New Link Functions for Distribution–Specific Quantile Regression Based on Vector Generalized Linear and Additive Models

V F Miranda-Soberanis,T W Yee

doi:10.1155/2019/3493628

Abstract

In the usual quantile regression setting, the distribution of the response given the explanatory variables is unspecified. In this work, the distribution is specified and we introduce new link functions to directly model specified quantiles of seven 1–parameter continuous distributions. Using the vector generalized linear and additive model (VGLM/VGAM) framework, we transform certain prespecified quantiles to become linear or additive predictors. Our parametric quantile regression approach adopts VGLMs/VGAMs because they can handle multiple linear predictors and encompass many distributions beyond the exponential family. Coupled with the ability to fit smoothers, the underlying strong assumption of the distribution can be relaxed so as to offer a semiparametric–type analysis. By allowing multiple linear and additive predictors simultaneously, the quantile crossing problem can be avoided by enforcing parallelism constraint matrices. This article gives details of a software implementation called the VGAMextra package for R. Both the data and recently developed software used in this paper are freely downloadable from the internet.

Highlights

The VGAMextra Manual and Miranda-Soberanis [13] give further details about the derivation of the quantile links, whilst Yee [6] describes in the iteratively reweighted least squares (IRLS) and Fisher scoring algorithms for estimating VGLMs and VGAMs
Keeping the levels of white blood cell count (WBC) constant, for patients at either the 25% or 75% percentiles, the time to death for AG–negatives compared to AG–positives is multiplicative by a factor of exp(−1.02) ≈ 0.361, i.e., a 63.9% reduction in lifetime
This work in parametric quantile regression is blighted by the strong assumption of the assumed distribution

Summary

Introduction

Our approach uses the vector generalized linear and additive model (VGLM/VGAM; [6, 7]) framework. Equations (2)–(3) state that the conditional distribution of the response at a given value of x has a distribution involving a parameter θ and that the transformed quantile of the distribution becomes a linear predictor of the form (5). This can be achieved by defining link functions that connect (3) to (5). VGAMs are estimated by IRLS, where the difference with respect to VGLMs is that a vector additive model is fitted to the pseudo–response z푖 with explanatory variables x푖 and working weight matrices W푖 at each IRLS iteration. The basic penalty approach adopted here is described in Green and Silverman Green and Silverman [10]

Methodology

Examples

Findings

Discussion and Future