A smooth covariate rank transformation for use in regression models with a sigmoid dose-response function

Area of Science:

• Biostatistics
• Statistical Modeling
• Regression Analysis

Background:

• Sigmoid-type regression relationships are common but challenging to model.
• Existing smoothers like fractional polynomials and splines struggle with doubly asymptotic curves.
• Accurate modeling of singly asymptotic curves can also be problematic.

Purpose of the Study:

• To develop a practical and parsimonious method for representing sigmoid-type regression relationships.
• To address limitations of current smoothers in modeling asymptotic curves.
• To introduce a new command, 'acd', for implementing the proposed method.

Main Methods:

• Applying a preliminary scaled rank transformation to compress covariate tails.
• Approximating the empirical cumulative distribution function (ECDF) of the covariate.
• Using fractional polynomial regression on the outcome with smoothed, scaled ranks as the covariate.
• Identifying sigmoid functions when the resulting fractional polynomial is monotone.

Main Results:

• The approximate cumulative distribution transformation effectively models sigmoid relationships.
• The method accurately represents curves with single and double asymptotes.
• Demonstrated practical applications and ability to model unusual functional forms.

Conclusions:

• The proposed scaled rank transformation and fractional polynomial approach offers a robust solution for modeling sigmoid relationships.
• This method overcomes limitations of existing smoothers for asymptotic curves.
• The 'acd' command provides a practical tool for researchers to implement this technique.