Marginal and dynamic models for recurrent events and clustered survival data

We shall consider observation of clustered survival data or processes with recurrent events as defined in the introduction to Chapter 7. In the case of recurrent events we focus on the concept of dynamic models, which represent an attempt to understand in explicit terms how the past influences the present and the future. We may think of this as causal influences, but statistical dependence on the past may also be a reflection of heterogeneity. Instead of setting up a random effects, or frailty, model, one may alternatively condition with respect to past events to get a counting process model with a suitable intensity process. Frailty will induce dependence, such that, for example, the rate of a new event is increased if many events have been observed previously for this individual, since this would indicate a high frailty. The formulas (7.16) and (7.17) can be seen as expressions transforming the frailty structure into conditional representations given the past.

The existence of dynamic models follows from a general theorem for semimartingales, namely the Doob-Meyer decomposition, which states, essentially, that any semimartingale can be decomposed into a martingale and a compensator (Section 2.2.3). The martingale represents the “noise” or unpredictable changes, while the compensator represents the influence of the past history. A counting process is a submartingale, and hence a semimartingale, and the compensator is just the cumulative intensity process.