Scholarly papers describing the methodology

Transformation models have been around for more than 50 years, starting with the seminal paper introducing “Box-Cox” power-transformations published by George Box and Sir David Cox in 1964. Later developments focused on a semiparametric understanding of these models, most importantly the partial likelihood approach to parameter estimation in the Cox proportional hazards model.

During the last decade, fully parametric versions of transformation models have been studied. Model inference is much simpler once all components of the models have been parametrised appropriately. Research on transformation models implemented in the mlt add-on package started with a gradient-boosting algorithm for conditional transformation models (Hothorn, Kneib, and Bühlmann, 2014). This algorithm optimises the Brier score for model estimation. It turned out that maximum likelihood estimation is computationally and conceptionally much simpler and also helps to estimate models for discrete or censored data (Hothorn, Möst, and Bühlmann, 2018). So-called most likely transformations are implemented in the mlt add-on package (Hothorn, 2018a).

A generalisation of binary logistic regression models to continuous outcomes featuring parameters interpretable as log-odds ratios were described in (Lohse, Rohrmann, Faeh, and Hothorn, 2017). Simple transformation models as well as more complex transformation models (for example transformation trees and forests) for body mass index distributions are discussed in (Hothorn, 2018b). Two likelihood-based boosting methods for transformation models are introduced in (Hothorn, 2019).


[1] T. Hothorn, T. Kneib, and P. Bühlmann. “Conditional Transformation Models”. In: Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76.1 (2014), pp. 3–27. DOI: 10.1111/rssb.12017.

[2] T. Lohse, S. Rohrmann, D. Faeh, et al. “Continuous Outcome Logistic Regression for Analyzing Body Mass Index Distributions”. In: F1000Research 6 (2017), p. 1933. DOI: 10.12688/f1000research.12934.1.

[3] T. Hothorn. “Most Likely Transformations: The mlt Package”. In: Journal of Statistical Software (2018). Accepted 2018-03-05. URL:

[4] T. Hothorn. “Top-Down Transformation Choice”. In: Statistical Modelling 18.3–4 (2018), pp. 274–298. DOI: 10.1177/1471082X17748081.

[5] T. Hothorn, L. Möst, and P. Bühlmann. “Most Likely Transformations”. In: Scandinavian Journal of Statistics 45.1 (2018), pp. 110–134. DOI: 10.1111/sjos.12291.

[6] T. Hothorn. “Transformation Boosting Machines”. In: Statistics and Computing (2019). DOI: 10.1007/s11222-019-09870-4.