Scikit Survival

# 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