Variance of the Kaplan-Meier estimate for dependent observations Can someone help me find a way to estimate the variance of the Kaplan-Meier estimate with dependent observations? Specifically, I have failure time data from patients with several different observations for each patient (and different patients may have different number of observations). The observations for different patients are assumed to be independent but observations from the same patient are expected to be dependent.
I was suggested two publications, "Kaplan-Meier Analysis of Dental Implant Survival: A Strategy for Estimating Survival with Clustered Observations" and "The Kaplan-Meier Estimate for Dependent Failure Time Observations", the second of which is cited by the first.
However I was unable to make sense of these. The first has errors and the second seems far more rigorous but the equation for the variance does not make sense (double integration on a 1-form).
 A: In this situation, multiple events of the same type can occur in parallel within the same group. For the first reference, grouping is of multiple implanted teeth within the same individual. For the second reference, by Ying and Wei, litters are the groups, and a multiple event is more than 1 member of a litter developing a tumor.
This within-group dependence can be taken into account with an infinitesimal jackknife variance estimator, which can be thought of as related to removing one group at a time from the model.* In the R survival package, that's done for a simple Kaplan-Meier estimate by specifying an id variable for each group.
The data used in the example by Ying and Wei turn out to be a subset of the rats data included in the survival package (untreated females). The standard errors (SE) with the jackknife are very close to those reported by Ying and Wei.




time
KM estimate
SE, uncorrected
SE, Ying and Wei
SE, jackknife




70
0.9190
0.028
0.026
0.0265


80
0.8733
0.034
0.032
0.0320


90
0.8227
0.041
0.046
0.0466


100
0.8074
0.043
0.047
0.0477




Code that does this in R:
> library(survival)
> fitJackknife <- survfit(Surv(time,status)~1,data=rats, subset=rx==0&sex=="f",id=litter)
> summary(fitJackknife,times=c(70,80,90,100))
Call: survfit(formula = Surv(time, status) ~ 1, data = rats, subset = rx == 
    0 & sex == "f", id = litter)

 time n.risk n.event survival std.err lower 95% CI upper 95% CI
   70     88       8    0.919  0.0265        0.868        0.972
   80     71       4    0.873  0.0320        0.813        0.938
   90     63       4    0.823  0.0466        0.736        0.919
  100     50       1    0.807  0.0477        0.719        0.907


*The infinitesimal jackknife is a limiting case of the original jackknife, in which one observation at a time is removed (weight = 0) while others are maintained (weights = 1). It's the liming situation as weights approach 0. Therneau and Grambsch explain in Section 7.2 how that's implemented to get variance estimates in survival models.
