Distribution-Free Location-Scale Regression

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.

Automated summary

We introduce a generalized additive model for location, scale, and shape (GAMLSS) for distribution-free and parsimonious regression modelling. The model replaces strict parametric distribution with a transformation function and limits the number of linear or smooth model terms. Likelihood and score functions are derived for various types of observations. Various algorithms are used for model estimation and parameter interpretability is connected to model selection. A novel best subset selection procedure is proposed for simpler interpretation. Numerous applications are provided as examples and all analyses are reproducible using the tram add-on package to the R system.