ARL and ATS








Average Run Length (ARL)

 
ARL represents the average number of SPC points/subgroups that will pass before a signal (special cause) is encountered.
  • ARL0 is the average run length before a Type 1 error is encountered (concluding that the process mean has shifted when it has not).
  • ARL1 is the average run length before a Type 2 error is encountered (concluding that the process mean has not shifted when it has).




Average Time to Signal (ATS)


ATS combines ARL with the time between subgroups to determine the time to an event.
 
If the subgroups are "h" hours apart, then ATS = ARL * h



Why should I care about ARL?

Balance economics of quality and manufacturing

The concepts of ARL and ATS can be used to design an SPC plan that balances:
  • costs of sampling from production
  • costs of testing
  • the risks associated with the amount of time that may pass before seeing a shift in the process (accumulated nonconforming product, and costs of sorting, rework, or scrap)
  • the risks associate with detecting and reacting to a false signal  
 
ARL (and ATS) is a useful metric for comparing the statistical power and performance of various SPC approaches.


Example:
  • For Phase 1 SPC (developing and qualifying a process), an X-bar and R chart is traditionally used, as it has characteristics that are generally favorable to finding gross shifts in the process mean in order to fix the root cause(s).
  • For Phase 2 SPC (long-term production), it might be assumed that any causes of gross shifts in the process mean have been identified and eliminated.  An X-bar and R chart is less statistically powerful for detecting the remaining potential small shifts in the process mean.  Instead, other SPC methods would be considered, such as a CUSUM chart, EWMA chart, or other. 


More information


Statistical Quality Control (Grant) chapter 10.

Statistical Quality Control (Montgomery) chapter 4.


NIST has useful information on ARL and Time To Detection (aka ATS)



The math behind ARL


The 'average' run length is based on the notion that each SPC subgroup is a Bernoulli trial, and that each subgroup is independent from all other subgroups (no autocorrelation).


For ARL:
  • H0 = the process mean is in control
  • H1 = the process mean is not in control

Definitions
  • Pr() is equivalent to "Probability of"
  • "|" (vertical bar or pipe) means "given" or "given that"
  • Example:  Pr(a|b) = probability of "a is true" given "b is true"

Type 1 and Type 2 Error
  • Type 1 error = alpha = Pr(reject H0 | H0 is true)
  • Type 2 error = beta = Pr(reject H0 | H0 is false)


The geometric distribution is used to characterize ARL.

Run Length ~ GEOM(p)                   [read "the Run Length has a Geometric distribution with parameter p"]
where p = Probability(reject H0)        [read "reject the null hypothesis given the alternate hypothesis is true]



ARL0 is the average run length of the X-bar chart when the process mean is in control. 
  • It looks for the number of subgroups to first (false) detection of a shift in the process mean when the process mean is in control. 
  • p = alpha
  • ARL0 = 1 / alpha

ARL1 is the average run length of the X-bar chart when the process mean is not in control. 
  • It looks for the number of subgroups to first detection of a shift in the process mean when the process mean is in control. 
  • p = (1 - beta)
  • ARL1 = 1 / (1 - beta)



 
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