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()kappa1 (or kappa2), a solutiINITIAL VALUES
The splines and the regression coefficients are initialized to 0.1. The program fits an adjusted Cox model to provide new initial values for the regression and the splines coefficients. The variances of the frailties are initialized to 0.1. Then, a shared frailty model with covariates with only subgroup frailty is fitted to give a new initial value for the variance of the subgroup frailty term. Then, a shared frailty model with covariates and only group frailty terms is fitted to give a new initial value for the variance of the group frailties. In a last step, a nested frailty model is fitted.
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