Bayesian Conditional Transformation Models

Abstract

Recent developments in statistical regression methodology shift away from pure mean regression toward 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 toward 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 parameterization 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. Supplementary materials for this article are available online.

KEYWORD:

• Conditional distribution function
• Distributional regression
• Hamiltonian Monotonicity constraint
• Monte Carlo
• No-U-turn sampler
• Penalized splines

Supplementary Materials

supplement.pdf This supplement contains the proof of Theorem 2.1 and further results for simulations and applications.

Code to reproduce the results from the applications is available on GitHub.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors would like to thank the Editor, Associate Editor, and two referees for many valuable comments that lead to a significant improvement of our original submission.

Additional information

Funding

The work of Manuel Carlan was supported by DFG via the research training group 1644. Thomas Kneib received financial support from the DFG within the research project KN 922/9-1. Nadja Klein gratefully acknowledges funding from the DFG through the Emmy Noether grant KL 3037/1-1.