February 2011
This month's newsletter examines the "g" control chart. The g control chart is used to analyze rare events. For example, health care facilities often are interested in tracking events that don't occur very often such as a certain type of infection. If the events are very rare, the traditional attribute control charts (p, np, c or u) may not be very useful. An earlier newsletter examined how the individuals control chart (XmR) could be used to monitor these types of rare events (click here for more information). This newsletter introduces the g control chart as another way these rare events can be tracked. This control chart has been added to the SPC for Excel software as a result of a customer requesting it.
In this issue:
 Introduction: Infections after Surgery
 Use a c Control Chart?
 The g Control Chart
 Interpreting the g Control Chart
 Summary
 Quick Links
There are a number of options with the g control chart. One option is to track the time (number of days) between rare events. There is a point plotted on the chart for each event. You would like the chart to indicate that the number of days between events is increasing. Another option is to track the number of units between events. For example, you might want to track the number of hospital admissions before an event occurs. Or you might want to track the number of units produced in a plant before a rare breakdown. This month's newsletter examines how to construct and interpret a g control chart.
Introduction: Infections after Surgery
A surgical facility is tracking the number of times a certain infection occurs after surgery each month. These do not happen very often so they are considered "rare" events. The facility's quality assurance group has data for the last two years. The monthly data is shown in the table below.
Month 
Number 
Month 
Number 

Jan09  0  Jan10  2  
Feb09  1  Feb10  0  
Mar09  0  Mar10  0  
Apr09  2  Apr10  1  
May09  1  May10  0  
Jun09  0  Jun10  2  
Jul09  0  Jul10  1  
Aug09  0  Aug10  1  
Sep09  2  Sep10  0  
Oct09  1  Oct10  1  
Nov09  1  Nov10  0  
Dec09  1  Dec10  1 
The first step the group must do is to decide how to track the data.
Use a c Control Chart?
The four basic attribute control charts are the p, np, c and u control charts. Of these four, the c control chart seems to the most likely candidate. You use the c control chart when there is a large opportunity for defects (infections in this example) to occur, but the actual number that do occur is small compared to this opportunity. You can get a summary of these four attribute control charts in our April 2009 newsletter.
The QA group developed a c control chart based on the data in the table. The resulting chart is shown below.
c Chart: Number of Infections per Month
The average number of infections per month is 0.75; the upper control limit is 3.35. Take a hard look at this chart. Is it very useful? If you make a change to the process that you hope will reduce the number of infections, how will you know from the chart that an improvement has occurred? You would need to wait for the control chart to signal an improvement has been made. You would need a run below the average  seven months in a row with no infections. Not a really useful chart when you think about it.
For the charting purists reading this, the control limits shown in the above chart were calculated using the c control limit equations:
These are not really the appropriate control limits in this case. This is actually the small sample case for the c control chart. If you use the approach outlined in our March 2008 newsletter, the upper control limit is equal to 5. Either way, this c control chart is not that useful.
The g Control Chart
A different approach is need. We can use the g control chart to examine the days between infections. The monthly data is not useful. Instead, we need the dates the infections occur. These dates are shown in the table below. The table also includes the days between infections.
Date 
Days Between Infections 
2/5/2009  
4/3/2009  57 
4/25/2009  22 
5/6/2009  11 
9/15/2009  132 
9/21/2009  6 
10/7/2009  16 
11/22/2009  46 
12/1/2009  9 
1/5/2010  35 
1/30/2010  25 
4/7/2010  67 
6/4/2010  58 
6/24/2010  20 
7/5/2010  11 
8/23/2010  49 
10/8/2010  46 
12/4/2010  57 
The first infection occurred on 2/5/2009. The next infection occurred on 4/3/2009. There are 57 days between 2/5/2009 and 4/3/2009. The next infection occurred on 4/25/2009. There are 22 days between 4/3/2009 and 4/25/2009. The days between infections are plotted on the g control chart.
After calculating the days between infections, the next step is to calculate the average g. This average is given by:
where the g_{i} values are the individual days between infections and k is the number of samples. Note that the first date listed in the table is not included in k. The average is 39.24. This means that, on average, there were 39 days between infections.
The control limits for the g control chart are shown below.
Substituting in the average gives the following results:
If the LCL is less than zero, there is no lower control. This is the usual case with the g control chart. The average and control limits are added to the g control chart. The g control chart is shown below.
All the points on this chart are within the control limits and there are no patterns. The "process" is in statistical control.
Interpreting the g Control Chart
You can see that the g control chart gives you a much better picture of what is happening in the infection process than the c control chart. You obviously want a point to be above the upper control limit. So, if you can go 158 days or more, you have significantly reduced the frequency of infections. A run of 7 or more in a row trending up is also an indication that the process is improving.
Summary
This newsletter has introduced a g control chart as another method of looking at rare events. It assumes that the data follows a geometrictype distribution. This newsletter showed an example of a g control chart in health care and tracked the number of days between infections. You can also use the g control chart to the number "produced" between events. For example, we could have track the number of admissions between infections or the number of surgeries.
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Thanks so much for reading our publication. We hope you find it informative and useful. Happy charting and may the data always support your position.
Sincerely,
Dr. Bill McNeese
BPI Consulting, LLC