The Frank copula as a latent model for ordinal variables: dependence structure, correlation attenuation, and statistical inference

Abstract

In this work, we consider the Frank copula, belonging to the Archimedean family and enjoying several interesting properties, and suggest its use as a bivariate and multivariate latent variable for modeling correlated ordinal data, exploring the relationship between its parameter, which models the dependence among the standard uniform components, and the implied (linear) correlation between any pair of ordinal random variables, also as a function of the assigned marginal ordinal distributions. Based on extensive numerical evidence, we conjecture that discretization of a Frank copula latent variable cannot inflate linear correlation, suggesting a behaviour that appears numerically consistent with the Gaussian extremal property. We discuss inference for the proposed model, which appears to be straightforward, although it requires numerical methods, and can be easily implemented in the programming environment. For assessing the goodness-of-fit of the model, we suggest using a parametric bootstrap procedure, which is particularly useful and preferable to the use of the asymptotic distribution of standard statistics when the observed contingency table presents several small counts. The methodology is illustrated through the analysis of two real datasets with ordinal responses, and compared with the customary latent Gaussian model. Despite its more limited modeling flexibility, due to the use of a single parameter controlling the correlation among variables, the advantage of this latent construction over the customary normal latent variable lies in the availability of a closed-form expression for the joint distribution function, which facilitates and accelerates the evaluation of the resulting joint probability mass function and correlation matrix, and obviously parameter estimation.