Weibull Model with Gamma Frailties

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Description

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

Usage

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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:

optimizer

a character string indicating which optimizer to use; options are "optim" (default) and "nlminb".

parscale

the parscale control argument for optim(), or the scale argument for nlminb(). It should be a numeric vector of length equal to the number of parameters. Default is 0.01 for all parameters.

maxit

the maximum number of iterations. Default is 500.

numeriDeriv

a character string indicating which type of numerical derivative to use to compute the Hessian matrix; options are "fd" denoting the forward difference approximation, and "cd" (default) denoting the central difference approximation.

eps.Hes

tolerance value used in the numerical derivative method. Default is 1e-03.

Details

The fitted model is defined as follows:

λ(t_i | ω_i) = λ_0(t_i) ω_i \exp(x_i^T β),

where i denotes the subject, λ(.) denotes the hazard function, conditionally on the frailty ω_i, x_i is a vector of covariates with corresponding regression coefficients β, and λ_0(.) is the Weibull baseline hazard defined as λ_0(t) = shape * scale * t^{shape -1}. Finally, for the frailties we assume ω_i ~ Gamma(η, η), with η^{-1} denoting the unknown variance of ω_i's.

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.

Note

weibull.frailty() currently supports only right-censored data.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

Examples

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

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