Statistical Process Control for Shifts in Mean or Variance Using a Changepoint Formulation

Abstract

Statistical process control (SPC) involves ongoing checks to ensure that neither the mean nor the variability of the process readings has changed. Conventionally, this is done by pairs of charts—Shewhart X and S (or R) charts, cumulative sum charts for mean and for variance, or exponentially weighted moving average charts for mean and variance. The traditional methods of calculating the statistical properties of control charts are based on the assumption that the in-control true mean and variance were known exactly, and use these assumed true values to set center lines, control limits, and decision intervals. The reality, however, is that true parameter values are seldom if ever known exactly; rather, they are commonly estimated from a phase I sample. The random errors in the estimates lead to uncertain run length distribution of the resulting charts. An attractive alternative to the traditional charting methods is a single chart using the unknown-parameter likelihood ratio test for a change in mean and/or variance in normally distributed data. This formulation gives a single diagnostic to detect a shift in mean, in variance, or in both, rather than two separate diagnostics. Using the unknown parameter formulation recognizes the reality that at best one has reasonable estimates of parameters and not their exact values. This description implies an immediate benefit of the formulation, that the run behavior is controlled despite the lack of a large phase I sample. We demonstrate another benefit, that the changepoint formulation is competitive with the best of traditional formulations for detecting step changes in parameters.

KEY WORDS :

• Control charts
• Generalized likelihood ratio
• Phase i
• Phase ii