Modeling Count Data via Copulas

TL;DR

This paper develops methods to measure dependence between count variables using copulas, deriving formulas for Spearman's rho, comparing it with Kendall's tau, and applying it to a cervical cancer dataset.

Contribution

It introduces a closed-form expression for Spearman's rho for discrete variables via copulas and proposes a new bivariate copula regression model for count data.

Findings

Derived Spearman's rho for discrete variables with copulas.

Compared Spearman's rho with Kendall's tau in various cases.

Applied the model to analyze cervical cancer count data.

Abstract

Copula models have been widely used to model the dependence between continuous random variables, but modeling count data via copulas has recently become popular in the statistics literature. Spearman's rho is an appropriate and effective tool to measure the degree of dependence between two random variables. In this paper, we derived the population version of Spearman's rho correlation via copulas when both random variables are discrete. The closed-form expressions of the Spearman correlation are obtained for some copulas of simple structure such as Archimedean copulas with different marginal distributions. We derive the upper bound and the lower bound of the Spearman's rho for Bernoulli random variables. Then, the proposed Spearman's rho correlations are compared with their corresponding Kendall's tau values. We characterize the functional relationship between these two measures of dependence in some special cases. An extensive simulation study is conducted to demonstrate the validity of our theoretical results. Finally, we propose a bivariate copula regression model to analyze the count data of a \emph{cervical cancer} dataset.