Sense and sensibility: on the diagnostic value of control chart rules for detection of shifts in time series data

The aim of this study was to quantify and compare the diagnostic value of The Western Electric (WE) statistical process control (SPC) chart rules and the Anhoej rules for detection of non-random variation in time series data in order to make recommendations for their application in practice.

SPC charts are point-and-line graphs showing a measure over time and employing statistical tests for identification of non-random variation. In this study we used simulated time series data with and without non-random variation introduced as shifts in process centre over time. The primary outcome was likelihood ratios of combined tests. Likelihood ratios are useful measures of a test's ability to discriminate between the true presence or absence of a specific condition.

With short data series (10 data points), the WE rules 1-4 combined and the Anhoej rules alone or combined with WE rule 1 perform well for identifying or excluding persistent shifts in the order of 2 SD. For longer data series, the Anhoej rules alone or in combination with the WE rule 1 seem to perform slightly better than the WE rules combined. However, the choice of which and how many rules to apply in a given situation should be made deliberately depending on the specific purpose of the SPC analysis and the number of available data points.

Based on these results and our own practical experience, we suggest a stepwise approach to SPC analysis: Start with a run chart using the Anhoej rules and with the median as process centre. If, and only if, the process shows random variation at the desired level, apply the 3-sigma rule in addition to the Anhoej rules using the mean as process centre.