frailtypack (version 2.7.1)

multivPenal: Fit a multivariate frailty model for two types of recurrent events and a terminal event.

Description

Fit a multivariate frailty model for two types of recurrent events with a terminal event using a penalized likelihood estimation on the hazard function or a parametric estimation. Right-censored data are allowed. Left-truncated data and stratified analysis are not possible. Multivariate frailty models allow studying, with a joint model, three survival dependent processes for two types of recurrent events and a terminal event. Multivariate joint frailty models are applicable in mainly two settings. First, when focus is on the terminal event and we wish to account for the effect of previous endogenous recurrent event. Second, when focus is on a recurrent event and we wish to correct for informative censoring. The multivariate frailty model for two types of recurrent events with a terminal event is (in the calendar or time-to-event timescale): $$\left{ \begin{array}{lll} r_{i}^{(1)}(t|u_i,v_i) &= r_0^{(1)}(t)\exp({{\beta_1^{'}}}Z_{i}(t)+u_i) &\quad \mbox{(rec. of type 1)}\ r_{i}^{(2)}(t|u_i,v_i) &= r_0^{(2)}(t)\exp({{\beta_2^{'}}}Z_{i}(t)+v_i) &\quad \mbox{(rec. of type 2)}\ \lambda_i(t|u_i,v_i) &= \lambda_0(t)\exp({{\beta_3^{'}}}Z_{i}(t)+\alpha_1u_i+\alpha_2v_i) &\quad \mbox{(death)}\ \end{array} \right.$$ where $r_0^{(l)}(t)$, $l\in{1,2}$ and $\lambda_0(t)$ are respectively the recurrent and terminal event baseline hazard functions, and $\beta_1,\beta_2,\beta_3$ the regression coefficient vectors associated with $Z_{i}(t)$ the covariate vector. The covariates could be different for the different event hazard functions and may be time-dependent. We consider that death stops new occurrences of recurrent events of any type, hence given $t>D$, $dN^{R(l)*}(t), l\in{1,2}$ takes the value 0. Thus, the terminal and the two recurrent event processes are not independent or even conditional upon frailties and covariates. We consider the hazard functions of recurrent events among individuals still alive.The three components in the above multivariate frailty model are linked together by two Gaussian and correlated random effects $u_i,v_i$: $(u_i,v_i)^{T}\sim\mathcal{N}\left({{0}},\Sigma_{uv}\right)$, with $$\Sigma_{uv}=\left(\begin{array}{cc} \theta_1 & \rho\sqrt{\theta_1\theta_2} \ \rho\sqrt{\theta_1\theta_2}&\theta_2 \end{array}\right)$$ Dependencies between these three types of event are taken into account by two correlated random effects and parameters $\theta_1,\theta_2$ the variance of the random effects and $\alpha_1,\alpha_2$ the coefficients for these random effects into the terminal event part. If $\alpha_1$ and $\theta_1$ are both significantly different from 0, then the recurrent events of type 1 and death are significantly associated (the sign of the association is the sign of $\alpha_1$). If $\alpha_2$ and $\theta_2$ are both significantly different from 0, then the recurrent events of type 2 and death are significantly associated (the sign of the association is the sign of $\alpha_2$). If $\rho$, the correlation between the two random effects, is significantly different from 0, then the recurrent events of type 1 and the recurrent events of type 2 are significantly associated (the sign of the association is the sign of $\rho$).

Usage

multivPenal(formula, formula.Event2, formula.terminalEvent, data,
            initialize = TRUE, recurrentAG = FALSE, n.knots, kappa,
            maxit = 350, hazard = "Splines", nb.int, 
            print.times = TRUE)

Arguments

formula

a formula object, with the response for the first recurrent event 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.Event2

a formula object, with the response for the second recurrent event 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, with the response for the terminal event 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.

data

a 'data.frame' with the variables used in 'formula', 'formula.Event2' and 'formula.terminalEvent'.

initialize

Logical value to initialize regression coefficients and baseline hazard functions parameters. When the estimation is semi-parametric with splines, this initialization produces also values for smoothing parameters (by cross validation). When initialization

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.

n.knots

integer vector of length 3 (for the three outcomes) giving the number of knots to use. First is for the recurrent of type 1, second is for the recurrent of type 2 and third is for the terminal event hazard function. Value required in the penalized likelih

kappa

vector of length 3 (for the three outcomes) for positive smoothing parameters in the penalized likelihood estimation. First is for the recurrent of type 1, second is for the recurrent of type 2 and third is for the terminal event hazard function. The coef

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 hazard function using equ

nb.int

An integer vector of length 3 (for the three outcomes). First is the Number of intervals (between 1 and 20) for the recurrent of type 1 parametrical hazard functions ("Piecewise-per", "Piecewise-equi"). Second is the Number of intervals (between 1 and 20)

print.times

a logical parameter to print iteration process. Default is TRUE.

Value

• Parameters estimates of a multivariate 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 'multivPenal' object for multivariate Joint frailty models.
• b
  sequence of the corresponding estimation of the splines coefficients, the random effects variances, the coefficients of the frailties and the regression coefficients.
• call
  The code used for fitting the model.
• n
  the number of observations used in the fit.
• groups
  the number of subjects used in the fit.
• n.events
  the number of recurrent events of type 1 observed in the fit.
• n.events2
  the number of the recurrent events of type 2 observed in the fit.
• n.deaths
  the number of deaths observed in the fit.
• loglikPenal
  the complete marginal penalized log-likelihood in the semi-parametrical case.
• loglik
  the marginal log-likelihood in the parametrical case.
• 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.$$AIC=\frac{1}{n}(np - l(.))$$
• theta1
  variance of the frailty parameter for recurrences of type 1 $(\bold{Var}(u_i))$
• theta2
  variance of the frailty parameter for recurrences of type 2 $(\bold{Var}(v_i))$
• alpha1
  the coefficient associated with the frailty parameter $u_i$ in the terminal hazard function.
• alpha2
  the coefficient associated with the frailty parameter $v_i$ in the terminal hazard function.
• rho
  the correlation coefficient between $u_i$ and $v_i$
• npar
  number of parameters.
• coef
  the regression coefficients.
• nvar
  A vector with the number of covariates of each type of hazard function as components.
• 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).
• formula
  the formula part of the code used for the model for the recurrent event.
• formula.Event2
  the formula part of the code used for the model for the second recurrent event.
• formula.terminalEvent
  the formula part of the code used for the model for the terminal event.
• x1
  vector of times for hazard functions of the recurrent events of type 1 are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times.
• lam1
  matrix of hazard estimates and confidence bands for recurrent events of type 1.
• xSu1
  vector of times for the survival function of the recurrent event of type 1.
• surv1
  matrix of baseline survival estimates and confidence bands for recurrent events of type 1.
• x2
  vector of times for the recurrent event of type 2 (see x1 value).
• lam2
  the same value as lam1 for the recurrent event of type 2.
• xSu2
  vector of times for the survival function of the recurrent event of type 2
• surv2
  the same value as surv1 for the recurrent event of type 2.
• xEnd
  vector of times for the terminal event (see x1 value).
• lamEnd
  the same value as lam1 for the terminal event.
• xSuEnd
  vector of times for the survival function of the terminal event
• survEnd
  the same value as surv1 for the terminal event.
• type.of.Piecewise
  Type of Piecewise hazard functions (1:"percentile", 0:"equidistant").
• n.iter
  number of iterations needed to converge.
• type.of.hazard
  Type of hazard functions (0:"Splines", "1:Piecewise", "2:Weibull").
• n.knots
  a vector with number of knots for estimating the baseline functions.
• kappa
  a vector with the smoothing parameters in the penalized likelihood estimation corresponding to each baseline function as components.
• n.knots.temp
  initial value for the number of knots.
• zi
  splines knots.
• time
  knots for Piecewise hazard function for the recurrent event of type 1.
• timedc
  knots for Piecewise hazard function for the terminal event.
• time2
  knots for Piecewise hazard function for the recurrent event of type 2.
• noVar
  indicator vector for reccurrent, death and recurrent 2 explanatory variables.
• nvarRec
  number of the recurrent of type 1 explanatory variables.
• nvarEnd
  number of death explanatory variables.
• nvarRec2
  number of the recurrent of type 2 explanatory variables.
• nbintervR
  Number of intervals (between 1 and 20) for the the recurrent of type 1 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").
• nbintervR2
  Number of intervals (between 1 and 20) for the the recurrent of type 2 parametrical hazard functions ("Piecewise-per", "Piecewise-equi").
• istop
  Vector of the convergence criteria.
• 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 of type 1).
• martingale2.res
  martingale residuals for each cluster (recurrent of type 2).
• martingaledeath.res
  martingale residuals for each cluster (death).
• frailty.pred
  empirical Bayes prediction of the first frailty term.
• frailty2.pred
  empirical Bayes prediction of the second frailty term.
• frailty.var
  variance of the empirical Bayes prediction of the first frailty term.
• frailty2.var
  variance of the empirical Bayes prediction of the second frailty term.
• frailty.corr
  Correlation between the empirical Bayes prediction of the two frailty.
• linear.pred
  linear predictor: uses Beta'X + ui in the multivariate frailty models.
• linear2.pred
  linear predictor: uses Beta'X + vi in the multivariate frailty models.
• lineardeath.pred
  linear predictor for the terminal part form the multivariate frailty models: Beta'X + alpha1 ui + alpha2 vi
• global_chisq
  Recurrent event of type 1: a vector with the values of each multivariate Wald test.
• dof_chisq
  Recurrent event of type 1: a vector with the degree of freedom for each multivariate Wald test.
• global_chisq.test
  Recurrent event of type 1: a binary variable equals to 0 when no multivariate Wald is given, 1 otherwise.
• p.global_chisq
  Recurrent event of type 1: a vector with the p-values for each global multivariate Wald test.
• names.factor
  Recurrent event of type 1: Names of the "as.factor" variables.
• global_chisq2
  Recurrent event of type 2: a vector with the values of each multivariate Wald test.
• dof_chisq2
  Recurrent event of type 2: a vector with the degree of freedom for each multivariate Wald test.
• global_chisq.test2
  Recurrent event of type 2: a binary variable equals to 0 when no multivariate Wald is given, 1 otherwise.
• p.global_chisq2
  Recurrent event of type 2: a vector with the p_values for each global multivariate Wald test.
• names.factor2
  Recurrent event of type 2: Names of the "as.factor" variables.
• global_chisq_d
  Terminal event: a vector with the values of each multivariate Wald test.
• dof_chisq_d
  Terminal event: a vector with the degree of freedom for each multivariate Wald test.
• global_chisq.test_d
  Terminal event: a binary variable equals to 0 when no multivariate Wald is given, 1 otherwise.
• p.global_chisq_d
  Terminal event: a vector with the p-values for each global multivariate Wald test.
• names.factordc
  Terminal event: Names of the "as.factor" variables.

References

Mazroui Y., Mathoulin-Pellissier S., MacGrogan G., Brouste V., Rondeau V. (2013). Multivariate frailty models for two types of recurrent events with an informative terminal event : Application to breast cancer data. Biometrical journal, 55(6), 866-884.

See Also

terminal,event2, print.multivPenal,summary.multivPenal,plot.multivPenal

Examples

###--- Multivariate Frailty model ---###

data(dataMultiv)

modMultiv.spli <- multivPenal(Surv(TIMEGAP,INDICREC)~cluster(PATIENT)+v1+v2+
             event2(INDICMETA)+terminal(INDICDEATH),formula.Event2=~v1+v2+v3,
             formula.terminalEvent=~v1,data=dataMultiv,n.knots=c(8,8,8),
             kappa=c(1,1,1),initialize=FALSE)

print(modMultiv.spli)

modMultiv.weib <- multivPenal(Surv(TIMEGAP,INDICREC)~cluster(PATIENT)+v1+v2+
             event2(INDICMETA)+terminal(INDICDEATH),formula.Event2=~v1+v2+v3,
             formula.terminalEvent=~v1,data=dataMultiv,hazard="Weibull")

print(modMultiv.weib)

modMultiv.cpm <- multivPenal(Surv(TIMEGAP,INDICREC)~cluster(PATIENT)+v1+v2+
             event2(INDICMETA)+terminal(INDICDEATH),formula.Event2=~v1+v2+v3,
             formula.terminalEvent=~v1,data=dataMultiv,hazard="Piecewise-per",
             nb.int=c(6,6,6))

print(modMultiv.cpm)