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references/cox-models.md

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
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