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Monitoring Bernoulli Processes: Geometric Control Chart and CUSUM Charts

This article discusses the monitoring of Bernoulli processes using the geometric control chart and Bernoulli or equivalent geometric CUSUM charts. It explores the effect of estimation errors on these charts and compares different methods based on their steady-state performance. Generalizations and conclusions are also provided.

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Monitoring Bernoulli Processes: Geometric Control Chart and CUSUM Charts

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  1. Monitoring Bernoulli ProcessesWilliam H. WoodallVirginia Techbwoodall@vt.edu

  2. Outline • Introduction to Bernoulli processes • Geometric control chart • Effect of estimation error on geometric control chart • Bernoulli and geometric CUSUM charts • Comparing methods using steady state performance • Effect of estimation error on the CUSUM charts • Generalizations • Conclusions

  3. Overview • Consider monitoring attribute data where each item is classified as “nonconforming” or “conforming.” This shows up in industrial and in health–related applications. • We may have 100% inspection of items. • This data stream is often modeled as a sequence of independent Bernoulli random variablesX1, X2, X3, …, with P(ith item is nonconforming) = P(Xi = 1) = p.

  4. Monitoring Bernoulli Processes • When the nonconforming rate, p, is small, traditional methods such as the Shewhart p–chart are inadequate. • It is inefficient to artificially group items into samples of size n when data are available successively on individual items. • Many methods have been proposed, including the geometric control chart. • The Bernoulli or equivalent geometric CUSUM charts have the best performance in detecting a sustained shift in p.

  5. Note that points plotted below the LCL of the geometric chart indicate process deterioration, whereas points above the UCL indicate process improvement. Generally p0 will be unknown and must be estimated using a Phase I sample. Three approaches have been proposed – binomial sampling, negative binomial sampling, and the use of a Bayes estimator.

  6. Performance Metrics • Charts should be compared based on the average number of observations until a signal occurs, the ANOS. Charts are designed so that the in-control ANOS, ANOS0, is a specified value. • Equivalently we can consider the average run length (ARL), the average number of points plotted until a signal occurs, since ANOS = ARL/p. • When p0 is estimated, the actual in-control ANOS value becomes a random variable. We would like the average value to be the specified ANOS0, but we also need the standard deviation of the in-control ANOS to be low enough that practitioners are confident in getting the desired value. • Previous work on the effect of estimation on the geometric chart has considered only the average in-control ANOS value, not the standard deviation.

  7. Figure 1: a) the In-Control ARLavg and b) the In-Control SDARL. Desired ARL0 is 370.4.

  8. The in-control ARLavg value converges relatively quickly to the desired value, but the in-control SDARL converges relatively slowly to zero. This means that for the geometric chart to have reliable and predictable in-control performance, the Phase I sample size must be quite large. The required sample size can be an order of magnitude higher than previously recognized.

  9. Performance with sequential sampling

  10. Geometric and Bernoulli CUSUM Charts The upper sided geometric CUSUM statistics are Si = max(0, Si-1 – Yi + kG), i = 1, 2, 3,…, where S0 = 0, Yi is the ith geometric count and kG is determined based on a likelihood ratio to detect a shift from p0 to p1 = δ p0. A signal is given when Si > hG.

  11. The upper sided Bernoulli CUSUM statistics are Bi = max(0, Bi-1+Xi - kB), i = 1, 2, 3,…, where B0 = 0, Xi is the ith Bernoulli observation and kB = 1/kGis determined based on a likelihood ratio to detect a shift from p0 to p1. A signal is given when Bi > hB. The geometric and Bernoulli CUSUM charts are equivalent if B0 = 1 – kB and hB = (hG+ kB– 1) / kG.

  12. ALARM: CUSUM ≥ 2.7 B B h Moves up when a malformation occurs. Moves down (or stays at zero) for a normal birth.

  13. Performance Metrics • Charts should be compared based on the average number of observations until a signal occurs, the ANOS. • Steady–state random–shift models should be used when comparing methods. Under this model a shift in p occurs after the process monitoring has been underway and the shift in p may occur at any time.

  14. Differences in Steady-State Models • Fixed–shift model. • Random–shift model.

  15. Misconceptions about the Geometric CUSUM Chart • For a zero–state analysis, a natural headstart feature is present for the geometric CUSUM chart. • For a steady–state analysis, the geometric CUSUM chart is considered better than the Bernoulli CUSUM chart in some cases, but only because the fixed–shift model is used.

  16. Fixed vs. Random–Shift Model Conclusions are much different based on the type of model used for the geometric CUSUM chart.

  17. The effects of parameter estimation are more significant when the tuned shift size δ is smaller, when the desired ANOS0 value is larger, and when the p0 value is smaller. (b) (a)

  18. TABLE 1. Values of ANOSavg (blue) and SDANOS (black)

  19. Two Generalizations of Bernoulli CUSUM Chart • Steiner et al. (2000) used a logistic regression model to let p0 vary from item to item. This is widely used in the risk-adjusted monitoring of surgical outcomes where the health characteristics of patients can vary widely. • Ryan et al. (2011) extended the approach to the case where there are more than two outcomes, e.g., manufactured items are classified as good, fair, or bad.

  20. http://www.sfar.org/scores2/parsonnet2.html

  21. Example of a two-sided risk-adjusted CUSUM chart (provided by Stefan H. Steiner)

  22. Conclusions • It is important to consider the variation in performance as well as the expected in-control performance of control charts when parameters are estimated. • The Phase I sample sizes required for reliable in-control performance of geometric control charts can be impractically large. • The steady-state performance of methods for monitoring with Bernoulli data should be evaluated using the random–shift model. • Using a fixed–shift model has led to conclusions that the geometric CUSUM chart is better than the Bernoulli CUSUM chart for detecting an increase in p when the methods can be designed to be equivalent.

  23. Conclusions (continued) • The Bernoulli CUSUM chart is much more adversely affected by estimation error than the geometric control chart, requiring much larger Phase I sample sizes. • Because required sample sizes can be too large to be practical, the method of Steiner and MacKay (2004) is highly recommended for identifying continuous product or process variables to monitor in place of the attribute approach. This can lead to more information, much smaller Phase I sample sizes, and greater ability to detect process changes and to improve the process.

  24. References • Jensen, W. A., Jones-Farmer, L. A., Champ, C. W., and Woodall, W. H. (2006). “Effect of Parameter Estimation on Control Chart Properties: A Literature Review”. Journal of Quality Technology 38(4), 349-364. • Lee, J., Wang, N., Xu, L., Schuh, A., and Woodall, W. H. (2011), “The Effect of Parameter Estimation on the Upper-Sided Bernoulli CUSUM Charts”, to be submitted to Journal of Quality Technology. • Quesenberry, C. P. (1995). “Geometric Q-chart for High Quality Processes”, Journal of Quality Technology 27, 304-315. • Reynolds, M. R., Jr. and Stoumbos, Z. G. (1999). “A CUSUM Chart for Monitoring a Proportion When Inspecting Continuously". Journal of Quality Technology 31(1), 87-108. • Ryan, A. G., Wells, L.J., and Woodall, W. H. (2011), “Methods for Monitoring Multiple Proportions When Inspecting Continuously”, to appear in the Journal of Quality Technology. • Steiner, S. H., Cook, R. J., Farewell, V. T., and Treasure, T. (2000). “Monitoring Surgical Performance Using Risk-Adjusted Cumulative Sum Charts”. Biostatistics 1, 441-452.

  25. References (continued) • Steiner, S. H. and MacKay, R. J. (2004). Effective Monitoring of Processes with Parts Per Million Defective – A Hard Problem! In H. J. Lenz and P.Th. Wilrich (Eds.), Frontiers in Statistical Quality Control7. Heidelberg, Germany: Springer-Verlag. • Szarka, J. L., III, and Woodall, W. H. (2011), “Performance Evaluations and Comparisons with the Bernoulli CUSUM Chart”, to appear in Journal of Quality Technology. • Szarka, J. L., III, and Woodall, W. H. (2011), “A Review and Perspective on Surveillance of High Quality Bernoulli Processes”, submitted to Technometrics. • Tang, L. C. and Cheong, W. T. (2004). “Cumulative Conformance Count Chart with Sequentially Updated Parameters”. IIE Transactions 36, 841-853. • Yang, Z., Xie, M., Kuralmani, V., and Tsui, K-L. (2002). “On the Performance of Geometric Charts with Estimated Control Limits”. Journal of Quality Technology 34(4), 448-458. • Zhang, M., Peng, Y., Schuh, A., Megahed, F. M., and Woodall, W. H. (2011), “A Reconsideration of Geometric Charts with Estimated Parameters”, submitted to Journal of Quality Technology.

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