ModalCens: Parametric Modal Regression with Right Censoring

ModalCens is an R package for fitting parametric modal regression models to continuous positive random variables. 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). It extends beyond traditional exponential family distributions to include highly flexible models such as the Log-Logistic and Birnbaum-Saunders distributions, as detailed in our preprint Galarza & Lachos (2026).

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 (\(\lambda > 3M_i\)) *
  "lognormal"	Log	\((0, \infty)\)	Multiplicative / log-symmetric data
  "loglogistic"	Log	\((0, \infty)\)	Non-monotonic hazard rates, heavy tails
  "bisa"	Log	\((0, \infty)\)	Fatigue life, wear-out processes

  * Note on Inverse Gaussian: The condition \(\lambda > 3M_i\) required to ensure a valid mode parameterization is a strong parametric restriction. This can make the Inverse Gaussian distribution less plausible in many routine modeling contexts compared to the other families available.

• 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: Multi-Family Comparison

Censoring is simulated for didactic purposes. Each family is fitted on its natural domain: long-tail families (Gamma, Weibull, InvGauss) on the original scale, and Beta on the \((0,1)\) scale.

library(ModalCens)
data(trees)
set.seed(007)

# Simulate 15% right-censoring (cens = 1: censored | cens = 0: observed)
q85      <- quantile(trees$Volume, 0.85)
cens_ind <- as.integer(trees$Volume > q85)

# (0,1) rescaling for Beta only
ys     <- (trees$Volume - min(trees$Volume) + 1) / (max(trees$Volume) - min(trees$Volume) + 2)
q85b   <- quantile(ys, 0.85)
cens_b <- as.integer(ys > q85b)

# Original scale for long-tail families; (0,1) scale for Beta
df_orig <- data.frame(y = pmin(trees$Volume, q85), cens = cens_ind,
                      Girth = trees$Girth, Height = trees$Height)
df_beta <- data.frame(y = pmin(ys, q85b), cens = cens_b,
                      Girth = trees$Girth, Height = trees$Height)

# Fit all families
datasets <- list(gamma = df_orig, weibull = df_orig,
                 invgauss = df_orig, lognormal = df_orig, beta = df_beta,
                 loglogistic = df_orig, bisa = df_orig)
models <- list(); aic_values <- list()

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

# Select best model by AIC
best <- names(which.min(aic_values))
cat("Best family:", best, "| AIC =", round(aic_values[[best]], 3), "\n")

# Full analysis of the winner
fit <- models[[best]]
summary(fit)
plot(fit)
confint(fit)

Methodology

Mode Reparameterization

The package reparameterizes exponential family densities of the form

\[ f(y_i \mid \theta_i, \phi) = \exp\left\lbrace\frac{y_i \theta_i - b(\theta_i)}{a(\phi)} + c(y_i, \phi)\right\rbrace \]

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 — Escuela Superior Politécnica del Litoral, Ecuador · chedgala@espol.edu.ec
• Víctor H. Lachos — University of Connecticut, USA · hlachos@uconn.edu

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

• Galarza, C. E., & Lachos, V. H. (2026). Parametric Modal Regression for Positive Distributions. arXiv preprint, arXiv:2603.07099. doi:10.48550/arXiv.2603.07099

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