Conditional Transformation Models

Abstract

The ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables. This goal is, however, seldom achieved because most established regression models only estimate the conditional mean as a function of the explanatory variables and assume that higher moments are not affected by the regressors. The underlying reason for such a restriction is the assumption of additivity of signal and noise. We propose to relax this common assumption in the framework of transformation models. The novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables. These transformation functions are estimated by regularised optimisation of scoring rules for probabilistic forecasts, e.g. the continuous ranked probability score. The corresponding estimated conditional distribution functions are consistent. Conditional transformation models are potentially useful for describing possible heteroscedasticity, comparing spatially varying distributions, identifying extreme events, deriving prediction intervals and selecting variables beyond mean regression effects. An empirical investigation based on a heteroscedastic varying coefficient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more beneficial than kernel-based non-parametric approaches or parametric generalised additive models for location, scale and shape.

Reconsidering the Income-Illness Relationship Using Distributional Regression An Application to Germany

“…Thus a flexible non-parametric estimate of the conditional distribution can be obtained. This form of model is termed 'distribution regression' in the econometric literature and is closely related to the so-called 'conditional transformation models' (Hothorn et al, 2014) in the statistical literature.…”

Conditional Transformation Models for Survivor Function Estimation

“…Hothorn et al [11] explain thoroughly how CTMs are estimated by the minimization of the continuous ranked probability score (CRPS) (see [48]) using a component-wise boosting algorithm. The CRPS was chosen because it constitutes a proper scoring rule for distributional and probabilistic forecasts [11].…”

“…Alternatively, the Brier score or the absolute error loss can be chosen as an appropriate loss function [11,48,49].…”

“…The CRPS was chosen because it constitutes a proper scoring rule for distributional and probabilistic forecasts [11]. When we estimated CTMs for survival data, we also used a component-wise boosting algorithm to minimize an appropriate integrated loss function.…”

“…CTMs have been presented recently in Hothorn et al [11] and allow the direct and semiparametric estimation of the conditional distribution function PðT tjX ¼ xÞ under rather weak assumptions. The general model class includes both the proportional odds model and the proportional hazards model as special cases.…”