Fit a mixed-effects beta interval regression model

Fits a beta interval-censored mixed model with Gaussian random intercepts using marginal maximum likelihood. The current implementation supports random = ~ 1 | group and uses a Laplace approximation for group-specific integrals.

Usage

brsmm(
  formula,
  random = ~1 | id,
  data,
  link = "logit",
  link_phi = "logit",
  repar = 2L,
  ncuts = 100L,
  lim = 0.5,
  int_method = c("laplace", "aghq", "qmc"),
  n_points = 11L,
  qmc_points = 1024L,
  start = NULL,
  method = c("BFGS", "L-BFGS-B"),
  hessian_method = c("numDeriv", "optim"),
  control = list(maxit = 2000L),
  seed = NULL
)

Arguments

formula: Model formula. Supports one- or two-part formulas: y ~ x1 + x2 or y ~ x1 + x2 | z1 + z2.
random: Random-effects specification. The supported format is ~ 1 | group.
data: Data frame.
link: Mean link function.
link_phi: Precision link function.
repar: Beta reparameterization code (0, 1, 2).
ncuts: Number of categories on the original scale.
lim: Half-width used to construct interval endpoints.
int_method: Integration method. Currently, only "laplace" is implemented.
n_points: Reserved for future AGHQ support.
qmc_points: Reserved for future QMC support.
start: Optional numeric vector of starting values (beta, gamma, log_sigma_b).
method: Optimizer passed to optim.
hessian_method: "numDeriv" (default) or "optim".
control: Control list for optim.
seed: Optional seed used by integration methods that depend on randomized points (reserved for future use).

Value

An object of class "brsmm".

Details

The conditional contribution for each observation follows the same mixed censoring likelihood used by brs:

\(\delta=0\): exact contribution via beta density,
\(\delta=1\): left-censored contribution via beta CDF,
\(\delta=2\): right-censored contribution via survival CDF,
\(\delta=3\): interval contribution via CDF difference.

For group \(i\), the random intercept \(b_i \sim N(0, \sigma_b^2)\) is integrated out numerically via Laplace approximation.

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.

Examples

# \donttest{
set.seed(123)
g <- 15
ni <- 8
id <- factor(rep(seq_len(g), each = ni))
n <- length(id)
x1 <- rnorm(n)
b <- rnorm(g, sd = 0.5)
eta_mu <- 0.2 + 0.6 * x1 + b[as.integer(id)]
mu <- plogis(eta_mu)
phi <- plogis(-0.2 + 0.2 * x1)
shp <- brs_repar(mu = mu, phi = phi, repar = 2)
y <- round(stats::rbeta(n, shp$shape1, shp$shape2) * 100)
d <- data.frame(y = y, x1 = x1, id = id)

fit_mm <- brsmm(y ~ x1, random = ~ 1 | id, data = d, repar = 2)
fit_mm
#> 
#> Call:
#> brsmm(formula = y ~ x1, random = ~1 | id, data = d, repar = 2)
#> 
#> Mixed beta interval model (Laplace)
#> Observations: 120  | Groups: 15 
#> Log-likelihood: -499.3888 
#> Random SD: 0.2206 
#> Convergence code: 0 
# }