choose.pweibull: Choose a Partition for a Power Piecewise Weibull Model

Description

Selects a time partition for the power piecewise Weibull model, given a maximum number of allowed partitions. For choose.pweibull, the user may specify whether the $\lambda$'s are equal (default FALSE), whether the $\alpha$'s are equal (default FALSE), and whether $\alpha$ is fixed (default FALSE). For choose2.pweibull, the procedure selects the best combination among:

$\lambda$'s different and $\alpha$'s different,
$\lambda$'s equal and $\alpha$'s different,
$\lambda$'s different and $\alpha$'s equal,
$\lambda$'s different and $\alpha = 1$ (piecewise exponential distribution),
$\lambda$'s different and $\alpha = 2$ (piecewise Rayleigh distribution).

Usage

choose.pweibull(formula, data, criteria = "AIC", L.max = 5, t = NULL, 
    prec = 1e-04, max.iter = 1000, lambda.identical = FALSE, 
    alpha.identical = FALSE, alpha.fixed = FALSE)
choose2.pweibull(formula, data, criteria = "AIC", L.max = 5, t = NULL, 
    prec = 1e-04, max.iter = 1000, alpha.fixed = c(1, 2))

Value

A list with components:

estimate: A matrix of parameter estimates and standard errors for the selected partition.
logLik: Log-likelihood evaluated at the parameter estimates.
t: Selected time partition.
AIC: Akaike Information Criterion.
BIC: Bayesian Information Criterion.
L.sel: Number of selected partitions.
AIC.L: AIC values for $L = 1, \ldots, L.max$.
BIC.L: BIC values for $L = 1, \ldots, L.max$.

Arguments

formula: A model formula of class "formula" describing the survival model to be fitted. Details about model specification are given in Details.
data: An optional data frame, list, or environment containing the variables in the model. If not found in data, variables are taken from environment(formula).
criteria: Model selection criterion: "AIC" (default) or "BIC".
L.max: Maximum number of partitions to consider (default 5).
t: Optional fixed time partition. If provided, both choose.pweibull and choose2.pweibull evaluate only the model combinations with the specified partition.
prec: Numerical tolerance for the estimation algorithm (default 1e-4).
max.iter: Maximum number of iterations for the estimation algorithm (default 1000).
lambda.identical: Logical; should the $\lambda$'s be constrained to be equal? (default FALSE).
alpha.identical: Logical; should the $\alpha$'s be constrained to be equal? (default FALSE).
alpha.fixed: If FALSE (default), $\alpha$ is estimated. If a positive numeric value is supplied, all $\alpha$'s are fixed at that value. For choose2.pweibull, this may be a vector of fixed values.

Author

Diego I. Gallardo, Yolanda M. Gomez, Hector W. Gomez, and Barry C. Arnold.

Details

The hazard function of the power piecewise Weibull model is

$$ h(t \mid \boldsymbol{\lambda}, \boldsymbol{\alpha}) = \lambda_\ell \alpha_\ell t^{\alpha_\ell - 1}, \qquad t \in (a_{\ell-1}, a_\ell),\; \ell = 1,\ldots,L, $$

where $0 = a_0 < a_1 < \cdots < a_L < \infty$ is the time partition, $\boldsymbol{\lambda} = (\lambda_1,\ldots,\lambda_L)$ and $\boldsymbol{\alpha} = (\alpha_1,\ldots,\alpha_L)$.

The special cases include:

$\alpha_1 = \cdots = \alpha_L = 1$: the piecewise exponential model (Feigl and Zelen, 1965; Friedman, 1982),
$\alpha_1 = \cdots = \alpha_L = 2$: a piecewise Rayleigh model.

References

Feigl P., Zelen M. (1965). Estimation of exponential survival probabilities with concomitant information. Biometrics, 21, 826-838.

Friedman M. (1982). Piecewise exponential models for survival data with covariates. Annals of Statistics, 10, 101-113.

Gomez Y. M., Gallardo D. I., Arnold B. C. (2018). The power piecewise exponential model. Journal of Statistical Computation and Simulation, 88, 825-840.

Examples

Run this code

# \donttest{

library(survival)
set.seed(3100)

n  <- 200
x1 <- rnorm(n)
x2 <- rnorm(n)

## drawing covariates
lambda <- c(0.05, 0.03)
rate   <- exp(cbind(x1, x2) %*% c(0.5, -0.5))

time2 = c()
for (i in 1:n)
  time2[i] <- rpweibull(1, rate = lambda * rate[i], alpha = c(1, 1), t = c(0, 10))

delta <- rbinom(n, size = 1, prob = 0.75)
cc    <- runif(n, 0, max(time2))
time  <- ifelse(delta == 1, time2, cc)

data  <- data.frame(time = time, x1 = x1, x2 = x2, delta = delta)

choose.pweibull(survival::Surv(time, delta) ~ x1 + x2, data = data, L.max = 3)
# }

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