GWRM: An R Package for Identifying Sources of Variation in Overdispersed Count Data

Abstract

Understanding why a random variable is actually random has been in the core of Statistics from its beginnings. The generalized Waring regression model for count data explains that inherent variability is given by three possible sources: randomness, liability and proneness. The model extends the negative binomial regression model and it is not included in the family of generalized linear models. In order to avoid that shortcoming, we developed the R package for fitting, describing and validating the model. The version we introduce in this communication provides a new design of the modelling function as well as new methods operating on the associated fitted model objects, so that the new software integrates easily into the computational toolbox for modelling count data in R. The release of a plug-in in order to use the package from the interface R Commander tries to contribute to the spreading of the model among non-advanced users. We illustrate the usage and the possibilities of the software with two examples from the fields of health and sport.

Modeling COVID-19 daily cases in Senegal using a generalized Waring regression model

“…In this model, we make the following distributional assumption for given : where is a gamma distributed variable: Then, using straightforward calculations we obtain (see, for example, [14] , [19] or [20] ): where . Its follows, for the NBRM, that the conditional mean is given by: Remark that this model is a Poisson-gamma mixture model and, for the gamma distribution, the parameter does not depend on the covariates contrary to .…”

“…The variance of the model is now given by: The decomposition of this variance is interpreted as follow: the first term of the model variance represents the variability due to randomness inherent in any random phenomenon and the second to differences between the days between March and October in our practical application. For more details on the interpretation of the variance decomposition see [14] or [20] .…”

“…The GWMR is an extension of NBRM. Contrary to the last model, it allows to distinguish the variability due to individual differences related to the covariates included in the model, to the one only due to individual differences not related to the model covariates [14] , [20] . Next, we present a brief background on the genesis of the GWRM and some properties of the model.…”

“…In this model we assume that given has a univariate generalized Waring distribution: This implies that the probability mass function of is given by [16] : where and for and is the gamma function defined by , for . Then, the conditional mean is given by: In addition, linking the conditional mean with the explanatory variables as , we have: Furthermore, the variance of the GWRM is given by [14] , [20] : The decomposition of this variance is interpreted as follow:…”

An Over and Underdispersed Biparametric Extension of the Waring Distribution

“…For this reason, the UGW distribution and the related regression model [5][6][7] have been widely applied for modelling overdispersed count data sets in different fields, such as lexicology [8], the number of authors in scientific articles [9], the evolution of the number of links in the World Wide Web [10], accident theory [11], clustered data [12], sources of variance in motor vehicle crash analysis [13], completeness errors in geographic data sets [14] or agriculture [15].…”