Run-Length Distributions of Special-Cause Control Charts for Correlated Processes

Abstract

We derive run-length distributions of the special-cause control chart proposed by Alwan and Roberts for correlated observations, given that the assignable cause to be detected is a shift in the process mean. Both recursive and closed-form solutions are derived for the run-length distribution, average run length (ARL), and standard deviation of the run length (SRL) for any AR(p) process, and approximate solutions are derived for the more general ARMA(p,q) processes. The expressions derived do not depend on the type of shift in the process mean. Numerical results are illustrated for the ARL and SRL of the ARMA(l,l) model, given that the shift in the mean is a step shift. These results show that the ARL and SRL of the specialcause control chart are relatively smaller when the process is negatively rather than positively autocorrelated. Regardless of the sign of the autocorrelation, the shape of the probability mass function of the run length reveals that the probability of detecting shifts very early is substantially higher for the special-cause chart than for more traditional control charts. Early detection makes the cause of the signal easier to identify, resulting in a more rapid rate of continuous quality improvement. There are some cases, however, when traditional charts, which are simpler to implement, should be considered even when the process is autocorrelated.

KEY WORDS:

• Autocorrelation
• Average run length
• Statistical process control
• Time series