Package 'AlphaPowerHazard'

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

x, q	numeric vector of quantiles or values.
alpha	positive numeric (> 1); alpha shape parameter.
beta	positive numeric (> 0); beta shape parameter.
log, log.p	logical; if TRUE, probabilities/densities are returned on the log scale.
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise $P[X > x]$.
p	numeric vector of probabilities.
n	integer; number of random variates to generate.

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

t	positive numeric vector of time points.
eta	numeric vector of linear predictors $\eta_j = Z_j^\top \theta$.
theta_frailty	positive numeric; frailty variance parameter (if NULL or 0, no frailty).
par	numeric vector of baseline parameters c(alpha, beta).
model	character; one of "M1", "M2", "M3", "M4".

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

formula	a formula object. The left-hand side must be a Surv(time, status) or Surv(time, time2, status, type = "interval") object.
data	a data frame containing the variables in the formula.
model	character; one of "M1", "M2", "M3", "M4". Default is "M1".
frailty	logical; whether to include frailty. Default is FALSE.
time2	numeric vector; upper bound for interval-censored observations.
prog_cen	integer vector; progressive censoring scheme.
init	numeric vector; initial parameter values.

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

t	positive numeric vector of time points.
par	numeric vector of baseline parameters c(alpha, beta).

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

fit	An object of class "alpha_power_fit".
B	integer; number of bootstrap iterations (default 200).

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

...	fitted objects of class "alpha_power_fit".
B_waic	integer; bootstrap replicates for WAIC calculation.

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

time	positive numeric vector of event/censoring times.
status	integer vector of censoring indicators.
x	numeric matrix of covariates.
model	character; model specification "M1", "M2", "M3", or "M4".
frailty	logical; whether to include frailty.
K	integer; number of folds (default 5).

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

fit	An object of class "alpha_power_fit".

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

time	positive numeric vector of event or censoring times.
status	integer vector indicating observation type: 1 = exact event, 0 = right-censored, 2 = left-censored, 3 = interval-censored.
x	numeric matrix or data frame of covariates.
model	character; one of "M1", "M2", "M3", "M4". Default is "M1".
frailty	logical; whether to include a gamma frailty parameter $\theta$. Default is FALSE.
time2	numeric vector; upper bound for interval-censored observations.
prog_cen	integer vector; progressive censoring scheme.
init	numeric vector of initial parameter values on estimation scale.

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

x	positive numeric vector of failure time observations.
method	character; estimation method. One of "mle", "lse", "wlse", "mpse", "cme".
init	numeric vector of initial parameters c(alpha, beta). If NULL, heuristic initial values are computed.

Value

An object of class "alpha_power_base_fit" containing:

par	Estimated parameters c(alpha, beta).
method	Estimation method name.
value	Optimized objective function value.
convergence	Integer code (0 = success).
hessian	Hessian matrix at minimum/maximum.
se	Standard errors of parameters (if available).

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

alpha	positive numeric (> 1); alpha parameter.
beta	positive numeric in (0, 1); beta shape parameter.

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

fit	An object of class "alpha_power_fit".

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

par_all	numeric vector of parameters on estimation scale.
time	numeric vector of event or censoring times.
status	integer vector: 1 = exact event, 0 = right-censored, 2 = left-censored, 3 = interval-censored.
x	numeric matrix of covariates.
model	character; one of "M1", "M2", "M3", "M4".
has_frailty	logical; whether frailty parameter is included as last element.
time2	numeric vector; upper bound for interval censoring.
prog_cen	integer vector; progressive censoring scheme.

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

k	integer >= 1; maximum moment order to compute.
alpha	positive numeric (> 1); alpha parameter.
beta	positive numeric (> 0); beta parameter.

Value

A list with components:

raw_moments	Numeric vector of raw moments $\mu_1, \dots, \mu_k$.
mean	First raw moment $\mu_1$.
variance	Variance $\sigma^2 = \mu_2 - \mu_1^2$.
skewness	Coefficient of skewness $S = \mu_3^* / \sigma^3$.
kurtosis	Coefficient of kurtosis $K = \mu_4^* / \sigma^4$.
central_moments	Numeric vector of central moments $\mu_1^*, \dots, \mu_k^*$.

---

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

x	numeric vector of values.
r	integer; rank of order statistic ($1 \le r \le n$).
n	integer; sample size.
alpha	positive numeric (> 1); alpha parameter.
beta	positive numeric (> 0); beta parameter.
type	character; either "pdf" or "cdf".

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

fit	An object of class "alpha_power_fit".
ask	logical; if TRUE (the default in interactive sessions) R pauses and waits for the user to press Enter before drawing each new plot. Set ask = FALSE to cycle through all plots without pausing.
save_to_file	character or NULL (default). If a file path is supplied (e.g., "diagnostics.pdf" or "plot.png"), the 8 plots are written to that file instead of the screen. The file type is detected from the extension (.pdf, .png, .svg, .jpeg / .jpg). When NULL (default) the plots go to the active device only.

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

par	numeric vector of baseline parameters c(alpha, beta).
t_range	numeric vector of length 2 giving the time range.
n_grid	integer; number of grid points (default 300).

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters passed to plot.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
...	additional graphical parameters.

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

fit	An object of class "alpha_power_fit".
newdata	numeric matrix or data frame of covariates for prediction.
newtime	numeric vector of time points for prediction.
type	character; prediction type: "survival", "hazard", "median", "expected", "risk", "marginal", "forecast".
window	numeric vector of length 2 giving time interval for type = "expected".
time_points	numeric vector of time points for survival_at.
horizon	positive numeric; time horizon for forecast_alpha_power.
n_grid	integer; number of grid points for forecasting.

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

alpha	positive numeric (> 1); alpha parameter.
beta	positive numeric (> 0); beta parameter.

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

n	integer; sample size.
par	numeric vector of baseline parameters c(alpha, beta).
theta	positive numeric; frailty variance parameter (optional).
x	numeric matrix of covariates.
beta	numeric vector of regression coefficients.
model	character; one of "M1", "M2", "M3", "M4". Default is "M1".
cen_type	character; censoring type: "none", "right", "left", "interval".
cen_rate	positive numeric; censoring rate parameter for exponential censoring distribution.
time2	numeric vector; upper bound for interval censoring.

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

fit	An object of class "alpha_power_fit".

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

n_sim	integer; number of replicates.
n	integer; sample size.
par	numeric vector of true baseline parameters c(alpha, beta).
beta	numeric vector of true regression coefficients.
model	character; model specification.
frailty	logical; whether frailty is included.
theta	positive numeric; true frailty variance.
cen_rate	numeric; exponential censoring rate.

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

n_sim	integer; number of Monte Carlo replicates per scenario.
n_vec	integer vector; sample sizes to evaluate.
par	numeric vector of true baseline parameters c(alpha, beta).
beta	numeric vector of true regression coefficients.
model	character; model specification "M1", "M2", "M3", or "M4".
frailty	logical; whether frailty parameter is included.
theta	positive numeric; true frailty variance (if frailty = TRUE).
cen_type	character; censoring type: "none", "right", "left", "interval".
cen_rate	positive numeric; censoring rate.
conf_level	numeric; nominal coverage probability.
verbose	logical; print progress.

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

alpha1, alpha2	positive numeric (> 1); alpha parameters.
beta1, beta2	positive numeric (> 0); beta parameters.
x	numeric vector of positive evaluation points.

Value

A list with F1, F2, and logical is_ordered.

---