Development of a Robust Shrinkage Empirical-Bayes P-Chart for Heterogeneous Proportion Data
Keywords:
Empirical Bayes, Quality control, P-chart, Shrinkage.Abstract
This study develops a Robust Shrinkage Empirical-Bayes P-Chart (RSEB p-chart) for monitoring proportions when subgroup sizes differ and observed proportions are heterogeneous. The classical p-chart assumes binomial variation around a stable process proportion, but real educational, health, and service data often show extra-binomial variation and unstable small-sample proportions. The proposed method combines a robust center line based on the median proportion with empirical-Bayes shrinkage of subgroup proportions. The empirical illustration uses the public STAR98 educational assessment dataset available through statsmodels. Results show that the proposed chart stabilizes proportions across subgroup sizes and provides interpretable control limits. The article contributes a transparent p-chart development for proportion monitoring in heterogeneous public-sector data.
References
[1] NIST/SEMATECH, e-Handbook of Statistical Methods: Proportions Control Charts. Gaithersburg, MD, USA: National Institute of Standards and Technology.
[2] D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed. Hoboken, NJ, USA: Wiley, 2019.
[3] W. A. Shewhart, Economic Control of Quality of Manufactured Product. New York, NY, USA: D. Van Nostrand, 1931.
[4] W. H. Woodall, "Control charts based on attribute data: Bibliography and review," Journal of Quality Technology, vol. 29, no. 2, pp. 172-183, 1997.
[5] T. P. Ryan, Statistical Methods for Quality Improvement, 3rd ed. Hoboken, NJ, USA: Wiley, 2011.
[6] W. H. Woodall and D. C. Montgomery, "Research issues and ideas in statistical process control," Journal of Quality Technology, vol. 31, no. 4, pp. 376-386, 1999.
[7] D. B. Laney, "Improved control charts for attributes," Quality Engineering, vol. 14, no. 4, pp. 531-537, 2002.
[8] A. Agresti and B. A. Coull, "Approximate is better than exact for interval estimation of binomial proportions," The American Statistician, vol. 52, no. 2, pp. 119-126, 1998.
[9] L. D. Brown, T. T. Cai, and A. DasGupta, "Interval estimation for a binomial proportion," Statistical Science, vol. 16, no. 2, pp. 101-133, 2001.
[10] D. Williams, "Extra-binomial variation in logistic linear models," Applied Statistics, vol. 31, no. 2, pp. 144-148, 1982.
[11] R. E. Tarone, "Testing the goodness of fit of the binomial distribution," Biometrika, vol. 66, no. 3, pp. 585-590, 1979.
[12] E. B. Wilson, "Probable inference, the law of succession, and statistical inference," Journal of the American Statistical Association, vol. 22, no. 158, pp. 209-212, 1927.
[13] Jeffreys, "An invariant form for the prior probability in estimation problems," Proceedings of the Royal Society A, vol. 186, no. 1007, pp. 453-461, 1946.
[14] J. C. Benneyan, "Use and interpretation of statistical quality control charts," International Journal for Quality in Health Care, vol. 10, no. 1, pp. 69-73, 1998.
[15] C. Alwan and H. V. Roberts, "The problem of misplaced control limits," Applied Statistics, vol. 44, no. 3, pp. 269-278, 1995.
[16] R. Quesenberry, "The effect of sample size on estimated limits for X-bar and X control charts," Journal of Quality Technology, vol. 25, no. 4, pp. 237-247, 1993.
[17] R. Quesenberry, SPC Methods for Quality Improvement. New York, NY, USA: Wiley, 1997.
[18] California Department of Education, STAR 1998 test performance data, public educational assessment dataset.
[19] Seabold and J. Perktold, "Statsmodels: Econometric and statistical modeling with Python," in Proc. 9th Python in Science Conf., 2010, pp. 92-96.
[20] D. Spiegelhalter, "Funnel plots for comparing institutional performance," Statistics in Medicine, vol. 24, no. 8, pp. 1185-1202, 2005, doi: 10.1002/sim.1970.
