Cox Proportional Hazards Models

Cox Proportional Hazards Models

Overview

Cox proportional hazards models are semi-parametric models that relate covariates to the time of an event. The hazard function for individual i is expressed as:

h_i(t) = h_0(t) × exp(β^T x_i)

where:

• h_0(t) is the baseline hazard function (unspecified)
• β is the vector of coefficients
• x_i is the covariate vector for individual i

The key assumption is that the hazard ratio between two individuals is constant over time (proportional hazards).

CoxPHSurvivalAnalysis

Basic Cox proportional hazards model for survival analysis.

When to Use

• Standard survival analysis with censored data
• Need interpretable coefficients (log hazard ratios)
• Proportional hazards assumption holds
• Dataset has relatively few features

Key Parameters

• alpha: Regularization parameter (default: 0, no regularization)
• ties: Method for handling tied event times ('breslow' or 'efron')
• n_iter: Maximum number of iterations for optimization

Example Usage

from sksurv.linear_model import CoxPHSurvivalAnalysis
from sksurv.datasets import load_gbsg2

# Load data
X, y = load_gbsg2()

# Fit Cox model
estimator = CoxPHSurvivalAnalysis()
estimator.fit(X, y)

# Get coefficients (log hazard ratios)
coefficients = estimator.coef_

# Predict risk scores
risk_scores = estimator.predict(X)

CoxnetSurvivalAnalysis

Cox model with elastic net penalty for feature selection and regularization.

When to Use

• High-dimensional data (many features)
• Need automatic feature selection
• Want to handle multicollinearity
• Require sparse models

Penalty Types

• Ridge (L2): alpha_min_ratio=1.0, l1_ratio=0

  • Shrinks all coefficients
  • Good when all features are relevant

• Lasso (L1): l1_ratio=1.0

  • Performs feature selection (sets coefficients to zero)
  • Good for sparse models

• Elastic Net: 0 < l1_ratio < 1

  • Combination of L1 and L2
  • Balances feature selection and grouping

Key Parameters

• l1_ratio: Balance between L1 and L2 penalty (0=Ridge, 1=Lasso)
• alpha_min_ratio: Ratio of smallest to largest penalty in regularization path
• n_alphas: Number of alphas along regularization path
• fit_baseline_model: Whether to fit unpenalized baseline model

Example Usage

from sksurv.linear_model import CoxnetSurvivalAnalysis

# Fit with elastic net penalty
estimator = CoxnetSurvivalAnalysis(l1_ratio=0.5, alpha_min_ratio=0.01)
estimator.fit(X, y)

# Access regularization path
alphas = estimator.alphas_
coefficients_path = estimator.coef_path_

# Predict with specific alpha
risk_scores = estimator.predict(X, alpha=0.1)

Cross-Validation for Alpha Selection

from sklearn.model_selection import GridSearchCV
from sksurv.metrics import concordance_index_censored

# Define parameter grid
param_grid = {'l1_ratio': [0.1, 0.5, 0.9],
              'alpha_min_ratio': [0.01, 0.001]}

# Grid search with C-index
cv = GridSearchCV(CoxnetSurvivalAnalysis(),
                  param_grid,
                  scoring='concordance_index_ipcw',
                  cv=5)
cv.fit(X, y)

# Best parameters
best_params = cv.best_params_

IPCRidge

Inverse probability of censoring weighted Ridge regression for accelerated failure time models.

When to Use

• Prefer accelerated failure time (AFT) framework over proportional hazards
• Need to model how features accelerate/decelerate survival time
• High censoring rates
• Want regularization with Ridge penalty

Key Difference from Cox Models

AFT models assume features multiply survival time by a constant factor, rather than multiplying the hazard rate. The model predicts log survival time directly.

Example Usage

from sksurv.linear_model import IPCRidge

# Fit IPCRidge model
estimator = IPCRidge(alpha=1.0)
estimator.fit(X, y)

# Predict log survival time
log_time = estimator.predict(X)

Model Comparison and Selection

Choosing Between Models

Use CoxPHSurvivalAnalysis when:

• Small to moderate number of features
• Want interpretable hazard ratios
• Standard survival analysis setting

Use CoxnetSurvivalAnalysis when:

• High-dimensional data (p >> n)
• Need feature selection
• Want to identify important predictors
• Presence of multicollinearity

Use IPCRidge when:

• AFT framework is more appropriate
• High censoring rates
• Want to model time directly rather than hazard

Checking Proportional Hazards Assumption

The proportional hazards assumption should be verified using:

• Schoenfeld residuals
• Log-log survival plots
• Statistical tests (available in other packages like lifelines)

If violated, consider:

• Stratification by violating covariates
• Time-varying coefficients
• Alternative models (AFT, parametric models)

Interpretation

Cox Model Coefficients

• Positive coefficient: increased hazard (shorter survival)
• Negative coefficient: decreased hazard (longer survival)
• Hazard ratio = exp(β) for one-unit increase in covariate
• Example: β=0.693 → HR=2.0 (doubles the hazard)

Risk Scores

• Higher risk score = higher risk of event = shorter expected survival
• Risk scores are relative; use survival functions for absolute predictions