Kaplan-Meier Survival Curve and Log-Rank Test Calculator

Comparing survival times between two groups calls for neither a t-test nor a comparison of means. The reason is censoring: some subjects are still alive at the end of the study, or are lost to follow-up — you only know they survived “at least this long,” not their exact time. Treating these subjects as “dying at the last follow-up” underestimates survival; dropping them outright overestimates it. Both approaches are wrong.

The Kaplan-Meier estimator and log-rank test were designed specifically to handle censoring.

Two steps

Kaplan-Meier：  S(t) = Π over each event time ti ≤ t  (1 − d_i / r_i)
                d_i = events at that time,  r_i = number at risk at that time

log-rank：      form a 2×2 table at each event time, then combine
                e_j = d_j · r0_j / r_j                        (expected events in group 0)
                v_j = r1_j · r0_j · d_j · (r_j − d_j) / [r_j² · (r_j − 1)]
                χ² = [Σ(o_j − e_j)]² / Σ v_j                  (df = 1)

Censored subjects remain in the risk set until the moment of censoring, then exit. That is the entire key: they contribute information about surviving to that point, but nothing about what follows.

Why median survival may be undefined

Median survival is the first time point at which S(t) ≤ 0.5. If the curve remains above 0.5 at the end of follow-up, median survival is not reached: it cannot be extrapolated, nor written as “greater than X” and used as a point estimate. This tool reports “not reached” in that situation.

Verification

Using the leukemia remission data from Cox & Oakes, Analysis of Survival Data, Table 1.1 (Freireich 1963, 6-MP vs placebo, 21 subjects per group, time in weeks):

The reference values are from the University of Michigan Biostatistics lecture notes (Yi Li), which provide a complete worked derivation of this table. The tool loads this dataset by default, so you can verify immediately.

The sign reversal in Σ(o − e) is correct: the lecture notes use the placebo group as reference; this tool uses Group 1 (6-MP in the default data). The absolute value is identical. Since χ² squares the sum, the choice of reference group does not affect it.

Matching intermediate quantities is a stronger check than matching only the final statistic — two errors can cancel in the final value, but matching both Σ(o−e) and Σv confirms every step is correct.

Scope and limitations

• Log-rank answers “is there a difference between the two curves,” not “how large is the difference.” No matter how small the p-value, it is not an effect size. A hazard ratio (HR) requires Cox proportional-hazards regression, which this tool does not perform. The reason is not merely convenience: using the leukemia data above, four defensible algorithms yield four different HRs — O/E ratio 0.239, Peto one-step 0.194, Cox with Breslow tie-handling 0.221, Cox with Efron 0.208 — a range of 23%. Even “run Cox regression” is not a single, unambiguous answer: the same model, varying only how simultaneous events (ties) are handled, differs by 6%. Reporting an HR therefore requires stating the algorithm used, and an online tool that just hands you a number cannot do that — which is why this tool only performs the test and reports no effect size.
• Log-rank assumes proportional hazards: the hazard ratio between the two groups should be roughly constant throughout follow-up. If the two curves cross, log-rank will cancel out the opposing differences before and after the crossing and conclude “no difference” — when in fact the difference may be substantial, just reversed in direction. When curves cross, consider the Renyi test or restricted mean survival time (RMST) instead.
• Only two-group comparisons are supported. Multiple groups require a generalized log-rank test (df = number of groups − 1), which this tool does not implement.
• Censoring must be non-informative (independent of prognosis). If sicker subjects are more likely to be lost to follow-up, KM will systematically overestimate survival — this cannot be corrected by the method itself and must be addressed in study design.
• The χ² approximation is unreliable with few events. This tool displays a warning when the total event count is below 10.