frailtypack (version 2.2-26)

frailtyPenal for Joint frailty models: Fit Joint Frailty model for recurrent and terminal events using semi-parametrical penalized likelihood estimation or a parametrical estimation

Description

Fit a joint frailty model for recurrent and terminal events using a penalized likelihood estimation on the hazard function or a parametrical estimation. 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 function for recurrent event $r_i(.)$ and death $\lambda_i(.)$ is : $$\left{ \begin{array}{ll} r_i(t|\omega_i)=\omega_ir_0(t)exp(\bold{\beta_1^{'}Z_i(t)}) \ \lambda_i(t|\omega_i)=\omega_i^{\alpha}\lambda_0(t)exp(\bold{\beta_2^{'}Z_i(t)}) \end{array} \right.$$ 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 in the penalized likelihood estimation? If so a search of the smoothing parameter using cross validation is done, with kappa1 as the seed.

n.knots

integer giving the number of knots to use. Value required in the penalized likelihood estimation. It corresponds to the (n.knots+2) splines functions for the approximation of the hazard or the survival functions. Number o

kappa1

positive smoothing parameter in the penalized likelihood estimation. 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

kappa2

positive smoothing parameter in the penalized likelihood estimation 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)

maxit

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

hazard

Type of hazard functions: "Splines" for semi-parametrical hazard functions with the penalized likelihood estimation, "Piecewise-per" for piecewise constant hazard function using percentile, "Piecewise-equi" for piecewise constant

nb.int1

Number of intervals (between 1 and 20) for the recurrent parametrical hazard functions ("Piecewise-per", "Piecewise-equi").

nb.int2

Number of intervals (between 1 and 20) for the death parametrical hazard functions ("Piecewise-per", "Piecewise-equi").

intcens

Not implemented for joint frailty model.

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.
• b
  sequence of the corresponding estimation of the splines coefficients, the random effects variances, the power of the frailties and the regression coefficients.
• alpha
  the coefficient associated with the frailty parameter in the terminal hazard function.
• call
  The code used for fitting the model.
• coef
  the regression coefficients.
• cross.Val
  Logical value. Is cross validation procedure used for estimating the smoothing parameters in the penalized likelihood estimation?
• 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 in the penalized likelihood estimation corresponding to each baseline function as components.
• lam
  matrix of hazard estimates and confidence bands.
• lam2
  the same value as lam for the second stratum.
• loglikPenal
  the complete marginal penalized log-likelihood in the semi-parametrical case.
• loglik
  the marginal log-likelihood in the parametrical case.
• n
  the number of observations used in the fit.
• n.events
  the number of events observed in the fit.
• n.knots
  number of knots for estimating the baseline functions.
• n.iter
  number of iterations needed to converge.
• surv
  matrix of baseline survival estimates and confidence bands.
• surv2
  the same value as surv for the the second stratum.
• theta
  variance of the frailty parameter $(\bold{Var}(\omega_i))$
• varH
  the 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.
• 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 functions for the recurrent events are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times.
• x2
  vector of times for the second stratum (see x1 value).
• type.of.hazard
  Type of hazard functions (0:"Splines", "1:Piecewise", "2:Weibull").
• type.of.Piecewise
  Type of Piecewise hazard functions (1:"percentile", 0:"equidistant").
• nbintervR
  Number of intervals (between 1 and 20) for the recurrent parametrical hazard functions ("Piecewise-per", "Piecewise-equi").
• nbintervDC
  Number of intervals (between 1 and 20) for the death parametrical hazard functions ("Piecewise-per", "Piecewise-equi").
• npar
  number of parameters.
• nvar
  A vector with the number of covariates of each type of hazard function as components.
• nvarRec
  number of recurrent explanatory vairables.
• nvarEnd
  number of death explanatory vairables.
• noVar1
  indicator of recurrent explanatory vairables.
• noVar2
  indicator of death explanatory vairables.
• LCV
  the approximated likelihood cross-validation criterion in the semi parametrical case (with H minus the converged hessien matrix, and l(.) the full log-likelihood.$$LCV=\frac{1}{n}(trace(H^{-1}_{pl}H) - l(.))$$
• AIC
  the Akaike information Criterion for the parametrical case.$$AIC=\frac{1}{n}(np - l(.))$$
• n.knots.temp
  initial value for the number of knots.
• shape.weib
  shape parameters for the weibull hazard function.
• scale.weib
  scale parameters for the weibull hazard function.
• martingale.res
  martingale residuals for each cluster (recurrent).
• martingaledeath.res
  martingale residuals for each cluster (death).
• frailty.pred
  empirical Bayes prediction of the frailty term.
• frailty.var
  variance of the empirical Bayes prediction of the frailty term.
• linear.pred
  linear predictor: uses "Beta'X + log w_i" in the joint Frailty models.
• lineardeath.pred
  linear predictor for the terminal part form the joint model: "Beta'X + alpha.log w_i".
• global_chisq
  a vector with the values of each multivariate Wald test.
• dof_chisq
  a vector with the degree of freedom for each multivariate Wald test.
• global_chisq.test
  a binary variable equals to 0 when no multivariate Wald is given, 1 otherwise.
• p.global_chisq
  a vector with the p_values for each global multivariate Wald test.
• names.factor
  Names of the "as.factor" variables.
• Xlevels
  vector of the values that factor might have taken for the recurrent part.
• contrasts
  type of contrast for factor variable for the recurrent part.
• Xlevels2
  vector of the values that factor might have taken for the death part.
• contrasts2
  type of contrast for factor variable for the death part.

synopsis

frailtyPenal(formula, formula.terminalEvent, data, Frailty = FALSE, joint = FALSE, recurrentAG = FALSE, cross.validation = FALSE, n.knots, kappa1, kappa2, maxit = 350,hazard = "Splines",nb.int1,nb.int2 intcens = FALSE)

Details

The estimated parameter are obtained by maximizing the semi-parametrical penalized loglikelihood estimation or a parametrical estimation 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 were 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, Y. Mazroui and J. R. Gonzalez (2012). Frailtypack: An R package for the analysis of correlated survival data with frailty models using penalized likelihood estimation or parametrical estimation. Journal of Statistical Software 47, 1-28. 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.

See Also

summary.jointPenal, print.jointPenal, plot.jointPenal, readmission, terminal, cluster

Examples

### 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 #