A method for monitoring a process from an out of control to an in control state: Application to the learning curve.

The cumulative summation (CUSUM) test is increasingly being used in medicine to monitor a wide variety of processes such as cardiac surgery or disease outbreaks. The CUSUM sequentially tests the null hypothesis that the process is in control, i.e. its mean is equal to a given target. Thus, it detects when the process changes to an out of control state. Conversely, monitoring the learning curve requires detecting the time when the process reaches an in control state. In this work we develop an alternative to the CUSUM, the learning curve CUSUM (LC-CUSUM), that serves to detect when a process deviates from an out of control state to an in control state. The test is based on a two one-sided tests procedure where the null hypothesis is that the process is out of control. This can be written as H(0): |mu-mu(0)|> or =delta tested against H(1): |mu-mu(0)|< delta. The null hypothesis is thus the union of two one-sided hypotheses and is rejected when both are rejected. A CUSUM test statistic is then constructed for each hypothesis in a traditional way. The properties of the test are investigated through numerical simulations, and are illustrated on the learning curve of an endoscopist performing endoscopic retrograde cholangiopancreatographies for biliary tract disorders.