Package {AlphaPowerHazard}

Type:	Package
Title:	Alpha-Power Hazard Regression Models for Survival Data
Version:	0.1.0
Description:	Implements the alpha-power hazard model and regression frameworks for survival data based on the flexible hazard rate function h(x; alpha, beta) = alpha^x + x^(beta-1) (Pal et al., 2026 <doi:10.1007/s41096-026-00297-5>). Provides standard distribution functions (d, p, q, r, h, H, s) and distributional properties including raw/central moments, variance, skewness, kurtosis, quantile statistics (Bowley's skewness, Moors's kurtosis), Lambert W hazard rate function minimum (Corless et al., 1996), order statistics, and stochastic ordering (Shaked & Shanthikumar, 1994). Computes five classical estimation methods for baseline parameters: Maximum Likelihood Estimation (Casella & Berger, 2002), Least Squares Estimation (Swain et al., 1988), Weighted Least Squares Estimation (Styan, 1973), Maximum Product of Spacings Estimation (Cheng & Amin, 1983 <doi:10.1111/j.2517-6161.1983.tb01241.x>), and Cramer-von Mises Estimation (Macdonald, 1971). Supports four hazard regression models (M1-M4) within proportional hazards and parametric frameworks across uncensored data, right censoring, left censoring, interval censoring, and progressive Type-I and Type-II censoring schemes (Lee & Wang, 2003; Lawless, 2011; Balakrishnan & Aggarwala, 2000). Includes comprehensive model diagnostics, Cox-Snell, martingale, deviance, standardized, and studentized residuals, leverage, Cook's distance, DFFITS, DFBETAS, model comparisons (AIC, BIC, WAIC), k-fold cross-validation, prediction suites, random data generators, and an eight-panel diagnostic visualization suite.
License:	GPL-3
Depends:	R (≥ 4.0.0)
Imports:	survival, maxLik, numDeriv, MASS, stats, graphics, grDevices, utils
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown
Config/testthat/edition:	3
Encoding:	UTF-8
Language:	en-US
RoxygenNote:	7.3.3
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-07-20 11:50:02 UTC; shikhar tyagi
Author:	Shikhar Tyagi [aut, cre], Arvind Pandey [aut], Bhupendra Singh [aut], Vrijesh Tripathi [aut]
Maintainer:	Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Repository:	CRAN
Date/Publication:	2026-07-29 16:50:09 UTC

AlphaPowerHazard: Alpha-Power Hazard Regression Models for Survival Data

Description

The AlphaPowerHazard package implements alpha-power hazard regression models for survival data based on the flexible hazard rate function h(x; \alpha, \beta) = \alpha^x + x^{\beta-1}.

Model specification

The baseline hazard rate function is:

h(x; \alpha, \beta) = \alpha^x + x^{\beta-1}, \quad \alpha > 1, \beta > 0, x > 0

The cumulative hazard function is:

H(x; \alpha, \beta) = \frac{\alpha^x - 1}{\log(\alpha)} + \frac{x^\beta}{\beta}

The survival function is:

S(x; \alpha, \beta) = \exp(-H(x; \alpha, \beta))

Regression models

Four regression models are supported:

M1

Baseline hazard with covariate multiplier: h(y | Z) = (\alpha^y + y^{\beta-1}) \exp(Z^\top \theta)

M2

Covariate-dependent polynomial exponent: h(y | Z) = \alpha^y + y^{\exp(Z^\top \theta)-1}

M3

Covariate-dependent exponential component: h(y | Z) = \alpha^{\exp(Z^\top \theta)} + y^{\beta-1}

M4

Dual covariate dependence: h(y | Z) = \alpha^{\exp(Z^\top \theta)} + y^{\exp(Z^\top \theta)-1}

Censoring types

The package handles

• Uncensored (complete) data (status = 1)

• Right censoring (status = 0)

• Left censoring (status = 2)

• Interval censoring (status = 3, requires time2)

• Progressive Type-I censoring (time-terminated)

• Progressive Type-II censoring (failure-terminated)

Main functions

alpha_power_hazard

Formula interface (like lm).

fit_alpha_power

Low-level MLE fitting.

r_alpha_power

Random data generation.

predict_alpha_power

Survival / hazard / median / risk predictions.

residuals_alpha_power

Cox-Snell, martingale, deviance, raw, standardized, studentized residuals.

influence_alpha_power

Leverage, Cook's D, DFFITS, DFBETAS.

compare_models

AIC / BIC / WAIC model comparison table.

simulation_study

Full Monte Carlo simulation study.

plot_all

Eight-panel diagnostic plot suite.

Author(s)

Your Name your.email@example.com

References

Pal, S., Tyagi, V., Chesneau, C., & Sharma, V. K. (2026). A New Flexible Hazard Regression Model for Survival Data Analysis Alternative to Cox and AFT Regression Models. Journal of the Indian Society for Probability and Statistics. doi:10.1007/s41096-026-00297-5

Alpha-Power Hazard Distribution Functions and Statistical Properties

Description

Probability density function (PDF), cumulative distribution function (CDF), quantile function, random variates generation, hazard function, cumulative hazard function, survival function, raw and central moments, quantile statistics (Bowley's skewness and Moors's kurtosis), hazard rate minimum location, order statistics, and stochastic ordering for the Alpha-Power Hazard distribution.

Usage

halpha_power(x, alpha, beta)

Halpha_power(x, alpha, beta)

salpha_power(x, alpha, beta)

dalpha_power(x, alpha, beta, log = FALSE)

palpha_power(q, alpha, beta, lower.tail = TRUE, log.p = FALSE)

qalpha_power(p, alpha, beta, lower.tail = TRUE, log.p = FALSE)

ralpha_power(n, alpha, beta)

r_alpha_power(n, alpha, beta)

Arguments

Details

The baseline hazard rate function is given by:

h(x; \alpha, \beta) = \alpha^x + x^{\beta-1}, \quad \alpha > 1, \beta > 0, x > 0.

The cumulative hazard function is:

H(x; \alpha, \beta) = \frac{\alpha^x - 1}{\log \alpha} + \frac{x^\beta}{\beta}.

The survival function is:

S(x; \alpha, \beta) = \exp\left(-H(x; \alpha, \beta)\right).

The probability density function is:

f(x; \alpha, \beta) = (\alpha^x + x^{\beta-1}) \exp\left(-\left[\frac{\alpha^x - 1}{\log \alpha} + \frac{x^\beta}{\beta}\right]\right).

Value

dalpha_power returns density values. palpha_power returns cumulative probabilities. qalpha_power returns quantiles. ralpha_power and r_alpha_power return random variates. halpha_power returns hazard values. Halpha_power returns cumulative hazard values. salpha_power returns survival probabilities.

References

Pal, S., Tyagi, V., Chesneau, C., & Sharma, V. K. (2026). A New Flexible Hazard Regression Model for Survival Data Analysis Alternative to Cox and AFT Regression Models. Journal of the Indian Society for Probability and Statistics. doi:10.1007/s41096-026-00297-5

Examples

dalpha_power(c(0.5, 1, 1.5), alpha = 1.5, beta = 0.8)
palpha_power(c(0.5, 1, 1.5), alpha = 1.5, beta = 0.8)
qalpha_power(c(0.25, 0.5, 0.75), alpha = 1.5, beta = 0.8)
set.seed(123)
ralpha_power(5, alpha = 1.5, beta = 0.8)

Alpha-Power Hazard Model Functions (Models M1-M4)

Description

Computes the survival function S(t), density f(t), hazard h(t), and cumulative hazard H(t) for regression models M1, M2, M3, and M4 with optional gamma frailty.

Usage

alpha_power_functions(
  t,
  eta = rep(0, length(t)),
  theta_frailty = NULL,
  par = c(1.5, 1.1),
  model = "M1"
)

Arguments

Value

A list with components S, f, h, H.

Alpha-Power Hazard Regression Model (Formula Interface)

Description

A formula-based interface to fit_alpha_power, modelled after lm and glm. The response must be a Surv object.

Usage

alpha_power_hazard(
  formula,
  data,
  model = c("M1", "M2", "M3", "M4"),
  frailty = FALSE,
  time2 = NULL,
  prog_cen = NULL,
  init = NULL
)

Arguments

Value

An object of class c("alpha_power", "alpha_power_fit") with an additional call and formula component.

Examples

set.seed(1)
dat <- data.frame(
  time   = c(1.2, 0.8, 2.3, 1.5, 3.1),
  status = c(1, 1, 0, 1, 0),
  X1     = c(0.5, -0.2, 0.8, 0.1, -0.5)
)
fit <- alpha_power_hazard(survival::Surv(time, status) ~ X1, data = dat, model = "M1")
summary(fit)

Baseline Cumulative Hazard and Hazard Functions

Description

Computes the baseline cumulative hazard H_0(t) and baseline hazard h_0(t) for the alpha-power hazard distribution.

Usage

baseline_hazard(t, par)

Arguments

Details

Alpha-Power Hazard (par = c(alpha, beta)): h_0(t) = \alpha^t + t^{\beta-1}, H_0(t) = (\alpha^t - 1)/\log(\alpha) + t^\beta/\beta.

Value

A list with components:

H0

Numeric vector of cumulative hazard values.

h0

Numeric vector of hazard values.

Examples

bh <- baseline_hazard(seq(0.1, 2, by = 0.1), par = c(1.5, 0.8))
plot(seq(0.1, 2, by = 0.1), bh$h0, type = "l", main = "Alpha-Power hazard")

Bootstrap WAIC for the Alpha-Power Hazard Model

Description

Approximates the Widely Applicable Information Criterion (WAIC) via bootstrap re-sampling of the log-likelihood.

Usage

bootstrap_waic(fit, B = 200)

Arguments

Value

A named list with components WAIC, lppd, p_waic, B_effective.

Model Comparison Table (AIC, BIC, WAIC)

Description

Compares multiple fitted alpha-power models in a single table.

Usage

compare_models(..., B_waic = 100)

Arguments

Value

A data frame summarizing model comparisons.

K-Fold Cross-Validation for the Alpha-Power Hazard Model

Description

Performs k-fold cross-validation and reports out-of-sample log-likelihood.

Usage

cv_alpha_power(
  time,
  status,
  x = matrix(nrow = length(time), ncol = 0),
  model = "M1",
  frailty = FALSE,
  K = 5L
)

Arguments

Value

A list with cross-validation performance metrics.

Summary Table of Diagnostics

Description

Creates a summary table of model diagnostic statistics.

Usage

diagnostics_table(fit)

Arguments

Value

A data frame of diagnostic metrics.

Fit an Alpha-Power Hazard Regression Model (Models M1-M4)

Description

Fits an alpha-power hazard regression model via maximum likelihood estimation (MLE). Supports models M1, M2, M3, and M4 from Pal et al. (2026). Handles uncensored and right-, left-, interval-, and progressive-censored data.

Usage

fit_alpha_power(
  time,
  status,
  x = matrix(nrow = length(time), ncol = 0),
  model = c("M1", "M2", "M3", "M4"),
  frailty = FALSE,
  time2 = NULL,
  prog_cen = NULL,
  init = NULL
)

Arguments

Value

An object of class "alpha_power_fit" containing parameter estimates, standard errors, log-likelihood, AIC, BIC, and model diagnostics.

Classical Baseline Parameter Estimation Methods

Description

Computes point estimates and standard errors for the parameters \alpha and \beta of the Alpha-Power Hazard distribution using five classical estimation methods: Maximum Likelihood Estimation (MLE), Least Squares Estimation (LSE), Weighted Least Squares Estimation (WLSE), Maximum Product of Spacings Estimation (MPSE), and Cramér-von Mises Estimation (CME).

Usage

fit_alpha_power_base(
  x,
  method = c("mle", "lse", "wlse", "mpse", "cme"),
  init = NULL
)

Arguments

Value

An object of class "alpha_power_base_fit" containing:

References

Pal, S., Tyagi, V., Chesneau, C., & Sharma, V. K. (2026). A New Flexible Hazard Regression Model for Survival Data Analysis Alternative to Cox and AFT Regression Models. Journal of the Indian Society for Probability and Statistics. doi:10.1007/s41096-026-00297-5

Examples

set.seed(123)
dat <- ralpha_power(50, alpha = 1.5, beta = 0.8)
fit_mle  <- fit_alpha_power_base(dat, method = "mle")
fit_lse  <- fit_alpha_power_base(dat, method = "lse")
fit_wlse <- fit_alpha_power_base(dat, method = "wlse")
fit_mpse <- fit_alpha_power_base(dat, method = "mpse")
fit_cme  <- fit_alpha_power_base(dat, method = "cme")
fit_mle$par

Hazard Rate Function Minimum Location and Value

Description

Computes the unique minimum point x_{\text{min}} and minimum hazard rate h(x_{\text{min}}) for the Alpha-Power Hazard distribution when 0 < \beta < 1.

Usage

hrf_min_alpha_power(alpha, beta)

Arguments

Value

A list containing xmin and hmin.

Influence Diagnostics for the Alpha-Power Hazard Model

Description

Computes leverage values, Cook's distance, DFFITS, and DFBETAS.

Usage

influence_alpha_power(fit)

Arguments

Value

A named list containing influence metrics.

Log-Likelihood for the Alpha-Power Hazard Regression Models

Description

Internal function computing total log-likelihood for uncensored and censored (right, left, interval, progressive) data under models M1-M4.

Usage

loglik_alpha_power(
  par_all,
  time,
  status,
  x,
  model = "M1",
  has_frailty = FALSE,
  time2 = NULL,
  prog_cen = NULL
)

Arguments

Value

Scalar log-likelihood value.

Statistical Moments of the Alpha-Power Hazard Distribution

Description

Computes the raw moments \mu_k, central moments, variance, coefficient of skewness, and coefficient of kurtosis for the Alpha-Power Hazard distribution.

Usage

moments_alpha_power(k = 4, alpha = 1.5, beta = 1.1)

Arguments

Value

A list with components:

Order Statistics of the Alpha-Power Hazard Distribution

Description

Computes the PDF or CDF of the r-th order statistic from an independent sample of size n drawn from the Alpha-Power Hazard distribution.

Usage

order_stats_alpha_power(x, r, n, alpha, beta, type = c("pdf", "cdf"))

Arguments

Value

Numeric vector of PDF or CDF values of X_{(r)}.

Plot All Diagnostics

Description

Generates all 8 diagnostic plots for a fitted alpha-power hazard model and displays them in the active R graphics device (e.g., the RStudio Plots pane). Plots are never saved to a file automatically. To save, wrap the call in pdf() / dev.off() yourself, or use the save_to_file argument.

Usage

plot_all(fit, ask = grDevices::dev.interactive(), save_to_file = NULL)

Arguments

Value

Invisibly returns fit.

Note

This function never saves plots automatically. The save_to_file argument exists purely for user convenience and is NULL by default.

Examples

set.seed(1)
dat <- r_alpha_power_reg(80, par = c(1.5, 0.8), theta = 0.3,
                     cen_type = "right", cen_rate = 0.2)
fit <- fit_alpha_power(dat$time, dat$status)
# Display all 8 plots in the R graphics window (no file saved):
 plot_all(fit, ask = FALSE) 

Plot Baseline Distribution Functions

Description

Plots the PDF, CDF, survival function, and hazard function for the alpha-power hazard distribution over a time grid.

Usage

plot_baseline(par, t_range = c(0.01, 3), n_grid = 300)

Arguments

Value

Invisibly returns a list with components t (time grid), f (PDF), F (CDF), S (survival), and h (hazard) evaluated over the time grid.

Examples

plot_baseline(par = c(1.5, 0.8), t_range = c(0.01, 3))

Coefficient Forest Plot

Description

Displays all parameter estimates with 95% confidence intervals as a horizontal forest (dot-and-whisker) plot.

Usage

plot_coef_forest(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

DFBETAS Dot Plot

Description

Plots DFBETAS values as a stem plot for each covariate, highlighting observations that exceed the 2/\sqrt{n} threshold in red.

Usage

plot_dfbetas(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Leverage Histogram

Description

Displays a histogram of leverage values with a threshold line at 2(p+1)/n.

Usage

plot_leverage(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Q-Q Plot of Cox-Snell Residuals

Description

Compares sorted Cox-Snell residuals against theoretical quantiles of the standard exponential distribution. Deviations from the diagonal indicate model misfit.

Usage

plot_qq_residuals(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Residuals vs Fitted Values Plot

Description

Plots deviance residuals against fitted survival probabilities with a LOWESS smoother to assess linearity and heteroscedasticity.

Usage

plot_residuals_fitted(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Residuals vs Leverage Plot

Description

Plots deviance residuals against leverage values with Cook's distance contours (D = 0.5 and D = 1) to identify influential points.

Usage

plot_residuals_leverage(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Scale-Location Plot

Description

Plots the square root of absolute deviance residuals against fitted survival probabilities to detect heteroscedasticity.

Usage

plot_scale_location(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Kaplan-Meier vs Model-Based Survival Plot

Description

Overlays the Kaplan-Meier curve (empirical) with the model-based survival curve evaluated at the mean covariate vector.

Usage

plot_survival_km(fit, ...)

Arguments

Value

No return value, called for side effects (plotting).

Predictions from an Alpha-Power Hazard Model

Description

Computes survival probabilities, hazard rates, median survival times, expected survival in a window, risk scores, marginal survival, and forecasted survival curves for a fitted alpha-power hazard model.

Usage

predict_alpha_power(
  fit,
  newdata = NULL,
  newtime = NULL,
  type = c("survival", "hazard", "median", "expected", "risk", "marginal", "forecast"),
  window = NULL
)

survival_at(fit, time_points, newdata = NULL)

risk_predict(fit, newdata = NULL)

forecast_alpha_power(fit, horizon, newdata = NULL, n_grid = 100)

Arguments

Value

Prediction outputs depending on type or helper function invoked.

Examples

set.seed(1)
dat <- r_alpha_power_reg(50, par = c(1.5, 0.8), cen_type = "right")
fit <- fit_alpha_power(dat$time, dat$status)
pred_s <- predict_alpha_power(fit, type = "survival")

Quantile Statistics of the Alpha-Power Hazard Distribution

Description

Computes key quantiles (0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875), Bowley's coefficient of skewness, and Moors's coefficient of kurtosis.

Usage

quantile_stats_alpha_power(alpha = 1.5, beta = 1.1)

Arguments

Value

A list with quantiles, Bowley's skewness (bowley_skewness), and Moors's kurtosis (moors_kurtosis).

Random Data Generation for Alpha-Power Hazard Regression Models

Description

Generates synthetic survival data under the Alpha-Power Hazard regression models (M1-M4) with optional gamma frailty and censoring schemes (right, left, interval, progressive Type-I & Type-II).

Usage

r_alpha_power_reg(
  n,
  par = c(1.5, 1.1),
  theta = NULL,
  x = matrix(nrow = n, ncol = 0),
  beta = numeric(0),
  model = "M1",
  cen_type = c("none", "right", "left", "interval"),
  cen_rate = 0.1,
  time2 = NULL
)

Arguments

Value

A data frame with columns time, status, covariates, and optionally time2.

Examples

set.seed(123)
dat <- r_alpha_power_reg(50, par = c(1.5, 0.8), x = matrix(rnorm(50), ncol = 1),
                         beta = 0.5, cen_type = "right", cen_rate = 0.2)
head(dat)

Residuals for the Alpha-Power Hazard Model

Description

Computes a comprehensive set of residuals and goodness-of-fit metrics for a fitted alpha-power hazard model.

Usage

residuals_alpha_power(fit)

Arguments

Value

A named list containing residuals, leverage, and fit metrics.

Quick Simulation Performance Check

Description

A lightweight single-scenario simulation to quickly verify MLE parameter recovery.

Usage

simulate_mle_performance(
  n_sim = 50L,
  n = 50L,
  par = c(1.5, 1.1),
  beta = numeric(0),
  model = "M1",
  frailty = FALSE,
  theta = 0.5,
  cen_rate = 0.1
)

Arguments

Value

A data frame of simulation summary metrics.

Monte Carlo Simulation Study for Alpha-Power Hazard Models

Description

Evaluates MLE performance (bias, MSE, coverage, and convergence rate) for a specified alpha-power hazard model across multiple sample sizes and censoring settings.

Usage

simulation_study(
  n_sim = 100L,
  n_vec = c(50L, 100L),
  par = c(1.5, 1.1),
  beta = numeric(0),
  model = "M1",
  frailty = FALSE,
  theta = 0.5,
  cen_type = "right",
  cen_rate = 0.2,
  conf_level = 0.95,
  verbose = TRUE
)

Arguments

Value

A data frame summarizing simulation metrics.

Stochastic Ordering Comparison

Description

Verifies stochastic ordering F(x; \alpha_1, \beta_1) \le F(x; \alpha_2, \beta_2) for \alpha_1 \le \alpha_2 and \beta_1 \le \beta_2.

Usage

stochastic_ordering_alpha_power(
  alpha1,
  beta1,
  alpha2,
  beta2,
  x = seq(0.1, 5, by = 0.1)
)

Arguments

Value

A list with F1, F2, and logical is_ordered.