Phase I monitoring of individual normal data: Design and implementation

Abstract

Control charts play an important role in Phase I or retrospective studies, which are essential to establish stability and generate reference data necessary for calculation of prospective (Phase II) control limits for process monitoring. The Shewhart-type Phase I chart for individual normally distributed data is one of the most popular monitoring methods. However, the control limits (charting constants) available in the literature do not account for either the effect of parameter estimation or use an appropriate chart performance metric. This can increase the rate of false alarms to an unacceptable level. In this paper, we consider the Shewhart-type Phase I chart for individual data for two common estimators of standard deviation and a suitable chart performance metric. We derive the formulas and tabulate the corrected (or adjusted) charting constants. Effects of violations of assumptions are examined in terms of in-control (IC) robustness to normality and presence of autocorrelation, via simulations. Some out-of-control (OOC) performance analysis is also presented. An illustration using some data is provided along with a summary and some recommendations. The accompanying R program may be used to calculate the corrected charting constants in practice on demand.

Keywords:

• autocorrelation
• univariate and multivariate normal distributions
• parameter estimation
• R program
• robustness
• statistical process control

Acknowledgments

The authors acknowledge the comments of two anonymous reviewers, which contributed to clarifications and improvements.

Additional information

Notes on contributors

Yuhui Yao

Yuhui Yao is a PhD student in Applied Statistics at the Department of Information Systems, Statistics and Management Science in the Culverhouse College of Commerce at the University of Alabama, Tuscaloosa. He received his BS in 2011 from Guangdong University of Technology, China and MS in 2016 from the University of Alabama, Tuscaloosa. His current research interests are in developing new quality monitoring methodologies and software development. He presented papers at the 2017 and 2018 Joint Statistical Meetings in Baltimore and Vancouver, respectively. He has worked for several communication companies and banks in China between 2011 and 2014 as a data analyst.

Subha Chakraborti

Subhabrata Chakraborti is Professor of Statistics and Morrow Faculty Fellow at the University of Alabama, Tuscaloosa, Alabama, USA. He is a Fellow of the American Statistical Association and an Elected Member of the International Statistical Institute. Professor Chakraborti has contributed in a number of research areas including censored data analysis, studies on income distribution, poverty, reliability, and general statistical inference. His current research interests include applications of statistical methods, including nonparametric methods, to the area of industrial statistics and statistical process control. He is the coauthor of the books Nonparametric Statistical Inference, sixth edition (2021) published by Taylor and Francis and Nonparametric Statistical Process Control (2019) published by John Wiley and Sons. Professor Chakraborti has supervised over twenty Master's and PhD students and has been cited for his contributions in mentoring and collaborative work with students and scholars from around the world.