Journal of Data Science

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable are commonly modeled using the generalized Pareto distribution, a peak-over-threshold method that often gives good results in practice. When data is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

Abstract: Response variables that are scored as counts, for example, number of mastitis cases in dairy cattle, often arise in quantitative genetic analysis. When the number of zeros exceeds the amount expected such as under the Poisson density, the zero-inflated Poisson (ZIP) model is more appropriate. In using the ZIP model in animal breeding studies, it is necessary to accommodate genetic and environmental covariances. For that, this study proposes to model the mixture and Poisson parameters hierarchically, each as a function of two random effects, representing the genetic and environmental sources of variability, respectively. The genetic random effects are allowed to be correlated, leading to a correlation within and between clusters. The environmental effects are introduced by independent residual terms, accounting for overdispersion above that caused by extra-zeros. In addition, an inter correlation structure between random genetic effects affecting mixture and Poisson parameters is used to infer pleiotropy, an expression of the extent to which these parameters are influenced by common genes. The methods described here are illustrated with data on number of mastitis cases from Norwegian Red cows. Bayesian analysis yields posterior distributions useful for studying environmental and genetic variability, as well as genetic correlation.

Abstract: Count data often have excess zeros in many clinical studies. These zeros usually represent “disease-free state”. Although disease (event) free at the time, some of them might be at a high risk of having the putative outcome while others may be at low or no such risk. We postulate these zeros as a one of the two types, either as ‘low risk’ or as ‘high risk’ zeros for the disease process in question. Low risk zeros can arise due to the absence of risk factors for disease initiation/progression and/or due to very early stage of the disease. High risk zeros can arise due to the presence of significant risk factors for disease initiation/ progression or could be, in rare situations, due to misclassification, more specific diagnostic tests, or below the level of detection. We use zero inflated models which allows us to assume that zeros arise from one of the two separate latent processes-one giving low-risk zeros and the other high-risk zeros and subsequently propose a strategy to identify and classify them as such. To illustrate, we use data on the number of involved nodes in breast cancer patients. Of the 1152 patients studied, 38.8% were node- negative (zeros). The model predicted that about a third (11.4%) of negative nodes are “high risk” and the remaining (27.4%) are at “low risk” of nodal positivity. Posterior probability based classification was more appropriate compared to other methods. Our approach indicates that some node negative patients may be re-assessed for their diagnosis about nodal positivity and/or for future clinical management of their disease. The approach developed here is applicable to any scenario where the disease or outcome can be characterized by count-data.

We propose a method of spatial prediction using count data that can be reasonably modeled assuming the Conway-Maxwell Poisson distribution (COM-Poisson). The COM-Poisson model is a two parameter generalization of the Poisson distribution that allows for the flexibility needed to model count data that are either over or under-dispersed. The computationally limiting factor of the COM-Poisson distribution is that the likelihood function contains multiple intractable normalizing constants and is not always feasible when using Markov Chain Monte Carlo (MCMC) techniques. Thus, we develop a prior distribution of the parameters associated with the COM-Poisson that avoids the intractable normalizing constant. Also, allowing for spatial random effects induces additional variability that makes it unclear if a spatially correlated Conway-Maxwell Poisson random variable is over or under-dispersed. We propose a computationally efficient hierarchical Bayesian model that addresses these issues. In particular, in our model, the parameters associated with the COM-Poisson do not include spatial random effects (leading to additional variability that changes the dispersion properties of the data), and are then spatially smoothed in subsequent levels of the Bayesian hierarchical model. Furthermore, the spatially smoothed parameters have a simple regression interpretation that facilitates computation. We demonstrate the applicability of our approach using simulated examples, and a motivating application using 2016 US presidential election voting data in the state of Florida obtained from the Florida Division of Elections.