Parametric Modal Regression with Right Censoring.

ModalCens: Parametric Modal Regression with Right Censoring

ModalCens is an R package for fitting parametric modal regression models to continuous positive random variables within the exponential family. While traditional Generalized Linear Models (GLMs) target the conditional mean, ModalCens directly models the conditional mode $M_i$, providing a more robust and meaningful measure of central tendency for asymmetric and heavy-tailed distributions — particularly under right censoring.

The methodology is grounded in the framework of modal linear regression introduced by Yao & Li (2014), with connections to recent advances surveyed in Xiang, Yao & Cui (2026), extended here to parametric exponential family distributions under right censoring.

Key Features

• Mode-Link Framework: Analytical reparameterization of exponential family densities to link the conditional mode $M_i$ directly to a linear predictor via $g(M_i) = \mathbf{X}_i^\top \boldsymbol{\gamma}$.

• Right-Censoring Support: Right-censored observations (cens = 1) contribute their survival probability $S(y_i)$ to the likelihood, while fully observed values (cens = 0) contribute their density $f(y_i)$.

• Strict Data Validation: The function stops with an informative error if cens or any variable in data contains missing values (NA). No imputation is performed — the user is responsible for supplying a complete dataset.

• Supported Distributions:

  Family	Link	Domain	Typical Use
  "gamma"	Log	$(0, \infty)$	Skewed positive data
  "beta"	Logit	$(0, 1)$	Proportions and rates
  "weibull"	Log	$(0, \infty)$	Survival and reliability
  "invgauss"	Log	$(0, \infty)$	Highly skewed data ($\lambda > 3M_i$)

• Asymptotic Inference: Standard errors derived from the observed Fisher information matrix (inverse Hessian at the MLE).

• Randomized Quantile Residuals: Implements the Dunn–Smyth method for model diagnostics, valid under any continuous distribution and under censoring.

Installation

# Development version from GitHub
# install.packages("devtools")
devtools::install_github("chedgala/ModalCens")

Quick Start

This example uses the built-in trees dataset with simulated censoring for illustration purposes only. In practice, supply your own dataset with a real censoring indicator.

library(ModalCens)

# 1. Prepare data: scale Volume to (0, 1) for multi-family compatibility
data(trees)
y_scaled <- (trees$Volume - min(trees$Volume) + 1) /
            (max(trees$Volume) - min(trees$Volume) + 2)

# 2. Simulate a censoring indicator (for illustration only)
#    cens = 1: right-censored  |  cens = 0: fully observed
set.seed(2026)
cut_point <- quantile(y_scaled, 0.85)
cens_ind  <- ifelse(y_scaled > cut_point, 1, 0)
y_obs     <- pmin(y_scaled, cut_point)

df <- data.frame(
  y      = y_obs,
  cens   = cens_ind,
  Girth  = trees$Girth,
  Height = trees$Height
)

# 3. Fit a Gamma modal regression
fit_gamma <- modal_cens(y ~ Girth + Height, data = df,
                        cens = df$cens, family = "gamma")

# 4. Summary and diagnostics
summary(fit_gamma)
plot(fit_gamma)

Multi-Family Comparison

families   <- c("gamma", "beta", "weibull", "invgauss")
models     <- list()
aic_values <- numeric()

for (f in families) {
  mod <- try(
    modal_cens(y ~ Girth + Height, data = df, cens = df$cens, family = f),
    silent = TRUE
  )
  if (!inherits(mod, "try-error")) {
    models[[f]]     <- mod
    aic_values[[f]] <- AIC(mod)
  }
}

best_family <- names(which.min(aic_values))
cat("Best model:", best_family, "| AIC =", min(aic_values), "\n")

summary(models[[best_family]])
plot(models[[best_family]])

Methodology

Mode Reparameterization

The package reparameterizes exponential family densities of the form

$$f(y_i \mid \theta_i, \phi) = \exp!\left{\frac{y_i \theta_i - b(\theta_i)}{a(\phi)} + c(y_i, \phi)\right}$$

by solving the first-order condition $\left.\dfrac{\partial \log f(y_i)}{\partial y_i}\right|_{y_i = M_i} = 0$, which expresses the canonical parameter $\theta_i$ as an explicit function of the conditional mode $M_i$ and the dispersion $\phi$. This yields the mode-link models:

Censored Log-Likelihood

For a sample of $n$ observations with censoring indicator $\delta_i \in {0, 1}$ (where $\delta_i = 1$ denotes right-censored):

$$\ell(\boldsymbol{\gamma}, \phi) = \sum_{i=1}^{n} \left[(1 - \delta_i) \log f(y_i \mid M_i, \phi) + \delta_i \log S(y_i \mid M_i, \phi)\right]$$

where $S(\cdot) = 1 - F(\cdot)$ is the survival function. Maximization is performed via the BFGS algorithm.

Randomized Quantile Residuals

For each observation, define $u_i = F(y_i \mid \hat{M}_i, \hat{\phi})$. For right-censored observations ($\delta_i = 1$), $u_i$ is drawn uniformly on $[F(y_i), 1]$. The residual is $r_i = \Phi^{-1}(u_i)$, which follows $\mathcal{N}(0,1)$ under the correct model (Dunn & Smyth, 1996).

Main Function

modal_cens(formula, data, cens, family = "gamma")

The function stops with an error if any NA is detected in cens or in the variables referenced by formula. No imputation is performed.

The function returns an object of class "ModalCens" with methods for summary(), plot(), AIC(), BIC(), logLik(), coef(), vcov(), and residuals().

Authors

• Christian E. Galarza — ESPOL, Ecuador · [email protected]
• Víctor H. Lachos — University of Connecticut · [email protected]

References

• Yao, W. & Li, L. (2014). A new regression model: modal linear regression. Scandinavian Journal of Statistics, 41(3), 656–671. https://doi.org/10.1111/sjos.12054

• Xiang, S., Yao, W. & Cui, X. (2026). Advances in modal regression: from theoretical foundations to practical implementations. Wiley Interdisciplinary Reviews: Computational Statistics, e70014. https://doi.org/10.1002/wics.70014

• Dunn, P. K. & Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5(3), 236–244.