How to calculate the confidence interval of the survival curve based on Kaplan-Meier estimation? The general used formula doesn't seem to be the correct one. As the time goes on, the sample size will shrink, which widen the CI range dramatically. When refer to the picture, the black dot lines are the CI given by statistical software, while the red crosses are the calculated from the estimated survival and standard error.
You should wake up and be alarmed after seeing that your presumed "CI" includes negative values.
Readings:
Newcombe, Robert G. "Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods," Statistics in Medicine, 17, 857-872 (1998).
http://www.barkerstats.com/PDFs/tas.pdf
http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_lifetest_a0000000263.htm
http://www.uni-kiel.de/psychologie/rexrepos/posts/survivalKM.html
and, surely,
https://www.google.de/search?q=confidence+interval+kaplan+meier
You should give some information about your data and the formulae/softwares used. The gap between the curves on your figure is very large, but I wonder if the display might show confidence intervals (for the estimation of the "mean" curve) and prediction intervals (for the population).
You should wake up and be alarmed after seeing that your presumed "CI" includes negative values.
Readings:
Newcombe, Robert G. "Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods," Statistics in Medicine, 17, 857-872 (1998).
http://www.barkerstats.com/PDFs/tas.pdf
http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_lifetest_a0000000263.htm
http://www.uni-kiel.de/psychologie/rexrepos/posts/survivalKM.html
and, surely,
https://www.google.de/search?q=confidence+interval+kaplan+meier
Following on Jochen's point, the confidence interval calculation for a proportion that you see in a textbook is hopeless, and will give negative values when these are undefined and include zero when this value is impossible.
The big paper that informed current practice was Brown and Dasgupta http://projecteuclid.org/euclid.ss/1009213286 which is a slightly later paper than Jochen's.
Stata's Kaplan-Meier command uses the method of Kalbfleisch and Prentice, 2002, The Statistical Analysis of Failure Time Data, page 18
I have studied this issue with Erica Brittain in a recent paper in Statistics in Medicine (2016, DOI: 10.1002/sim.6905, Finite sample pointwise confidence intervals for a survival distribution with right-censored data). We deal with the fact that as time goes on the sample size is shrinking. Also, when there is no censoring our confidence interval for the survival function at time t, say S(t), gives the exact binomial confidence interval (i.e., the Clopper-Pearson interval). We also have an R package called bpcp that does all the calculations.
Please refer the attached file as an additional information.
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