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Cox regression is commonly extended to estimate repeated events processes (for a quick review see [ 1 ] and [ 2 ]).

In Clark et al's first article in their excellent review series of survival analysis [ 3 ] cumulative hazard is explained in the following manner:

The interpretation of H(t) is difficult, but perhaps the easiest way to think of H(t) is as the cumulative force of mortality, or the number of events that would be expected for each individual by time t if the event were a repeatable process.

In light of [ 3 ]: It seems logical to conclude that in the setting of Cox regression of repeated events, the cumulative hazard represents the expected number of events given the covariates.

I am especially interested in the relationship between the Nelson-Aalen estimator of the cumulative hazard and the mean cumulative function or mean cumulative count [ 4 ]. Assume there are no competing risks.

What is the relationship, if any, between the mean cumulative function and cumulative hazard in the setting of Cox regression for repeated events?

  1. Modelling recurrent events: a tutorial for analysis in epidemiology. Amorim LD, Cai J. Int J Epidemiol. 2015 Feb;44(1):324-33. doi: 10.1093/ije/dyu222.
  2. Survival analysis for recurrent event data: an application to childhood infectious diseases. Kelly PJ, Lim LL. Stat Med. 2000 Jan 15;19(1):13-33.
  3. Survival analysis part I: basic concepts and first analyses. Clark TG, Bradburn MJ, Love SB, Altman DG. Br J Cancer. 2003 Jul 21;89(2):232-8.
  4. Estimating the Burden of Recurrent Events in the Presence of Competing Risks: The Method of Mean Cumulative Count. Dong et al. Am J Epidemiol. 2015 Apr 1; 181(7): 532–540.
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  • $\begingroup$ "cumulative hazard represents the expected number of events given the covariates" this is not what the cumulative hazard means. The actual interpretation of the cumulative hazard depends heavily on whether subjects who experience events but still at risk for recurrent events, are re-entered into analysis at time 0 or the event time, and whether any measure of incidence is adjusted as a covariate in models. $\endgroup$
    – AdamO
    Apr 19 '21 at 21:54
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Therneau's main R survival vignette explains this pretty clearly in Chapter 2, "Survival curves," for the situation without covariates. Section 2.2, "Repeated events," covers "the case of a single event type, with the possibility of multiple events per subject." That's the situation posited in this question. Quoting (emphasis in the original):

In multi-event data, the cumulative hazard is an estimate of the expected number of events for a unit that has been observed for the given amount of time, whereas the survival $S$ estimates the probability that a unit has had 0 repairs. The cumulative hazard is the more natural quantity to plot in such studies; in reliability analysis it is also known as the mean cumulative function.

That is, the mean cumulative function is the cumulative hazard function in this situation.

Section 3.2 of the vignette extends this non-parametric display of repeated-events data to semi-parametric Cox proportional-hazards models. That allows the cumulative hazard function, the expected number of events as a function of time, to be estimated for any specified set of covariate values. The identity of the "cumulative hazard function" and "mean cumulative count" is clear here, too:

Perhaps more interesting in this situation is the expected number of infections, rather than the probability of having at least 1. The former is estimated by the cumulative hazard...

as illustrated in the final figure of that section, a plot of the expected number of infections in the chronic granulotomous disease data set over time for 4 different combinations of covariate values.

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