A Novel Regression Model for Fractiles: Formulation, Computational Aspects, and Applications to Medical Data

Abstract

Covariate-related response variables that are measured on the unit interval frequently arise in diverse studies when index and proportion data are of interest. A regression on the mean is commonly used to model this relationship. Instead of relying on the mean, which is sensitive to atypical data and less general, we can estimate such a relation using fractile regression. A fractile is a point on a probability density curve such that the area under the curve between that point and the origin is equal to a specified fraction. Fractile or quantile regression modeling has been considered for some statistical distributions. Our objective in the present article is to formulate a novel quantile regression model which is based on a parametric distribution. Our fractile regression is developed reparameterizing the initial distribution. Then, we introduce a functional form based on regression through a link function. The main features of the new distribution, as well as the density, distribution, and quantile functions, are obtained. We consider a brand-new distribution to model the fractiles of a continuous dependent variable (response) bounded to the interval (0, 1). We discuss an R package with random number generators and functions for probability density, cumulative distribution, and quantile, in addition to estimation and model checking. Instead of the original distribution-free quantile regression, parametric fractile regression has lately been employed in several investigations. We use the R package to fit the model and apply it to two case studies using COVID-19 and medical data from Brazil and the United States for illustration.

Keywords

• GLM
• Marshall–Olkin distribution
• maximum likelihood estimation methods
• Monte–Carlo simulation methods
• quantile function
• <tt>R</tt> statistical software