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I have taken the survival analysis class a few years ago, and we have learned both Kaplan-Meier estiamtes for survival curves, and the log-rank test for inferencing the difference of hazard rate between 2 groups.

That being said, I can use software (such as proc lifetest in SAS or survival package in R) to do KM estimates or log-rank test seaprately.

My question is though, what is the theoretical link between these 2? Is there any connection in between? e.g., is the test statsitic of (stratified) log-rank test based on the KM-estimate?

The reason I am asking is because, recently I see in a document (for confidential purpose I could not mention from where), that a stratified log-rank test is based on KM method, but I do not understand why, as the document did not give any further detail exept referring to the literature by Kaplan and Meier back in 1958 (Nonparametric estimation from incomplete observations).

I checked the referred literature (KM 1958) but it only layed out the deriation of KM estimates, without mentioning anything about log-rank test.

Could someone help?

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The Kaplan-Meier method is a popular way to estimate survival over time for a group of individuals when some survival times might be censored. It doesn't provide a test per se.

The log-rank test is one of several related ways to test whether there is are survival differences between two or more groups of individuals. These tests are based on differences between the observed numbers of events at event times and the number expected if, at each event time, all groups had the same hazard. The tests differ in the weight placed on different times; the log-rank test weights all times equally. See this answer for some details on the log-rank version.

The derivation by Klein and Moeschberger in Section 7.3 is based on a different survival estimator, the Nelson-Aalen estimator. The log-rank test is nevertheless a typical default for evaluating differences among survival curves however they are estimated.

You might be recalling that the log-rank test is equivalent to the score test for differences between groups in a Cox proportional hazards model. As the log-rank test is limited to mutually exclusive categories, doesn't allow for covariate adjustment, and doesn't extend to Bayesian analysis, Frank Harrell says that "we no longer need to be using or teaching the log-rank test."

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  • $\begingroup$ Thanks for the help, and I can definitely echo with your explanation in the first paragraph of the answer. That is my understanding initially as well, that KM is a method for survival estimates only, and log-rank test is for test of the difference, until I saw in a statistical analysis plan that I was recently reviewing, where it mentions that a significance test is to be performed with "stratified" log-rank test based on KM methods, which makes me wonder whether theory or equation-wise there is some connection between the two. $\endgroup$ Aug 17, 2022 at 1:40
  • $\begingroup$ @battousai1983 there is a "stratified" form of the log-rank test in which you stratify by a covariate that isn't of primary interest, do log-rank tests separately within each stratum for the predictor of primary interest, then combine results. I found a brief explanation here. I don't know that it's "based on KM methods" except that KM curves are inherently stratified. $\endgroup$
    – EdM
    Aug 17, 2022 at 12:56

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