Log-Rank Test Calculator (Mantel-Cox Survival Analysis)
Calculate the Mantel-Cox log-rank test statistic (chi-square, df = 1), p-value, observed vs. expected events, hypergeometric variance, and hazard ratio from censored survival data.
Open toolKaplan-Meier Survival Calculator (Product-Limit & Greenwood CI)
Paste right-censored time-to-event data (`time, status`) to compute the Kaplan-Meier product-limit survival curve `S(t)`, Greenwood standard error, 95% log-log confidence intervals, restricted mean survival time (RMST), and median survival time `t_50`.
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How the Kaplan-Meier Product-Limit Calculator Works
Introduced by Edward L. Kaplan and Paul Meier in 1958, the product-limit estimator calculates non-parametric survival probabilities from incomplete (right-censored) follow-up records without assuming an exponential or Weibull distribution.
S(t) = prod_{t_i <= t} ((n_i - d_i) / n_i); Greenwood SE = S(t) * sqrt(sum d_i / (n_i * (n_i - d_i))); RMST(tau) = sum S(t_{i-1}) * (t_i - t_{i-1}).
If you need to compare two treatment arms statistically, use our Log-Rank Test Calculator (/tools/log-rank-test-calculator) alongside this curve estimator.
Kaplan-Meier Product-Limit & Greenwood Formulas Reference Table
Core survival formulas, tie-breaking rules, and exact outputs for the default 10-subject clinical follow-up cohort.
| Biostatistics Metric | Exact Formula / Rule | Default Cohort Output (`N = 10`, `6` Events) | Clinical / Engineering Meaning |
|---|---|---|---|
| Product-Limit Survival `S(t)` | `prod_{t_i <= t} (1 - d_i / n_i)` | `S(12) = 0.5486` (`54.86%`) | Probability of surviving past landmark time `t* = 12` |
| Greenwood Standard Error `SE(S(t))` | `S(t) * sqrt(sum [d_i / (n_i * (n_i - d_i))])` | `SE(12) = 0.1724` at `t* = 12` | Sampling uncertainty of the step survival estimate |
| 95% Complementary Log-Log CI | `S(t)^{exp(+- 1.96 * sigma_u)}` | `[0.1950, 0.8065]` at `t* = 12` | Range-preserving confidence interval bounded in `(0, 1)` |
| Median Survival Time (`t_50`) | `min { t_i : S(t_i) <= 0.50 }` | `t_50 = 15` time units (`S(15) = 0.4114`) | Earliest timestamp where half the cohort has experienced the event |
| Restricted Mean Survival (RMST) | `integral_0^{t_max} S(u) du` | `14.36` time units (up to `t = 24`) | Area under the Kaplan-Meier staircase up to horizon `t_max` |
How to Compute a Kaplan-Meier Survival Table in 3 Steps
Generate a publication-ready survival table and landmark confidence intervals in seconds.
Enter follow-up times and censoring indicators (1 = event, 0 = censored)
Paste one subject per row as 'time, status' where status = 1 indicates an observed event (death, relapse, component failure, churn) and status = 0 indicates a right-censored record.
Select Log-Log or Linear Greenwood confidence bounds and landmark time t*
Choose between the complementary log-log confidence interval (default in R survfit and SAS) or plain linear Greenwood bounds, and specify a landmark time t* to query.
Inspect median survival, RMST, and copy the step-by-step risk table
Review the landmark survival S(t*), median survival time t_50, restricted mean survival time (RMST), and export the complete step table showing n_i, d_i, c_i, S(t_i), SE, and CI.
Worked Kaplan-Meier Survival Examples
Verify the product-limit steps and Greenwood variance on real biostatistics scenarios.
Baseline example
Free Kaplan-Meier survival calculator. Compute product-limit survival S(t), Greenwood variance, log-log or linear confidence intervals, median survival time, and risk-set tables in your browser.
Comparison scenario
S(t) = prod_{t_i <= t} ((n_i - d_i) / n_i); Greenwood SE = S(t) * sqrt(sum d_i / (n_i * (n_i - d_i))); RMST(tau) = sum S(t_{i-1}) * (t_i - t_{i-1}).
Boundary check before exporting
If you need to compare two treatment arms statistically, use our Log-Rank Test Calculator (/tools/log-rank-test-calculator) alongside this curve estimator.
How to Interpret S(t), Greenwood SE, and Median Survival
Combine point survival estimates with the at-risk sample size n_i when reading the tail of a survival curve.
First, watch the risk set n_i in the final columns of follow-up. When heavy censoring reduces n_i below 5 subjects, a single late event causes a dramatic vertical drop in S(t) and widens the Greenwood 95% confidence interval.
Second, when fewer than 50% of subjects experience the event before the study ends, the median survival time t_50 is not reached; in that case, report the Restricted Mean Survival Time (RMST) up to the study horizon tau and the landmark survival probability S(tau).
Statistical Assumptions, Censoring Rules & Related Tools
Ensure your time-to-event dataset satisfies these three non-parametric assumptions.
1. Independent (Non-Informative) Censoring: Right-censored subjects at time t must have the same future hazard rate as subjects who remain under observation at time t.
2. Exact Tie-Handling Convention: When an event (1) and a censoring (0) share the exact same timestamp t, the event is evaluated first (so the censored subject contributes to the risk set n_i at time t) before being removed for t > t_i.
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Sample Size Calculator for Means
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Open toolFrequently Asked Questions About Kaplan-Meier Survival Analysis
Answers about using Kaplan-Meier Survival Calculator (Product-Limit & Greenwood CI) and understanding its results.
How does the Kaplan-Meier product-limit estimator treat right-censored subjects?
Subjects who are right-censored (status = 0, such as patients still alive at study close or lost to follow-up) remain in the at-risk denominator n_i for every event time up to and including their censoring timestamp. Immediately after their censoring time, they are removed from n_i without reducing the survival probability S(t).
Why is the log-log confidence interval preferred over the linear Greenwood formula?
Plain linear bounds S(t) +- z * SE(S(t)) can drop below 0.0 or exceed 1.0 near the start or tail of follow-up. Transforming survival onto the complementary log-log scale u = ln(-ln(S(t))) and exponentiating back ensures both lower and upper confidence bounds always remain strictly within (0, 1).
How is median survival time defined on a Kaplan-Meier step curve?
Median survival time t_50 is the smallest event time t_i at which the estimated survival probability S(t_i) falls to 0.50 (50%) or below. If S(t) remains strictly above 0.50 at the final observation, median survival is reported as 'Not reached' (> max follow-up).
When should I use the Kaplan-Meier Calculator vs. the Log-Rank Test Calculator?
Use this Kaplan-Meier Calculator to estimate the survival curve S(t), Greenwood confidence bands, and median survival for a single cohort. To test whether two cohorts' survival curves differ significantly, pass both groups into our Log-Rank Test Calculator (/tools/log-rank-test-calculator).
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