JM (version 0.8-1)

weibull.frailty: Weibull Model with Gamma Frailties

Description

Fits a Weibull model with Gamma frailties for multivariate survival data under maximum likelihood

Usage

weibull.frailty(formula = formula(data), data = parent.frame(), 
    id = "id", subset, na.action, init, control = list())

Arguments

formula

an object of class formula: a symbolic description of the model to be fitted. The response must be a survival object as returned by function Surv().

data

an optional data frame containing the variables specified in the model.

id

either a character string denoting a variable name in data or a numeric vector specifying which event times belong to the same cluster (e.g., hospital, patient, etc.).

subset

an optional vector specifying a subset of observations to be used in the fitting process.

na.action

what to do with missing values.

init

a numeric vector of length $p + 3$ of initial values. The first $p$ elements should correspond to the regression coefficients for the covariates, and the last $3$ to log-scale, log-shape, and log-frailty-variance, respectively. See Details.

control

a list of control values with components: [object Object],[object Object],[object Object],[object Object],[object Object]

Value

• an object of class weibull.frailty with components:
• coefficients
  a list with the estimated coefficients values. The components of this list are: betas, scale, shape, and var.frailty, and correspond to the coefficients with the same name.
• hessian
  the hessian matrix at convergence. For the shape, scale, and var-frailty parameters the Hessian is computed on the log scale.
• logLik
  the log-likelihood value.
• control
  a copy of the control argument.
• y
  an object of class Surv containing the observed event times and the censoring indicator.
• x
  the design matrix of the model.
• id
  a numeric vector specifying which event times belong to the same cluster.
• nam.id
  the value of argument id, if that was a character string.
• terms
  the term component of the fitted model.
• data
  a copy of data or the created model.frame.
• call
  the matched call.

Details

The fitted model is defined as follows: $$\lambda(t_i | \omega_i) = \lambda_0(t_i) \omega_i \exp(x_i^T \beta),$$ where $i$ denotes the subject, $\lambda(\cdot)$ denotes the hazard function, conditionally on the frailty $\omega_i$, $x_i$ is a vector of covariates with corresponding regression coefficients $\beta$, and $\lambda_0(\cdot)$ is the Weibull baseline hazard defined as $\lambda_0(t) = shape * scale * t^{shape -1}$. Finally, for the frailties we assume $\omega_i \sim Gamma(\eta, \eta)$, with $\eta^{-1}$ denoting the unknown variance of $\omega_i$'s.

Examples

weibull.frailty(Surv(time, status) ~ age + sex, kidney)