## Parametric Frailty Models in R

F.Rotolo, M.Munda

Frailty models [1, 2] 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

• 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 tρ-1,

with ρ,λ>0.

The Gompertz model is

h(t; γ, λ) = λ eγt,

with γ,λ>0.

The lognormal model is

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

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 tκ-1 } / { 1 + exp(α) tκ},

with α∈R and κ>0.

#### Frailty distributions

The gamma distribution is

f(u;, θ) = { u1/θ-1 e-u/θ }/{ Γ(1/θ) θ1/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) = -Σk=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) = (2π)-1/2 u-1 exp{-(log u)2 / 2σ2 },

with σ>0.

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

### References

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

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

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

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

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