timereg (version 1.3-0)

two.stage: Fit Clayton-Oakes-Glidden Two-Stage model

Description

Fit Clayton-Oakes-Glidden Two-Stage model with Cox-Aalen marginals and regression on the variance parameters.

The model specifikatin allows a regression structure on the variance of the random effects, such it is allowed to depend on covariates fixed within clusters $$\theta_{k} = Q_{k}^T \nu$$. This is particularly useful to model jointly different groups and to compare their variances.

Fits an Cox-Aalen survival model. Time dependent variables and counting process data (multiple events per subject) are not possible !

The marginal baselines are on the Cox-Aalen form $$\lambda_{ki}(t) = Y_{ki}(t) ( X_{ki}^T(t) \alpha(t) ) \exp(Z_{ki}^T \beta )$$

The model thus contains the Cox's regression model and the additive hazards model as special cases. (see cox.aalen function for more on this).

The modelling formula uses the standard survival modelling given in the survival package.

Usage

two.stage(formula=formula(data),data=sys.parent(),beta=0,Nit=60,
detail=0,start.time=0,max.time=NULL,id=NULL,clusters=NULL,
robust=1,rate.sim=1,beta.fixed=0,theta=NULL,theta.des=NULL,var.link=0,step=1)

Arguments

formula

a formula object with the response on the left of a '~' operator, and the independent terms on the right as regressors. The response must be a survival object as returned by the `Surv' function. Terms with a proportional effect are specified by the

data

a data.frame with the variables.

start.time

start of observation period where estimates are computed.

max.time

end of observation period where estimates are computed. Estimates thus computed from [start.time, max.time]. Default is max of data.

robust

to compute robust variances and construct processes for resampling. May be set to 0 to save memory.

id

For timevarying covariates the variable must associate each record with the id of a subject.

clusters

cluster variable for computation of robust variances.

beta

starting value for relative risk estimates

Nit

number of iterations for Newton-Raphson algorithm.

detail

if 0 no details is printed during iterations, if 1 details are given.

rate.sim

rate.sim=1 such that resampling of residuals is based on estimated martingales and thus valid in rate case, rate.sim=0 means that resampling is based on counting processes and thus only valid in intensity case.

beta.fixed

option for keeping beta in the Cox-Aalen model fixed.

theta

starting values for the frailty variance (default=0.1).

theta.des

design for regression for variances. The defauls is NULL that is equivalent to just one theta and the design with only a baseline.

var.link

default "0" is that the regression design on the variances is without a link, and "1" uses the link function exp.

step

step size for Newton-Raphson.

Value

• returns an object of type "two.stage". With the following arguments:
• cum
  cumulative timevarying regression coefficient estimates are computed within the estimation interval.
• var.cum
  the martingale based pointwise variance estimates.
• robvar.cum
  robust pointwise variances estimates.
• gamma
  estimate of parametric components of model.
• var.gamma
  variance for gamma.
• robvar.gamma
  robust variance for gamma.
• D2linv
  inverse of the derivative of the score function.
• score
  value of score for final estimates.
• theta
  estimate of Gamma variance for frailty.
• var.theta
  estimate of variance of theta.
• S.theta
  estimate of derivative of score of theta.
• theta.score
  score for theta parameters.

Details

The data for a subject is presented as multiple rows or 'observations', each of which applies to an interval of observation (start, stop]. For counting process data with the )start,stop] notation is used the 'id' variable is needed to identify the records for each subject. Only one record per subject is allowed in the current implementation for the estimation of theta. The program assumes that there are no ties, and if such are present random noise is added to break the ties.

References

Glidden (2000), A Two-Stage estimator of the dependence parameter for the Clayton Oakes model.

Martinussen and Scheike, Dynamic Regression Models for Survival Data, Springer (2006).

Examples

library(survival)
data(diabetes)
# Marginal Cox model  with treat as covariate
fit<-two.stage(Surv(time,status) ~ prop(treat) + cluster(id),
diabetes,Nit=40,theta=1)
summary(fit)

# Stratification after adult 
theta.des<-model.matrix(~-1+factor(adult),diabetes);
des.t<-model.matrix(~-1+factor(treat),diabetes);
design.treat<-cbind(des.t[,-1]*(diabetes$adult==1),
                    des.t[,-1]*(diabetes$adult==2))
fit.s<-two.stage(Surv(time,status) ~
-1+factor(adult)+prop(design.treat)+cluster(id),
data=diabetes,Nit=40,theta=1,theta.des=theta.des)
summary(fit.s)

# test for common baselines 
fit.s1<-cox.aalen(Surv(time,status) ~ 
factor(adult)+prop(design.treat)+cluster(id),data=diabetes)
summary(fit.s1)

# with common baselines  and common treatment effect (although test reject this)
fit.s2<-two.stage(Surv(time,status) ~+1+prop(treat) + cluster(id),
data=diabetes,Nit=40,theta=1,theta.des=theta.des)
summary(fit.s2)

# test for same variance among the two strata
theta.des<-model.matrix(~factor(adult),diabetes);
fit.s3<-two.stage(Surv(time,status) ~+1+prop(treat)+cluster(id),
data=diabetes,Nit=40,theta=1,theta.des=theta.des)
summary(fit.s3)

# to fit model without covariates, beta.fixed=1, but still need prop term !
fit<-two.stage(Surv(time,status) ~ prop(treat) + cluster(id),
data=diabetes,theta=0.95,detail=0,beta.fixed=1)
summary(fit)