When it comes to statistics, we usually expect to be informed about what happens "on average". But sometimes the key information lies in the deviations from that mean: How likely is heavy rain, and how likely is it to remain dry? So-called distributional regression describes not only the mean but also the distribution of possible values. An international team of authors, including three researchers from the University of Innsbruck, demonstrates in Nature Reviews Methods Primers how versatile this method is and how
Classical regression models estimate how explanatory factors affect the mean of a dependent variable. However, many relevant phenomena-from rain forecasts to modeling the progression of diseases-occur precisely beyond the mean and in the tails of a distribution. Distributional regression therefore models not only the mean but the entire expected distribution around that mean. "If I perform a standard mean regression, I get a rainfall forecast of 3 mm for the next day, for example," explains statistician Achim Zeileis. "While that's useful, it would be even better to know what the expected distribution around that mean looks like. It could be that there will almost certainly be precipitation between 2 mm and 4 mm. But it could also be that, with high probability, it will remain completely dry, or-with a small probability-there will be heavy rain."
The so-called GAMLSS-Framework (Generalized Additive Models for Location, Scale and Shape) is a key approach in this field. It allows all parameters of a statistical distribution to depend jointly and flexibly on the available explanatory variables. In addition to Zeileis, Nikolaus Umlauf and Reto Stauffer from the Department of Statistics are also involved in the work. Stauffer is also affiliated with the Digital Science Center . All three have already published methodological papers, applications in various disciplines, and open-source software on the topic.
Wide range of applications
The range of possible applications is broad: In medicine, for example, GAMLSS is used to detect abnormal levels of alkaline phosphatase (AP) relative to gestational age, thereby enabling the early identification of liver disease or severe pregnancy complications. In environmental science, it can be used to model the effects of water temperatures and nutrient concentrations on the likelihood of algal blooms in lakes. And in weather forecasting, such models demonstrate greater accuracy in precipitation modeling than traditional linear methods.
Strengths and limitations of distributional regression
"In order for GAMLSS to reach its full potential, several aspects need to be considered when specifying the model: Which distribution is appropriate? Which predictor variables are suitable for modeling the mean or the deviation from it?" explains Nikolaus Umlauf. In general, larger sample sizes allow more complex models to be fitted, which in turn requires greater computational power and memory. Model selection is therefore always a balancing act between the research question, the available data, and computational resources. "The good news is that there is already free and easy-to-use open-source software available that allows anyone to explore these methods in their own field," emphasizes Reto Stauffer.
Integration of machine learning
The review article in the journal Nature Reviews Methods Primers concludes with a look to the future: The integration of machine learning methods into the GAMLSS framework is an active area of research. The goal is to further develop the models so that high predictive quality can be more easily combined with calibrated uncertainty estimates, interpretable effects, and scientifically sound conclusions.
Publication: Distributional regression using generalized additive models for location, scale and shape. Merder et al. Nature Reviews Methods Primers 2026 DOI: 10.1038/s43586-026-00498-z
