Survival function estimator for correlated recurrence time data under a Gamma Frailty Model

Share:

Description

Estimation of survival function for correlated recurrence time data under a Gamma Frailty model using the maximum likelihood criterion. The resulting object of class "survfitr" is plotted by ‘plot.survfitr’, before it is returned.

Usage

1
2
mlefrailty.fit(x,tvals, lambda=NULL, alpha=NULL, alpha.min, alpha.max, 
tol=1e-07, maxiter=500,alpha.console=TRUE)

Arguments

x

a survival recurrent event object.

tvals

vector of times where the survival function can be estimated.

lambda

optional vector of baseline hazard probabilities at t (see details). Default is numdeaths/apply(AtRisk,2,sum).

alpha

optional parameter of shape and scale for the frailty distribution. If this parameter is unknown is estimate via EM algorithm. In order to obtain the convergence of this algorithm a seed is calculated (see details).

alpha.min

optional left bound of the alpha parameter in order to obtain a seed to estimate alpha parameter. Default value is 0.5.

alpha.max

optional rigth bound of the alpha parameter in order to obtain a seed to estimate alpha parameter. Default value is the maximum of distinct times of events.

tol

optional tolerance of the EM algorithm used to estimate the alpha parameter. Default is 10e-7

maxiter

optional maximum number of iterations of the EM algorithm used to estimate the alpha parameter. Default is 500.

alpha.console

if TRUE prints in the console the estimates initial value for alpha and the alpha estimate via the EM algorithm, if FALSE not.

Details

The product limit estimator developed by Pe<f1>a, Strawderman and Hollander (2001) are valid when the interoccurrence times are assumed to represent an IID sample from some underlying distribution F. This assumption is clearly restrictive in biomedical applications, and one obvious generalization that allows association between interocurrence times is a frailty model.

A common and convenient choice of frailty distribution is a gamma distribution with shape and scale parameters set equal to an unknown parameter α. The common marginal survival function can be written as following

1-F(t)=(alpha/(alpha+Lambda_0(t))^alpha

The parameter α controls the degree of association between interoccurrence times within a unit. Pe<f1>a, Strawderman and Hollander (2001) showed that the estimation of α and Λ_0 can be obtained via the maximisation of the marginal likelihood function and the expectation-maximisation (EM) algorithm. For details and the theory behind this estimator, please refer to Pe<f1>a, Strawderman and Hollander (2001, JASA).

In order to obtain a good convergence, α is estimated previously. This estimation is used as a initial value in the EM procedure and it's carried out by the maximisation of the profile likelihood for α. In this case the arguments of mlefrailty.fit function called alpha.min and alpha.max are the boundaries of this maximisation. The maximum is obtained using the golden section search method.

Value

If the convergence of EM algorithm is not obtained, the initial value of alpha can be used as a alpha.min argument and recalculate.

n

number of unit or subjects observed.

m

vector of number of recurrences in each subject (length n).

failed

vector of number of recurrences in each subject (length n*m). Vector ordered (e.g. times of first unit, times of second unit , ..., times of n-unit).

censored

vector of times of censorship for each subject (length n).

numdistinct

number of distinct failures times.

distinct

vector of distinct failures times.

status

0 if the estimation is can be provided and 1 if not depending if alpha could be estimate or not.

alpha

parameter of Gamma Frailty Model.

lambda

Estimates of the hazard probabilities at distinct failures times.

survfunc

vector of survival estimated in distinct times.

tvals

copy of argument.

MLEAttvals

vector of survival estimated in tvals times.

References

Pe<f1>a, E.A., Strawderman, R. and Hollander, M. (2001). Nonparametric Estimation with Recurrent Event Data. J. Amer. Statist. Assoc 96, 1299-1315.

See Also

survfitr Survr

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
data(MMC)
fit<-mlefrailty.fit(Survr(MMC$id,MMC$time,MMC$event))
fit
plot(fit)

# compare with pena-straderman-hollander

fit<-psh.fit(Survr(MMC$id,MMC$time,MMC$event))
fit
lines(fit,lty=2)

# and with wang-chang

fit<-wc.fit(Survr(MMC$id,MMC$time,MMC$event))
fit
lines(fit,lty=3)