Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models

Abstract

Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. Modeling bimodal behavior in discrete data is useful for various purposes, from comparing shapes of different samples (or survey questions) to predicting future ratings by new raters. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However, it is suitable only for modeling equi-dispersed distributions and is limited in its ability to capture bimodality. The Conway-Maxwell-Poisson (CMP) distribution is a two-parameter generalization of the Poisson distribution that allows for over-and under-dispersion. In this work, we propose a mixture of CMPs for capturing a wide range of truncated discrete data, which can exhibit unimodal and bimodal behavior. We present methods for estimating the parameters of a mixture of two CMP distributions using an EM approach. Our approach introduces a special two-step optimization within the M step to estimate multiple parameters. We examine computational and theoretical issues. The methods are illustrated for modeling ordered rating data as well as truncated count data, using simulated and real examples.

Dual random utility maximisation

“…As a consequence, in each state there is some alternative that has a higher utility than a in A [ B, and therefore a must be chosen with probability zero 5 E.g. Frolich, Oppenheimer and Moore [16] and Dufwenberg and Muren [9] (choices in a dictator games concentrated on giving nothing or 50/50), Sura, Shmuelib, Bosec and Dubeyc [35] (bimodal distributions in ratings, such as Amazon), Plerou, Gopikrishnan, and Stanley [30] (phase transition to bimodal demand "bulls and bears"-in financial markets), Engelmann and Normann [13] (bimodality on maximum and minimum effort levels in minimum effort games), McClelland, Schulze and Coursey [26] (bimodal beliefs for unlikely events and willingness to insure).…”

A Bimodal Discrete Shifted Poisson Distribution. A Case Study of Tourists’ Length of Stay

“…Parameter θ controls the unimodality or bimodality of the family given in (1). Here f Y (y; µ, σ) is the parent distribution from which we can construct a distribution that can be unimodal or bimodal.…”

“…Bimodal and multimodal distributions are found in many continuous and discrete data sets; for example, in aggregate counts of responses to Likert scale questions, as in online ratings of movies or hotels [1]; in the durations of intervals between the eruptions of certain geysers; in the distributions of male and female body weights; in student test scores, distinguishing between those who studied for the test and those who did not; and in tourism analysis, regarding the number of nights that tourists spend at a given destination [2,3]. However, these distributions have received little attention in theoretical and empirical literature, with the exceptions of classical distributions based on continuous data, such as exponential or normal distributions [4]; discrete data frameworks, such as censored count data (where an additional mode might be used for the highest category; see [5]); latent class models for count data which account for heterogeneity using a finite mixture of unimodal Poisson distributions (i.e., the latent class truncated Poisson regression [6]); and flexible models that capture both over and underdispersion, such as the mixed Conway-Maxwell-Poisson distribution, which can reflect a wide range of truncated discrete data, and can exhibit either unimodal or bimodal behaviour [1] (the Conway-Maxwell-Poisson (CMP) distribution is a two-parameter generalisation of the Poisson distribution that allows for either over or underdispersion.). An important feature of multimodal data sets is that they can reveal when two or more types of individuals are represented in a data set (for example, consumer segments and preferences).…”

Low-Income Housing Tax Credit Developments and Neighborhood Property Conditions

“…For additional applications, seeLord, Geedipally, and Guikema (2010);Lord, Guikema, and Geedipally (2008);Sur et al (2015), andFrancis et al (2012).…”