Beta Regression for Interval-Censored Scale-Derived Outcomes • betaregscale

Beta Regression for Interval-Censored Scale-Derived Outcomes

The betaregscale package provides maximum-likelihood estimation of beta regression models for responses derived from bounded rating scales (e.g., NRS-11, NRS-21, NRS-101, Likert scales). Observations are treated as interval-censored on (0, 1) after a scale-to-unit transformation, and the likelihood is built from the beta CDF evaluated at the interval endpoints.

The package is designed for situations where the recorded score carries measurement uncertainty inherent to the instrument. For example, a pain score of 6 on a (0-10) NRS scale is interpreted as lying in the interval [5.5, 6.5] after rescaling to (0, 1). The complete likelihood (Lopes, 2024, Eq. 2.24) supports mixed censoring types: uncensored, left-censored, right-censored, and interval-censored within the same dataset.

Key features

Mixed censoring support: the complete likelihood handles four censoring types simultaneously: exact observations ( $\delta=0$ ), left-censored ( $\delta=1$ ), right-censored ( $\delta=2$ ), and interval-censored ( $\delta=3$ ).
Fixed and variable dispersion: model a scalar $\phi$ or let it depend on covariates via a second linear predictor (y ~ x1 + x2 | z1).
High-performance C++ backend: the log-likelihood and analytical gradient are compiled via Rcpp/RcppArmadillo for fast, numerically stable estimation.
Flexible link functions: logit, probit, cauchit, cloglog for the mean; logit, log, identity, sqrt, inverse, and others for the dispersion.
Three reparameterizations: direct (0), Ferrari–Cribari-Neto precision (1), and mean–variance (2).
betareg-style S3 interface: print, summary, coef, vcov, logLik, AIC, BIC, nobs, formula, model.matrix, fitted, residuals, predict, confint, and plot methods. The coef() and vcov() methods accept model = c("full", "mean", "precision").
Diagnostic plots: six residual diagnostic panels with both base R and ggplot2 backends, including a half-normal envelope plot.
Censoring summary: brs_cens() provides visual and tabular summaries of the censoring structure.
Simulation toolkit: brs_sim() supports both fixed- and variable-dispersion Monte Carlo studies via one- or two-part formulas.

Installation

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

Quick start

Fixed dispersion model

library(betaregscale)

# Simulate interval-censored data from a fixed-dispersion model
set.seed(42)
n <- 200
dat <- data.frame(x1 = rnorm(n), x2 = rnorm(n))
sim <- brs_sim(
  formula = ~ x1 + x2, data = dat,
  beta = c(0.3, -0.6, 0.4), phi = 1/10,
  link = "logit", link_phi = "logit",
  ncuts = 100, repar = 2
)
head(sim)

# Fit the model
fit <- brs(y ~ x1 + x2, data = sim, repar = 2)
summary(fit)

Variable dispersion model

# Simulate data with covariate-dependent dispersion
set.seed(2222)
n <- 200
dat <- data.frame(
  x1 = rnorm(n), x2 = rnorm(n),
  z1 = rnorm(n), z2 = rnorm(n)
)
sim_z <- brs_sim(
  formula = ~ x1 + x2 | z1,
  data = dat,
  beta = c(0.2, -0.6, 0.2),
  zeta = c(0.2, -0.8),
  link = "logit", link_phi = "logit",
  ncuts = 100, repar = 2
)

# Fit variable-dispersion model (pipe notation)
fit_z <- brs(y ~ x1 + x2 | z1, data = sim_z, repar = 2)
summary(fit_z)

Comparing link functions

links <- c("logit", "probit", "cauchit", "cloglog")
fits <- lapply(setNames(links, links), function(lnk) {
  brs(y ~ x1 + x2, data = sim, link = lnk, repar = 2)
})

# Goodness-of-fit comparison
do.call(rbind, lapply(fits, brs_gof))

S3 methods

# Coefficients by submodel
coef(fit)                          # full parameter vector
coef(fit, model = "mean")         # mean submodel only
coef(fit, model = "precision")    # precision submodel only

# Variance-covariance matrix
vcov(fit, model = "mean")

# Wald confidence intervals
confint(fit)

# Predictions
predict(fit, type = "response")    # fitted means
predict(fit, type = "precision")   # fitted dispersion
predict(fit, type = "variance")    # conditional variance
predict(fit, type = "quantile", at = c(0.25, 0.5, 0.75))

# Residuals
residuals(fit, type = "pearson")
residuals(fit, type = "rqr")       # randomized quantile residuals

# Diagnostic plots (base R)
plot(fit)

# Diagnostic plots (ggplot2, if installed)
plot(fit, gg = TRUE)

# Censoring structure summary
brs_cens(fit)

Model details

Complete likelihood

The complete log-likelihood for mixed censoring (Lopes, 2024, Eq. 2.24) combines four observation types:

$\ell(\boldsymbol{\theta}) = \sum_{i:\,\delta_i=0} \log f(y_i) + \sum_{i:\,\delta_i=1} \log F(u_i) + \sum_{i:\,\delta_i=2} \log [1 - F(l_i)] + \sum_{i:\,\delta_i=3} \log [F(u_i) - F(l_i)]$

where $f(\cdot)$ and $F(\cdot)$ are the beta density and CDF, $[l_i, u_i]$ are the interval endpoints, and $\delta_i$ indicates the censoring type.

Reparameterizations

Code	Name	Shape parameters
0	Direct	$a = \mu,\; b = \phi$
1	Precision (Ferrari & Cribari-Neto)	$a = \mu\phi,\; b = (1-\mu)\phi$
2	Mean–variance (Bayer)	$a = \mu(1-\phi)/\phi,\; b = (1-\mu)(1-\phi)/\phi$

Interval construction

Scale observations are mapped to (0, 1) with midpoint uncertainty intervals:

$y_t = y/K, \quad \text{interval } [y_t - h/K,\; y_t + h/K]$

where $K$ is the number of scale categories (ncuts) and $h$ is the half-width (lim, default 0.5).

References

Lopes, J. E. (2024). Beta Regression for Interval-Censored Scale-Derived Outcomes. MSc Dissertation, PPGMNE/UFPR.
Ferrari, S. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815.

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