  14  12  413  20  19  1928  27  26 
3  11  103  10  20  257  19  31  623  30 
4  5  1147  5  7  5963  7  9  28525  9 
The full table of analysis is then presented. Each row represents a theoretical level
the count has shifted to when the situation is out of control. Column 1 represents
the count out of control, and the rest of the table the average run length (number of measurements)
before it is detected. There are 3 sets (yellow, orange, and red), each
occupies 3 columns, the arl (+) for reaching h above the line, the arl () for
reaching h below the line, and the arl (±) for reaching either.
For example, if the situation is out of control and the count shifts from
the bench mark (2.5) to 4 (last line), then it will take 5 sets of measurements
to trigger a yellow alert. If the increase is only 0.5 to 3 (second last line)
then on average it takes 11 measurements to trigger a yellow alert.
Also, at the count of 4 (last line), the false positive rate for yellow alert is 1 in
1147, and at count of 3 1 in 103.
count  arlY+  arlO+  arlR+ 
3  11  20  31 
4  5  7  9 
Calculations are made for all levels of interest, for all 3 levels of alert, for
arl of true and false positives. However, as we are only interested in detecting
an increase in the count, only the last two rows, and the arl needed to detect the change
are of interest, that part of the total table as shown to the left.
The plot of the counts is the same as that shown in the Shewhart chart example,
and will not be repeated here. The data represents the number of complaints received
for each month for 80 months.
Please note that, although the parameters are set to 2.5 for the in control count
and 4 for out of control count, the data actually used averaged 2.6 per month
in the first 30 months, and 2.9 per month the next 50 months.
The CUSUM is seen in the plot to the right. Although a significant increase
did occur from the 31st month, the increase is smaller than that set in the
parameters, so it takes longer for CUSUM to reach the red alert line.
This plot shows that, providing there is a persistent change from the bench mark,
the CUSUM will increase progressively. The setting of the parameters merely
control how quickly the alarm will be triggered.
The plot also shows episodic decrease in the monthly counts, but these are
likely to be part of the random variations. As we are only interested in
an increase in count, we would modify the graph to delete all data below 0
before presentation.
Hawkins DM, Olwell DH (1997) Cumulative sum charts and charting for
quality improvement. SpringerVerlag New York. ISBN 0387983651 p110120
The program from StatsToDo uses the algorithm from the following references. Both references refer to the same program
Computer program to calculate CUSUM decision limits can be downloaded from
http://www.stat.umn.edu/cusum/software.htm
White CH and Keats JB (July 1966) ARLs and higherorder runlength moments
for the Poisson CUSUM. Journal of Quality Technology 28:3 p.363369
