USING RUN‐LENGTH DISTRIBUTIONS OF CONTROL CHARTS TO DETECT FALSE ALARMS

Abstract

Run‐length distributions for various statistical process‐control charts and techniques for computing them recently have been reported in the literature. The real advantages of knowing the run‐length distribution for a process‐control chart versus knowing only the associated average‐run length of the chart have not been exploited. Our purpose is to use knowledge of the run‐length distribution as an aid in deciding if an out‐of‐control signal is a true signal or merely a false alarm. The ability to distinguish between true and false signals is important, especially in operations where it is costly to investigate the causes of out‐of‐control conditions. Knowledge of the run‐length distribution allows us to compute likelihood ratios, which are simple to calculate and to interpret and which are used to determine the odds of obtaining an out‐of‐control signal at a particular run length when a shift in the process mean actually has occurred vis‐a‐vis no such shift. We extend our analysis in a Bayesian sense by incorporating prior information on the distribution of the shift size of the process mean, combined with the likelihood ratio obtained from the run‐length distribution, to determine if a shift larger than a critical size has occurred. We give examples for the Shewhart chart, the exponentially weighted moving‐average chart, and the special‐cause control chart for processes with autocorrelated observations. The examples show that the current recommended usage of the average‐run length alone as a guide for determining whether a signal is a false alarm or otherwise can be misleading. We also show that the performance of the traditional charts, in terms of their average‐run length, can be enhanced in many instances by using the likelihood‐ratio procedure.