Statistical process control of overdispersed count data based on one‐parameter Poisson mixture models

Abstract

The Poisson distribution is a discrete model widely used to analyze count data. Statistical control charts based on this distribution, such as the c$c$ and u$u$ charts, are relatively well‐established in the literature. Nevertheless, many studies suggest the need for alternative approaches that allow for modeling overdispersion, a phenomenon that can be observed in several fields, including biology, ecology, healthcare, marketing, economics, and industry. The one‐parameter Poisson mixture distributions, whose literature is extensive and essential, can model extra‐Poisson variability, accommodating different overdispersion levels. The distributions belonging to this class of models, including the Poisson‐Lindley (PL), Poisson‐Shanker (PSh), and Poisson‐Sujatha (PSu) models, can thus be used as interesting alternatives to the usual Poisson and COM‐Poisson distributions for analyzing count data in several areas. In this paper, we consider the class of probabilistic models mentioned above (as well as the cited three members of such a class) to develop novel and useful statistical control charts for counting processes, monitoring count data that exhibit overdispersion. The performance of the so‐called one‐parameter Poisson mixture charts, namely the PLc$\text{PL}_c$‐PLu$\text{PL}_u$, PShc$\text{PSh}_c$‐PShu$\text{PSh}_u$, and PSuc$\text{PSu}_c$‐PSuu$\text{PSu}_u$ charts, is measured by the average run length in exhaustive numerical simulations. Some data sets are used to illustrate the applicability of the proposed methodology.

Enhancing Control: Unveiling the Performance of Poisson EWMA Charts through Simulation with Poisson Mixture DATA

“…This approach was adopted as a precautionary measure to mitigate the effects of overdispersion. In view of this concept, Jesus et al [21] suggested in their study that handling overdispersion with the Poisson mixture distribution would eliminate the problem and increase the ARL performance. However, the charts they deal with are not EWMA charts as similar to Boaventura et al [27], but for the c and u charts developed for attribute data.…”

“…By capitalizing on the equality between the variance and mean of a Poisson distribution, we make a substitution of α i instead of σ 2 i , leading to the derivation of the formula presented in Equation (21).…”

“…Although the Poisson distribution is frequently used to describe count data, it assumes that variance is equal to the mean (equi dispersion), which could lead to incorrect inferences in the case of overdispersion [20]. One of the most common methods to cope with different levels of overdispersion is to use Poisson mixture distributions [21]. Several various control charts have been proposed for different types of Poisson distributions in ref.…”

A nonparametric control chart for monitoring count data mean

“…Saghir and Lin 15 provided an excellent overview on different approaches to construct control charts for count data that are either over‐ or under‐dispersed. Recently, several control charts have been proposed for count data using the BerG distribution, 16 the Touchard distribution, 17 and one‐parameter Poisson mixture distributions 18 to account for over‐ or under‐dispersion in count data.…”

Novel nonparametric control charts for monitoring dispersion of count data

“…Saghir and Lin, 9,10 among others, further extended that control chart to EWMA and CUSUM charts, respectively. Other existing control charts for monitoring under-or over-dispersed count data used the Katz distribution family, 11 the generalized Poisson model, 12 the BerG distribution, 13 the Touchard distribution, 14 and one parameter Poisson mixture models, 15 to list a few. Readers are referred to Saghir and Lin 16 for an overview of control charts for monitoring under-or over-dispersed count data.…”