A Smooth Covariate Rank Transformation for Use in Regression Models with a Sigmoid Dose–Response Function

Abstract

We consider how to represent sigmoid-type regression relationships in a practical and parsimonious way. A pure sigmoid relationship has an asymptote at both ends of the range of a continuous covariate. Curves with a single asymptote are also important in practice. Many smoothers, such as fractional polynomials and restricted cubic regression splines, cannot accurately represent doubly asymptotic curves. Such smoothers may struggle even with singly asymptotic curves. Our approach to modeling sigmoid relationships involves applying a preliminary scaled rank transformation to compress the tails of the observed distribution of a continuous covariate. We include a step that provides a smooth approximation to the empirical cumulative distribution function of the covariate via the scaled ranks. The procedure defines the approximate cumulative distribution transformation of the covariate. To fit the substantive model, we apply fractional polynomial regression to the outcome with the smoothed, scaled ranks as the covariate. When the resulting fractional polynomial function is monotone, we have a sigmoid function. We demonstrate several practical applications of the approximate cumulative distribution transformation while also illustrating its ability to model some unusual functional forms. We describe a command, acd, that implements it.