parfm, Parametric Frailty Models in R

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. 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.