[R] Expected number of events, Andersen-Gill model fit via coxph in package survival

[Terry Therneau]

> I am interested in producing the expected number of events, in a
> recurring events setting. I am using the Andersen-Gill model, as fit
> by the function "coxph" in the package "survival."
>
> I need to produce expected numbers of events for a cohort,
> cumulatively, at several fixed times. My ultimate goal is: To fit an
> AG model to a reference sample, then use that fitted model to generate
> expected numbers of events for a new cohort; then, comparing the
> expected vs. the observed numbers of events would give us some idea of
> whether the new cohort differs from the reference one.
>
>> From my reading of the documentation and the text by Therneau and
> Grambsch, it seems that the function "survexp" is what I need. But
> using it I am not able to obtain expected numbers of events that match
> reasonably well the observed numbers *even for the same reference
> population.* So, I think I am misunderstanding something quite badly.
>

  You've hit a common confusion.  Observed versus expected events computations are done on 
a cumulative hazard scale H, not the surivival scale S; S = exp(-H).  Relating this back 
to simple Poisson models H(t) would be the expected number of events by time t and S(t) 
the probability of "no events before time t".  G. Berry (Biometrics 1983) has a classic 
ane readable article on this (especially if you ignore the proofs).

   Using your example:

 > cphfit <- coxph(Surv(start,stop,event)~rx+number+size+cluster(id),data=bladder2)
 > zz <- predict(cphfit, type='expected')
 > c(sum(zz), sum(bladder2$event))
[1] 112 112

 > tdata <- bladder2[1:10]   #new data set (lazy way)
 > predict(cphfit, type='expected', newdata=tdata)
  [1] 0.0324089 0.3226540 0.4213402 1.0560768 0.6702130 0.2163531 0.6490665
  [8] 0.8864808 0.2932915 0.5190647


  You can also do this using survexp and the cohort=FALSE argument, which would return 
S(t) for each subject and we would then use -log(result) to get H.  This is how it was 
done when I wrote the book, but the newer predict function is easier.

Terry Therneau