Conway-Maxwell-Poisson profile monitoring with rk-Shewhart control chart: a comparative study

Year 2024, , 121 - 132, 30.06.2024

Ulduz Mammadova

https://doi.org/10.59313/jsr-a.1323676

Abstract

A control chart is an essential tool in Statistical Quality Control for monitoring the production process. It provides a visual means of identifying process irregularities. In this study, we focus on the Shewhart control chart based on the rk-deviance residuals, namely rk-Shewhart control charts to examine the Conway–Maxwell–Poisson (COM-Poisson) profile, which is used to model the count data with varying degrees of dispersion. The primary goal of this study is to identify the biasing parameter that produces the best result among newly presented biasing parameters developed based on existing ones. It provides a short overview of the COM-Poisson distribution, its modeling, and rk parameter estimation in the case of multicollinearity, as well as the construction of the deviance-residual-based Shewhart chart. To evaluate the performance of the rk-Shewhart, we conduct an analysis using a real-life data set, considering various shift sizes. By employing different biasing parameters, we examine the effectiveness of the rk-Shewhart control chart. The performance evaluation outcomes of the rk-Shewhart charts are compared to the ML-deviance-based Shewhart chart and within themselves based on the biasing parameters. The results demonstrate the advantage of the rk-Shewhart charts over the ML-deviance-based control chart in detecting out-of-control signals. Among the considered biasing parameters, the rk-Shewhart chart utilizing the adjusted biasing parameter k_4 shows the best performance based on the ARL metric.

Keywords

Control chart, Conway-Maxwell-Poisson model, Multicollinearity, Process monitoring, r-k estimation

References

• [1] K.R. Skinner, D.C. Montgomery, and G.C. Runger, “Process monitoring for multiple count data using generalized linear model-based control charts,” Int. J. Prod. Res., vol. 41, no. 6, pp. 1167–1180, Jun. 2003, doi: 10.1080/00207540210163964.
• [2] K.R. Skinner, D.C. Montgomery, and G.C. Runger, “Generalized linear model-based control charts for discrete semiconductor process data,” Qual. Reliab. Eng. Int., vol. 20, no. 8, pp. 777–786, 2004, doi: 10.1002/qre.603.
• [3] D. Jearkpaporn, D.C. Montgomery, G.C. Runger, and C.M. Borror, “Process monitoring for correlated gamma-distributed data using generalized-linear-model-based control charts,” Qual. Reliab. Eng. Int., vol. 19, no. 6, pp. 477–491, 2003, doi: 10.1002/qre.521.
• [4] A. Amiri, M. Koosha, and A. Azhdari, “Profile monitoring for Poisson responses,” in IEEE Int. Conf. Ind. Eng. Eng. Manag., 2011, pp. 1481–1484, doi: 10.1109/IEEM.2011.6118163
• [5] A. Asgari, A. Amiri, and S.T.A. Niaki, “A new link function in GLM-based control charts to improve monitoring of two-stage processes with Poisson response,” Int. J. Adv. Manuf. Technol., vol. 72, no. 9, pp. 1243–1256, 2014, doi: 10.1007/s00170-014-5692-z.
• [6] W.Y. Hwang, “Quantile-based control charts for Poisson and gamma distributed data,” J. Korean Stat. Soc., pp. 1–18, 2021, doi: 10.1007/s42952-021-00108-6.
• [7] H. Wen, L. Liu, and X. Yan, “Regression-adjusted Poisson EWMA control chart,” Qual. Reliab. Eng. Int., vol. 37, no. 5, pp. 1956–1964, 2021, doi: 10.1002/qre.2840.
• [8] D. Marcondes Filho and A.M.O. Sant’Anna, “Principal component regression – based control charts for monitoring count data,” Int. J. Adv. Manuf. Technol., vol. 85, no. 5, pp. 1565–1574, 2016, doi: 10.1007/s00170-015-8054-6.
• [9] U. Mammadova and M.R. Özkale, “Profile monitoring for count data using Poisson and Conway – Maxwell – Poisson regression–based control charts under multicollinearity problem,” J. Comput. Appl. Math., vol. 388, p. 113275, 2021, doi: 10.1016/j.cam.2020.113275.
• [10] U. Mammadova and M.R. Özkale, “Comparison of deviance and ridge deviance residual-based control charts for monitoring Poisson profiles,” Commun. Stat. – Simul. Comput., vol. 52, no. 3, pp. 826–853, 2023, doi: 10.1080.
• [11] R.W. Conway and W.L. Maxwell, “A queuing model with state-dependent service rates,” J. Ind. Eng., vol. 12, no. 2, pp. 132–136, 1962. [Online]. Available: https://tinyurl.com/28cv3brs
• [12] S.D. Guikema and J.P. Coffelt, “A flexible count data regression model for risk analysis,” Risk Anal.: An Int. J., vol. 28, no. 1, pp. 213–223, 2008, doi: 10.1111/j.1539-6924.2008.01014.x.
• [13] D. Lord, S.D. Guikema, and S.R. Geedipally, “Application of the Conway–Maxwell–Poisson generalized linear model for analyzing motor vehicle crashes,” Accident Anal. & Prev., vol. 40, no. 3, pp. 1123–1134, 2008, doi: 10.1016/j.aap.2007.12.003.
• [14] V. Jowaheer and N.M. Khan, “Estimating regression effects in COM–Poisson generalized linear model,” Int. J. Comput. Math. Sci., vol. 1, no. 2, pp. 59-63, 2009, doi: 10.5281/zenodo.1075883.
• [15] K.F. Sellers and G. Shmueli, “A flexible regression model for count data,” Ann. Appl. Stat., vol. 4, no. 2, pp. 943–961, 2010, doi: 10.2139/ssrn.1127359
• [16] K. Park, J.M. Kim, and D. Jung, “GLM-based statistical control r-charts for dispersed count data with multicollinearity between input variables,” Qual. Reliab. Eng. Int., vol. 34, no. 6, pp. 1103–1109, 2018, doi: 10.1002/qre.2310.
• [17] K. Park, D. Jung, and J.M. Kim, “Control charts based on randomized quantile residuals,” Appl. Stoch. Models Bus. Ind., vol. 36, no. 4, pp. 716–729, 2020, doi: 10.1002/asmb.2527.
• [18] A. Jamal et al., “GLM-based flexible monitoring methods: An application to real-time highway safety surveillance,” Symmetry, vol. 13, no. 2, p. 362, 2021, doi: 10.3390/sym13020362.
• [19] U. Mammadova and M.R. Özkale, “Conway–Maxwell Poisson regression-based control charts under iterative Liu estimator for monitoring count data,” Appl. Stoch. Models Bus. Ind., vol. 38, no. 4, pp. 695-725, 2022, doi: 10.1002/asmb.2682.
• [20] U. Mammadova and M.R. Özkale, “Detecting shifts in Conway–Maxwell–Poisson profile with deviance residual-based CUSUM and EWMA charts under multicollinearity,” Stat. Papers, pp. 1–47, 2023, doi: 10.1007/s00362-023-01399-z.
• [21] G. Shmueli, T.P.Minka, J.B. Kadane, S. Borle, and P. Boatwright, “A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution,” J. R. Stat. Soc.: Series C (Appl. Stat.), vol. 54, no. 1, pp. 127–142, 2005, doi: 10.1111/j.1467-9876.2005.00474.x.
• [22] R.A. Francis, S.R. Geedipally, S.D. Guikema, S.S. Dhavala, D. Lord, and S. LaRocca, “Characterizing the performance of the Conway-Maxwell-Poisson generalized linear model,” Risk Anal.: An Int. J., vol. 32, no. 1, pp. 167–183, 2012, doi: 10.1111/j.1539-6924.2011.01659.x.
• [23] M.R. Baye and D.F. Parker, “Combining ridge and principal component regression: a money demand illustration,” Commun. Stat.-Theory Methods, vol. 13, no. 2, pp. 197–205, 1984, doi: 10.1080/03610928408828675.
• [24] A. Abbasi and M.R. Özkale, “The r-k class estimator in generalized linear models applicable with sim. and empirical study using a Poisson and gamma responses,” Hacettepe J. Math. Stat., vol. 50, no. 2, pp. 594–611, 2021, doi: 10.15672/hujms.715206.
• [25] I.T. Jolliffe, Principal Components in Regression Analysis. New York: Springer, 2002.
• [26] A.M. Aguilera, M. Escabias, and M.J. Valderrama, “Using principal components for estimating logistic regression with high-dimensional multicollinear data,” Comput. Stat. Data Anal., vol. 50, no. 8, pp. 1905–1924, 2006, doi: 10.1016/j.csda.2005.03.011.
• [27] A.E. Hoerl and R.W. Kennard, “Ridge regression: Biased estimation for nonorthogonal problems,” Technometrics, vol. 12, no. 1, pp. 55–67, 1970, doi: 10.2307/1267351.
• [28] A.E. Hoerl, R.W. Kannard, and K.F. Baldwin, “Ridge regression: some simulations,” Commun. Stat.-Theory Methods, vol. 4, no. 2, pp. 105–123, 1975, doi: 10.1080/03610927508827232.
• [29] B.M.G. Kibria, “Performance of some new ridge regression estimators,” Commun. Stat.-Simulation Comput., vol. 32, no. 2, pp. 419–435, 2003, doi: 10.1081/SAC-120017499.
• [30] M.A. Alkhamisi and G. Shukur, “A Monte Carlo study of recent ridge parameters,” Commun. Stat.-Simulation Comput., vol. 36, no. 3, pp. 535–547, 2007, doi: 10.1080/03610910701208619.
• [31] G. Muniz and B.M.G. Kibria, “On some ridge regression estimators: An empirical comparison,” Commun. Stat.-Simulation Comput., vol. 38, no. 3, pp. 621–630, 2009, doi: 10.1080/03610910802592838.
• [32] Y. Asar, A. Karaibrahimoglu, and G. Aşır, “Modified ridge regression parameters: A comparative Monte Carlo study,” Hacettepe J. Math. Stat., vol. 43, no. 5, pp. 827–841, 2014. [Online]. Available: https://dergipark.org.tr/en/pub/hujms/issue/43879/536843
• [33] Y. Asar and A. Genç, “A note on some new modifications of ridge estimators,” Kuwait J. Sci., vol. 44, no. 3, 2017. [Online]. Available: https://journalskuwait.org/kjs/index.php/KJS/article/view/1110
• [34] M. Qasim et al., “Biased adjusted Poisson ridge estimators – method and application,” Iran. J. Sci. Technol. Trans. A Sci., vol. 44, pp. 1775–1789, 2020, doi: 10.1007/s40995-020-00974-5.
• [35] K. Mansson and G. Shukur, “A Poisson ridge regression estimator,” Econ. Model., vol. 28, no. 4, pp. 1475–1481, 2011, doi: 10.1016/j.econmod.2011.02.030.
• [36] B.M.G. Kibria, K. Mansson, and G. Shukur, “A simulation study of some biasing parameters for the ridge type estimation of Poisson regression,” Commun. Stat.-Simulation Comput., vol. 44, no. 4, pp. 943–957, 2015, doi: 10.1080/03610918.2013.796981.
• [37] A. Zaldivar, “On the performance of some Poisson ridge regression estimators” (Master's thesis). Florida International University, 2018, doi: 10.25148/etd.fidc006538.
• [38] Z.Y. Algamal and M.M. Alanaz, “Proposed methods in estimating the ridge regression parameter in Poisson regression model,” Electron. J. Appl. Stat. Anal., vol. 11, no. 2, pp. 506–515, 2018, doi: 10.1285/i20705948v11n2p506.
• [39] F. Sami, M. Amin, and M.M. Butt, “On the ridge estimation of the Conway–Maxwell–Poisson regression model with multicollinearity: Methods and applications,” Concurrency Comput. Pract. Exp., vol. 34, no. 1, e6477, 2022, doi: 10.1002/cpe.6477.
• [40] B.M.G. Kibria, “More than hundred (100) estimators for estimating the shrinkage parameter in a linear and generalized linear ridge regression models,” J. Econom. Stat., vol. 2, no. 2, pp. 233–252, 2022, doi: 10.47509/JES.2022.v02i02.06.
• [41] W.A. Shewhart, “Some applications of statistical methods to the analysis of physical and engineering data,” Bell Syst. Tech. J., vol. 3, no. 1, pp. 43–87, 1924, doi: 10.1002/j.1538-7305.1924.tb01347.x.
• [42] D.A. Pierce and D.W. Schafer, “Residuals in generalized linear models,” J. Am. Stat. Assoc., vol. 81, no. 396, pp. 977–986, 1986, doi: 10.1080/01621459.1986.10478361.
• [43] T. Park, C.S. Davis, and N. Li, “Alternative GEE estimation procedures for discrete longitudinal data,” Comput. Stat. Data Anal., vol. 28, no. 3, pp. 243–256, 1998, doi: 10.1016/s0167-9473(98)00039-5.
• [44] U. Mammadova and M.R. Özkale, "Deviance residual-based Shewhart control chart for monitoring Conway-Maxwell-Poisson profile under the r-k class estimator," Qual. Tech. Quan. Manag. pp. 1-22, 2023, doi: 10.1080/16843703.2023.2259589.
• [45] M. McCann and A. Johnston, "Dataset: SECOM." UC Irvine Machine Learning Repository, https://archive.ics.uci.edu/dataset/179/secom.
• [46] K. Sellers, T. Lotze, A. Raim, and M.A. Raim, “COMPoissonReg (R package version 3.0).” A comprehensive R archieve network, https://cran.r-project.org/web/packages/COMPoissonReg/index.html
• [47] M.J. Mackinnon and M.L. Puterman, "Collinearity in generalized linear models," Commun. Stat.-Theory Methods, vol. 18, no. 9, pp. 3463–3472, 1989, doi: 10.1080/03610928908830102.
• [48] D.C. Montgomery, Introduction to Statistical Quality Control. John Wiley & Sons, New Jersey, 2012.