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.
Hub for Trials Methodology Research
MRC Clinical Trials Unit
and University College London
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acd, continuous covariate, sigmoid function, fractional polynomials, regression models
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