Computes the baseline cumulative hazard function.

Description

Provides the baseline cumulative hazard function (\Lambda_0) for an object with extrafrail class.

Usage

baseCH(t, fit)

Arguments

Details

Provides the baseline cumulative hazard function. When the baseline distribution is assumed as the Weibull model, this function is \Lambda_0(t)=\lambda t^{\rho}. For the piecewise exponential model, this function is \Lambda_0(t)=\sum_{j=1}^L \lambda_j \nabla_j(t), where \nabla_j(t)=0, if t<a_{j-1}, \nabla_j(t)=t-a_{j-1}, if a_{j-1}\leq t < a_{j} and \nabla_j(t)=a_j-a_{j-1}, if t\geq a_{j}, with a=(a_0=0, a_1, \ldots, a_{j-1}), the corresponding partition time.

Value

a vector with the same length that t, including the baseline cumulative hazard function related to t.

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Gallardo, D.I., Bourguignon, M., Santibanez, J.L. (2025) The multivariate weighted Lindley frailty model for cluster failure time data. Biometrical Journal, 67(2), e70044.

Examples

#require(frailtypack)
require(survival)
data(rats, package="frailtyHL")
#Example for WL frailty model
fit.WL <- frailty.fit(survival::Surv(time, status) ~ rx + survival::cluster(litter), 
dist.frail="WL", data = rats)
baseCH(c(80,90,100),fit.WL)

Fitted different shared frailty models

Description

frailty.fit computes the maximum likelihood estimates based on the EM algorithm for the shared gamma, inverse gaussian, weighted Lindley, Birnbaum-Saunders, truncated normal, mixture of inverse gaussian and mixture of Birbaum-Saunders frailty models.

Usage

frailty.fit(formula, data, dist.frail="gamma", dist = "np", prec = 1e-04, 
        max.iter = 1000, part=NULL)

Arguments

Details

The multiplicative frailty models are defined in the following way. Consider the time to the event of an individual to be denoted by a random variable T, where the latent effect Z acts multiplicative in the baseline hazard function, whose hazard function, conditional on a latent variable Z, satisfies

\lambda(t|Z)=Z \,\lambda_0(t), t >0,

where \lambda_0(t) is a baseline hazard function and Z is the frailty of the individual, and \Lambda_0(t) = \int_{0}^{t}\lambda_0(w)\textrm{d}w is the cumulative baseline hazard function.

Let Z_i > 0, i=1,\ldots,n, be the latent random variable representing the frailty term associated with the i-th cluster, which has n_i observations (note that the n_i's can be different, i.e., the cluster can be unbalanced). The case n_i=1, \forall i=1,\ldots,n, is known in the literature as the univariate case. In a multiplicative hazards framework, given Z_i = z_i and a vector of p covariates (without intercept term), say \mathbf{x}_i=(x_{i1},\ldots,x_{ip}), the conditional hazard function for the observations in the i-th cluster is given by

\lambda(t_{i1},\ldots,t_{in_i}\mid z_i, \mathbf{X}_i)=z_i \sum_{j=1}^{n_i} \lambda_0(t) \exp(\mathbf{x}_{ij}^\top {\bm \beta}), \quad i=1,\ldots,n,

where {\bm \beta}^\top=(\beta_1,\ldots,\beta_p) are the regression parameters, respectively, and the distribution of Z corresponds to a nonnegative random variable. In this package, it is considered Z such as \mathbb{E}(Z)=1 and Var(Z)=\theta. The options avaliable for the distribution of Z and the corresponding probability density function are:

- gamma:

f(z;\theta)=\frac{1}{\theta^{1\theta}\Gamma(1/\theta)}z^{1/\theta-1}e^{-z/\theta}.

- IG:

f(z;\theta)=\sqrt{\frac{1}{2\pi\theta z^3}}\exp\left[-\frac{(z-1)^2}{2\theta z}\right].

- WL (Mota et al; 2021 for the univariate case; Tyagi et al., 2021 for the bivariate case, Gallardo et al. 2025 for the general case):

f(z;\theta)=\frac{\theta}{2\Gamma(b_\theta)}a_\theta^{-b_\theta-1}z^{b_\theta-1}(1+z)\exp\left(-\frac{z}{a_\theta}\right),

where a_\theta=\theta(\theta+4)/[2(\theta+2)] and b_\theta=4/[\theta(\theta+4)].

- BS (Leao et al; 2021 for the univariate case; Gallardo et al., 2024 for the general case):

f(z;\theta)=\frac{\exp(\phi_\theta/2)\sqrt{\phi_\theta+1}}{4z^{3/2}\sqrt{\pi}}\left(z+\frac{\phi_\theta}{\phi_\theta+1}\right)\exp\left[-\frac{\phi_\theta}{4}\left(\frac{z(\phi_\theta+1)}{\phi_\theta}+\frac{\phi_\theta}{z(\phi_\theta+1)}\right)\right],

where \phi_\theta=(1+\sqrt{3\theta+1}-\theta)/\theta.

The marginal survival function for the times in a same cluster can be expressed in terms of the Laplace transform as

S(t_{i1},\ldots,t_{in_i}\mid {\bm X}_i)=\mathcal{L}_Z\left(\sum_{j=1}^{n_i}\exp(\mathbf{x}_i^\top {\bm \beta} \Lambda_0(t_{ij}))\right).

For the different models considered here, the Laplace transform is given by

- gamma:

\mathcal{L}_Z(s)=(1+\theta s)^{-1/\theta}.

- IG:

\mathcal{L}_Z(s)=\exp\left[\frac{1}{\theta}\left(1-\sqrt{1+2\theta s}\right)\right].

- WL:

\mathcal{L}_Z(s)=(1+a_\theta s)^{-b_\theta-1}(1+\theta s/2).

- BS:

\mathcal{L}_Z(s)=\frac{1}{2}\left[1+\left(1+\frac{4s}{\phi_\theta+1}\right)^{-1/2}\right]\exp\left[\frac{\phi_\theta}{2}\left(1-\left(1+\frac{4s}{\phi_\theta+1}\right)^{1/2}\right)\right].

The estimation is performed based on the EM algorithm. For the weibull, exponential and piecewise exponential distributions as the basal model (\Lambda_0(\cdot)), the M1-step is performed using the optim function. For the non-parametric case, the M1-step is based on the coxph function from the survival package.

Value

an object of class "extrafrail" is returned. The object returned for this functions is a list containing the following components:

Author(s)

Diego Gallardo, Marcelo Bourguignon and John Santibanez.

References

Gallardo, D.I., Bourguignon, M., Santibanez, J.L. (2025) The shared weighted Lindley frailty model for cluster failure time data. Biometrical Journal, 67, e70044.

Gallardo, D.I., Bourguignon, M., Romeo, J. (2024) Birnbaum-Saunders frailty regression models for clustered survival data. Statistics and Computing, 34, 141.

Leao, J., Leiva, V., Saulo, H., Tomazella, V. (2017). Birnbaum-Saunders frailty regression models: Diagnostics and application to medical data. Biometrical journal, 59, 291-317.

Mota, A., Milani, E., Calsavara, V., Tomazella, V., Leao, J., Ramos, P., Ferreira, P., F., L. (2021). Weighted lindley frailty model: estimation and application to lung cancer data. Lifetime Data Analysis, 27, 561-587.

Tyagi, S., Pandey, A., Agiwal, V., Chesneau, C. (2021). Weighted Lindley multiplicative regression frailty models under random censored data. Computational and Applied Mathematics, 40, 265.

Examples

require(survival)
#require(frailtyHL)
data(rats, package="frailtyHL")
#Fit for WL frailty model
fit.WL <- frailty.fit(survival::Surv(time, status)~ rx+ survival::cluster(litter), 
dist.frail="WL", data = rats)
summary(fit.WL)
#Fit for gamma frailty model
fit.GA <- frailty.fit(survival::Surv(time, status) ~ rx + survival::cluster(litter), 
dist.frail="gamma", data = rats)
summary(fit.GA)

Generated random variables from the weighted Lindley distribution.

Description

Generated random variables from the weighted Lindley distribution with mean 1.

Usage

rWL(n, theta = 1)

Arguments

Details

The weighted Lindley distribution has probability density function

f(z;\theta)=\frac{\theta}{2\Gamma(\theta)}a_{\theta}^{-b_{\theta}-1}z^{b_{\theta}-1}(1+z)\exp\left(-\frac{z}{a_{\theta}}\right), \quad z, \theta>0,

where a_{\theta}=\frac{\theta(\theta+4)}{2(\theta+2)} and b_{\theta}=\frac{4}{\theta(\theta+4)}. Under this parametrization, E(Z)=1 and Var(Z)=\theta.

Value

a vector of length n with the generated values.

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Gallardo, D.I., Bourguignon, M. (2022) The multivariate weighted Lindley frailty model for cluster failure time data. Submitted.

Examples

rWL(10, theta=0.5)

Print a summary for a object of the "extrafrail" class.

Description

Summarizes the results for a object of the "extrafrail" class.

Usage

## S3 method for class 'extrafrail'
summary(object, ...)

Arguments

Details

Supported frailty models are: - gamma frailty model - inverse gaussian frailty model - weighted frailty model - Birnbaum-Saunders frailty model - Truncated normal frailty model - Mixture of inverse gaussian frailty model - Mixture of Birnbaum-Saunders frailty model

Value

A complete summary for the coefficients extracted from a "extrafrail" object.

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Gallardo and Bourguignon (2022).

Examples

#require(frailtyHL)
require(survival)
data(rats, package="frailtyHL")
fit <- frailty.fit(survival::Surv(time, status) ~ rx + survival::cluster(litter), 
dist.frail="WL", data = rats)
summary(fit)