Cox Proportional Hazards Model

The Cox model is a semi-parametric regression model for survival data that relates covariates to the hazard function without specifying the baseline hazard.

Cox Model Specification

Hazard Ratios

Proportional Hazards Assumption

The model assumes the ratio of hazards between any two individuals is constant over time.

Testing PH Assumption

1. Schoenfeld residuals: Plot residuals against time; should show no pattern

2. Log-log survival plots: Parallel curves indicate PH holds

3. Statistical test: Test correlation of Schoenfeld residuals with time

Partial Likelihood

Cox's key insight: the baseline hazard drops out of the likelihood.

Confidence Intervals for HR

Python Implementation

import numpy as np

import pandas as pd

from lifelines import CoxPHFitter

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate survival data with covariates

n = 300

age = np.random.normal(60, 10, n)

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

beta_true = [0.03, -0.5]  # age increases risk, treatment decreases risk



# Generate survival times

U = np.random.uniform(0, 1, n)

linpred = beta_true[0]*age + beta_true[1]*treatment

time = -np.log(U) / np.exp(linpred) * 100

censored_time = np.random.uniform(50, 150, n)

event = (time <= censored_time).astype(int)

observed_time = np.minimum(time, censored_time)



# Create DataFrame

df = pd.DataFrame({

    'duration': observed_time,

    'event': event,

    'age': age,

    'treatment': treatment

})



# Fit Cox model

cph = CoxPHFitter()

cph.fit(df, duration_col='duration', event_col='event')

print(cph.summary[['coef', 'exp(coef)', 'se(coef)', 'p', 'exp(coef) lower 95%', 'exp(coef) upper 95%']])



# Plot hazard ratios

cph.plot()

plt.title('Cox Model - Hazard Ratios')

plt.show()



# Concordance index

print(f"\nConcordance index: {cph.concordance_index_:.3f}")

Worked Example

Model Evaluation

| Metric | Description |

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

| Concordance index | Probability that a randomly chosen event occurs at a shorter time for higher-risk individual (0.5 = random, 1.0 = perfect) |

| Partial AIC | For model comparison (lower is better) |

| Schoenfeld residuals | Check PH assumption |

| Likelihood ratio test | Test overall model significance |

Key Takeaways

Related Topics

• See Kaplan-Meier Estimator for non-parametric survival estimation

• See Logistic Regression for binary outcomes

• See Mediation Analysis for causal pathway analysis