Kaplan-Meier Estimator — Survival Function

The Kaplan-Meier estimator is a non-parametric method for estimating the survival function from time-to-event data, even when observations are censored.

Censoring

| Type | Description |

|------|------------|

| Right-censored | Event not observed before study ends |

| Left-censored | Event occurred before study began |

| Interval-censored | Event known to occur in an interval |

Kaplan-Meier Formula

The estimator is a step function that drops at each event time.

Standard Error

The 95% confidence interval is:

Log-Rank Test

The log-rank test compares survival curves between two or more groups.

| Hypothesis | Meaning |

|-----------|---------|

| : | No difference in survival between groups |

| : | Survival curves differ |

Median Survival Time

The median survival is the smallest time at which .

Python Implementation

import numpy as np

import pandas as pd

from lifelines import KaplanMeierFitter

from lifelines.statistics import logrank_test

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate survival data

n = 200

treatment = np.random.binomial(1, 0.5, n)

time = np.where(treatment,

                np.random.exponential(12, n),  # Treatment: longer survival

                np.random.exponential(8, n))    # Control: shorter survival

censored = np.random.binomial(1, 0.2, n)       # 20% censoring

event = 1 - censored



# Kaplan-Meier curves

kmf_treat = KaplanMeierFitter()

kmf_control = KaplanMeierFitter()



mask_treat = treatment == 1

kmf_treat.fit(time[mask_treat], event[mask_treat], label='Treatment')

kmf_control.fit(time[~mask_treat], event[~mask_treat], label='Control')



# Plot

fig, ax = plt.subplots(figsize=(8, 5))

kmf_treat.plot_survival_function(ax=ax)

kmf_control.plot_survival_function(ax=ax)

ax.set_title('Kaplan-Meier Survival Curves')

ax.set_xlabel('Time')

ax.set_ylabel('Survival Probability')

plt.show()



# Median survival

print(f"Treatment median: {kmf_treat.median_survival_time_:.1f}")

print(f"Control median: {kmf_control.median_survival_time_:.1f}")



# Log-rank test

result = logrank_test(time[mask_treat], time[~mask_treat],

                      event_observed_A=event[mask_treat],

                      event_observed_B=event[~mask_treat])

print(f"\nLog-rank test: ?²={result.test_statistic:.2f}, p={result.p_value:.4f}")

Worked Example

Key Takeaways

Related Topics

• See Cox Proportional Hazards for regression with covariates

• See Hypothesis Testing for the log-rank test framework

• See Missing Data for handling different censoring mechanisms