frailtyPenal for Joint frailty models: Fit Joint Frailty model for recurrent and terminal events using penalized likelihood estimation

Description

Fit a joint frailty model for recurrent and terminal events using a Penalized Likelihood on the hazard function. Left-truncated and right-censored data are allowed. Stratified analysis is not possible. Joint frailty models allow studying, jointly, survival processes of recurrent and terminal events, by considering the terminal event as an informative censoring. A common frailty term $(\omega_i)$ for the two rates will take into account the heterogeneity in the data, associated with unobserved covariates. The frailty term acts differently for the two rates ( $\omega_i$ for the recurrent rate and $\omega_i^{\alpha}$ for death rate). The covariates could be different for the recurrent rate and death rate. The joint model for the hazard functions for recurrent event $r_i(.)$ and death $\lambda_i(.)$ is : $$r_i(t|\omega_i)=\omega_ir_0(t)exp(\bold{\beta_1^{'}Z_i(t)})$$ $$\lambda_i(t|\omega_i)=\omega_i^{\alpha}\lambda_0(t)exp(\bold{\beta_2^{'}Z_i(t)})$$ where $r_0(t)$ (resp. $\lambda_0(t)$) is the recurrent (resp. terminal) event baseline hazard function, $\bold{\beta_1}$ (resp. $\bold{\beta_2}$) the regression coefficient vector, $\bold{Z_i(t)}$ the covariate vector. And where $\omega_i\sim\bold{\Gamma}(\frac{1}{\theta},\frac{1}{\theta})$ is iid.

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

a formula object, only requires terms on the right to indicate which variables are modelling the terminal event.

data

a 'data.frame' in which to interpret the variables named in the 'formula' and 'formula.terminalEvent'.

Frailty

Logical value. Is model with frailties fitted? If so, variance of frailty parameter is estimated. The default is FALSE

joint

Logical value. Is joint model fitted? If so, 'formula.terminalEvent' is required. The default is FALSE

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 cov

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 implem

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 a good value for kappa1, a solution is to fit the correspondi

kappa2

positive smoothing parameter for the terminal event rate. To obtain a good value for kappa2, a solution is to fit the corresponding Cox model using cross-validation (See cross.validation) with the death indicator as the event of interest. Se

maxit

maximum number of iterations for the Marquardt algorithm. Default is 350

Value

Parameters estimates of a joint frailty model, more generally a 'fraityPenal' object. Methods defined for 'frailtyPenal' objects are provided for print, plot and summary. The following components are included in a 'frailtyPenal' object for Joint frailty models.
alphathe coefficient associated with the frailty parameter in the terminal hazard function
callThe code used for fitting the model
coefthe coefficients of the linear predictor, which multiply the columns of the model matrix.
cross.ValLogical value. Is cross validation procedure used for estimating the smoothing parameters?
formulathe formula part of the code used for the model
groupsthe maximum number of groups used in the fit
kappaA vector with the smoothing parameters corresponding to each baseline function as components
lammatrix of hazard estimates at x1 times and confidence bands.
lam2the same value as lam for the second stratum
LCVthe approximation likelihood cross-validation criterion in the semi-parametric/parametric case
nthe number of observations used in the fit.
n.eventsthe number of events observed in the fit
n.iternumber of iterations needed to converge
nvarA vector with the number of covariates of each type of hazard function as components
survmatrix of baseline survival estimates at x1 times and confidence bands.
surv2the same value as surv for the the second stratum
thetavariance of the frailty parameter $(\bold{Var}(\omega_i))$
varHthe variance matrix of all parameters before positivity constraint transformation (theta, the regression coefficients and the spline coefficients). Thenafter, the delta method is needed to obtain the estimated variance parameters.
varHIHthe robust estimation of the variance matrix of all parameters (theta, the regression coefficients and the spline coefficients).
x1vector of times where both survival and hazard function for the recurrent events are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times.
x2see x1 value for the terminal events survival and hazard function
nparnumber of parameters.
nvarRecnumber of recurrent vairables.
nvarEndnumber of death variable.
noVar1indicator of recurrent variable.
noVar2indicator of death variable.
LCVthe approximate likelihood cross-validation criterion in the semi parametric case.
n.knots.tempinitial value of n.knots.

synopsis

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

Details

The estimated parameter are obtained by maximizing the penalized loglikelihood 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. The inverse of the hessian matrix is the variance estimator and to deal with thepositivity constraint of the variance component, a squared transformation is used and the Standard errors are computed by the $\Delta$-method (Knight & Xekalaki, 2000). The smooth parameter can be chosen by maximizing a likelihood cross validation criterion (Joly and other, 1998). The integrations in the full log likelihood werre evaluated using Gaussian quadrature. Laguerre polynomials with 20 points were used to treat the integrations on $[0,\infty[$. INITIAL VALUES The splines and the regression coefficients are initialized to 0.5. The program fits an adjusted Cox model to have new initial values for the regression and the splines coefficients. The variance of the frailty term $\theta$ and the coefficient $\alpha$ associated in the death hazard function are initialized to 1. Then, it fits a joint frailty model.

References

V. Rondeau, S. Mathoulin-Pellissier, H. Jacqmin-Gadda, V. Brouste, P. Soubeyran (2007). Joint frailty models for recurring events and death using maximum penalized likelihood estimation:application on cancer events. Biostatistics, 8,4, 708-721. 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.

Examples

Run this code

### Joint model (recurrent and terminal events) with 2 covariates ###
### on a simulated dataset ###


data(readmission)

## Gap-time ##
modJoint_gap<-frailtyPenal(Surv(time,event)~cluster(id)+sex+as.factor(dukes)
		+as.factor(charlson)+terminal(death),
		formula.terminalEvent=~sex+as.factor(dukes)+as.factor(charlson),
		data=readmission,n.knots=14,kappa1=9550000000,
		kappa2=1410000000000,Frailty=TRUE,joint=TRUE,recurrentAG=FALSE)

## Calendar time ##
modJoint_calendar<-frailtyPenal(Surv(t.start,t.stop,event)~cluster(id)+sex
		+as.factor(dukes)+as.factor(charlson)+terminal(death),
		formula.terminalEvent=~sex+as.factor(dukes)+as.factor(charlson),
		data=readmission,n.knots=10,kappa1=9550000000,
		kappa2=1410000000000,Frailty=TRUE,joint=TRUE,recurrentAG=TRUE)
    
  print(modJoint_gap)
  summary(modJoint_gap)
  plot(modJoint_gap)
  
  print(modJoint_calendar)
  summary(modJoint_calendar)
  plot(modJoint_calendar)

# A model takes around 1 minute to converge #

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