A Julia package for distributional regression models (DRMs): model how predictors change a response's average, variability, or probability of zero. Use DRModels.jl directly in Julia; no R installation is needed.
An ordinary regression often asks how the average response changes with a predictor. Distributional regression also asks whether other features of the response change, such as its spread or probability of being zero. The letters DR in DRModels refer to distributional regression.
For example, warmer conditions might change both average body size and how much individuals differ in size. In psychology, a treatment might change both average reaction time and its variability. These are separate questions, and a change in one does not imply a change in the other.
DRModels.jl is a standalone Julia package for fitting these models to one or two responses. It is general-purpose; the tutorials draw mainly on ecology, evolution, environmental science, and evidence synthesis.
Experimental software
Check the supported models and current limits before choosing a model, and check the fitted model before interpreting it. Validation is specific to the model and method used.
Start with the scientific question, then choose one complete route:
Does a response's average or variability change with predictors? Begin with Getting started. It fits a model for a continuous response, with separate formulas for the average and the variation left after accounting for that average.
Are observations related through a phylogeny? Begin with Phylogenetic structured effects. It shows how to supply a tree, account for shared ancestry, and interpret the results.
What is the average effect across studies, and why do effects differ? Begin with Mean effects and residual heterogeneity. This example combines study estimates with their known sampling variances.
For another response type or a more specialised structure, use What can I fit today? after one of these examples.
This example creates a continuous response whose average and spread both increase with a predictor. Think of x as temperature relative to its average and y as a centred body-size measurement. These are simulated data, so the example illustrates the method rather than a biological finding.
After installing DRModels.jl, run:
using DRModels, Random
Random.seed!(20260610)
x = randn(400)
y = 1.0 .+ 0.5 .* x .+ exp.(-0.4 .+ 0.3 .* x) .* randn(400)
dat = (; y, x)
# Separate formulas for the average response and its remaining spread:
fit = drm(bf(@formula(y ~ x), @formula(sigma ~ x)), Gaussian(); data = dat)
is_converged(fit) # did the fitting algorithm finish successfully?
coef(fit, :mu) # effects on the average response
exp(coef(fit, :sigma)[2]) # ratio of standard deviations per unit increase in xThe first formula describes the average response. The second describes its remaining spread, measured here by the standard deviation (sigma). In these simulated data, the average rises by 0.5 per unit of x, while the standard deviation is multiplied by exp(0.3), about 1.35. Fitted values will differ because the data include random variation.
The first-model tutorial explains the formulas, how to read estimates and confidence intervals, and what to check before reporting results. This type of model is often called a location–scale model: location describes the average and scale describes the spread.
Use supported models and current limits to check which combinations have been tested. After fitting, use Checking and using fitted models to decide what to inspect and report. A method being available does not guarantee that it will work well for every data set. In particular, the accuracy of confidence intervals depends on the model, the data, and the method used to calculate them.
The R package drmTMB also fits distributional regression models. DRModels.jl uses related formula conventions, but you can install and use it entirely within Julia. The Rosetta page compares the two syntaxes directly.
For R users who want to call Julia from R, the optional Coming from R explains the optional engine = "julia" in drmTMB. That bridge is experimental; the two packages do not support every model in the same way.
DRModels.jl is independently written Julia software, available under the MIT license. For models of many responses together, see GLLVModels.jl.