[R] Frailty by strata interactions in coxph (or coxme)‏

[Christos Argyropoulos]

Hello,
I was hoping that someone could answer a few questions for me (the background is given below):
1) Can the coxph accept an interaction between a covariate and a frailty term
2) If so, is it possible to
a) test the model in which the covariate and the frailty appear as main terms using the penalized likelihood (for gaussian/t frailties)
b)augment model 1) by stratifying on the variable that appers in the frailty term ?

Thanks in advance! I apologise for sending this as html earlier today.

Chris Argyropoulos
University of Pittsburgh Medical Center


Background
-------------
I am trying to fit a cox proportional hazards model tothe data from a two arm (treatment indicator is 0/1)
multicenter trial, adjusting for the presence of covariates and testing for the presence of a center
(and possibly a treatment X center interaction effect).

I would like to contrast the different approaches discussed in Glidden D.V. and Eric Vittinghoff. Modelling clustered survival data from multicentre clinical trials. Stat Med 2004, 23:369-388 (DOI: 10.1002/sim. 1599). in this paper the authors present a decomposition of the hazard function for the ith individual, receiving Z(k,i) treatment (0-1) in the kth center as:

lambda(k,i)=lambda(0,k)*exp(b*Z(k,i)+om(k)*Z(k,i))

where om(k) = center specific effect - mean treatment effect ~ distributed as a (log-) gamma or normal frailty.


I coded this model in R as:

r1<-coxph(Surv(d,e)~treat*frailty(center)+othercovars+strata(center))

coxph produces a table with the coefficients and standard errors for:
a) the treatment variable
b) the covariates as well as
c) the treatment x frailty interaction effect.

but the main effect of frailty is estimated to be zero (which I expected, since it is "absorbed" into the baseline hazards for the strata).

Am I correct to assume that coxph can in fact fit the model from the Stat Med paper?

If coxph cannot fit such models, could I simply fit:
coxph(Surv(d,e)~treat*frailty(center)+othercovars),

and interpret the coefficient for interaction between treatment and frailty { treat:frailty(center) }as in the first model?
If so, could one view the coefficient of the main effect of the frailty component, as a random intercept and the interaction term
as a random slope in a general mixed effects Cox model?
Can coxme be used to fit these models in a cleaner way?
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