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))

Arguments

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.

Value

A list with components:

Author(s)

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

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

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)

Parameter Estimation for the Power Piecewise Weibull Model

Description

Fits the power piecewise Weibull model and returns parameter estimates, standard errors, likelihood measures, and the selected time partition.

Usage

fit.pweibull(formula, data, L = 1, t = NULL, prec = 1e-04, max.iter = 1000,
    lambda.identical = FALSE, alpha.identical = FALSE, alpha.fixed = FALSE)

Arguments

Details

The hazard function of the power piecewise Weibull model is given by

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 defines the time partition, \boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_L) and \boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_L).

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.

Value

A list with components:

Author(s)

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

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

library(survival)
set.seed(3100)

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

## design matrix
x  <- model.matrix(~ x1 + x2)[, -1]

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

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

delta <- rep(1, n)

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

fit.pweibull(survival::Surv(time, delta) ~ x1 + x2, data = data, L = 2)

The Power Piecewise Weibull Distribution

Description

Density, distribution function, quantile function, and random generation for the power piecewise Weibull distribution.

Usage

dpweibull(x, rate = 1, alpha = 1, t = 0, log = FALSE)

ppweibull(q, rate = 1, alpha = 1, t = 0, lower.tail = TRUE, log.p = FALSE)

qpweibull(p, rate = 1, alpha = 1, t = 0, lower.tail = TRUE, log.p = FALSE)

rpweibull(n = 1, rate = 1, alpha = 1, t = 0)

Arguments

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 - 1} < a_L < \infty defines the partition of time, \boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_L), and \boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_L).

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.

Value

dpweibull returns the density, ppweibull returns the distribution function, qpweibull returns the quantile function, and rpweibull generates random deviates.

For rpweibull, the result has length n. For the other functions, the result has length equal to the maximum of the lengths of the numerical arguments. Arguments are recycled as needed.

Only the first elements of the logical arguments log, lower.tail, and log.p are used.

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

set.seed(3100)
## Random sample
rpweibull(n = 10, rate = c(0.05, 0.03), alpha = c(1, 1.5), t = c(0, 10))
## Distribution function
ppweibull(c(5, 10, 20), rate = c(0.05, 0.03), alpha = c(1, 1.5), t = c(0, 10))