In the context of Statistical Process Control (SPC), control charts are used to monitor a process over time and detect shifts in the mean or variation. The traditional Shewhart $\bar{X}$ charts makes use of (rational) group means to monitor the process mean over time. However, one of the assumptions is that there is no between-group variation. While this is the ideal scenario for a process, it does not seem to be realistic in the real world. To be precise, the underlying model that is used in traditional charts is $$X_{ij} = \mu + \varepsilon_{ij} \tag{M1},$$ where $\mu$ is the process mean and $\varepsilon_{ij} \sim N(0,\sigma^2)$ is a random variable. This model fails to account for between-group variation. The article Statistical process control with several components of common cause variability (Woodall and Thomas, 1995) discusses this topic, along with an example. They propose the model $$X_{ij} = \mu + \beta_{i}+\varepsilon_{ij} \tag{M2},$$ where now $\beta_i \sim N(0,\sigma_{B}^2)$ is a random variable that accounts for the between-group variation.They also refer to other specific examples presented by other authors. However, I was wondering whether there have been larger and/or more recent studies regarding this topic. Ideally, studies that use data from the industry to show that between-group variation is usually present. I would greatly appreciate any references or ideas!


1 Answer 1


These models are referred to as linear mixed effect or random effects models in general.

A search for articles in this area yield a couple of potential references:

  1. Monitoring of Linear Profiles Using Linear Mixed Model in the Presence of Measurement Errors
  2. Statistical Process Control Charts as a Tool for Analyzing Big Data
  3. Nonparametric Profile Monitoring by Mixed Effects Modeling
  4. Statistical Process Control Methods for Network Monitoring Using Generalized Linear Mixed Models
  5. Statistical process control methods for network monitoring using generalized linear mixed models

Another method is to look at papers that cite the paper you reference:

  1. Daewon Yang, Jinsu Park, Hayang Park, Sungki Hong, Jongmin Kim, Seonghui Huh, Eunkyung Kim, Jaeyong Jeong & Yeonseung Chung. (2023) Robust control chart based on mixed-effects modeling framework: A case study in NAND flash memory industry. Quality Engineering 0:0, pages 1-11.
  2. Rob Goedhart & William H. Woodall. (2022) Monitoring proportions with two components of common cause variation. Journal of Quality Technology 54:3, pages 324-337.
  3. T. C. Chang & F. F. Gan. (2004) Shewhart Charts for Monitoring the Variance Components. Journal of Quality Technology 36:3, pages 293-308.
  4. Stephen M. Scariano & Jaimie L. Hebert. (2003) Adapting EWMA Control Charts for Batch-Correlated Data. Quality Engineering 15:4, pages 545-556.
  5. Thomas C. Maness, Robert A. Kozak & Christina Staudhammer. (2003) Applying Real-Time Statistical Process Control to Manufacturing Processes Exhibiting Between and Within Part Size Variability in the Wood Products Industry. Quality Engineering 16:1, pages 113-125.

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