Phase I process monitoring: The case of the balanced one‐way random effects model

Abstract

Phase I is crucial for the success of the overall statistical process control (SPC) and monitoring regime. Shewhart‐type charts are recommended in this phase because of their broader shift detection ability. In this paper, a Phase I Shewhart‐type trueX¯ chart is considered for the balanced random effects (also called a variance components) model. The proposed methodology takes proper account of the effects of parameter estimation and uses the false alarm probability (FAP) metric to design the chart. In the sequel, the corrected (adjusted) charting constants are calculated and tabulated. The constant can be found, on demand, from an accompanying R package. Motivations and illustrations with some real data are provided. Performance of the chart is examined in terms of in‐control robustness and detection of nonhomogeneity (out‐of‐control). The proposed chart is shown to be easily adaptable to more general models, with more variance components and nested factors, and can accommodate various estimators of variance. Thus, it enables a broader Phase I process monitoring strategy, under normality, which can be applied within the ANOVA framework applicable for many DOE models. A summary and some recommendations are provided.

Phase II process monitoring for the balanced one‐way random effects model

“…From Table 6, it is seen that the batch effect is significant, so this effect should be considered in the construction of a Phase I control chart to monitor the retrospective data. Hence the Phase I Shewhart chart for the balanced one‐way random effects model, proposed by Yao and Chakraborti 32 is applied here. Their chart requires the standard deviation estimator using Table 6 and the Phase I charting constants are and for and , respectively, found using the R package, PH1XBAR (Yao and Chakraborti 29 ).…”

“…This includes what are now known as the retrospective phase (Phase I) and the prospective phase (Phase II) of SPC, where the goals of process monitoring are somewhat different. The Phase I aspect has been discussed in our recent paper (Yao and Chakraborti 32 ), so in this study, we focus on Phase II. In both of these phases, there is now a wealth of knowledge about the effects of parameter estimation, which is most common while setting up control charts in practice, how it affects chart performance and how to adjust the control limits in order to achieve nominal in‐control (IC).…”

“…Thus we find the “corrected” or “adjusted” for parameter estimation charting constants for some specified in‐control chart performance (such as the average) and set up the proper control limits. In a follow‐up to an earlier paper by Yao and Chakraborti, 32 for the Phase I setting, here we consider the Phase II setting and using two different perspectives, and a popular performance metric, find the adjusted charting constants that take proper account of the effects of parameter estimation. Our results provide both theoretical insights and practical results to conduct appropriate Phase II monitoring using control charts for linear models beyond the basic Shewhart model.…”

A phase II EWMA chart for the balanced one‐way random effects model

“…Otherwise, one runs the risk of a high number of false alarms which can diminish the value of the overall monitoring regime. Shewhart‐type $$\bar{X}$$ charts for Phases I and II applications of the balanced random effects model have been discussed in two recent papers (Yao and Chakraborti 43,44 ). In this paper, we go a step further, proposing and studying the Phase II Exponentially Weighted Moving Average (EWMA) chart for the balanced random effects model, which has two sources of common cause variation, the variation between and the variation within the batches.…”