# Parametric Frailty Models

Fits Parametric Frailty Models by maximum marginal likelihood. Possible baseline hazards: exponential, Weibull, inverse Weibull (Fréchet), Gompertz, lognormal, log-skew-normal, and loglogistic. Possible Frailty distributions: gamma, positive stable, inverse Gaussian and lognormal.

Federico Rotolo and Marco Munda

## Description

Fits Parametric Frailty Models by maximum marginal likelihood. Possible baseline hazards: exponential, Weibull, inverse Weibull (Fréchet), Gompertz, lognormal, log-skew-normal, and loglogistic. Possible Frailty distributions: gamma, positive stable, inverse Gaussian and lognormal.

## Details

Frailty models are survival models for clustered or overdispersed time-to-event data. They consist in proportional hazards Cox's models with the addition of a random effect, accounting for different risk levels.

When the form of the baseline hazard is somehow known in advance, the parametric estimation approach can be used advantageously. The parfm package provides a wide range of parametric frailty models in R. The following baseline hazard families are implemented

• exponential,

• Weibull,

• inverse Weibull (Fréchet),

• Gompertz,

• lognormal,

• log-skew-normal,

• loglogistic,

together with the frailty distributions

• gamma,

• positive stable,

• inverse Gaussian, and

• lognormal.

Parameter estimation is done by maximising the marginal log-likelihood, with right-censored and possibly left-truncated data.

## Parametrisations

### Baseline hazards

The exponential hazard is $$h(t; \lambda) = \lambda,$$ with $\lambda > 0$.

The Weibull hazard is $$h(t; \rho, \lambda) = \rho \lambda t^{\rho-1},$$ with $\rho,\lambda > 0$.

The inverse Weibull (or Fréchet) hazard is $$h(t; \rho, \lambda) = \frac{\rho \lambda t^{-\rho - 1}}{\exp(\lambda t^{-\rho}) - 1}$$ with $\rho, \lambda > 0$.

$$h(t; \rho, \lambda) = \rho \lambda t^{\rho-1},$$ with $\rho,\lambda > 0$.

The Gompertz hazard is $$h(t; \gamma, \lambda) = \lambda e^{\gamma t},$$ with $\gamma,\lambda > 0$.

The lognormal hazard is $$h(t; \mu, \sigma) = { \phi([log t -\mu]/\sigma)} / { \sigma t [1-\Phi([log t -\mu]/\sigma)]},$$ with $\mu\in\mathbb R$, $\sigma > 0$ and $\phi(\cdot)$ and $\Phi(\cdot)$ the density and distribution functions of a standard Normal.

The log-skew-normal hazard is obtained as the ratio between the density and the cumulative distribution function of a log-skew normal random variable (Azzalini, 1985), which has density $$f(t; \xi, \omega, \alpha) = \frac{2}{\omega t} \phi\left(\frac{\log(t) - \xi}{\omega}\right) \Phi\left(\alpha\frac{\log(t)-\xi}{\omega}\right)$$ with $\xi \in {R}, \omega > 0, \alpha \in {R}$, and where $\phi(\cdot)$ and $\Phi(\cdot)$ are the density and distribution functions of a standard Normal random variable. Of note, if $alpha=0$ then the log-skew-normal boils down to the log-normal distribution, with $\mu=\xi$ and $\sigma=\omega$.

The loglogistic hazard is $$h(t; \alpha, \kappa) = {exp(\alpha) \kappa t^{\kappa-1} } / { 1 + exp(\alpha) t^{\kappa}},$$ with $\alpha\in\mathbb R$ and $\kappa>0$.

### Frailty distributions

The gamma frailty distribution is $$f ( u ) = \frac{\theta^{-\frac{1}{\theta}} u^{\frac{1}{\theta} - 1} \exp \left( - u / \theta \right)} {\Gamma ( 1 / \theta )}$$ with $\theta > 0$ and where $\Gamma(\cdot)$ is the gamma function.

The inverse Gaussian frailty distribution is $$f(u) = \frac1{\sqrt{2 \pi \theta}} u^{- \frac32} \exp \left( - \frac{(u-1)^2}{2 \theta u} \right)$$ with $\theta > 0$.

The positive stable frailty distribution is $$f(u) = f(u) = - \frac1{\pi u} \sum_{k=1}^{\infty} \frac{\Gamma ( k (1 - \nu ) + 1 )}{k!} \left( - u^{ \nu - 1} \right)^{k} \sin ( ( 1 - \nu ) k \pi )$$ with $0 < \nu < 1$.

The lognormal frailty distribution is $$f(u) = \frac1{\sqrt{2 \pi \theta}} u^{-1} \exp \left( - \frac{\log(u)^2}{2 \theta} \right)$$ with $\theta > 0$. As the Laplace tranform of the lognormal frailties does not exist in closed form, the saddlepoint approximation is used (Goutis and Casella, 1999).

## References

Azzalini A (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12(2):171-178. URL [http://www.jstor.org/stable/4615982]

Cox DR (1972). Regression models and life-tables. Journal of the Royal Statistical Society. Series B (Methodological), 34:187–220.

Duchateau L, Janssen P (2008). The frailty model. Springer.

Goutis C, Casella G (1999). Explaining the Saddlepoint Approximation. The American Statistician, 53(3):216-224. http://dx.doi.org/10.1080/00031305.1999.10474463.

Munda M, Rotolo F, Legrand C (2012). parfm: Parametric Frailty Models in R. Journal of Statistical Software, 51(11):1-20. DOI: 10.18637/jss.v051.i11

Wienke A (2010). Frailty Models in Survival Analysis. Chapman & Hall/CRC biostatistics series. Taylor and Francis.

# News

Changes in Version 2.7.4 (February 2017) o the observed Fisher information matrix is now returned as attribute ('FisherI') of the returned object

Changes in Version 2.7.3 (January 2017) o New baseline hazard family: inverse Weibull (Fréchet)

Changes in Version 2.7 (January 2017) o optimization is now based on the optimx package o the default optimizer is now nlminb (from the stats package) o fixed issue in the likelihood with lognormal frailties o new function coeff()

Changes in Version 2.6.0 (November 2016) o new baseline distribution: log-skew normal

Changes in Version 2.5.10 (September 2015) o added Kendall's tau for the lognormal frailty distrubution

Changes in Version 2.5 (October 2012) o new frailty distrubution available: lognormal

Changes in Version 2.01 (June 2012) o Added function anova.parfm() to compare models

Changes in Version 1.01 (March 2012) o Added stratification

Changes in Version 0.66-2 (February 2012) o Added example datasets

# Reference manual

install.packages("parfm")

2.7.6 by Federico Rotolo, 3 years ago

Browse source code at https://github.com/cran/parfm

Authors: Federico Rotolo [aut, cre] , Marco Munda [aut] , Andrea Callegaro [ctb]

Documentation:   PDF Manual