DRM.jl

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Proportions and success rates ​

Status — Stable (beta + beta-binomial + zero-one-inflated beta)

Mirrors drmTMB's Proportions and success rates. In DRM.jl today: the beta family Beta() for continuous proportions in (0,1), the beta-binomial family BetaBinomial() for counts of successes out of known trials (cbind(successes, failures) ~ …), and the zero-one-inflated beta ZeroOneBeta() for proportions on [0,1].

Proportions, rates, and probabilities live on (0,1) — bounded at both ends, so a Gaussian model can predict impossible values. The beta family models the mean on the logit scale and a precision separately.

A beta regression ​

The mean uses a logit link; the sigma slot carries the dispersion, mapped to the beta precision φ = 1/σ² (drmTMB's convention) — larger φ means tighter spread around the mean:

julia
using DRM, Random
import Distributions          # `Beta` below is DRM's family; qualify the distribution
Random.seed!(20260617)

n = 3000
x = randn(n)
φ = 15.0
μ = 1 ./ (1 .+ exp.(-(0.3 .+ 0.8 .* x)))         # logit-linear mean
y = Float64.([rand(Distributions.Beta(μi * φ, (1 - μi) * φ)) for μi in μ])
dat = (; y, x)

fit = drm(bf(@formula(y ~ x), @formula(sigma ~ 1)), Beta(); data = dat)
coef(fit, :mu)              # coefficients on the logit-mean
2-element Vector{Float64}:
 0.29931081535354864
 0.8067298348726738

The slope is a log odds ratio on the mean proportion; recover the precision from the sigma coefficient:

julia
exp(-2 * coef(fit, :sigma)[1])     # precision φ = 1/σ² (≈ 15)
14.481589827982711

Fitted values come back as proportions on (0,1):

julia
extrema(fitted(fit))
(0.0720714426570787, 0.952919062202302)

Precision, not SD

The sigma slot is a dispersion handle: φ = 1/σ². A large φ (small σ) means observations hug the mean; a small φ means they spread toward 0 and 1.

Successes out of trials: the beta-binomial family ​

When the data are counts of successes out of a known number of trials (not a continuous proportion), keep the denominator: give a two-column response cbind(successes, failures) — exactly as drmTMB — and use BetaBinomial(). It adds extra-binomial overdispersion through the same sigma precision (φ = 1/σ²):

julia
Random.seed!(20260623)
N = 3000; ntr = fill(25, N)
φbb = 12.0
μbb = 1 ./ (1 .+ exp.(-(0.2 .+ 0.7 .* x)))
succ = [rand(Distributions.BetaBinomial(ntr[i], μbb[i] * φbb, (1 - μbb[i]) * φbb)) for i in 1:N]
datbb = (; s = Float64.(succ), fail = Float64.(ntr .- succ), x)

fitbb = drm(bf(@formula(cbind(s, fail) ~ x), @formula(sigma ~ 1)), BetaBinomial(); data = datbb)
exp(-2 * coef(fitbb, :sigma)[1])     # recovered precision φ (≈ 12)
11.908086999428532

coef(fitbb, :mu) is the success probability on the logit scale; ordinary binomial is the no-overdispersion limit (φ → ∞).

Proportions that hit 0 or 1: zero-one-inflated beta ​

A plain beta lives on the open (0,1) — it cannot represent an exact 0 or 1. When the data pile up at the boundaries (none-detected, all-detected), ZeroOneBeta() adds two logit parameters: zoi = P(the value is a boundary) and coi = P(it is 1, given a boundary). The interior is the usual Beta(μ, φ):

julia
Random.seed!(20260624)
zoi = 0.25; coi = 0.4; φz = 12.0
μz = 1 ./ (1 .+ exp.(-(0.2 .+ 0.6 .* x)))
yz = map(μi -> rand() < zoi ? (rand() < coi ? 1.0 : 0.0) :
                rand(Distributions.Beta(μi * φz, (1 - μi) * φz)), μz)
datz = (; y = yz, x)

fitz = drm(bf(@formula(y ~ x), @formula(sigma ~ 1), @formula(zoi ~ 1), @formula(coi ~ 1)),
    ZeroOneBeta(); data = datz)
(zoi = 1 / (1 + exp(-coef(fitz, :zoi)[1])),     # recovered P(boundary) ≈ 0.25
 coi = 1 / (1 + exp(-coef(fitz, :coi)[1])))     # recovered P(1 | boundary) ≈ 0.40
(zoi = 0.24300000000000027, coi = 0.41289437585733885)

fitted(fitz) returns the unconditional mean (1 - zoi)·μ + zoi·coi.

See also ​