Pattern detection in phase I monitoring using runs-based tests

Abstract

Control charts are useful to monitor if a process is in a state of statistical control (in-control) or if changes have occurred due to the presence of any assignable causes. To this end, the pattern of points displayed on a control chart plays an important role. A process is declared as in-control when the plotted points display a random pattern. On the other hand, when the points display a nonrandom pattern, with or without one or more points falling beyond the control limits, the process may be declared out-of-control. Thus, the constellation of points, along with the type of the displayed pattern in a control chart can provide useful clues about the possible presence and diagnosis of assignable causes, which can then be dealt with in an appropriate manner. In this work, we propose a methodology using randomness tests based on the theory of runs that can be applied in a supplementary manner in order to assess the statistical significance of a pattern on a Phase I Shewhart X ¯ chart in an objective way. Five common nonrandom patterns with corresponding tests are considered. The performance of the tests is evaluated in terms of their false alarm rate and power, via simulation. An illustration based on some real data is provided. Conclusions and practical recommendations are offered.