Log-Rank Test

Created:December 10, 2025

Last Updated:December 10, 2025

The Log-Rank Test (also known as the Mantel-Cox test) is a non-parametric hypothesis test used to compare survival distributions between two or more groups. It tests the null hypothesis that there is no difference in survival between groups, making it essential for comparing treatment effects in clinical trials and other time-to-event studies.

What You'll Get:

Test Statistic: Chi-squared value measuring the difference between groups
P-value: Statistical significance of the group difference
Degrees of Freedom: Number of groups minus one
Survival Curves: Interactive plot showing survival by group
Group Statistics: Sample size, events, and median survival for each group
Expected vs Observed Events: Comparison table for each group
APA-Formatted Report: Professional results ready for publication

💡 Pro Tip: Your data must include a time column (time to event or censoring), an event column (1 = event occurred, 0 = censored), and a group column (to identify which group each observation belongs to). The test requires at least 2 groups. For survival curve estimation without group comparison, use Kaplan-Meier Estimator. For modeling multiple covariates, consider Cox Proportional Hazards Model.

Ready to compare survival curves between groups? (clinical trial comparing Treatment vs. Control) to see the Log-Rank Test in action, or upload your own time-to-event data with group labels.

Calculator

1. Load Your Data

2. Select Columns & Options

Time Column *

Time to event or censoring

Event Column *

1 = event, 0 = censored

Group Column *

Required for group comparison

Confidence Level:

Related Calculators

Kaplan-Meier Estimator Calculator

Cox Proportional Hazards Model Calculator

Independent Samples t-Test Calculator

Learn More

Definition

The Log-Rank Test (also called the Mantel-Cox test or Mantel-Haenszel test) is a non-parametric hypothesis test that compares the survival distributions of two or more groups. It tests whether there are significant differences in the survival experiences between groups by comparing the observed number of events in each group to what would be expected if survival was the same across all groups.

The Log-Rank test statistic:

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}

where $O_i$ is the observed number of events in group $i$ , $E_i$ is the expected number of events, and $k$ is the number of groups. The statistic follows a chi-squared distribution with $k-1$ degrees of freedom.

Data Format Requirements

Correct Data Format

Time	Event	Group	Meaning
5	1	Treatment	Event occurred
8	0	Treatment	Censored
12	1	Control	Event occurred
15	0	Control	Censored
18	1	Treatment	Event occurred

Time: Any non-negative number (days, months, years, etc.)

Event: 1 = event occurred, 0 = censored

Group: Any text labels to identify groups (e.g., Treatment, Control, Drug A, Drug B)

Common Data Issues

Event column has wrong values

Time	Event	Group
5	yes	Treatment
8	no	Control

❌ Event must be 1 (event) or 0 (censored)

Negative time values

Time	Event	Group
-5	1	Treatment
8	0	Control

❌ Time values must be zero or positive

Missing required columns

Time	Event	Group
5	—	Treatment
8	—	Control

❌ Missing Event column

Visual Explanation: Comparing Groups

Log-Rank Test: Comparing Survival Curves

How the Log-Rank Test Works:

1. Compares at each event time: At each time an event occurs, counts how many events happened in each group

2. Expected vs Observed: Calculates how many events we'd expect if groups were identical, then compares to actual observations

3. Chi-squared statistic: Sums up differences across all event times: Σ(Observed - Expected)² / Expected

4. p-value: If curves diverge significantly (like above), p-value is small → groups differ significantly

When to Use Log-Rank Test

Use the Log-Rank Test when:

Comparing two or more groups: You have survival data from multiple groups and want to test if they differ
Time-to-event data: You're analyzing survival, failure, or duration outcomes
Censored observations: Some subjects have incomplete follow-up
Non-parametric test needed: You don't want to assume a specific distribution
Proportional hazards assumption: The hazard ratio between groups is relatively constant over time
Overall survival comparison: You want to test the entire survival experience, not specific time points

Common Applications

Clinical trials: Comparing survival between treatment and control groups
Drug effectiveness: Testing if a new drug improves survival compared to standard treatment
Prognostic factors: Comparing survival between different patient subgroups (e.g., disease stages)
Quality control: Comparing product lifetimes across manufacturing batches
A/B testing: Comparing time to conversion or churn between user groups
Epidemiology: Comparing disease-free survival across populations

Requirements & Assumptions

Independent groups: Observations in different groups should be independent
Independent censoring: Censoring should be unrelated to the event probability
Proportional hazards: Hazard ratio should be relatively constant over time (though the test is robust to mild violations)
At least 2 groups: Need at least two groups to compare
Sufficient events: Each group should have a reasonable number of events for reliable results
No tied event times within risk sets: Or use appropriate correction methods

Interpreting Results

Anatomy of a Survival Curve

Survival Curve: Steps down when events occur, stays flat when observations are censored

Confidence Band: Shows uncertainty in survival estimates (widens over time)

Median Survival: Time when survival probability reaches 50%

Test Statistic (χ²):

Larger values indicate greater differences between groups
Follows a chi-squared distribution with (k-1) degrees of freedom

P-value:

p < 0.05: Survival curves differ significantly between groups (reject null hypothesis)
p ≥ 0.05: No significant difference detected (fail to reject null hypothesis)

Expected vs. Observed Events:

If observed events exceed expected: Group has worse survival than average
If observed events are less than expected: Group has better survival than average

Visual Inspection:

Non-overlapping or clearly separated curves suggest significant differences
Crossing curves may indicate violation of proportional hazards assumption

Example Code

R Code

library(survival)
library(survminer)
library(tidyverse)

# Clinical trial data: time to recovery (days), event (1=recovered, 0=censored)
data <- tibble(
  time = c(5, 8, 12, 15, 18, 23, 25, 30, 35, 40, 45, 48, 50, 55, 60),
  event = c(1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1),
  group = c('Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment',
            'Treatment', 'Treatment', 'Treatment', 'Control', 'Control',
            'Control', 'Control', 'Control', 'Control', 'Control')
)

# Perform Log-Rank Test (also known as Mantel-Cox test)
logrank_result <- survdiff(Surv(time, event) ~ group, data = data)

# Print results
print(logrank_result)

# Extract key statistics
chi_squared <- logrank_result$chisq
df <- length(logrank_result$n) - 1
p_value <- 1 - pchisq(chi_squared, df)

cat("\nLog-Rank Test Results:\n")
cat("Chi-squared statistic:", round(chi_squared, 3), "\n")
cat("Degrees of freedom:", df, "\n")
cat("p-value:", format.pval(p_value, digits = 3), "\n")

# Visualize survival curves by group
km_fit <- survfit(Surv(time, event) ~ group, data = data)

ggsurvplot(km_fit,
           conf.int = TRUE,
           pval = TRUE,
           risk.table = TRUE,
           xlab = "Time (days)",
           ylab = "Survival Probability",
           title = "Survival Curves by Group",
           legend.title = "Group",
           legend.labs = levels(factor(data$group)))

Python Code

Python

import numpy as np
import pandas as pd
from lifelines import KaplanMeierFitter
from lifelines.statistics import logrank_test, multivariate_logrank_test
import matplotlib.pyplot as plt

# Clinical trial data: time to recovery (days), event (1=recovered, 0=censored)
data = pd.DataFrame({
    'time': [5, 8, 12, 15, 18, 23, 25, 30, 35, 40, 45, 48, 50, 55, 60],
    'event': [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1],
    'group': ['Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment',
              'Treatment', 'Treatment', 'Treatment', 'Control', 'Control',
              'Control', 'Control', 'Control', 'Control', 'Control']
})

# Log-Rank Test for two groups
treatment_data = data[data['group'] == 'Treatment']
control_data = data[data['group'] == 'Control']

results = logrank_test(
    treatment_data['time'], control_data['time'],
    treatment_data['event'], control_data['event']
)

print("Log-Rank Test Results:")
print(f"Test statistic: {results.test_statistic:.3f}")
print(f"p-value: {results.p_value:.4f}")
print(results.summary)

# For more than 2 groups, use multivariate_logrank_test
# result = multivariate_logrank_test(data['time'], data['group'], data['event'])
# print(result.summary)

# Plot survival curves by group
fig, ax = plt.subplots(figsize=(10, 6))

for group in data['group'].unique():
    group_data = data[data['group'] == group]
    kmf = KaplanMeierFitter()
    kmf.fit(
        durations=group_data['time'],
        event_observed=group_data['event'],
        label=group
    )
    kmf.plot_survival_function(ax=ax, ci_show=True)

plt.title(f'Survival Curves by Group (p={results.p_value:.4f})')
plt.xlabel('Time (days)')
plt.ylabel('Survival Probability')
plt.ylim(0, 1)
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()

Choosing the Right Test

Which Survival Analysis Test Should I Use?

I have time-to-event data

What is my goal?

Describe/estimate survival

Kaplan-Meier Estimator

Get survival curves & median survival

Compare 2+ groups

Log-Rank Test

Test if groups differ significantly

Model multiple factors

Cox Regression

Adjust for covariates & hazard ratios

Kaplan-Meier

• Estimate survival curves
• Find median survival time
• Visualize survival by group
• No formal hypothesis test

Log-Rank Test

• Compare 2+ groups
• Get p-value for difference
• Test overall survival
• Single grouping variable

Cox Regression

• Multiple covariates
• Hazard ratios
• Adjust for confounders
• Most comprehensive

Quick Decision Guide:

Start with Kaplan-Meier to visualize your data and understand survival patterns
Use Log-Rank Test when you have 2+ groups and want to test if they differ (e.g., treatment vs control)
Use Cox Regression when you have multiple variables (age, treatment, stage, etc.) and want to model their effects

Log-Rank Test

Created:December 10, 2025

Last Updated:December 10, 2025

What You'll Get:

Test Statistic: Chi-squared value measuring the difference between groups
P-value: Statistical significance of the group difference
Degrees of Freedom: Number of groups minus one
Survival Curves: Interactive plot showing survival by group
Group Statistics: Sample size, events, and median survival for each group
Expected vs Observed Events: Comparison table for each group
APA-Formatted Report: Professional results ready for publication

Ready to compare survival curves between groups? (clinical trial comparing Treatment vs. Control) to see the Log-Rank Test in action, or upload your own time-to-event data with group labels.

Calculator

1. Load Your Data

2. Select Columns & Options

Time Column *

Time to event or censoring

Event Column *

1 = event, 0 = censored

Group Column *

Required for group comparison

Confidence Level:

Related Calculators

Kaplan-Meier Estimator Calculator

Cox Proportional Hazards Model Calculator

Independent Samples t-Test Calculator

Learn More

Definition

The Log-Rank test statistic:

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}

Data Format Requirements

Correct Data Format

Time	Event	Group	Meaning
5	1	Treatment	Event occurred
8	0	Treatment	Censored
12	1	Control	Event occurred
15	0	Control	Censored
18	1	Treatment	Event occurred

Time: Any non-negative number (days, months, years, etc.)

Event: 1 = event occurred, 0 = censored

Group: Any text labels to identify groups (e.g., Treatment, Control, Drug A, Drug B)

Common Data Issues

Event column has wrong values

Time	Event	Group
5	yes	Treatment
8	no	Control

❌ Event must be 1 (event) or 0 (censored)

Negative time values

Time	Event	Group
-5	1	Treatment
8	0	Control

❌ Time values must be zero or positive

Missing required columns

Time	Event	Group
5	—	Treatment
8	—	Control

❌ Missing Event column

Visual Explanation: Comparing Groups

Log-Rank Test: Comparing Survival Curves

How the Log-Rank Test Works:

1. Compares at each event time: At each time an event occurs, counts how many events happened in each group

2. Expected vs Observed: Calculates how many events we'd expect if groups were identical, then compares to actual observations

3. Chi-squared statistic: Sums up differences across all event times: Σ(Observed - Expected)² / Expected

4. p-value: If curves diverge significantly (like above), p-value is small → groups differ significantly

When to Use Log-Rank Test

Use the Log-Rank Test when:

Comparing two or more groups: You have survival data from multiple groups and want to test if they differ
Time-to-event data: You're analyzing survival, failure, or duration outcomes
Censored observations: Some subjects have incomplete follow-up
Non-parametric test needed: You don't want to assume a specific distribution
Proportional hazards assumption: The hazard ratio between groups is relatively constant over time
Overall survival comparison: You want to test the entire survival experience, not specific time points

Common Applications

Clinical trials: Comparing survival between treatment and control groups
Drug effectiveness: Testing if a new drug improves survival compared to standard treatment
Prognostic factors: Comparing survival between different patient subgroups (e.g., disease stages)
Quality control: Comparing product lifetimes across manufacturing batches
A/B testing: Comparing time to conversion or churn between user groups
Epidemiology: Comparing disease-free survival across populations

Requirements & Assumptions

Independent groups: Observations in different groups should be independent
Independent censoring: Censoring should be unrelated to the event probability
Proportional hazards: Hazard ratio should be relatively constant over time (though the test is robust to mild violations)
At least 2 groups: Need at least two groups to compare
Sufficient events: Each group should have a reasonable number of events for reliable results
No tied event times within risk sets: Or use appropriate correction methods

Interpreting Results

Anatomy of a Survival Curve

Survival Curve: Steps down when events occur, stays flat when observations are censored

Confidence Band: Shows uncertainty in survival estimates (widens over time)

Median Survival: Time when survival probability reaches 50%

Test Statistic (χ²):

Larger values indicate greater differences between groups
Follows a chi-squared distribution with (k-1) degrees of freedom

P-value:

p < 0.05: Survival curves differ significantly between groups (reject null hypothesis)
p ≥ 0.05: No significant difference detected (fail to reject null hypothesis)

Expected vs. Observed Events:

If observed events exceed expected: Group has worse survival than average
If observed events are less than expected: Group has better survival than average

Visual Inspection:

Non-overlapping or clearly separated curves suggest significant differences
Crossing curves may indicate violation of proportional hazards assumption

Example Code

R Code

library(survival)
library(survminer)
library(tidyverse)

# Clinical trial data: time to recovery (days), event (1=recovered, 0=censored)
data <- tibble(
  time = c(5, 8, 12, 15, 18, 23, 25, 30, 35, 40, 45, 48, 50, 55, 60),
  event = c(1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1),
  group = c('Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment',
            'Treatment', 'Treatment', 'Treatment', 'Control', 'Control',
            'Control', 'Control', 'Control', 'Control', 'Control')
)

# Perform Log-Rank Test (also known as Mantel-Cox test)
logrank_result <- survdiff(Surv(time, event) ~ group, data = data)

# Print results
print(logrank_result)

# Extract key statistics
chi_squared <- logrank_result$chisq
df <- length(logrank_result$n) - 1
p_value <- 1 - pchisq(chi_squared, df)

cat("\nLog-Rank Test Results:\n")
cat("Chi-squared statistic:", round(chi_squared, 3), "\n")
cat("Degrees of freedom:", df, "\n")
cat("p-value:", format.pval(p_value, digits = 3), "\n")

# Visualize survival curves by group
km_fit <- survfit(Surv(time, event) ~ group, data = data)

ggsurvplot(km_fit,
           conf.int = TRUE,
           pval = TRUE,
           risk.table = TRUE,
           xlab = "Time (days)",
           ylab = "Survival Probability",
           title = "Survival Curves by Group",
           legend.title = "Group",
           legend.labs = levels(factor(data$group)))

Python Code

Python

import numpy as np
import pandas as pd
from lifelines import KaplanMeierFitter
from lifelines.statistics import logrank_test, multivariate_logrank_test
import matplotlib.pyplot as plt

# Clinical trial data: time to recovery (days), event (1=recovered, 0=censored)
data = pd.DataFrame({
    'time': [5, 8, 12, 15, 18, 23, 25, 30, 35, 40, 45, 48, 50, 55, 60],
    'event': [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1],
    'group': ['Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment',
              'Treatment', 'Treatment', 'Treatment', 'Control', 'Control',
              'Control', 'Control', 'Control', 'Control', 'Control']
})

# Log-Rank Test for two groups
treatment_data = data[data['group'] == 'Treatment']
control_data = data[data['group'] == 'Control']

results = logrank_test(
    treatment_data['time'], control_data['time'],
    treatment_data['event'], control_data['event']
)

print("Log-Rank Test Results:")
print(f"Test statistic: {results.test_statistic:.3f}")
print(f"p-value: {results.p_value:.4f}")
print(results.summary)

# For more than 2 groups, use multivariate_logrank_test
# result = multivariate_logrank_test(data['time'], data['group'], data['event'])
# print(result.summary)

# Plot survival curves by group
fig, ax = plt.subplots(figsize=(10, 6))

for group in data['group'].unique():
    group_data = data[data['group'] == group]
    kmf = KaplanMeierFitter()
    kmf.fit(
        durations=group_data['time'],
        event_observed=group_data['event'],
        label=group
    )
    kmf.plot_survival_function(ax=ax, ci_show=True)

plt.title(f'Survival Curves by Group (p={results.p_value:.4f})')
plt.xlabel('Time (days)')
plt.ylabel('Survival Probability')
plt.ylim(0, 1)
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()

Choosing the Right Test

Which Survival Analysis Test Should I Use?

I have time-to-event data

What is my goal?

Describe/estimate survival

Kaplan-Meier Estimator

Get survival curves & median survival

Compare 2+ groups

Log-Rank Test

Test if groups differ significantly

Model multiple factors

Cox Regression

Adjust for covariates & hazard ratios

Kaplan-Meier

• Estimate survival curves
• Find median survival time
• Visualize survival by group
• No formal hypothesis test

Log-Rank Test

• Compare 2+ groups
• Get p-value for difference
• Test overall survival
• Single grouping variable

Cox Regression

• Multiple covariates
• Hazard ratios
• Adjust for confounders
• Most comprehensive

Quick Decision Guide:

Start with Kaplan-Meier to visualize your data and understand survival patterns
Use Log-Rank Test when you have 2+ groups and want to test if they differ (e.g., treatment vs control)
Use Cox Regression when you have multiple variables (age, treatment, stage, etc.) and want to model their effects

Time

Event

Group

Meaning

Treatment

Event occurred

Treatment

Censored

Control

Event occurred

Control

Censored

Treatment

Event occurred

Time

Event

Group

yes

Treatment

Control

Time

Event

Group

-5

Treatment

Control

Time

Event

Group

—

Treatment

—

Control

library(survival) library(survminer) library(tidyverse) # Clinical trial data: time to recovery (days), event (1=recovered, 0=censored) data <- tibble( time = c(5, 8, 12, 15, 18, 23, 25, 30, 35, 40, 45, 48, 50, 55, 60), event = c(1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1), group = c('Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Control', 'Control', 'Control', 'Control', 'Control', 'Control', 'Control') ) # Perform Log-Rank Test (also known as Mantel-Cox test) logrank_result <- survdiff(Surv(time, event) ~ group, data = data) # Print results print(logrank_result) # Extract key statistics chi_squared <- logrank_result$chisq df <- length(logrank_result$n) - 1 p_value <- 1 - pchisq(chi_squared, df) cat("\nLog-Rank Test Results:\n") cat("Chi-squared statistic:", round(chi_squared, 3), "\n") cat("Degrees of freedom:", df, "\n") cat("p-value:", format.pval(p_value, digits = 3), "\n") # Visualize survival curves by group km_fit <- survfit(Surv(time, event) ~ group, data = data) ggsurvplot(km_fit, conf.int = TRUE, pval = TRUE, risk.table = TRUE, xlab = "Time (days)", ylab = "Survival Probability", title = "Survival Curves by Group", legend.title = "Group", legend.labs = levels(factor(data$group)))

import numpy as np import pandas as pd from lifelines import KaplanMeierFitter from lifelines.statistics import logrank_test, multivariate_logrank_test import matplotlib.pyplot as plt # Clinical trial data: time to recovery (days), event (1=recovered, 0=censored) data = pd.DataFrame({ 'time': [5, 8, 12, 15, 18, 23, 25, 30, 35, 40, 45, 48, 50, 55, 60], 'event': [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1], 'group': ['Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Control', 'Control', 'Control', 'Control', 'Control', 'Control', 'Control'] }) # Log-Rank Test for two groups treatment_data = data[data['group'] == 'Treatment'] control_data = data[data['group'] == 'Control'] results = logrank_test( treatment_data['time'], control_data['time'], treatment_data['event'], control_data['event'] ) print("Log-Rank Test Results:") print(f"Test statistic: {results.test_statistic:.3f}") print(f"p-value: {results.p_value:.4f}") print(results.summary) # For more than 2 groups, use multivariate_logrank_test # result = multivariate_logrank_test(data['time'], data['group'], data['event']) # print(result.summary) # Plot survival curves by group fig, ax = plt.subplots(figsize=(10, 6)) for group in data['group'].unique(): group_data = data[data['group'] == group] kmf = KaplanMeierFitter() kmf.fit( durations=group_data['time'], event_observed=group_data['event'], label=group ) kmf.plot_survival_function(ax=ax, ci_show=True) plt.title(f'Survival Curves by Group (p={results.p_value:.4f})') plt.xlabel('Time (days)') plt.ylabel('Survival Probability') plt.ylim(0, 1) plt.grid(True, alpha=0.3) plt.legend() plt.show()

Time

Event

Group

Meaning

Treatment

Event occurred

Treatment

Censored

Control

Event occurred

Control

Censored

Treatment

Event occurred

Time

Event

Group

yes

Treatment

Control

Time

Event

Group

-5

Treatment

Control

Time

Event

Group

—

Treatment

—

Control