Package {BetaDanish}

BetaDanish: The Beta-Danish Distribution for Lifetime Data Analysis

Description

Implements the four-parameter Beta-Danish distribution and its three-parameter submodel for survival and reliability analysis, based on Ahmad and Danish (2025) doi:10.2478/jamsi-2025-0010. Provides functions for density, distribution, quantile, hazard, and random generation. Includes maximum likelihood estimation for complete and right-censored data, goodness-of-fit assessment, comparison with standard lifetime distributions, and publication-quality visualizations. Advanced modules support Accelerated Failure Time (AFT) regression, mixture and promotion-time cure models, and competing risks analysis.

Details

The 'BetaDanish' package provides a comprehensive suite of tools for survival and reliability analysis using the Beta-Danish distribution. It includes core mathematical functions (density, distribution, quantile, random generation, and hazard), robust Maximum Likelihood Estimation (MLE) for both complete and right-censored data, and advanced modeling capabilities including Accelerated Failure Time (AFT) regression, cure-rate models, and competing risks analysis.

Author(s)

Maintainer: Bilal Ahmad bilalahmad.imcbh9@gmail.com

Authors:

• Bilal Ahmad bilalahmad.imcbh9@gmail.com

• Muhammad Yameen Danish yameen.danish@aiou.edu.pk

References

Ahmad, B. (2026). Modeling Diverse Survival Patterns: The Development and Characterization of a New Four-Parameter Lifetime Distribution (Ph.D. Thesis). Allama Iqbal Open University, Islamabad.

See Also

Useful links:

• https://github.com/bilal-aiou/BetaDanish

• Report bugs at https://github.com/bilal-aiou/BetaDanish/issues

The Beta-Danish Distribution

Description

Density, distribution function, quantile function, hazard function, and random generation for the four-parameter Beta-Danish distribution.

Usage

dbetadanish(x, a, b, c, k, log = FALSE)

pbetadanish(q, a, b, c, k, lower.tail = TRUE, log.p = FALSE)

qbetadanish(p, a, b, c, k, lower.tail = TRUE, log.p = FALSE)

rbetadanish(n, a, b, c, k)

hbetadanish(x, a, b, c, k, log = FALSE)

Arguments

Details

The Beta-Danish distribution is a highly flexible lifetime distribution capable of modeling decreasing, increasing, unimodal, and bathtub-shaped hazard rates.

Value

'dbetadanish' gives the density, 'pbetadanish' gives the distribution function, 'qbetadanish' gives the quantile function, 'hbetadanish' gives the hazard function, and 'rbetadanish' generates random deviates.

References

Ahmad, B., & Danish, M. Y. (2026). Development and Characterization of a Flexible Three-Parameter Lifetime Distribution.

Examples

# Density
dbetadanish(x = 2, a = 1.5, b = 2, c = 3, k = 0.5)

# CDF
pbetadanish(q = 2, a = 1.5, b = 2, c = 3, k = 0.5)

# Hazard
hbetadanish(x = 2, a = 1.5, b = 2, c = 3, k = 0.5)

# Random generation
rbetadanish(n = 10, a = 1.5, b = 2, c = 3, k = 0.5)

Aarset Device Failure Times

Description

Times to failure of 50 devices, exhibiting a classic bathtub-shaped hazard rate. This is a standard benchmark dataset in reliability engineering.

Usage

aarset

Format

A data frame with 50 rows and 2 columns:

time

Failure time

status

Event indicator (1 = event occurred)

Source

Aarset, M. V. (1987). How to Identify a Bathtub Hazard Rate. IEEE Transactions on Reliability, R-36(1), 106-108.

Examples

data(aarset)

fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = aarset)
plot(fit, type = "hazard")

Comprehensive Beta-Danish Analysis Pipeline

Description

Runs a complete end-to-end analysis: reads data, fits the 4-parameter and 3-parameter models, compares them, benchmarks against standard distributions, and generates diagnostic plots.

Usage

analyze_betadanish(file, time_col, status_col = NULL)

Arguments

Value

Invisibly returns a list containing the fitted full model and submodel objects. The function is mainly called for its side effects of printing an analysis report and producing diagnostic plots.

Brain Cancer Survival Data

Description

A comprehensive dataset of 500 brain cancer patients, including survival times, censoring status, and multiple clinical covariates. This dataset was used to demonstrate Accelerated Failure Time (AFT) regression and Cure-Rate models using the Beta-Danish distribution.

Usage

brain_cancer

Format

A data frame with 500 rows and 16 columns:

ID

Patient identifier

Gender

Patient gender (1 = Male, 0 = Female)

Age

Age group (1 = Young, 2 = Middle, 3 = Old)

Area

Geographic area (1 = Urban, 0 = Rural)

FH

Family history of cancer (1 = Yes, 0 = No)

CMH

Comorbid history (1 = Yes, 0 = No)

Grade

Tumor grade (1 = I/II, 2 = III, 3 = IV)

Surgery

Surgical intervention (1 = Yes, 0 = No)

Radiotherapy

Radiotherapy treatment (1 = Yes, 0 = No)

Chemotherapy

Chemotherapy treatment (1 = Yes, 0 = No)

Treatment

Treatment type

Morphology

Tumor morphology

Survstatus

Survival status (1 = Event/Death, 0 = Censored)

Survtime

Survival time in months

Types

Tumor types classification

Morphology1

Alternative morphology classification

Source

Atomic Energy Cancer Hospital (NORI), Islamabad, Pakistan.

Examples

data(brain_cancer)

# Fit an AFT model using the brain cancer data
fit <- fit_bd_aft(survival::Surv(Survtime, Survstatus) ~ Age + Grade + Surgery,
                  data = brain_cancer, n_starts = 2)
summary(fit)

Breaking Stress of Carbon Fibres

Description

Breaking stress (in Gba) of 100 carbon fibre specimens. This dataset exhibits a unimodal (increasing-then-decreasing) hazard pattern that classical distributions like the Weibull cannot adequately capture.

Usage

carbon_fibres

Format

A data frame with 100 rows and 2 columns:

time

Breaking stress in Gba

status

Event indicator (1 = event occurred)

Source

Nichols, M. D., & Padgett, W. J. (2006). A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International, 22(2), 141-151.

Examples

data(carbon_fibres)

fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = carbon_fibres)

Compute Cumulative Incidence Function (CIF)

Description

Computes the CIF for a specific cause from a fitted Beta-Danish competing risks model using numerical integration.

Usage

cif_betadanish(fit, tvec, cause_idx)

Arguments

Value

A numeric vector of CIF probabilities corresponding to 'tvec'.

Extract Coefficients

Description

Extract Coefficients

Usage

## S3 method for class 'betadanish'
coef(object, ...)

Arguments

Value

A named numeric vector of maximum likelihood parameter estimates.

Compare Beta-Danish with Standard Distributions

Description

Fits standard lifetime distributions (Weibull, Log-Normal, Log-Logistic, Gamma, Exponential) using the 'flexsurv' package and compares them to the Beta-Danish fit.

Usage

compare_distributions(object)

Arguments

Value

A ranked data frame of model comparisons.

Compare Nested Beta-Danish Models

Description

Performs a Likelihood Ratio Test (LRT) between the 4-parameter full model and the 3-parameter submodel.

Usage

compare_models(full_model, sub_model)

Arguments

Value

A data frame with the test statistic and p-value.

Fit Beta-Danish AFT Regression Model

Description

Fits an Accelerated Failure Time (AFT) regression model using the Complementary Exponentiated Danish (CED) baseline (Beta-Danish with a=1).

Usage

fit_bd_aft(formula, data, n_starts = 10, method = "BFGS")

Arguments

Details

To ensure identifiability, the shape parameter 'a' is fixed to 1. The scale parameter 'k' is linked to covariates via 'k_i = exp(X_i Positive coefficients in 'delta' indicate a larger 'k', which corresponds to shorter survival times (accelerated failure).

Value

An object of class 'bd_aft'.

Fit Beta-Danish Competing Risks Model

Description

Fits a parametric competing risks model assuming independent latent failure times, where each cause-specific baseline follows a 4-parameter Beta-Danish distribution.

Usage

fit_bd_competing(time, cause, n_starts = 5, method = "BFGS")

Arguments

Details

Under the assumption of independent latent failure times, the joint likelihood factorizes. The function first fits independent Beta-Danish models for each cause (treating other causes as censored) to find robust starting values, then optimizes the joint likelihood.

Value

An object of class 'bd_competing'.

Fit Beta-Danish Cure Models

Description

Fits mixture and promotion-time (non-mixture) cure models using the Beta-Danish AFT baseline.

Usage

fit_bd_cure(
  formula_aft,
  formula_cure,
  data,
  type = c("mixture", "promotion"),
  n_starts = 10,
  method = "BFGS"
)

Arguments

Details

In the **mixture** model, the population is split into susceptible and cured fractions. The susceptible probability is modeled via logistic regression: 'pi = exp(Z

In the **promotion-time** (non-mixture) model, the cure fraction is derived from a latent Poisson process of clonogenic cells: 'theta = exp(Z The cure fraction is 'exp(-theta)'.

Value

An object of class 'bd_cure'.

Fit the Beta-Danish Distribution to Survival Data

Description

Fits the Beta-Danish distribution using Maximum Likelihood Estimation (MLE). Supports both complete and right-censored data via 'survival::Surv' objects.

Usage

fit_betadanish(formula, data, submodel = FALSE, n_starts = 10, method = "BFGS")

Arguments

Details

The optimization is performed on the log-transformed parameters to strictly enforce positivity constraints. The returned coefficients and variance-covariance matrix are transformed back to the natural scale using the Delta method.

Value

An object of S3 class '"betadanish"', containing the parameter estimates, log-likelihood, variance-covariance matrix, and convergence diagnostics.

Examples

# Simulate some data
set.seed(123)
sim_time <- rbetadanish(100, a = 1.5, b = 2, c = 3, k = 0.5)
sim_status <- sample(c(0, 1), 100, replace = TRUE, prob = c(0.2, 0.8))
dat <- data.frame(time = sim_time, status = sim_status)

# Fit the 4-parameter model
fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = dat)
summary(fit)

# Fit the 3-parameter submodel
fit_sub <- fit_betadanish(survival::Surv(time, status) ~ 1, data = dat, submodel = TRUE)
summary(fit_sub)

Goodness-of-Fit Statistics for Beta-Danish Models

Description

Computes Information Criteria (AIC, BIC, HQIC, AICC) and the Kolmogorov-Smirnov (K-S) statistic for a fitted Beta-Danish model.

Usage

gof_betadanish(object)

Arguments

Value

A list containing the Information Criteria and the K-S statistic.

Acute Myelogenous Leukemia Survival

Description

Survival times (in weeks) for 23 patients with acute myelogenous leukemia. A classic, small dataset perfect for fast testing of censored data workflows.

Usage

leukemia

Format

A data frame with 23 rows and 3 columns:

time

Survival time in weeks

status

Event indicator (1 = event, 0 = censored)

group

Treatment group (Maintained vs Non-maintained)

Source

Miller, R. G. (1997). Survival Analysis. Wiley.

Examples

data(leukemia)
fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = leukemia)

Extract Log-Likelihood

Description

Extract Log-Likelihood

Usage

## S3 method for class 'betadanish'
logLik(object, ...)

Arguments

Value

An object of class logLik containing the maximized log-likelihood value, with degrees of freedom and number of observations stored as attributes.

Malignant Melanoma Survival After Surgery

Description

Survival times for 205 patients with malignant melanoma after surgery. This rich clinical dataset includes multiple covariates and heavy censoring.

Usage

melanoma

Format

A data frame with 205 rows and 6 columns:

time

Survival time in days

status

Event indicator (1 = died from melanoma, 0 = alive, 2 = other death)

thickness

Tumor thickness in mm

sex

Patient sex (1 = male, 0 = female)

age

Patient age in years

ulcer

Ulceration indicator (1 = present, 0 = absent)

year

Year of operation

Source

Andersen, P. K., Borgan, O., Gill, R. D., & Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer.

Examples

data(melanoma)

# Treat status 1 as event, others as censored
melanoma$event <- ifelse(melanoma$status == 1, 1, 0)
fit <- fit_betadanish(survival::Surv(time, event) ~ age + thickness, data = melanoma)

Plot Diagnostics for Beta-Danish Fit

Description

Generates diagnostic plots for a fitted Beta-Danish model, including survival, hazard, density, PP, and QQ plots.

Usage

## S3 method for class 'betadanish'
plot(x, type = c("survival", "hazard", "density", "pp", "qq", "all"), ...)

Arguments

Value

Invisibly returns the input betadanish object. Called mainly for its side effect of producing diagnostic plots.

Print Method for Beta-Danish Fit

Description

Print Method for Beta-Danish Fit

Usage

## S3 method for class 'betadanish'
print(x, ...)

Arguments

Value

Invisibly returns the input betadanish object. Called mainly for its side effect of printing the fitted model summary.

Print Summary Method for Beta-Danish Fit

Description

Print Summary Method for Beta-Danish Fit

Usage

## S3 method for class 'summary.betadanish'
print(x, ...)

Arguments

Value

Invisibly returns the input summary.betadanish object. Called mainly for its side effect of printing the coefficient table and fit statistics.

Read and Prepare Survival Data

Description

A helper function to read survival data from CSV or Excel files and prepare it for analysis with the Beta-Danish package. It automatically handles missing status columns by assuming all observations are complete (uncensored).

Usage

read_survival_data(file, time_col, status_col = NULL, covar_cols = NULL)

Arguments

Details

The function checks the file extension to determine how to read the data. For '.xlsx' or '.xls' files, the 'readxl' package must be installed. Missing values ('NA') in the specified columns will cause those rows to be dropped with a warning.

Value

A clean 'data.frame' containing the 'time', 'status', and any specified covariates, ready to be passed to 'fit_betadanish()'.

Examples

tmp <- tempfile(fileext = ".csv")
example_data <- data.frame(
  survival_time = c(5, 8, 12, 16),
  status = c(1, 1, 0, 1)
)
write.csv(example_data, tmp, row.names = FALSE)
dat <- read_survival_data(tmp, time_col = "survival_time", status_col = "status")
unlink(tmp)

Bladder Cancer Remission Times

Description

Remission times (in months) for 128 bladder cancer patients. This is a complete (uncensored) sample widely used in lifetime distribution literature to demonstrate decreasing or right-skewed hazard rates.

Usage

remission

Format

A data frame with 128 rows and 2 columns:

time

Remission time in months

status

Event indicator (1 = event occurred)

Source

Lee, E. T., & Wang, J. W. (2003). Statistical Methods for Survival Data Analysis (3rd ed.). Wiley.

Examples

data(remission)

fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = remission)
summary(fit)

Create a compact report from a BetaDanish model fit

Description

Create a compact report from a BetaDanish model fit

Usage

report_betadanish(fit)

Arguments

Value

A list containing model summary information.

Simulate Beta-Danish Cure Data

Description

Generates synthetic survival data from a Beta-Danish mixture or promotion-time cure model, incorporating covariates.

Usage

simulate_bd_cure_data(
  n,
  type = c("mixture", "promotion"),
  a = 1,
  b = 2,
  c = 1.5,
  delta,
  gamma,
  X,
  Z,
  target_censor = 0.3,
  seed = NULL
)

Arguments

Value

A list containing the simulated 'data' (time, status), the 'cured' indicator, and the true parameters.

Simulate Data from the Beta-Danish Distribution

Description

Generates synthetic survival data from the Beta-Danish distribution, with optional right-censoring.

Usage

simulate_bd_data(n, a, b, c, k, censor_rate = 0, seed = NULL)

Arguments

Value

A data frame with columns 'time' and 'status'.

Examples

# Simulate complete data
dat <- simulate_bd_data(n = 100, a = 1.5, b = 2, c = 3, k = 0.5)

# Simulate censored data
dat_cens <- simulate_bd_data(n = 100, a = 1.5, b = 2, c = 3, k = 0.5, censor_rate = 0.1)

Summary Method for Beta-Danish Fit

Description

Summary Method for Beta-Danish Fit

Usage

## S3 method for class 'betadanish'
summary(object, ...)

Arguments

Value

An object of class summary.betadanish containing coefficient estimates, standard errors, test statistics, p-values, log-likelihood, and model selection criteria.

Bone Marrow Transplant Survival

Description

Survival times for 91 patients with refractory acute lymphoblastic leukemia who received either an allogeneic or autologous bone marrow transplant. This dataset includes right-censoring and a treatment covariate, making it ideal for demonstrating cure-rate models and AFT regression.

Usage

transplant

Format

A data frame with 91 rows and 3 columns:

time

Survival time in days

status

Event indicator (1 = death/relapse, 0 = censored)

group

Treatment group (0 = Allogeneic, 1 = Autologous)

Source

Klein, J. P., & Moeschberger, M. L. (2003). Survival Analysis: Techniques for Censored and Truncated Data (2nd ed.). Springer.

Examples

data(transplant)

# Fit a model with a covariate
fit <- fit_bd_aft(survival::Surv(time, status) ~ group, data = transplant)

Extract Variance-Covariance Matrix

Description

Extract Variance-Covariance Matrix

Usage

## S3 method for class 'betadanish'
vcov(object, ...)

Arguments

Value

A numeric variance-covariance matrix for the estimated model parameters.