Test whether two right-censored time-to-event curves differ significantly: compute pooled risk sets, observed vs. expected events (`O_1`, `E_1`, `O_2`, `E_2`), exact hypergeometric variance (`V_1`), Mantel-Cox `chi^2` (`df = 1`), two-sided `p`-value, and Hazard Ratio.
Enter the inputs, then select Calculate Log-Rank Test.
Result will appear here
The log-rank test treats each distinct failure timestamp `t_j` as a `2 x 2` contingency table conditioned on the number of subjects still at risk just prior to `t_j`.
At each distinct event time t_j: Expected E_{1j} = d_j * (n_{1j} / n_j), Hypergeometric Variance V_j = (n_{1j} * n_{2j} * d_j * (n_j - d_j)) / (n_j^2 * (n_j - 1)); Log-Rank Statistic chi^2 = (sum(O_{1j} - E_{1j}))^2 / sum(V_j) with df = 1; Mantel-Haenszel Hazard Ratio HR = (O_1 / E_1) / (O_2 / E_2).
Requires exactly two groups with positive follow-up times (t > 0) and at least one uncensored event (status = 1). Assumes non-informative right censoring and proportional hazards across follow-up.
Chi-square critical thresholds (`df = 1`), hazard ratio interpretation, and exact risk-set outputs for the default 12-subject dataset.
| Metric / Scenario | Formula / Rule | Default Dataset Value (`Group A` vs `Group B`) | Statistical Interpretation |
|---|---|---|---|
| **Group 1 Observed vs. Expected (`O_1`, `E_1`)** | `O_1 = sum(d_{1j})`, `E_1 = sum(d_j * n_{1j} / n_j)` | `O_1 = 5.00`, `E_1 = 2.1069` | Group A had 2.8931 more events than expected under `H_0` |
| **Group 2 Observed vs. Expected (`O_2`, `E_2`)** | `O_2 = sum(d_{2j})`, `E_2 = sum(d_j * n_{2j} / n_j)` | `O_2 = 3.00`, `E_2 = 5.8931` | Group B had 2.8931 fewer events than expected under `H_0` |
| **Pooled Hypergeometric Variance (`V_1`)** | `sum((n_{1j}*n_{2j}*d_j*(n_j-d_j)) / (n_j^2*(n_j-1)))` | `V_1 = 1.2509` across `7` distinct event times | Accounts for tied failures at `t = 10` (`d = 2`) |
| **Mantel-Cox Chi-Square & p-Value** | `chi^2 = (O_1 - E_1)^2 / V_1`, `p = erfc(sqrt(chi^2 / 2))` | `chi^2 = 6.6911` (`df = 1`), `p = 0.0097` | Rejects `H_0` at both `alpha = 0.05` and `alpha = 0.01` |
| **Pike / Mantel-Haenszel Hazard Ratio (`HR`)** | `HR = (O_1 / E_1) / (O_2 / E_2)` | `HR = 4.6617` (`A vs. B`) | Group A hazard rate is `4.66x` higher than Group B |
Enter the values for your scenario, verify the units, and run the calculation.
Start by entering the required fields in the Log-Rank Test Calculator (Mantel-Cox Survival Analysis) form above.
Match each value to the unit or format shown next to the input box before running the calculation.
Select Calculate Log-Rank Test, inspect the primary metric and breakdown cards, then use Copy result or Download result.txt.
Compare baseline and alternative runs for Log-Rank Test Calculator (Mantel-Cox Survival Analysis) before exporting the result.
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.
At each distinct event time t_j: Expected E_{1j} = d_j * (n_{1j} / n_j), Hypergeometric Variance V_j = (n_{1j} * n_{2j} * d_j * (n_j - d_j)) / (n_j^2 * (n_j - 1)); Log-Rank Statistic chi^2 = (sum(O_{1j} - E_{1j}))^2 / sum(V_j) with df = 1; Mantel-Haenszel Hazard Ratio HR = (O_1 / E_1) / (O_2 / E_2).
Requires exactly two groups with positive follow-up times (t > 0) and at least one uncensored event (status = 1). Assumes non-informative right censoring and proportional hazards across follow-up.
Review the calculated output alongside your entered parameters and unit conventions.
Requires exactly two groups with positive follow-up times (t > 0) and at least one uncensored event (status = 1). Assumes non-informative right censoring and proportional hazards across follow-up.
Confirm these three formatting rules before running your test.
At each distinct event time t_j: Expected E_{1j} = d_j * (n_{1j} / n_j), Hypergeometric Variance V_j = (n_{1j} * n_{2j} * d_j * (n_j - d_j)) / (n_j^2 * (n_j - 1)); Log-Rank Statistic chi^2 = (sum(O_{1j} - E_{1j}))^2 / sum(V_j) with df = 1; Mantel-Haenszel Hazard Ratio HR = (O_1 / E_1) / (O_2 / E_2). Requires exactly two groups with positive follow-up times (t > 0) and at least one uncensored event (status = 1). Assumes non-informative right censoring and proportional hazards across follow-up.
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Open toolAnswers about using Log-Rank Test Calculator (Mantel-Cox Survival Analysis) and understanding its results.
Right-censored subjects (patients lost to follow-up or still event-free when the study ends, coded with status = 0) do not trigger an event row of their own, but they remain in the at-risk set n_{1j} or n_{2j} for every event time t_j less than or equal to their censoring time. Once time passes their censored timestamp, they drop out of subsequent at-risk denominators.
The exact Mantel-Cox variance V_j = (n_{1j} * n_{2j} * d_j * (n_j - d_j)) / (n_j^2 * (n_j - 1)) accounts for tied events at the same timestamp t_j and finite-population sampling without replacement from the risk set n_j. Summing V_j across all event times reproduces the exact test statistic output by R survival::survdiff and SAS PROC LIFETEST.
The ratio O_1 / E_1 measures the relative event rate in Group 1 compared to the null expectation under identical survival curves. Dividing (O_1 / E_1) by (O_2 / E_2) gives the Pike / Mantel-Haenszel hazard ratio estimate: HR > 1 means Group 1 experiences events at a higher hazard rate than Group 2, while HR < 1 indicates a lower hazard rate in Group 1.
Because the log-rank test weights every distinct event time equally (weight w_j = 1), it is maximally powerful when the two hazard functions stay proportional over time. If the two survival curves cross mid-study (for example, high early surgical risk followed by better late survival), positive and negative (O_{1j} - E_{1j}) terms cancel out.
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