Example: The bladder cancer study: 86 patients with superficial bladder tumours

1. Borgan and Henderson: Event History Methodology Lancaster, September 2006Session 6.1: Recurrent event data Intensity processes and rate functionsRobust variance estimation in Cox regression

2. Recurrent event data Example: The bladder cancer study: • 86 patients with superficial bladder tumours • Tumours were removed, and the patients randomized to placebo or treatment by thiotepa • Patients were followed up, and the recurrence of tumours were registered (truncated at 4)

4. Choice of time-scale Important distinction between models is the choice of time scale: • Time measured from an initial event (e.g. birth, onset of disease) • Poisson processes and their generalizations • Time measured between events • Renewal processes and their generalizations

5. A Cox-Poisson regression model Let Ni(t) count therecurrent events for individual i as a function of (study) time t Assume that Ni(t) has intensity process at risk indicator intensity for individual i depending on fixed covariates The events for individual i occur according to a Poisson process with intensity

6. Data file for analysis of bladder data: id trt fu no size intno start stop rec 1 1 1 1 3 1 0 1 0 2 1 4 2 1 1 0 4 0 ···· 5 1 10 4 1 1 0 6 1 5 1 10 4 1 2 6 10 0 ···· 59 2 21 1 1 1 0 17 1 59 2 21 1 1 2 17 19 1 59 2 21 1 1 3 19 21 0 ···· Note: One line for each time interval between events

7. Coding: • id patient number • trt treatment (1=placebo; 2=thiotepa) • fu total time of follow-up • no initial number of tumors (8 denotes 8+) • size initial size of largest tumor (in cm) • intno number of the time interval • start start of the time interval • stop end of the time interval • rec recurrence at end of the time interval (0=no; 1=yes)

8. R code: coxph(Surv(start,stop,rec)~trt, data=bladder) coef se(coef) p trt -0.373 0.198 0.059 Standard error and p-value are valid provided we have an appropriate model for the intensity processes. Looking at the data, there seems to be more variation among the individuals than can be described by the fixed covariates, so the Cox-Poisson model may be too simplistic.

9. Modelling options for recurrent event data In addition to the Cox-Poisson model, we have the following modelling options for recurrent event data: • Frailty model • Marginal model • Dynamic model (cf. session 6.2)

10. Frailty model for recurrent events Model the differences between individuals (not accounted for by the fixed covariates) by frailties Zi (e.g. gamma distributed). Given Zi assume that the recurrent events for individual i occur according to a Poisson process with intensity May integrate out Zito obtain the intensityprocess of Ni(t).

11. R code for gamma frailty model: coxph(Surv(start,stop,rec)~trt+frailty(id), data=bladder) coef se(coef) p trt -0.396 0.335 0.24 Note that the standard error for the frailty model becomes substantially larger (and more appropriate) than the one for the Cox-Poisson model

12. Marginal model for recurrent events The intensity process of Ni(t) is given by Note that the intensity process depends on all information available "just before" time t, i.e. fixed covariates and events in [0,t). One may try to model this dependence using information on previous events as dynamic covariates (cf. session 6.2)

13. If the main interest is on the effect of the fixed covariates, another option is to adopt a marginal model. A marginal model is defined by means of the rate function ri(t) given by Note that we only condition on fixed covariates and at risk status.Thus we average (marginalize) over all possible event patterns in [0,t).

14. We assume a proportional effect of the fixed covariates on the rate functions: Note: The interpretation of the regression coefficients are not the same for the frailty model and the marginal model. For the frailty model they measure the effects of the covatiates for a given individual (i.e. a given value of the frailty), while for the marginal model they measure the effects of the covariates averaged over all individuals at risk.

15. For the marginal model we may still estimate the regression coefficients using Cox's partial likelihood (even though this is not a proper partial likelihood for marginal models) The usual estimator for the covariance matrix based on the observed information does not apply, however, and has to be replaced by a robust "sandwich estimator" (cf. below)

16. R code for marginal model: coxph(Surv(start,stop,rec)~trt+cluster(id),data=bladder) coef se(coef) robust se p trt -0.373 0.198 0.283 0.19 Note that the robust standard error (which is the one to be used for the marginal model) becomes substantially larger than for the Cox-Poisson model.

17. Robust variance estimation For the marginal model we may write but is not a martingale increment. Therefore the martigale based proofs of the large sample properties (cf. session 3)no longer apply, and the usual likelihood based methods cannot be used.

18. By the argument in session 3, we have with the observed information and

19. Assuming an iid model, is a sum of iid variables. By the (ordinary) cental limit theorem is approximately normally distributed with mean zero and a covariance matrix that may be estimated by It follows by (*) that is approximately normally distributed around with a covariance matrix that may be estimated by the (robust) "sandwich estimator"