Competing Risks Analysis

Overview

Competing risks occur when subjects can experience one of several mutually exclusive events (event types). When one event occurs, it prevents ("competes with") the occurrence of other events.

Examples of Competing Risks

Medical Research:

Death from cancer vs. death from cardiovascular disease vs. death from other causes
Relapse vs. death without relapse in cancer studies
Different types of infections in transplant patients

Other Applications:

Job termination: retirement vs. resignation vs. termination for cause
Equipment failure: different failure modes
Customer churn: different reasons for leaving

Key Concept: Cumulative Incidence Function (CIF)

The Cumulative Incidence Function (CIF) represents the probability of experiencing a specific event type by time t, accounting for the presence of competing risks.

CIF_k(t) = P(T ≤ t, event type = k)

This differs from the Kaplan-Meier estimator, which would overestimate event probabilities when competing risks are present.

When to Use Competing Risks Analysis

Use competing risks when:

Multiple mutually exclusive event types exist
Occurrence of one event prevents others
Need to estimate probability of specific event types
Want to understand how covariates affect different event types

Don't use when:

Only one event type of interest (standard survival analysis)
Events are not mutually exclusive (use recurrent events methods)
Competing events are extremely rare (can treat as censoring)

Cumulative Incidence with Competing Risks

cumulative_incidence_competing_risks Function

Estimates the cumulative incidence function for each event type.

python

from sksurv.nonparametric import cumulative_incidence_competing_risks
from sksurv.datasets import load_leukemia

# Load data with competing risks
X, y = load_leukemia()
# y has event types: 0=censored, 1=relapse, 2=death

# Compute cumulative incidence for each event type
# Returns: time points, CIF for event 1, CIF for event 2, ...
time_points, cif_1, cif_2 = cumulative_incidence_competing_risks(y)

# Plot cumulative incidence functions
import matplotlib.pyplot as plt

plt.figure(figsize=(10, 6))
plt.step(time_points, cif_1, where='post', label='Relapse', linewidth=2)
plt.step(time_points, cif_2, where='post', label='Death in remission', linewidth=2)
plt.xlabel('Time (weeks)')
plt.ylabel('Cumulative Incidence')
plt.title('Competing Risks: Relapse vs Death')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

Interpretation

CIF at time t: Probability of experiencing that specific event by time t
Sum of all CIFs: Total probability of experiencing any event (all cause)
1 - sum of CIFs: Probability of being event-free and uncensored

Data Format for Competing Risks

Creating Structured Array with Event Types

python

import numpy as np
from sksurv.util import Surv

# Event types: 0 = censored, 1 = event type 1, 2 = event type 2
event_types = np.array([0, 1, 2, 1, 0, 2, 1])
times = np.array([10.2, 5.3, 8.1, 3.7, 12.5, 6.8, 4.2])

# Create survival array
# For competing risks: event=True if any event occurred
# Store event type separately or encode in the event field
y = Surv.from_arrays(
    event=(event_types > 0),  # True if any event
    time=times
)

# Keep event_types for distinguishing between event types

Converting Data with Event Types

python

import pandas as pd
from sksurv.util import Surv

# Assume data has: time, event_type columns
# event_type: 0=censored, 1=type1, 2=type2, etc.

df = pd.read_csv('competing_risks_data.csv')

# Create survival outcome
y = Surv.from_arrays(
    event=(df['event_type'] > 0),
    time=df['time']
)

# Store event types
event_types = df['event_type'].values

Comparing Cumulative Incidence Between Groups

Stratified Analysis

python

from sksurv.nonparametric import cumulative_incidence_competing_risks
import matplotlib.pyplot as plt

# Split by treatment group
mask_treatment = X['treatment'] == 'A'
mask_control = X['treatment'] == 'B'

y_treatment = y[mask_treatment]
y_control = y[mask_control]

# Compute CIF for each group
time_trt, cif1_trt, cif2_trt = cumulative_incidence_competing_risks(y_treatment)
time_ctl, cif1_ctl, cif2_ctl = cumulative_incidence_competing_risks(y_control)

# Plot comparison
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Event type 1
ax1.step(time_trt, cif1_trt, where='post', label='Treatment', linewidth=2)
ax1.step(time_ctl, cif1_ctl, where='post', label='Control', linewidth=2)
ax1.set_xlabel('Time')
ax1.set_ylabel('Cumulative Incidence')
ax1.set_title('Event Type 1')
ax1.legend()
ax1.grid(True, alpha=0.3)

# Event type 2
ax2.step(time_trt, cif2_trt, where='post', label='Treatment', linewidth=2)
ax2.step(time_ctl, cif2_ctl, where='post', label='Control', linewidth=2)
ax2.set_xlabel('Time')
ax2.set_ylabel('Cumulative Incidence')
ax2.set_title('Event Type 2')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

Statistical Testing with Competing Risks

Gray's Test

Compare cumulative incidence functions between groups using Gray's test (available in other packages like lifelines).

python

# Note: Gray's test not directly available in scikit-survival
# Consider using lifelines or other packages

# from lifelines.statistics import multivariate_logrank_test
# result = multivariate_logrank_test(times, groups, events, event_of_interest=1)

Modeling with Competing Risks

Approach 1: Cause-Specific Hazard Models

Fit separate Cox models for each event type, treating other event types as censored.

python

from sksurv.linear_model import CoxPHSurvivalAnalysis
from sksurv.util import Surv

# Separate outcome for each event type
# Event type 1: treat type 2 as censored
y_event1 = Surv.from_arrays(
    event=(event_types == 1),
    time=times
)

# Event type 2: treat type 1 as censored
y_event2 = Surv.from_arrays(
    event=(event_types == 2),
    time=times
)

# Fit cause-specific models
cox_event1 = CoxPHSurvivalAnalysis()
cox_event1.fit(X, y_event1)

cox_event2 = CoxPHSurvivalAnalysis()
cox_event2.fit(X, y_event2)

# Interpret coefficients for each event type
print("Event Type 1 (e.g., Relapse):")
print(cox_event1.coef_)

print("\nEvent Type 2 (e.g., Death):")
print(cox_event2.coef_)

Interpretation:

Separate model for each competing event
Coefficients show effect on cause-specific hazard for that event type
A covariate may increase risk for one event type but decrease for another

Approach 2: Fine-Gray Sub-distribution Hazard Model

Models the cumulative incidence directly (not available directly in scikit-survival, but can use other packages).

python

# Note: Fine-Gray model not directly in scikit-survival
# Consider using lifelines or rpy2 to access R's cmprsk package

# from lifelines import CRCSplineFitter
# crc = CRCSplineFitter()
# crc.fit(df, event_col='event', duration_col='time')

Practical Example: Complete Competing Risks Analysis

python

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sksurv.nonparametric import cumulative_incidence_competing_risks
from sksurv.linear_model import CoxPHSurvivalAnalysis
from sksurv.util import Surv

# Simulate competing risks data
np.random.seed(42)
n = 200

# Create features
age = np.random.normal(60, 10, n)
treatment = np.random.choice(['A', 'B'], n)

# Simulate event times and types
# Event types: 0=censored, 1=relapse, 2=death
times = np.random.exponential(100, n)
event_types = np.zeros(n, dtype=int)

# Higher age increases both events, treatment A reduces relapse
for i in range(n):
    if times[i] < 150:  # Event occurred
        # Probability of each event type
        p_relapse = 0.6 if treatment[i] == 'B' else 0.4
        event_types[i] = 1 if np.random.rand() < p_relapse else 2
    else:
        times[i] = 150  # Censored at study end

# Create DataFrame
df = pd.DataFrame({
    'time': times,
    'event_type': event_types,
    'age': age,
    'treatment': treatment
})

# Encode treatment
df['treatment_A'] = (df['treatment'] == 'A').astype(int)

# 1. OVERALL CUMULATIVE INCIDENCE
print("=" * 60)
print("OVERALL CUMULATIVE INCIDENCE")
print("=" * 60)

y_all = Surv.from_arrays(event=(df['event_type'] > 0), time=df['time'])
time_points, cif_relapse, cif_death = cumulative_incidence_competing_risks(y_all)

plt.figure(figsize=(10, 6))
plt.step(time_points, cif_relapse, where='post', label='Relapse', linewidth=2)
plt.step(time_points, cif_death, where='post', label='Death', linewidth=2)
plt.xlabel('Time (days)')
plt.ylabel('Cumulative Incidence')
plt.title('Competing Risks: Relapse vs Death')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

print(f"5-year relapse incidence: {cif_relapse[-1]:.2%}")
print(f"5-year death incidence: {cif_death[-1]:.2%}")

# 2. STRATIFIED BY TREATMENT
print("\n" + "=" * 60)
print("CUMULATIVE INCIDENCE BY TREATMENT")
print("=" * 60)

for trt in ['A', 'B']:
    mask = df['treatment'] == trt
    y_trt = Surv.from_arrays(
        event=(df.loc[mask, 'event_type'] > 0),
        time=df.loc[mask, 'time']
    )
    time_trt, cif1_trt, cif2_trt = cumulative_incidence_competing_risks(y_trt)
    print(f"\nTreatment {trt}:")
    print(f"  5-year relapse: {cif1_trt[-1]:.2%}")
    print(f"  5-year death: {cif2_trt[-1]:.2%}")

# 3. CAUSE-SPECIFIC MODELS
print("\n" + "=" * 60)
print("CAUSE-SPECIFIC HAZARD MODELS")
print("=" * 60)

X = df[['age', 'treatment_A']]

# Model for relapse (event type 1)
y_relapse = Surv.from_arrays(
    event=(df['event_type'] == 1),
    time=df['time']
)
cox_relapse = CoxPHSurvivalAnalysis()
cox_relapse.fit(X, y_relapse)

print("\nRelapse Model:")
print(f"  Age:        HR = {np.exp(cox_relapse.coef_[0]):.3f}")
print(f"  Treatment A: HR = {np.exp(cox_relapse.coef_[1]):.3f}")

# Model for death (event type 2)
y_death = Surv.from_arrays(
    event=(df['event_type'] == 2),
    time=df['time']
)
cox_death = CoxPHSurvivalAnalysis()
cox_death.fit(X, y_death)

print("\nDeath Model:")
print(f"  Age:        HR = {np.exp(cox_death.coef_[0]):.3f}")
print(f"  Treatment A: HR = {np.exp(cox_death.coef_[1]):.3f}")

print("\n" + "=" * 60)

Important Considerations

Censoring in Competing Risks

Administrative censoring: Subject still at risk at end of study
Loss to follow-up: Subject leaves study before event
Competing event: Other event occurred - NOT censored for CIF, but censored for cause-specific models

Choosing Between Cause-Specific and Sub-distribution Models

Cause-Specific Hazard Models:

Easier to interpret
Direct effect on hazard rate
Better for understanding etiology
Can fit with scikit-survival

Fine-Gray Sub-distribution Models:

Models cumulative incidence directly
Better for prediction and risk stratification
More appropriate for clinical decision-making
Requires other packages

Common Mistakes

Mistake 1: Using Kaplan-Meier to estimate event-specific probabilities

Wrong: Kaplan-Meier for event type 1, treating type 2 as censored
Correct: Cumulative incidence function accounting for competing risks

Mistake 2: Ignoring competing risks when they're substantial

If competing event rate > 10-20%, should use competing risks methods

Mistake 3: Confusing cause-specific and sub-distribution hazards

They answer different questions
Use appropriate model for your research question

Summary

Key Functions:

cumulative_incidence_competing_risks: Estimate CIF for each event type
Fit separate Cox models for cause-specific hazards
Use stratified analysis to compare groups

Best Practices:

Always plot cumulative incidence functions
Report both event-specific and overall incidence
Use cause-specific models in scikit-survival
Consider Fine-Gray models (other packages) for prediction
Be explicit about which events are competing vs censored