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Similar to this question, I would like a good metric for the distance between multiple ecdfs. However, in this case I have survival functions that are right-censored.

Question

Is there a metric that is commonly used with survival functions?

  • If not, is there any metric that weights divergence at earlier values of $t$ more than later values? Early failures are much less desirable than later ones, so it seems natural to devise a metric with such a property.

    • If not, does it make sense to devise one, and does anyone have any pointers for doing this? I have some intuition that cdfs might naturally penalize early failures, but I don't know enough about the subject to get from there to a metric.
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    $\begingroup$ Have you considered having a look at the log-rank test that compares two Kaplan-Meier curves? $\endgroup$ – ocram Mar 24 '13 at 15:11
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I've discovered that the Peto and Peto generalization of the Wilcoxon test fits this purpose. It weights divergence at earlier values of t more than later values. You can use the test statistic as the metric, or the p-value if you need to compare pairs of curves with different sample sizes.

This metric has the added bonus of being readily available in R as the survdiff function in the survival package. Some citations (and the R implemenation) point at the Harrington and Fleming $G^\rho$ family of tests, where $\rho = 1$ gives the Peto-Peto test described above.


[1] Peto, R., & Peto, J. (1972). Asymptotically efficient rank invariant test procedures. Journal of the Royal Statistical Society. Series A (General), 185-207.

[2] Harrington, D. P., & Fleming, T. R. (1982). A class of rank test procedures for censored survival data. Biometrika, 69(3), 553-566.

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