This page provides explanations and example R codes for CUSUM quality control charts, for detecting changes in counts that conforms to the Poisson Distribution.

CUSUM Generally

CUSUM is a set of statistical procedures used in quality control. CUSUM stands for Cumulative Sum of Deviations.

In any ongoing process, be it manufacture or delivery of services and products, once the process is established and running, the outcome should be stable and within defined limits near a benchmark. The situation is said to be In Control

When things go wrong, the outcomes depart from the defined benchmark. The situation is then said to be Out of Control

In some cases, things go catastrophically wrong, and the outcomes departure from the benchmark in a dramatic and obvious manner, so that investigation and remedy follows. For example, the gear in an engine may fracture, causing the machine to seize. An example in health care is the employment of an unqualified fraud as a surgeon, followed by sudden and massive increase in mortality and morbidity.

The detection of catastrophic departure from the benchmark is usually by the Shewhart Chart, not covered on this site. Usually, some statistically improbable outcome, such as two consecutive measurements outside 3 Standard Deviations, or 3 consecutive measurements outside 2 Standard Deviations, is used to trigger an alarm that all is not well.

In many instances however, the departures from outcome benchmark are gradual and small in scale, and these are difficult to detect. Examples of this are changes in size and shape of products caused by progressive wearing out of machinery parts, reduced success rates over time when experienced staff are gradually replaced by novices in a work team, increases in client complaints to a service department following a loss of adequate supervision.

CUSUM is a statistical process of sampling outcome, and summing departures from benchmarks. When the situation is in control, the departures caused by random variations cancel each other numerically. In the out of control situation, departures from benchmark tend to be unidirectional, so that the sum of departures accumulates until it becomes statistically identifiable.

The mathematical process for CUSUM is in 2 parts. The common part is the summation of depertures from the bench mark (CUSUM, and graphically demonstrating it. The unique part is the calculation of the decision interval abbreviated as DI or h, and the reference value, abbreviated as k, which continuously adjustes the CUSUM and itd variance. The two values of h and k depends on the following parameters

• The in control values
• The out of control values
• The Type I Error or false positive rate, expressed as the Average Run Length, abbreviated as ARL, the number of samples expected for a false positve decision when the situation is in control. ARL is the inverse of false positive rate. A false positive rate of 1% would have ARL=100

CUSUM for Poisson Distributed Counts

The Poisson distribution is specific for counting data within a defined environment, and is used frequently in quality control. Examples are the number of eggs each day from a farm, the number of absentees per term in a school, the number of asthma attacks per 100 child month, the number of pregnancies per 100 women year using a particular contraception, the number of complaints from clients per month in a store, the number of falls per year in an age care facility, and so on.

This procedure is statistically precise and powerful, but requires that the counts have a Poisson distribution, that the variance is the same as the mean. When this assumption cannot be met, the less powerful but more tolerant model of counts based on the Negative Binomial Distribution should be used. Details can be found in CUSUM for Negative Binomial Distributed Counts Explained Page

The parameters required are

• The expected count when the situation is in control.
• The count when the situation is out of control. This is not so much the expected value, but the departure that is big enough to warrant investigation and intervention
• The Average Run Length (ARL). This depends on a balance between the importance of detecting deviation against the cost of disruption in case of a false positive. Please note: that the algorithm on this page is intended for a one tail monitoring, either an increase or a decrease in the value. If the user intends a two tail monitoring, to detect either increase of decrease, the two CUSUM charts should be created, each with half the ARL that of a one tail CUSUM.

Details of how the analysis is done and the results are describer in the panel R Code Explained

References

CUSUM : Hawkins DM, Olwell DH (1997) Cumulative sum charts and charting for quality improvement. Springer-Verlag New York. ISBN 0-387-98365-1 p 47-74, 141-142

Hawkins DM (1992) Evaluation of average run lengths of cumulative sum charts for an arbitrary data distribution. Journal Communications in Statistics - Simulation and Computation Volume 21, - Issue 4 Pages 1001-1020

https://cran.r-project.org/web/packages/CUSUMdesign/index.html

https://cran.r-project.org/web/packages/CUSUMdesign/CUSUMdesign.pdf

This section presents the R code in total so it can be copied and pasted into RStudio as a template. Detailed explanations for each step are in the next panel

For those unfamiliar with R, basic information on setting up R and common R procedures can be found in the R Explained Page

#              Poission Distributed Counts
# Step 1: parameters and data
inControlCount = 3
outOfControlCount = 5
arl = 100 
theModel = "F"   #F for FIR,   Z for zero, S for steady state 
dat=c(3,3,2,5,3,5,1,5,1,3,5,4,3,8,6,1,8,2,5,7)

# Step 2: Calculate k and h
#install.packages("CUSUMdesign")   # if not already installed
library(CUSUMdesign)
result <- getH(distr=3, ICmean=inControlCount, OOCmean=outOfControlCount, ARL=arl, type=theModel)
k = result$ref
h = result$DI
if(outOfControlCount<inControlCount)
{
  h = -h
}
cat("Reference Value k=",k,"\tDecision Interval h=", h, "\n")

# Step 3: Calculate and plot CUSUM
# Step 3a: Calculate CUSUM
cusum <- vector()
cusumValue = 0
if(theModel=="F")
{
  cusumValue = h / 2
}
for(i in 1 : length(dat))
{
  cusumValue = cusumValue + dat[i] - k
  if(outOfControlCount>inControlCount) # Up
  {
    if(cusumValue<0)
    {
      cusumValue = 0
    }
  }
  else                               # down
  {
    if(cusumValue>0)
    {
      cusumValue = 0
    }
  }
  cusum[i] = cusumValue
}

# Step 3b: Plot CUSUM
plot(cusum,type="l")
abline(h=h)

# Step 4: Optional export of results
#myDataFrame <- data.frame(dat,cusum)         #combine dat and cusum to dataframe
#myDataFrame                                  #display dataframe
#write.csv(myDataFrame, "CusumPoisson.csv")   # write dataframe to .csv file

The example is a made up one to demonstrate the numerical process, and the data is generated by the computer. It purports to be from a quality control exercise in a large age care facility looking after bedridden residents, where the quality of care is measured weekly by the number of residents suffering from bed sores.

• From records in the past, we expect 3 bedsores per week, and this can be capped if the staff are well trained and motivated
• With time however, experienced staff leave and replaced by the less experienced and trained. The standard of care would gradually deteriorate, resulting in an increase in the weekly bed sores numbers.
• We would like to trigger an alarm and refresh the training of the staff when the weekly number of bed sores reached 5 or more.
• As training is expensive and dusrupts services, we would like any false alarm to be no more frequent than once every 100 weeks, so the average run length ARL = 100

The data is entered in step 1. This is the only part of the program that needs any editing

# Step 1: parameters and data
inControlCount = 3
outOfControlCount = 5
arl = 100 
theModel = "F"   #F for FIR,   Z for zero, S for steady state 
dat=c(3,3,2,5,3,5,1,5,1,3,5,4,3,8,6,1,8,2,5,7)

Step 1 contains the parameters and the data. This is the part where the user can edit, and change the values to that required in his/her own analysis

The first 3 lines sets the parameters required for the analysis.

The 4th line, the model has 3 options, which sets the first value of the CUSUM

• F means Fast Initial Response, where the initial CUSUM value is set at half of the Decision Interval h. The rationale is that, if the situation is in control then CUSUM will gradually drift towards zero, but if the situation is already out of control, an alarm would be triggered early. The down side is that a false alarm is slightly more likely early on in the monitoring. As FIR is recommended by Hawkins, it is set as the default option
• Z is for zero, and CUSUM starts at the baseline value of 0. This will lower the risk of false alarm in the early stages of monitoring, but will detect the out of control situation slower if it already exists at the begining.
• S is for steady state, intended for when monitoring is already ongoing, and a new plot is being constructed. The CUSUM starts at the value when the previous chart ends.
• Each model will make minor changes to the value of the decision interval h. The setting of the initial values is mostly intended to determine how quickly an alarm can be triggered if the out of control situation exists from the beginning.

The last part, c() is a function that creates a vector (array) containing the comma separated values within the bracket. In this case, dat is the name of the vector which contains weekly bedsore counts for 20 consecutive weeks.

The remainder of the program does not require any editing or change by the user, unless he/she wishes to alter the program for specific purposes.

# Step 2: Calculate k and h
#install.packages("CUSUMdesign")   # if not already installed
library(CUSUMdesign)
result <- getH(distr=3, ICmean=inControlCount, OOCmean=outOfControlCount, ARL=arl, type=theModel)
k = result$ref
h = result$DI
if(outOfControlCount<inControlCount)
{
  h = -h
}
cat("Reference Value k=",k,"\tDecision Interval h=", h, "\n")

Step 2 performs that statistical calculations using the parameters entered. The package CUSUMdesign needs to be alrady installed, and the library activated each time the program is used.

result is the object that contains the results of the analysis. The result required for this program are the reference value (k) and decision interval h. Please note that h is calculated as a positive value. If the CUSUM is designed to detect a decrease from in control value, then h needs to be changed to a negative value.

The last line displays the results we need

Reference Value k= 3.9 	Decision Interval h= 5.6 

Step 3 is divided into 2 parts. Step 3a calculates the cusum vector, and 3b plots the vector and h in a graph.

# Step 3a: Calculate CUSUM
cusum <- vector()
cusumValue = 0
if(theModel=="F")
{
  cusumValue = h / 2
}
for(i in 1 : length(dat))
{
  cusumValue = cusumValue + dat[i] - k
  if(outOfControlCount>inControlCount) # Up
  {
    if(cusumValue<0)
    {
      cusumValue = 0
    }
  }
  else                               # down
  {
    if(cusumValue>0)
    {
      cusumValue = 0
    }
  }
  cusum[i] = cusumValue
}

The first 6 lines of code in step 3a creates the empty cusum vector and sets the initial cusum value. The remaining codes calculates the cusum value for each measurement, and places it in the cusum vector

# Step 3b: Plot the cusum vector and h
plot(cusum,type="l")
abline(h=h)

In step 3b, the first line plots the cusum vector, and the second line the decision interval h. The result plot is shown to the left.

# Step 4: Optional export of results
#myDataFrame <- data.frame(dat,cusum)         #combine dat and cusum to dataframe
#myDataFrame                                  #display dataframe
#write.csv(myDataFrame, "CusumPoisson.csv")   # write dataframe to .csv file

Step 4 is optional, and in fact commented out, and included as a template only. Each line can be activated by removing the #

The first line places the two vectors, dat and cusum together into a dataframe

The second line displays the data, along with row numbers, in the console, which can then be copied and pasted into other applications for further processing

The third line saves the dataframe as a comma delimited .csv file. This is needed if the data is too large to handle by copy and paste from the console.

Please note: R Studio write files to User/Document/ folder by default. The path needs to be reset if the user wishes to save files using a specific folder. Discussion in the file I/O panel of the R Explained Page

<