Survival analysis studies the time-to-event for various subjects. In the biological and medical sciences, interest can focus on patient time to death due to various (competing) causes. In engineering reliability, one may study the time to component failure due to analogous factors or stimuli. Cure rate models serve a particular interest because, with advancements in associated disciplines, subjects can be viewed as “cured meaning that they do not show any recurrence of a disease (in biomedical studies) or subsequent manufacturing error (in engineering) following a treatment. This chapter generalizes two classical cure-rate models via the development of a COM–Poisson cure rate model. The chapter first describes the COM–Poisson cure rate model framework and general notation, and then details the model framework assuming right and interval censoring, respectively. The chapter then describes the broader destructive COM–Poisson cure rate model which allows for the number of competing risks to diminish via damage or eradication. Finally, the chapter details the various lifetime distributions considered in the literature to date for COM–Poisson-based cure rate modeling.

This chapter defines the COM–Poisson distribution in greater detail, discussing its associated attributes and computing tools available for analysis. This chapter first details how the COM–Poisson distribution was derived, and then describes the probability distribution, and introduces computing functions available in R that can be used to determine various probabilistic quantities of interest, including the normalizing constant, probability and cumulative distribution functions, random number generation, mean, and variance. The chapter then outlines the distributional and statistical properties associated with this model, and discusses parameter estimation and statistical inference associated with the COM–Poisson model. Various processes for generating random data are then discussed, along with associated available R computing tools. Continued discussion provides reparametrizations of the density function that serve as alternative forms for statistical analyses and model development, considers the COM–Poisson as a weighted Poisson distribution, and details discussion describing the various ways to approximate the COM–Poisson normalizing function.

This chapter is an overview summarizing relevant established and well-studied distributions for count data that motivate consideration of the Conway–Maxwell–Poisson distribution. Each of the discussed models provides an improved flexibility and computational ability for analyzing count data, yet associated restrictions help readers to appreciate the need for and usefulness of the Conway–Maxwell–Poisson distribution, thus resulting in an explosion of research relating to this model. For completeness of discussion, each of these sections includes discussion of the relevant R packages and their contained functionality to serve as a starting point for forthcoming discussions throughout subsequent chapters. Along with the R discussion, illustrative examples aid readers in understanding distribution qualities and related statistical computational output. This background provides insights regarding the real implications of apparent data dispersion in count data models, and the need to properly address it.

A multivariate Poisson distribution is a natural choice for modeling count data stemming from correlated random variables; however, it is limited by the underlying univariate model assumption that the data are equi-dispersed. Alternative models include a multivariate negative binomial and a multivariate generalized Poisson distribution, which themselves suffer from analogous limitations as described in Chapter 1. While the aforementioned distributions motivate the need to instead consider a multivariate analog of the univariate COM–Poisson, such model development varies in order to take into account (or results in) certain distributional qualities. This chapter summarizes such efforts where, for each approach, readers will first learn about any bivariate COM–Poisson distribution formulations, followed by any multivariate analogs. Accordingly, because these models are multidimensional generalizations of the univariate COM–Poisson, they each contain their analogous forms of the Poisson, Bernoulli, and geometric distributions as special cases. The methods discussed in this chapter are the trivariate reduction, compounding, Sarmanov family of distributions, and copulas.

While the Poisson model motivated much of the classical control chart theory for count data, several works note the constraining equi-dispersion assumption. Dispersion must be addressed because over-dispersed data can produce false out-of-control detections when using Poisson limits, while under-dispersed data will produce Poisson limits that are too broad, resulting in potential false negatives and out-of-control states requiring a longer study period for detection. Section 6.1 introduces the Shewhart COM–Poisson control chart, demonstrating its flexibility in assessing in- or out-of-control status, along with advancements made to this chart. These initial works lead to a wellspring of flexible control chart development motivated by the COM–Poisson distribution. Section 6.2 describes a generalized exponentially weighted moving average control chart, and Section 6.3 describes the cumulative sum charts for monitoring COM–Poisson processes. Meanwhile, Section 6.4 introduces generally weighted moving average charts based on the COM-Poisson, and Section 6.5 presents the Conway–Maxwell–Poisson chart via the progressive mean statistic. Finally, the chapter concludes with discussion.

This chapter introduces the Conway–Maxwell–Poisson regression model, along with adaptations of the model to account for zero-inflation, censoring, and data clustering. Section 5.1 motivates the consideration and development of the various COM–Poisson regressions. Section 5.2 introduces the regression model and discusses related issues including parameter estimation, hypothesis testing, and statistical computing in R. Section 5.3 advances that work to address excess zeroes, while Section 5.4 describes COM–Poisson models that incorporate repeated measures and longitudinal studies. Section 5.5 focuses attention on the R statistical packages and functionality associated with regression analysis that accommodates excess zeros and/or clustered data as described in the two previous sections. Section 5.6 considers a general additive model based on COM–Poisson. Finally, Section 5.7 informs readers of other statistical computing softwares that are also available to conduct COM–Poisson regression, discussing their associated functionality. The chapter concludes with discussion.

The Conway–Maxwell–Poisson distribution has garnered interest in and development of other flexible alternatives to classical distributions. This chapter introduces various distributional extensions and generalities motivated by functions of COM–Poisson random variables, including Conway–Maxwell-inspired generalizations of the Skellam distribution, binomial distribution, negative binomial distribution, the Katz class of distributions, two flexible series system life length distributions, and generalizations of the negative hypergeometric distribution.

This chapter considers various models that focus largely on serially dependent variables and the respective methodologies developed with a COM–Poisson underpinning. This chapter first introduces the reader to the various stochastic processes that have been established, including a homogeneous COM–Poisson process, a copula-based COM–Poisson Markov model, and a COM–Poisson hidden Markov model. Meanwhile, there are two approaches for conducting time series analysis on time-dependent count data. One approach assumes that the time dependence occurs with respect to the intensity vector. Under this framework, the usual time series models that assume a continuous variable can be applied. Alternatively, the time series model can be applied directly to the outcomes themselves. Maintaining the discrete nature of the observations, however, requires a different approach referred to as a thinning-based method. Different thinning-based operators can be considered for such models. The chapter then broadens the discussion of dependence to consider COM–Poisson-based spatio-temporal models, thus allowing both for serial and spatial dependence among variables.