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 (2.5-1.5 = 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 (2-2=0 but truncated at 1).

The average run length (arl) is the same as that from the CUSUM Charts for Means Explained Page, but the scale is the departure from the in-control 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. Springer-Verlag New York. ISBN 0-387-98365-1 p110-120

The program from StatTools 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 higher-order run-length moments for the Poisson CUSUM. Journal of Quality Technology 28:3 p.363-369