Log-Rank Test Critical Value Calculator
What Is the Log-Rank Test?
The Log-Rank Test is a non-parametric hypothesis test used to compare the survival distributions of two or more groups. It answers one question: do the groups experience events at different rates over time? The test is the standard method for analyzing time-to-event data in clinical trials, cancer research, and epidemiology.
The test works by comparing how many events were observed in each group to how many were expected under the null hypothesis that all groups share the same underlying survival function. If observed and expected counts are sufficiently different, the null hypothesis is rejected.
The resulting test statistic follows a chi-square distribution with df = k − 1 degrees of freedom (where k is the number of groups). This page gives you the critical values to compare against that statistic — and a calculator to do the lookup in seconds.
Also known as: The Log-Rank Test is also called the Mantel-Cox test, honoring Nathan Mantel and David Cox. Most published research uses "Log-Rank Test" while some statistical textbooks prefer "Mantel-Cox." The test statistic, critical values, and interpretation are identical regardless of naming convention.
Log-Rank Test Critical Value Table
All values below are chi-square critical values (χ²crit) for the Log-Rank Test. If your computed test statistic ≥ the value in the table for your df and α, reject H₀ and conclude the survival curves differ significantly. Click any cell to load it into the calculator above.
df = k − 1, where k = number of groups. For two-group comparisons (treatment vs. control), df = 1. Critical values are chi-square quantiles at the given upper-tail probability. Values match standard chi-square distribution tables used in survival analysis textbooks.
Kaplan-Meier Survival Curves and the Log-Rank Test
The Log-Rank Test is nearly always paired with Kaplan-Meier (KM) survival curves. KM plots show the estimated probability of surviving (or remaining event-free) past each time point for each group. The Log-Rank Test gives the formal p-value for whether the curves differ beyond chance.
Illustrative Kaplan-Meier Survival Curves — Treatment vs. Control
Illustrative only. Tick marks (|) on curves indicate censored observations — patients lost to follow-up or who left the study without experiencing the event. The Log-Rank Test accounts for censoring when computing expected events.
The KM curve for each group drops whenever an event occurs. The vertical distances between the curves represent differences in survival probability at each time point. The Log-Rank Test evaluates whether these differences are collectively large enough to be statistically significant — not just at one time point, but across all event times simultaneously. For more on survival probability estimation, the National Cancer Institute's statistical training resources provide an authoritative explanation with clinical examples.
How to Perform the Log-Rank Test: Step by Step
Understanding the mechanics behind the test makes the critical value table meaningful. Each step below corresponds to a concrete calculation, not just a procedural checkbox.
Step 1 — State the Hypotheses
The null hypothesis is H₀: S₁(t) = S₂(t) for all time points t — that is, the survival functions of the groups are identical. The alternative is H₁: S₁(t) ≠ S₂(t) for some t. Choose α before examining the data. Most clinical research uses α = 0.05, though oncology trials often prespecify α = 0.025 for one-sided tests.
Step 2 — Determine Degrees of Freedom
df = k − 1, where k is the number of groups being compared.
The vast majority of survival analyses compare two groups (e.g., treatment vs. placebo), making df = 1 by far the most common case. The table covers df = 1 through 20 for multi-group comparisons.
Step 3 — Compute Observed and Expected Events
At each distinct event time tj, record: dj = total number of events; nj = total number at risk; nij = number at risk in group i. The expected events for group i at time tj are:
Sum Eij across all event times to get the total expected events Ei for group i. Do the same for observed events Oi.
Step 4 — Compute the Chi-Square Statistic
The Log-Rank test statistic compares total observed to total expected events:
For two groups, only one group's contribution is needed because O₁ + O₂ = E₁ + E₂ by constraint. Statistical software computes the full matrix version for multi-group comparisons.
Step 5 — Compare to the Critical Value
If χ² < χ²crit → Fail to reject H₀ → No significant difference detected at α
Step 6 — Report the Full Result
A complete report includes: the chi-square statistic with df, the exact p-value, and the hazard ratio (HR) with 95% confidence interval. The Log-Rank Test alone does not estimate the magnitude of the difference. Pair it with Cox regression for the HR. Example: χ²(1) = 5.23, p = 0.022; HR = 0.61 (95% CI: 0.40–0.93), favoring treatment.
Worked Example: Cancer Clinical Trial
Scenario: A phase III oncology trial randomizes 100 patients to receive either a new chemotherapy agent (n = 50) or standard-of-care placebo (n = 50). The outcome is time to disease progression (months). After 24 months of follow-up, the Kaplan-Meier curves are generated and the Log-Rank Test is performed.
Event Table (Selected Time Points)
| Time (months) | At Risk: Treatment | Events: Treatment | At Risk: Control | Events: Control | E (Treatment) |
|---|---|---|---|---|---|
| 3 | 50 | 2 | 50 | 4 | 3.00 |
| 6 | 47 | 3 | 46 | 6 | 4.35 |
| 12 | 38 | 4 | 34 | 8 | 5.86 |
| 18 | 28 | 3 | 20 | 6 | 3.58 |
| Total | O₁ = 22 | O₂ = 36 | E₁ = 28.1 |
Calculation
| Step | Calculation | Result |
|---|---|---|
| 1 | Hypotheses & α | H₀: S₁(t) = S₂(t); α = 0.05, df = 1 |
| 2 | Critical value (df=1, α=0.05) | χ²crit = 3.841 |
| 3 | Observed and expected | O₁ = 22, E₁ = 28.1 |
| 4 | Chi-square statistic | (22 − 28.1)² / 28.1 = 37.21 / 28.1 = 1.32 (treatment group contribution) |
| 5 | Full χ² (both groups) | χ²(1) = 5.23 |
| 6 | Decision: 5.23 vs 3.841 | 5.23 > 3.841 → Reject H₀ ✓ |
Clinical Interpretation
The Log-Rank Test is statistically significant: χ²(1) = 5.23, p = 0.022. The treatment group experienced significantly fewer progression events (O = 22) than expected under equal survival (E = 28.1), while the control group experienced more (O = 36, E = 29.9). The Kaplan-Meier curves confirm the treatment group maintained higher survival probability throughout follow-up. A Cox regression model would estimate the hazard ratio to quantify how much the treatment reduced the risk of progression — a necessary next step for clinical reporting. For guidance on reporting standards, the CONSORT guidelines specify how survival outcomes should be reported in randomized clinical trials.
Log-Rank Test vs. Cox Regression: When to Use Each
Both tests analyze time-to-event data, but they answer different questions and serve different purposes. Choosing the right method depends on your research question and what you need to report.
| Characteristic | Log-Rank Test | Cox Proportional Hazards |
|---|---|---|
| Purpose | Test whether survival curves differ | Estimate effect of predictors on survival |
| Output | χ² statistic, p-value | Hazard ratio (HR), 95% CI, p-value |
| Covariates | No adjustment for covariates | Adjusts for multiple covariates |
| Key assumption | Proportional hazards over time | Proportional hazards over time |
| Effect size | Does not provide HR directly | HR with CI — full effect size |
| Best used for | Unadjusted group comparison; KM plot p-value | Adjusted analysis; multivariable survival modeling |
Both methods assume proportional hazards. If hazard curves cross or the log of negative log survival curves are not parallel, consider the weighted log-rank test or restricted mean survival time (RMST) as alternatives.
Statistical Significance vs. Clinical Significance
A significant Log-Rank Test means the survival curves differ beyond what chance alone predicts. It says nothing about whether the difference is clinically meaningful. A trial with n = 2,000 may reach p < 0.001 for a survival difference of two weeks — statistically significant but clinically marginal. Always report median survival times, the hazard ratio, and confidence intervals alongside the p-value. This distinction between statistical and practical significance is covered in depth in the hypothesis testing guide at Statistics Fundamentals.
Log-Rank Test Assumptions
The Log-Rank Test makes four assumptions. Violating them does not invalidate your data — but it may change which test is most appropriate.
1. Proportional Hazards
The hazard ratio between groups must remain constant over time. If survival curves cross, the hazard ratio changes over time and the Log-Rank Test loses power. Check with log(-log) plots. If violated, consider the weighted log-rank test or restricted mean survival time.
2. Independent Censoring
Censoring must be independent of the event outcome. Patients who drop out because they are too sick to continue are not independently censored — their withdrawal is related to prognosis. Informative censoring biases survival estimates and inflates or deflates the test statistic.
3. Well-Defined Event
The event must be clearly defined, binary, and consistent across all participants. In cancer trials, "overall survival" (death from any cause) is unambiguous. "Disease-free survival" requires careful pre-specification. Ambiguous event definitions introduce measurement error that can distort the log-rank statistic.
4. Time-Independent Groups
Group membership (e.g., treatment assignment) should be fixed at baseline. If participants switch between groups during the study, a simple log-rank comparison is no longer valid. Time-varying covariates require extended Cox models or other specialized methods.
How to Read the Log-Rank Test Critical Value Table
The critical value table shows the minimum chi-square statistic needed to reject H₀ at each combination of degrees of freedom and significance level. Three examples illustrate direct table use.
Example A: Two-Group Cancer Trial (df = 1, α = 0.05)
You compare overall survival between a new immunotherapy and standard chemotherapy in 80 patients. The Log-Rank statistic is χ²(1) = 4.12. Look up df = 1, α = 0.05 in the table → χ²crit = 3.841. Since 4.12 > 3.841, reject H₀. The immunotherapy group has a statistically significantly different survival distribution at α = 0.05 (p ≈ 0.042).
Example B: Three-Group Drug Trial (df = 2, α = 0.05)
A trial with low-dose, high-dose, and placebo arms computes χ²(2) = 5.67. Look up df = 2, α = 0.05 → χ²crit = 5.991. Since 5.67 < 5.991, fail to reject H₀. The three survival curves do not differ significantly at α = 0.05 (p ≈ 0.059). A post-hoc pairwise comparison would still be exploratory only.
Example C: Two-Group Trial, Stricter Threshold (df = 1, α = 0.01)
A regulatory submission requires α = 0.01. Your χ²(1) = 7.12. Look up df = 1, α = 0.01 → χ²crit = 6.635. Since 7.12 > 6.635, reject H₀ at the stricter α = 0.01 threshold (p ≈ 0.008). This meets the pre-specified regulatory significance criterion.
Survival Analysis Glossary
The following terms appear throughout survival analysis literature and in the interpretation of the Log-Rank Test. Each definition is self-contained for direct citation.
| Term | Definition | Formula / Notation |
|---|---|---|
| Log-Rank Test | Non-parametric test comparing survival distributions across k groups by examining O vs. E at each event time | χ² = Σ(O−E)²/E, df = k−1 |
| Kaplan-Meier Estimator | Non-parametric estimator of the survival function S(t) — the probability of surviving past time t | Ŝ(t) = Π[(nj−dj)/nj] |
| Censored Observation | A subject whose event time is unknown because follow-up ended before the event occurred (lost to follow-up, study end, or withdrawal) | Marked as "+" in event tables |
| Hazard Ratio (HR) | Ratio of the hazard rate in one group to the hazard rate in another; HR < 1 favors the first group (reduced risk) | HR = h₁(t)/h₂(t) |
| Observed Events (O) | The actual number of events recorded in a group during the study period | O = Σ dij |
| Expected Events (E) | The number of events that would be expected in a group under H₀ of equal survival, weighted by group size at risk | Eij = (nij/nj) × dj |
| Proportional Hazards | Assumption that the hazard ratio between groups remains constant over time — required for both the Log-Rank Test and Cox regression | h₁(t)/h₂(t) = constant |
| Chi-Square Distribution | The probability distribution that the Log-Rank statistic follows under H₀; parameterized by degrees of freedom df = k−1 | χ²(df) ~ χ²(k−1) |
| Mantel-Cox Test | Alternative name for the Log-Rank Test, honoring Nathan Mantel and David Cox; the test statistic and critical values are identical | = Log-Rank Test |
| Cox Regression | A multivariable survival model that estimates the hazard ratio for each predictor while adjusting for other covariates; extends the Log-Rank framework | h(t|X) = h₀(t)·exp(βX) |
5 Common Mistakes When Using the Log-Rank Test
These errors appear frequently in published clinical research and student analyses. Recognizing them protects both the validity of your conclusions and the integrity of reported findings.
Mistake 1: Ignoring the Proportional Hazards Assumption
If the Kaplan-Meier curves cross or converge, the hazard ratio is not constant and the Log-Rank Test loses power — it may miss a real difference. Always inspect the log(-log) survival plot before applying the test. Crossing curves are common in immunotherapy trials where the treatment effect takes time to emerge.
Mistake 2: Reporting Only the p-Value
A Log-Rank p-value answers only "are the curves different?" It does not say how different, or by how much. Always report median survival times and the hazard ratio with confidence interval alongside the chi-square statistic and p-value. p < 0.05 is not a complete result.
Mistake 3: Multiple Comparisons Without Adjustment
When comparing more than two groups (df > 1) and the overall test is significant, researchers sometimes perform all pairwise Log-Rank tests without adjusting α. This inflates the Type I error rate. Use Bonferroni correction (divide α by the number of comparisons) or another multiple-comparisons procedure. See the significance level guide for details.
Mistake 4: Treating Censoring as Random When It Is Not
If patients are censored because of disease severity (e.g., they are too ill to continue), the censoring is informative — it is related to the outcome. The Log-Rank Test assumes censoring is independent of the event. Informative censoring produces biased survival estimates and misleading p-values.
Mistake 5: Using the Wrong Degrees of Freedom
df = k − 1, where k is the number of groups. A two-group comparison uses df = 1, not df = 2, and not df = n − 1. Looking up the critical value with the wrong df gives a wrong threshold and a wrong significance decision. The number of patients does not determine df for the Log-Rank Test.
Running the Log-Rank Test in R, Python, and SPSS
In practice, the Log-Rank Test is computed by software rather than by hand. The critical value table confirms the software's p-value or allows manual verification. Below are the core commands for the three most common statistical environments.
R (survival package)
km_fit <- survfit(
Surv(time, event) ~ group,
data = your_data
)
survdiff(
Surv(time, event) ~ group,
data = your_data
)
survdiff() returns the χ² statistic and p-value. Use ggsurvplot() from the survminer package for publication-quality KM curves.
Python (lifelines)
logrank_test
results = logrank_test(
T1, T2,
event_observed_A=E1,
event_observed_B=E2
)
results.print_summary()
The lifelines library provides the test statistic, p-value, and confidence intervals. The KaplanMeierFitter class produces survival curves for visualization.
SPSS
Kaplan-Meier...
Time: [time variable]
Status: [event variable]
Factor: [group variable]
Compare Factor: Log Rank ✓
→ OK
SPSS outputs the Log-Rank chi-square, df, and significance (p-value) in the "Test of Equality of Survival Distributions" table alongside KM curves.
Frequently Asked Questions About the Log-Rank Test
What is the Log-Rank Test?
The Log-Rank Test is a non-parametric hypothesis test that compares the survival distributions of two or more groups by examining observed versus expected events at each event time. The test statistic follows a chi-square distribution with df = k − 1 under the null hypothesis of equal survival curves. It is the standard test for Kaplan-Meier survival analyses in clinical research.
How do you read the Log-Rank Test critical value table?
Determine your degrees of freedom (df = k − 1, where k = number of groups) and your significance level α. Find the row for your df and the column for your α. The cell value is χ²crit. If your computed Log-Rank statistic ≥ χ²crit, the result is statistically significant — reject H₀. If your computed value is less than χ²crit, fail to reject H₀.
Is the Log-Rank Test the same as the Mantel-Cox test?
Yes — completely. The Log-Rank Test and Mantel-Cox test refer to the same procedure. "Log-Rank" is the dominant term in clinical literature; "Mantel-Cox" appears in some statistical software outputs (including SPSS) and older textbooks. The chi-square test statistic, degrees of freedom, critical values, and interpretation are all identical regardless of which name you encounter.
What degrees of freedom does the Log-Rank Test use?
df = k − 1, where k is the number of groups. Two groups → df = 1. Three groups → df = 2. Four groups → df = 3. The sample size does not determine degrees of freedom in the Log-Rank Test — only the number of groups does. For the most common two-group clinical comparison, df = 1 always applies.
What does a significant Log-Rank Test actually mean?
A significant Log-Rank Test (p < α) means the observed difference between survival curves is larger than would be expected by chance under H₀ of equal survival. In practice: one group experienced events sooner than the other, or more overall. It does not quantify the size of that difference — for that, report the hazard ratio and confidence interval from Cox regression.
When should you not use the Log-Rank Test?
Avoid the Log-Rank Test when the proportional hazards assumption is violated (curves cross or converge substantially), when censoring is informative rather than independent, or when you need to adjust for confounders — in that case, use Cox regression. For very early treatment effects or delayed effects (common in immunotherapy), weighted log-rank tests or restricted mean survival time (RMST) are more appropriate alternatives.
How is the Log-Rank Test related to chi-square critical values?
The Log-Rank Test statistic follows a chi-square distribution under the null hypothesis. This means the critical values in the Log-Rank Test table are identical to the critical values in a standard chi-square critical value table at the same df and α. The difference is context: the chi-square table is general, while the Log-Rank table is interpreted specifically in terms of survival curves, observed events, and expected events.
What is the Log-Rank Test p-value of 0.05 exactly?
For df = 1, the chi-square value that corresponds exactly to p = 0.05 (two-sided) is χ² = 3.841. For df = 2, it is χ² = 5.991. A computed Log-Rank statistic that just meets these thresholds sits exactly on the boundary of α = 0.05. Standard practice is to report the exact p-value rather than just whether it crosses 0.05.
Sources & Further Reading
The critical values in this table are chi-square quantiles computed from the chi-square distribution, validated against the following authoritative sources:
Kleinbaum, D. G., & Klein, M. (2012). Survival Analysis: A Self-Learning Text (3rd ed.). Springer. Chapter 2 covers Kaplan-Meier estimation; Chapter 2 and 6 derive the Log-Rank Test statistic and its chi-square distribution. The standard graduate-level survival analysis reference used in biostatistics programs worldwide. doi:10.1007/978-1-4419-6646-9
Mantel, N. (1966). "Evaluation of survival data and two new rank order statistics arising in its consideration." Cancer Chemotherapy Reports, 50(3), 163–170. The original paper proposing the log-rank statistic for comparing survival data, establishing the chi-square test framework used in this table.
National Cancer Institute (2024). Survival Analysis in Clinical Trials. U.S. Department of Health and Human Services. seer.cancer.gov — Open-access NCI training resource covering Kaplan-Meier curves, the Log-Rank Test, and clinical interpretation with cancer research examples.
Collett, D. (2015). Modelling Survival Data in Medical Research (3rd ed.). CRC Press. Chapters 2–4 cover the log-rank and related tests with detailed derivations of the test statistic and its distribution. doi:10.1201/b18041
Penn State STAT 421: Applied Nonparametric Statistics. Lesson 13: Log-Rank Test. Pennsylvania State University. online.stat.psu.edu — Open-access course notes detailing the log-rank statistic, degrees of freedom, and step-by-step hypothesis testing procedure.
Related Statistical Tables & Resources
Understanding What the Log-Rank Test Table Tells You
Why df = 1 Is the Standard Case
The vast majority of survival analyses compare exactly two groups — a treatment arm and a control arm. With k = 2 groups, df = k − 1 = 1 always applies. This is why the row for df = 1 in the critical value table is the one most clinical researchers and graduate students reach for first. For df = 1 at α = 0.05, the threshold is χ² = 3.841 — memorizing this single value covers most two-group clinical comparisons.
How Censoring Affects the Test
Censoring is a feature of survival data, not a problem. The Log-Rank Test accounts for censoring by only comparing subjects at each time point among those still at risk. A censored observation contributes to the at-risk count until the time of censoring, then exits the analysis. This is why large censoring rates do not automatically invalidate the test — as long as censoring is independent of the event outcome. See the study design section for how censoring is handled in clinical trial protocols.
The Log-Rank Test and Chi-Square Tables
The Log-Rank Test statistic follows a chi-square distribution, so its critical values are identical to those in any chi-square critical value table at the corresponding df. The distinction is interpretation: a chi-square test compares frequencies in a contingency table, while the Log-Rank Test compares observed vs. expected survival events across time. The mathematics are the same; the research context is different. Both tables are available on Statistics Fundamentals.