Likelihood-Based EWMA Charts for Monitoring Poisson Count Data With Time-Varying Sample Sizes

Abstract

Many applications involve monitoring incidence rates of the Poisson distribution when the sample size varies over time. Recently, a couple of cumulative sum and exponentially weighted moving average (EWMA) control charts have been proposed to tackle this problem by taking the varying sample size into consideration. However, we argue that some of these charts, which perform quite well in terms of average run length (ARL), may not be appealing in practice because they have rather unsatisfactory run length distributions. With some charts, the specified in-control (IC) ARL is attained with elevated probabilities of very short and very long runs, as compared with a geometric distribution. This is reflected in a larger run length standard deviation than that of a geometric distribution and an elevated probability of false alarms with short runs, which, in turn, hurt an operator's confidence in valid alarms. Furthermore, with many charts, the IC ARL exhibits considerable variations with different patterns of sample sizes. Under the framework of weighted likelihood ratio test, this article suggests a new EWMA control chart which automatically integrates the varying sample sizes with the EWMA scheme. It is fast to compute, easy to construct, and quite efficient in detecting changes of Poisson rates. Two important features of the proposed method are that the IC run length distribution is similar to that of a geometric distribution and the IC ARL is robust to various patterns of sample size variation. Our simulation results show that the proposed chart is generally more effective and robust compared with existing EWMA charts. A health surveillance example based on mortality data from New Mexico is used to illustrate the implementation of the proposed method. This article has online supplementary materials.

KEY WORDS:

• Average run length
• EWMA
• Healthcare
• Poisson count data
• Run length distribution
• Short-run processes
• Statistical process control

Acknowledgments

The authors thank the editor, associate editor, and three anonymous referees for their many helpful comments that have resulted in significant improvements in the article. We also thank William Woodall for the useful discussions. This research was supported by the NNSF of China grants 11001138, 11071128, 11131002, 11101306, and 71172131 and the RFDP of China grant 20110031110002. Zhou's work is partially supported by PAPD of Jiangsu Higher Education Institutions. Zou thanks the support of the National Center for Theoretical Sciences, Math Division. This work was completed when Zhou was a doctoral student at Nankai University. The first two authors contributed equally to this work. Wang is the corresponding author.