frailtyPenal for Shared frailty models {frailtypack}R Documentation

Fit Shared Gamma Frailty model using penalized likelihood estimation

Description

Fit a shared gamma frailty model using a Penalized Likelihood on the hazard function. Left truncated and censored data and strata (max=2) are allowed. It allows to obtain a non-parametric smooth hazard of survival function. This approach is different from the partial penalized likelihood approach of Therneau et al.

The hazard function, conditionnal on the frailty term z_i, of a shared frailty model for the j^{th} subject in the i^{th} group:

λ_{ij}(t|z_i)=λ_0(t)z_iexp(β^{'}X_{ij})

z_isimΓ(frac{1}{theta},frac{1}{theta}) hspace{0.5cm} E(z_i)=1 hspace{0.5cm}Var(z_i)=theta

where λ_0(t) is the baseline hazard function, β the vector of the regression coefficient associated to the covariate vector X_{ij} for the j^{th} individual in the i^{th} group.

Usage


frailtyPenal(formula, formula.terminalEvent, data, Frailty = FALSE,
                 joint = FALSE, recurrentAG = FALSE, cross.validation =
                 FALSE, n.knots, kappa1, kappa2, maxit = 350)

Arguments

formula a formula object, with the response on the left of a texttildelow operator, and the terms on the right. The response must be a survival object as returned by the 'Surv' function like in survival package.
formula.terminalEvent Not required.
data a 'data.frame' in which to interpret the variables named in the 'formula'.
Frailty Logical value. Is model with frailties fitted? If so, variance of frailty parameter is estimated. If not, Cox proportional hazards model is estimated using Penalized Likelihood on the hazard function. The default is TRUE
joint Not required.
recurrentAG Logical value. Is Andersen-Gill model fitted? If so indicates that recurrent event times with the counting process approach of Andersen and Gill is used. This formulation can be used for dealing with time-dependent covariates. The default is FALSE.
cross.validation Logical value. Is cross validation procedure used for estimating smoothing parameter? If so a search of the smoothing parameter using cross validation is done, with kappa1 as the seed. The cross validation is not implemented for two strata. The cross validation has been implemented for a Cox proportional hazard model, with no covariates. The default is FALSE.
n.knots integer giving the number of knots to use. Value required. It corresponds to the (n.knots+2) splines functions for the approximation of the hazard or the survival functions. Number of knots must be between 4 and 20.(See Note)
kappa1 positive smoothing parameter. The coefficient kappa of the integral of the squared second derivative of hazard function in the fit (penalized log likelihood). To obtain an initial value for kappa1 (or kappa2), a solution is to fit the corresponding shared frailty model using cross validation (See cross.validation). We advise the user to identify several possible tuning parameters, note their defaults and look at the sensitivity of the results to varying them. Value required.(See Note)
kappa2 positive smoothing parameter for the second stratum, when data are stratified. See kappa1.
maxit maximum number of iterations for the Marquardt algorithm. Default is 350

Details

The estimated parameter are obtained using the robust Marquardt algorithm (Marquardt, 1963) which is a combination between a Newton-Raphson algorithm and a steepest descent algorithm. When frailty parameter is small, numerical problems may arise. To solve this problem, an alternative formula of the penalized log-likelihood is used (see Rondeau, 2003 for further details). Cubic M-splines of order 4 are used for the hazard function, and I-splines (integrated M-splines) are used for the cumulative hazard function.

INITIAL VALUES

The splines and the regression coefficients are initialized to 0.1. The program fits, firstly, an adjusted Cox model to give new initial values for the splines and the regression coefficients. The variance of the frailty term theta is initialized to 0.1. Then, a shared frailty model is fitted.

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): 5000
maximum number of subjects (nsujetmax): 20000
maximum number of parameters (npmax) :50
maximum number of covariates (nvarmax):50

If these parameters are not large enough (an error message will let you know this), you need to reset them in frailtypack.f and recompile. In particular, the statements defining these parameters are

Value

The following components are included in a 'frailtyPenal' object for shared frailty models.

call The code used for the model
coef the coefficients of the linear predictor, which multiply the columns of the model matrix.
cross.Val Logical value. Is cross validation procedure used for estimating the smoothing parameters?
DoF Degrees of freedom associated with the "kappa"
formula the formula part of the code used for the model
groups the maximum number of groups used in the fit
kappa A vector with the smoothing parameters corresponding to each baseline function as components
lam matrix of hazard estimates at x1 times and confidence bands.
lam2 the same value as lam for the second stratum
loglikPenal the complete marginal penalized log-likelihood
n the number of observations used in the fit.
n.events the number of events observed in the fit
n.iter number of iterations needed to converge
n.knots number of knots for estimating the baseline function
n.strat number of stratum
surv matrix of baseline survival estimates at x1 times and confidence bands.
surv2 the same value as surv for the second stratum
theta variance of frailty parameter
type charater string specifying the type of censoring. Possible values are "right", "left", "counting", "interval", "interval2". The default is "right" or "counting" depending on whether the 'time2' argument is absent or present, respectively.
varH the variance matrix of all parameters (theta, the regression coefficients and the spline coefficients).
varHIH the robust estimation of the variance matrix of all parameters (theta, the regression coefficients and the spline coefficients).
x1 vector of times where both survival and hazard function are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times.
x2 see x1 value for the second stratum

Note

"Kappa" (kappa1 and kappa2) and "n.knots" are the arguments that the user have to change if the fitted model does not converge. "n.knots" takes integer values between 4 and 20. But with n.knots=20, the model would take a long time to converge. So, usually, begin first with n.knots=7, and increase it step by step until it converges. "Kappa" only takes positive values. So, choose a value for Kappa (for instance 10000), and if it does not converge, multiply or devide this value by 10 or 5 until it converges.

References

V. Rondeau, D Commenges, and P. Joly (2003). Maximum penalized likelihood estimation in a gamma-frailty model. Lifetime Data Analysis 9, 139-153.

McGilchrist CA and Aisbett CW (1991). Regression with frailty in survival analysis. Biometrics 47, 461-466.

D. Marquardt (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal of Applied Mathematics, 431-441.

See Also

print.frailtyPenal, summary.frailtyPenal, plot.frailtyPenal, cluster, strata

Examples


 ###  Shared model  ###

data(kidney)
frailtyPenal(Surv(time,status)~ cluster(id)+sex+age,
             n.knots=12,kappa1=10000,data=kidney,Frailty=TRUE)

 ###  COX proportional hazard model (SHARED without frailties) ### 
 ### estimated with penalized likelihood ###

frailtyPenal(Surv(time,status)~sex+age+cluster(id),
             n.knots=12,kappa1=10000,data=kidney,Frailty=FALSE)

###  Shared model with truncated data ###

# Here is created a hypothetical truncated data
kidney$tt0<-rep(0,nrow(kidney))
kidney$tt0[1:3]<-c(2,9,13)

# then, we fit the model
frailtyPenal(Surv(tt0,time,status)~sex+age+cluster(id),
             n.knots=12,kappa1=10000,data=kidney,Frailty=TRUE)

###  Shared model - stratified analysis ###

data(readmission)
frailtyPenal(Surv(time,event)~as.factor(dukes)+cluster(id)+strata(sex),
             n.knots=10,kappa1=10000,kappa2=10000,data=readmission,Frailty=TRUE)

###  Shared model - recurrentAG=TRUE ###

frailtyPenal(Surv(t.start,t.stop,event)~as.factor(sex)+as.factor(dukes)+
             as.factor(charlson)+cluster(id),data=readmission, Frailty=TRUE,
             n.knots=6,kappa1=100000,recurrentAG=TRUE)

###  Shared model - cross.validation=TRUE ###

## Not run: 
 
frailtyPenal(Surv(t.start,t.stop,event)~as.factor(sex)+as.factor(dukes)+
             as.factor(charlson)+cluster(id),data=readmission, Frailty=TRUE,
             n.knots=6,kappa1=5000,recurrentAG=TRUE,cross.validation=TRUE)
## End(Not run)


[Package frailtypack version 2.2-12 Index]