Identifying the time of change in the mean of a two-stage nested process

Abstract

Statistical process control charts are used to distinguish between common cause and special cause sources of variability. Once a control chart signals, a search to find the special cause should be initiated. If process analysts had knowledge of the change point, the search to find the special cause could be easily facilitated. Relevant literature contains an array of solutions to the change-point problem; however, these solutions are most appropriate when the samples are assumed to be independent. Unfortunately, the assumption of independence is often violated in practice. This work considers one such case of non-independence that frequently occurs in practice as a result of multi-stage sampling. Due to its commonality in practice, we assume a two-stage nested random model as the underlying process model and derive and evaluate a maximum-likelihood estimator for the change point in the fixed-effects component of this model. The estimator is applied to electron microscopy data obtained following a genuine control chart signal and from a real machining process where the important quality characteristic is the size of the surface grains produced by the machining operation. We conduct a simulation study to compare relative performances between the proposed change-point estimator and a commonly used alternative developed under the assumption of independent observations. The results suggest that both estimators are approximately unbiased; however, the proposed estimator yields smaller variance. The implication is that the proposed estimator is more precise, and thus, the quality of the estimator is improved relative to the alternative.

Keywords:

• change-point estimation
• maximum-likelihood estimation
• mixed-effects model
• nested samples
• statistical process control
• variance component estimation

Acknowledgements

We would like to thank the Editor and anonymous reviewers for their insightful comments which led to improvements in this paper.Disclaimer: The views expressed in this article are those of the authors and do not reflect the official policy of the United States Air Force, Department of Defense, or the United States Government.

Notes

The word ‘chip’ refers to the material that has been removed from the surface during machining.

Δ _i does not depend on i except that it is an n _i ×n _i matrix.

If multiple chips are available, a single chip can be selected randomly from this population.