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27 April 2013

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This is a work in progress. Add links to your favorites in the comments, below, and I'll add them.

10 April 2013

Understanding the Rule of Seven

Context: Discussion of the Rule of Seven in statistical process control 

Problem: In some industries, one cannot wait for repeatable errors as defects and errors lead to loss of life. I was told that, in pharmaceuticals, a certain % of death is acceptable and almost expected. The waiting process or period of time before halting tests or evaluations is where I am stuck.
First, let's use the terms specified value and control limit rather than errors or defects.
Also, we would normally discuss this topic in terms of parameters that have numeric values such as temperature, weight, and speed. In this conversation, we want to deal with measurable characteristics long before they result in catastrophic failures.

Rule 1. Out-of-bound conditions

If you have just one value outside the control limits, such as a severe side effect from a drug, you stop the process and figure out what went wrong. 

Rule 2. Calibrating the results

The Rule of Seven (or Run of Seven) does not apply to parameters that go outside the control limits. The Run of Seven applies when seven consecutive, acceptable values lie on the same side of the specified value. Such a situation indicates that the average has deviated from the desired value and you need to recalibrate the process so that the average is close to the specified value.

Illustrating with a made-up scenario

Suppose the scientists at Schpooky Pharmaceuticals want to test an inoculation against the HG (Heebie Geebie) virus. In order to train the immune system to fight off a full invasion of Heebie Geebies when somebody sneezes on us, the inoculation has to cause a fever of at least 0.5 degree F. They calibrate the variables in making the vaccine and in the dosage to cause a 1.0 degree F fever, or a temperature of 99.6 degrees F.  In this test, they set an upper control limit of 3.0 degrees, or a temperature of 101.6 degrees F.
  • T = 99.6 degrees F, average
  • 99.1 degrees F < T < 101.6 degrees F
Using the first rule, just one person develops a temperature of 103.0 degrees F. We stop the tests to see what's gone wrong because 103.0 > 101.6, the upper control limit.
Using the second rule, if we have seven consecutive people develop fevers between 99.6 and 101.6, we stop the tests to see what's gone wrong. These temperatures are all acceptable, but they are all greater than the desired value.
The Run of Seven indicates a special cause -- that is, one or more variables in the process need to be controlled. Maybe the dose is too large and needs to be reduced. Perhaps the HG virus needs to be baked five minutes longer. So we make the adjustments and then resume the trial.
These rules and others like them serve to stop a process long before it reaches catastrophic failure such as the death of a patient.

But catastrophic failures do happen

You might wonder, What about the catastrophic failures? They do happen! What about the one in 10,000 who dies? How can that be acceptable?
This takes us to other techniques such as Decision Tree analysis (p. 299 of the Project Management Institutes Project Management Body of Knowledge (PMBOK) Guide, 4th edition).
Suppose withholding the vaccine results in 1,000 deaths per 10,000, but giving the vaccine causes one death in 10,000. If you distribute the vaccine to 10,000, you save 999 lives.
Unfortunately, many drug companies withhold such drugs because that one in 10,000 will sue them, and the juries will severely punish the companies.
This issue, tort reform, is one of the dividing lines between the political parties in the US. A project manager needs to use various decision-making methods and maintain awareness of a wide range of environmental factors.

The Difference between Accuracy and Precision

In technology and science, accuracy and precision are different, although they go together.

For the measured characteristic
  • Precision is described by the range of values.
  • Accuracy is described by the difference between the average value and the specified value.
Lets have a goal of making two batches of cookies and specify that they measure 5" across. Afterwards, we measure the cookies.

Chocolate cookie widths
  • Average = 5.10 "
  • Range = 0.25"
Cinnamon cookie widths
  • Average = 5.01"
  • Range = 0.50"
The chocolate cookies have a smaller range of sizes, so they are more precise. The average size of the Cinnamon cookies comes closer to the specified value, though, so they are more accurate.
Does that seem counterintuitive?
For a measurement or a measuring tool
  • Precision is described by how many significant figures the tool gives you.
  • Accuracy is described by how closely the measurement matches the value of a standard device and also by how accurate the standard is.
As another example, I have two 18" rulers.
  • One came from a store that sells drafting supplies. It is graduated in 16ths of an inch.
  • The other, I made for myself after I loaned the store-bought ruler to a neighbor kid. I based it on one cubit (18"), the length from my elbow to my finger tips. When I graduated the cubit ruler in 50ths of an inch, I eyeballed the measurements and marked it by hand.
 The cubit ruler is more precise, but the store-bought ruler is more accurate.