StatTools : CUSUM Introduction and explanation

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Related link :
CUSUM and Shewhart Charts for Poisson Distributed Counts Explained Page
CUSUM for Exponential Distributed Data Explained Page
CUSUM Charts for Means Explained Page
Probability of z Explained and Table Page

The Shewhart and CUSUM charts are commonly used quality control methods to detect deviations from bench mark values.

These are used after a process has already been put in place and functioning under control to a specified bench mark. Continuing sampling and charting the results can then provide early detection should the process becomes faulty (out of control).

In industrial production, the out of control situation is usually due to machine wear and tear, accidental damage, or malfunctioning somewhere in the assembly line. In clinical practice, the out of control situation may occur when practices changes, or when the underlying epidemiological pattern changes.

The Shewhart chart was developed first to quality control normally distributed measurements. The chart consists of continuing checking of values, and triggers the out of control alarm if 2 consecutive measurements exceeded 3 standard errors from the mean, or 3 consecutive measurements exceeding 2 standard errors from the mean. The Shewhart chart is therefore effective to detect sudden and large departures from the bench mark.

The Shewhart chart has been adapted to other measurements, such as Poisson distributed count of events or the binomially distributed proportion. Decision borders with similar probability properties can be drawn, although they are calculated according to Poisson or Binomial probabilities. Although the numbers used differ, the principle remains the same.

Although the Shewhart chart is sensitive to sudden and large changes in measurement, it is ineffective in detecting small but persistent departure from the bench mark. For this, the CUSUM chart is more appropriate.

CUSUM is short for cumulative sums. As measurements are taken, the difference between each measurement and the bench mark value is calculated, and this is cumulatively summed up (thus CUSUM). If the processes are in control, measurements do not deviate significantly from the bench mark, so measurements greater than the bench mark and those less than the bench mark averaged each other out, and the CUSUM value should vary narrowly around the bench mark level. If the processes are out of control, measurements will more likely to be on one side of the bench mark, so the CUSUM value will progressively depart from that of the bench mark.

CUSUM is therefore conceptually simple. The statistical calculations involved are needed to make the method usable in practice, and they are as follows.

  • The first is to make the CUSUM line stable. As the number of measurements are taken, the probability that the CUSUM value may drift into extreme values increases. This is corrected by adjusting the CUSUM by an out of control criteria k, and resetting the CUSUM value whenever it crosses the bench mark value.

    The consequence of this is that two sets of CUSUM values are available, one marks departures in excess of bench mark value, and the other marks departures below the bench mark value.

  • The second is when to decide that the process is out of control and raise the alarm. This is set by a decision value above or below the bench mark h, and the out of control decision can be made when the CUSUM value reaches h.

    This h really depends on how sensitive the user wishes the method to be. The more sensitive (the closer h is to the bench mark), the quicker will any departure from the bench mark be detected, but also the more likely a false alarm will occur.

    The sensitivity of h is calculated from the averaged run length (arl). This is the averaged number of measurements between each false alarm when the situation is still in control.

StatTools provides CUSUM programs for 4 types of measurements, two of which also includes a Shewhart charting component. Details and calculations are included in these specific programs.

  • The CUSUM and Shewhart Charts for Poisson Distributed Counts Explained Page is used for CUSUM and Shewhart charts when the measurements are counts of events within a defined environment. Examples are falls in old age care facilities, complaints about quality of care, number of defects coming off the production line, and so on.
  • The CUSUM Charts for Means Explained Page is used when the measurements are usually made with a gauge or a measuring instrument, and are continuous with a normal distribution. Examples are weights, lengths, volumes, and areas. Time is exponentially distributed (the longer the time the larger its variance), so the logarithm of time (Log(time)) can also use CUSUM for monitoring. So the duration of the waiting list, and the time required to perform a procedure can also be monitored.
  • The CUSUM for Proportions Explained Page is based on the Benulli distribution can be used to evaluate proportions. Examples are proportion of people who died in a particular illness, proportion of operations that are followed by complications, proportion of births by Caesarean Section, and so on.
  • The CUSUM for Exponential Distributed Data Explained Page , based on the Exponential distribution, can be used to evaluate data that are essentially ratios and time to events. This is a particularly useful method as many quality criteria in health care are time related. Examples are waiting time for appointments, duration of labour and operations, response time to emergencies.