The hazard function conditional on the two frailties $v_i$ and $w_{ij}$ for the $k^{th}$ individual of the $j^{th}$ subgroup of the $i^{th}$ group is :
$$\lambda_{ijk}(t|v_i,w_{ij})=v_iw_{ij}\lambda_0(t)exp(\bold{\beta^{'}X_{ijk}})$$
$$\small{ v_i\sim\Gamma\left(\frac{1}{\alpha},\frac{1}{\alpha}\right) \hspace{0.05cm}i.i.d. \hspace{0.2cm} \bold{E}(v_i)=1 \hspace{0.2cm}\bold{Var}(v_i)=\alpha \hspace{0.5cm} w_{ij}\sim\Gamma\left(\frac{1}{\eta},\frac{1}{\eta}\right)\hspace{0.05cm}i.i.d. \hspace{0.2cm} \bold{E}(w_{ij})=1 \hspace{0.2cm}\bold{Var}(w_{ij})=\eta}$$
where $\lambda_0(t)$ is the baseline hazard function, $X_{ijk}$ denotes the covariate vector and $\beta$ the corresponding vector of regression parameters.
subcluster()INITIAL VALUES
When a nested frailty model is fitted, at first, a Cox proportional hazards model with covariates (without random effects) is fitted and it provides initial values for the regression coefficients. Then, a model with covariates and only subgroup frailties is fitted, so it gives new initial values for the regression coefficients and for the subgroup frailty terms variance. Then, a model with covariates and only group frailty terms is fitted, so it gives new initial values for the regression coefficients and for the group frailties variance. The last step is the maximisation of the penalized log-likelihood for all parameters.
PARAMETERS LIMIT VALUES
As frailtypack is written in Fortran 77 some parameters had to be hard coded in. The default values of these parameters are, with the corresponding variable name in the fortran code between brackets.
maximum number of observations (ndatemax): 30000 maximum number of groups (ngmax): 1000 maximum number of subjects (nsujetmax): 15000 maximum number of parameters (npmax) :50 maximum number of covariates (nvarmax):50 maximum number of subgroups (nssgmax):5000 If these parameters are not large enough (an error message will let you know this), you need to reset them in nested.f and recompile.
V. Rondeau, D Commenges, and P. Joly (2003). Maximum penalized likelihood estimation in a gamma-frailty model. Lifetime Data Analysis 9, 139-153.
D. Marquardt (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal of Applied Mathematics, 431-441.
print.nestedPenal,
summary.nestedPenal,
plot.nestedPenal,
cluster,
subcluster,
strata### Nested model (or hierarchical model) with 2 covariates ###
data(dataNested)
modClu<-frailtyPenal(Surv(t1,t2,event)~cluster(group)+subcluster(subgroup)+
cov1+cov2,Frailty=TRUE,data=dataNested,n.knots=8,kappa1=50000)
# It takes around 24 minutes to converge (depends on the processor)#
print(modClu)
summary(modClu)
plot(modClu)
modClu.str<-frailtyPenal(Surv(t1,t2,event)~cluster(group)+subcluster(subgroup)+
cov1+strata(cov2)
,Frailty=TRUE,data=dataNested,n.knots=8,kappa1=20000,kappa2=20000)
# It takes around 8 minutes to converge (depends on the processor)#
print(modClu.str)
summary(modClu.str)
plot(modClu.str)Run the code above in your browser using DataLab