Package 'ppweibull'

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

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.

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:

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 $L = 1, \ldots, L.max$.

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

formula	A model formula of class "formula" describing the survival model to be fitted. Details on 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).
L	Number of partitions to be used. Ignored if t is specified.
t	A fixed time partition. Ignored if L is supplied.
prec	Numerical tolerance used in the estimation procedure (default 1e-4).
max.iter	Maximum number of iterations allowed in 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	Should $\alpha$ be fixed at a known value? If FALSE (default), $\alpha$ is estimated. If a positive number is provided, all $\alpha_\ell$ are fixed at that value.

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:

estimate	A matrix containing parameter estimates and corresponding standard errors.
logLik	Log-likelihood evaluated at the estimated parameters.
t	The time partition used.
AIC	Akaike Information Criterion.
BIC	Bayesian Information Criterion.

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

x	Vector of quantiles for the density function.
q	Vector of quantiles for the distribution function.
p	Vector of probabilities for the quantile function.
n	Number of observations to generate. If length(n) > 1, the length is taken as the number of observations.
rate	A numeric vector of length $L$ containing the rate parameters $\lambda_1, \ldots, \lambda_L$.
alpha	A numeric vector of length $L$ containing the shape parameters $\alpha_1, \ldots, \alpha_L$.
t	A non-decreasing vector defining the time partition $(0 = a_0, a_1, \ldots, a_L)$.
log	Logical; if TRUE, probabilities are returned on the log scale.
lower.tail	Logical; if TRUE (default), probabilities are $P(T \le q)$; otherwise $P(T > q)$.
log.p	Logical; if TRUE, probabilities p are given on the log scale.

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))

---