# Distribution-Free Location-Scale Regression

arXiv: https://arxiv.org/abs/2208.05302 · HTML: https://mlsanity.com/paper/2208.05302/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2022.

## Built on by (most cited first, 3 of 3)

- [Deep conditional transformation models for survival analysis](https://mlsanity.com/paper/2210.11366/) (2022)
- [How Inverse Conditional Flows Can Serve as a Substitute for Distributional Regression](https://mlsanity.com/paper/2405.05429/) (2024)
- [Conformal Prediction in Dynamic Biological Systems](https://mlsanity.com/paper/2409.02644/) (2024)

## Abstract (arXiv)

We introduce a generalized additive model for location, scale, and shape (GAMLSS) next of kin aiming at distribution-free and parsimonious regression modelling for arbitrary outcomes. We replace the strict parametric distribution formulating such a model by a transformation function, which in turn is estimated from data. Doing so not only makes the model distribution-free but also allows to limit the number of linear or smooth model terms to a pair of location-scale predictor functions. We derive the likelihood for continuous, discrete, and randomly censored observations, along with corresponding score functions. A plethora of existing algorithms is leveraged for model estimation, including constrained maximum-likelihood, the original GAMLSS algorithm, and transformation trees. Parameter interpretability in the resulting models is closely connected to model selection. We propose the application of a novel best subset selection procedure to achieve especially simple ways of interpretation. All techniques are motivated and illustrated by a collection of applications from different domains, including crossing and partial proportional hazards, complex count regression, non-linear ordinal regression, and growth curves. All analyses are reproducible with the help of the "tram" add-on package to the R system for statistical computing and graphics.