Beta Regression for Interval-Censored Scale-Derived Outcomes

Maximum-likelihood estimation of beta regression models for responses derived from bounded rating scales. Observations are treated as interval-censored on (0, 1) after a scale-to-unit transformation, and the likelihood is built from the difference of the beta CDF at the interval endpoints. The complete likelihood supports mixed censoring types: uncensored, left-censored, right-censored, and interval-censored observations. Both fixed- and variable-dispersion submodels are supported, with flexible link functions for the mean and precision components. A compiled C++ backend (via 'Rcpp' and 'RcppArmadillo') provides numerically stable, high-performance log-likelihood evaluation. Standard S3 methods (print(), summary(), coef(), fitted(), residuals(), predict(), plot(), confint(), vcov(), logLik(), AIC(), BIC(), simulate()) are available for fitted objects.

betaregscale

betaregscale fits regression models for scores on a bounded rating scale $0, 1, \ldots, K$ (pain NRS-11 or NRS-101, Likert items, quality ratings). Each score is the coarsened reading of a latent beta variable on $(0, 1)$, written in mean–dispersion form: mean $\mu$, dispersion $\sigma = 1/(1 + \phi) \in (0, 1)$ with $\phi$ the beta precision, and $\mathrm{Var}(Y) = \mu(1 - \mu)\sigma$. Both can depend on covariates through link functions. The latent scale is cut at cutpoints anchored on the score labels, so each score is an interval-censored observation and the likelihood is a probability mass function over the scores $0, \ldots, K$. The scale borders need no ad hoc squeezing: score 0 is left-censored and score $K$ right-censored. Fixed-effects models are fitted with brs() and mixed models with Gaussian random effects with brsmm(); the log-likelihood and its derivatives are compiled C++.

Installation

# Released version from CRAN
install.packages("betaregscale")

# Development version from GitHub
# install.packages("remotes")
remotes::install_github("evandeilton/betaregscale")

Example

Simulated NRS-11 scores (0 to 10) for 200 patients, with a continuous covariate and a treatment indicator:

library(betaregscale)

set.seed(1)
d <- data.frame(x1 = rnorm(200), trt = rbinom(200, 1, 0.5))
nrs <- brs_sim(~ x1 + trt, data = d, beta = c(-0.4, 0.5, -0.6),
               phi = qlogis(0.15), ncuts = 10)

# Mean with logit link, constant dispersion with logit link
fit <- brs(y ~ x1 + trt, data = nrs)
summary(fit)

# The fitted model is a pmf over the scores 0..10
round(head(brs_predict_scoreprob(fit), 3), 3)

# Expected score by arm at x1 = 0, and the difference between arms,
# with delta-method standard errors
nd <- data.frame(x1 = 0, trt = c(0, 1))
brs_marginal(fit, nd, type = "expected_score")
brs_marginal(fit, nd, type = "expected_score", contrast = c(2, 1))

# Diagnostic plots of randomized quantile residuals
plot(fit)

Features

Interval maps interval = "mid", "right" or "left", and estimable boundary cutpoints (cutpoints = "boundary") for scores that pile up at the ends of the scale, with a likelihood-ratio test against the fixed map.
Variable dispersion with a two-part formula y ~ x | z, and three parameterizations of the beta law (repar).
Mixed models (brsmm()) with random intercepts and slopes, fitted by Laplace approximation, adaptive Gauss–Hermite quadrature or quasi-Monte Carlo, with chi-bar-square likelihood-ratio tests for variance components.
Score-scale predictions: score probabilities (brs_predict_scoreprob()), and score probabilities, expected scores, exceedance probabilities and contrasts with standard errors (brs_marginal()), population-averaged for mixed models.
Diagnostics and validation: randomized quantile and cell deviance residuals, parametric bootstrap, k-fold and group-level cross-validation, and simulation (brs_sim(), brsmm_sim(), simulate()).

Documentation

Introduction: the model, interval maps, the likelihood and a first fit.
Mixed models: brsmm(), integration methods, random effects and marginal predictions.
Analyst tools: predictions, contrasts, residuals, bootstrap, cross-validation, tables and plots.
Advanced workflows: variable dispersion, boundary cutpoints and model comparison.

Function reference and changelog: https://evandeilton.github.io/betaregscale/. Bug reports and questions: https://github.com/evandeilton/betaregscale/issues.

Citation

citation("betaregscale") gives the reference for the package. The model is described in

Lopes, J. E. (2023). Modelos de regressão beta para dados de escala. Master's dissertation, Universidade Federal do Paraná, Curitiba. https://acervodigital.ufpr.br/xmlui/handle/1884/86624

Other references:

Bayer, F. M. (2011). Modelagem e inferência em regressão beta. PhD thesis, Universidade Federal de Pernambuco.
Ferrari, S. L. P., and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815. https://doi.org/10.1080/0266476042000214501