Bayesian Conditional Transformation Models

TL;DR

This paper introduces Bayesian Conditional Transformation Models (BCTMs) that infer entire conditional distributions of responses using spline-based transformations within a Bayesian framework, accommodating censored and discrete data.

Contribution

The paper develops a Bayesian approach to conditional transformation models with spline parametrization, extending their applicability to censored and discrete responses, and providing credible intervals.

Findings

BCTMs perform competitively against likelihood-based CTMs and other Bayesian models.

The approach effectively handles censored and discrete data.

Applications demonstrate the method's versatility in real-world problems.

Abstract

Recent developments in statistical regression methodology shift away from pure mean regression towards distributional regression models. One important strand thereof is that of conditional transformation models (CTMs). CTMs infer the entire conditional distribution directly by applying a transformation function to the response conditionally on a set of covariates towards a simple log-concave reference distribution. Thereby, CTMs allow not only variance, kurtosis or skewness but the complete conditional distribution to depend on the explanatory variables. We propose a Bayesian notion of conditional transformation models (BCTMs) focusing on exactly observed continuous responses, but also incorporating extensions to randomly censored and discrete responses. Rather than relying on Bernstein polynomials that have been considered in likelihood-based CTMs, we implement a spline-based parametrization for monotonic effects that are supplemented with smoothness priors. Furthermore, we are able to benefit from the Bayesian paradigm via easily obtainable credible intervals and other quantities without relying on large sample approximations. A simulation study demonstrates the competitiveness of our approach against its likelihood-based counterpart but also Bayesian additive models of location, scale and shape and Bayesian quantile regression. Two applications illustrate the versatility of BCTMs in problems involving real world data, again including the comparison with various types of competitors.