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

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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):

≤ft\{ \begin{array}{lll} r_{i}^{(1)}(t|u_i,v_i) &= r_0^{(1)}(t)\exp({{β_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({{β_2^{'}}}Z_{i}(t)+v_i) &\quad \mbox{(rec. of type 2)}\\ λ_i(t|u_i,v_i) &= λ_0(t)\exp({{β_3^{'}}}Z_{i}(t)+α_1u_i+α_2v_i) &\quad \mbox{(death)}\\ \end{array} \right.

where r_0^{(l)}(t), l\in{1,2} and λ_0(t) are respectively the recurrent and terminal event baseline hazard functions, and β_1,β_2,β_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}≤ft({{0}},Σ_{uv}\right), with

Σ_{uv}=≤ft(\begin{array}{cc} θ_1 & ρ√{θ_1θ_2} \\ ρ√{θ_1θ_2}&θ_2 \end{array}\right)

Dependencies between these three types of event are taken into account by two correlated random effects and parameters θ_1,θ_2 the variance of the random effects and α_1,α_2 the coefficients for these random effects into the terminal event part. If α_1 and θ_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 α_1). If α_2 and θ_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 α_2). If ρ, 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 ρ).

Usage

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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 \sim operator, and the terms on the right. The response must be a survival object as returned by the 'Surv' function like in survival package. Interactions are possible using * or :.

formula.Event2

a formula object, with the response for the second recurrent event on the left of a \sim operator, and the terms on the right. The response must be a survival object as returned by the 'Surv' function like in survival package. Interactions are possible using * or :.

formula.terminalEvent

a formula object, with the response for the terminal event on the left of a \sim 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 is requested, the program first fit two shared frailty models (for the two types of recurrent events) and a Cox proportional hazards model (for the terminal event). Default is TRUE.

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 likelihood estimation. 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)

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 coefficient kappa of the integral of the squared second derivative of hazard function in the fit (penalized log likelihood). Initial values for the kappas can be obtained with the option "initialize=TRUE". 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)

maxit

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

hazard

Type of hazard functions: "Splines" for semi-parametric hazard functions with the penalized likelihood estimation, "Piecewise-per" for piecewise constant hazard function using percentile, "Piecewise-equi" for piecewise constant hazard function using equidistant intervals, "Weibull" for parametric Weibull function. Default is "Splines".

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 parametric hazard functions ("Piecewise-per", "Piecewise-equi"). Second is the Number of intervals (between 1 and 20) for the recurrent of type 2 parametric hazard functions ("Piecewise-per", "Piecewise-equi"). Third is Number of intervals (between 1 and 20) for the death parametric hazard functions ("Piecewise-per", "Piecewise-equi")

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-parametric case.

loglik

the marginal log-likelihood in the parametric case.

LCV

the approximated likelihood cross-validation criterion in the semi parametric case (with H minus the converged Hessian matrix, and l(.) the full log-likelihood.

LCV=\frac{1}{n}(trace(H^{-1}_{pl}H) - l(.))

)

AIC

the Akaike information Criterion for the parametric 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). Then, 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 recurrent, 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 parametric hazard functions ("Piecewise-per", "Piecewise-equi").

nbintervDC

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

nbintervR2

Number of intervals (between 1 and 20) for the the recurrent of type 2 parametric 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.

Note

"kappa" (kappa[1], kappa[2] and kappa[3]) and "n.knots" (n.knots[1], n.knots[2] and n.knots[3]) are the arguments that the user has 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 will 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 divide this value by 10 or 5 until it converges. Moreover, it may be useful to change the value of the initialize argument.

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

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## Not run: 

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

data(dataMultiv)

# (computation takes around 60 minutes)
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)


## End(Not run)

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