Survival Analysis

Overview

Survival analysis studies time until an event occurs, handling censored data where events haven't happened for some subjects, enabling prediction of lifetimes and risk assessment.

Key Concepts

• Survival Time: Time until event
• Censoring: Event not observed (subject dropped out)
• Hazard: Instantaneous risk at time t
• Survival Curve: Probability of surviving past time t
• Hazard Ratio: Relative risk between groups

Common Models

• Kaplan-Meier: Non-parametric survival curves
• Cox Proportional Hazards: Semi-parametric regression
• Weibull/Exponential: Parametric models
• Log-rank Test: Comparing survival curves
• Competing Risks: Multiple event types

Implementation with Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from lifelines import KaplanMeierFitter, CoxPHFitter, WeibullAFTFitter
from lifelines.statistics import logrank_test
import warnings
warnings.filterwarnings('ignore')

# Generate sample survival data
np.random.seed(42)
n_patients = 200

# Time to event (in months)
event_times = np.random.exponential(scale=24, size=n_patients)
# Censoring indicator (1 = event occurred, 0 = censored)
event_observed = np.random.binomial(1, 0.7, n_patients)
# Group assignment (0 = control, 1 = treatment)
group = np.random.binomial(1, 0.5, n_patients)
# Age at baseline
age = np.random.uniform(30, 80, n_patients)
# Risk score
risk_score = np.random.uniform(0, 100, n_patients)

# Adjust event times based on group (simulate treatment effect)
event_times = event_times * (1 + group * 0.3)

df = pd.DataFrame({
    'time': event_times,
    'event': event_observed,
    'group': group,
    'age': age,
    'risk_score': risk_score,
})

print("Survival Data Summary:")
print(df.head(10))
print(f"\nTotal subjects: {len(df)}")
print(f"Events: {df['event'].sum()} ({df['event'].sum()/len(df)*100:.1f}%)")
print(f"Censored: {(1-df['event']).sum()} ({(1-df['event']).sum()/len(df)*100:.1f}%)")

# 1. Kaplan-Meier Estimation
kmf = KaplanMeierFitter()
kmf.fit(df['time'], df['event'], label='Overall')

print("\n1. Kaplan-Meier Survival Estimates:")
print(f"Median survival time: {kmf.median_survival_time_:.1f} months")
print(f"6-month survival: {kmf.predict(6):.1%}")
print(f"12-month survival: {kmf.predict(12):.1%}")
print(f"24-month survival: {kmf.predict(24):.1%}")

# 2. Group Comparison
fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Overall survival curve
ax = axes[0, 0]
kmf.plot_survival_function(ax=ax, linewidth=2)
ax.set_xlabel('Time (months)')
ax.set_ylabel('Survival Probability')
ax.set_title('Kaplan-Meier Survival Curve (Overall)')
ax.grid(True, alpha=0.3)

# Survival curves by group
ax = axes[0, 1]
for group_val in [0, 1]:
    mask = df['group'] == group_val
    kmf.fit(df[mask]['time'], df[mask]['event'],
           label=f'{"Control" if group_val == 0 else "Treatment"}')
    kmf.plot_survival_function(ax=ax, linewidth=2)

ax.set_xlabel('Time (months)')
ax.set_ylabel('Survival Probability')
ax.set_title('Kaplan-Meier Curves by Group')
ax.grid(True, alpha=0.3)

# 3. Log-Rank Test
mask_control = df['group'] == 0
mask_treatment = df['group'] == 1

results = logrank_test(
    df[mask_control]['time'],
    df[mask_treatment]['time'],
    df[mask_control]['event'],
    df[mask_treatment]['event']
)

print(f"\n3. Log-Rank Test:")
print(f"Test statistic: {results.test_statistic:.4f}