## Parametric Frailty Models in R

*F.Rotolo, M.Munda*
Information about the current plublic release on CRAN can be found here.

Information about the project development can be found here.

A published paper giving more details on the theory of frailty models and on the use of the **parfm** package can be found here [5].

Frailty models [1, 2] are survival models for clustered or overdispersed time-to-event data.
They consist in proportional hazards Cox's models [3] 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

- gamma,
- inverse Gaussian,
- positive stable and
- lognormal

frailty distributions can be specified,
together with five different baseline hazard families:
- Weibull,
- exponential,
- Gompertz,
- lognormal,
- loglogistic.

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

### Parametrisations

#### Baseline hazards

The **exponential** model is

*h*(*t; λ*)* = λ*,

with *λ>0*.

The **Weibull** model is

*h*(*t; ρ, λ*)* = ρ λ t*^{ρ-1},

with *ρ,λ>0*.

The **Gompertz** model is

*h*(*t; γ, λ*)* = λ *e^{γt},

with *γ,λ>0*.

The **lognormal** model is

*h*(*t; μ, σ*) =
{* φ*([log* t -μ*]*/σ*)}* / *{* σ t *[*1-Φ*([log* t -μ*]*/σ*)]},

with *μ∈***R**, *σ>0* and *φ*(*.*) and *Φ*(*.*) the density and distribution functions of a standard Normal.

The **loglogistic** model is

*h*(*t; α, κ*) =
{exp(*α*) *κ t*^{κ-1} } */* {
*1 +* exp(*α*) *t*^{κ}},

with *α∈***R** and *κ>0*.

#### Frailty distributions

The **gamma** distribution is

*f*(*u;, θ*) =
{* u*^{1/θ-1} e^{-u/θ} }*/*{* Γ*(*1/θ*)* θ*^{1/θ} },

with *θ>0* and *Γ*(*.*) the Gamma function.

The **inverse Gaussian** distribution is

*f*(*u; θ*) =
(*2θπ*)^{-1/2} u^{-3/2} exp{(*u-1*)^{2} / 2uθ },

with *θ>0*.

The **positive stable** distribution is

*f*(*u; ν*)* =
-*Σ_{k=1...∞}{*-u*^{-(1-ν)k} sin(*(1-ν)kπ*)* Γ*(*(1-ν)k+1*)*/k!*}* / πu*,

with *ν∈(0, 1)* and *Γ*(*.*) the Gamma function.

The **lognormal** distribution is

*f*(*u; ν*)* =
(**2σ*^{2}π)^{-1/2} u^{-1} exp{-(log *u*)^{2} / 2σ^{2} },

*
*with *σ>0*.

As the Laplace tranform of the lognormal frailties does not exist in closed form,
the saddlepoint approximation is used [4].

### References

[1] Duchateau L., Janssen P. (2008)
*The frailty model*.
Springer.

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

[3] Cox D.R. (1972)
Regression models and life-tables.
*Journal of the Royal Statistical Society. Series B (Methodological)* 34, 187–220.

[4] Goutis C., Casella G. (1999)
Explaining the Saddlepoint Approximation.
*The American Statistician* 53(3), 216-224.

[5] Munda M., Rotolo F. and Legrand C. (2012)
**parfm**: Parametric frailty models in **R**. *Journal of Statistical Software* 51(11).