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Robust continuous responses ​

Status — Stable

Mirrors drmTMB's Robust continuous responses. In DRM.jl today: the Student-t family Student() — a formula per parameter for the location μ, scale σ, and degrees of freedom ν. Fixed effects, maximum likelihood.

When a few observations sit far from the trend, a Gaussian fit chases them — the mean tilts and σ inflates. The Student-t family gives the residuals heavier tails, so outliers are downweighted instead of dominating. The shape parameter ν (degrees of freedom) controls how heavy: small ν is very robust, and ν → ∞ returns to Gaussian.

Fitting a Student-t model ​

Give each parameter its own formula, exactly like the Gaussian model — nu ~ 1 estimates a single degrees-of-freedom value:

julia
using DRM, Random
using Distributions: TDist
Random.seed!(20260614)

n = 2000
x = randn(n)
# location-scale t errors (heavy-tailed): true ν = 5
y = 0.5 .+ 0.7 .* x .+ 0.8 .* rand(TDist(5.0), n)
dat = (; y, x)

fit = drm(bf(@formula(y ~ x), @formula(sigma ~ 1), @formula(nu ~ 1)), Student(); data = dat)
coef(fit, :mu)              # location coefficients
2-element Vector{Float64}:
 0.501075755292613
 0.7116232787930291

σ is on the log scale, so exp returns it. ν uses a logm2 link, ν = 2 + exp(η), which keeps ν > 2 so the variance stays finite — read it back with 2 + exp(...), not exp(...):

julia
exp(coef(fit, :sigma)[1])       # residual scale
2 + exp(coef(fit, :nu)[1])      # degrees of freedom, ν = 2 + exp(η)  (≈ 5 ⇒ heavy tails)
5.336378088942376

A small ν (say < 10) is the signal that robustness is earning its keep; a large ν says the data are effectively Gaussian and you could simplify.

Scale, not variance

For the t the residual variance is σ² · ν/(ν−2) (and is undefined for ν ≤ 2). σ here is the scale parameter, not the SD — quote it as such when comparing to a Gaussian fit.

See also ​