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.