Cox 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 ```python 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 > 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