`parfm`

, Parametric Frailty Models in `R`

Federico Rotolo and Marco Munda

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.

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.

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 \(\kappa>0\).

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).

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.