Survival analysis is crucial in biostatistics, helping researchers understand time-to-event data. The Cox proportional hazards model is a powerful tool in this field, allowing us to examine how various factors affect survival rates over time.

This model estimates how different variables impact the risk of an event occurring, like death or disease recurrence. It's flexible, handling both continuous and categorical predictors, and doesn't require specifying a particular distribution for survival times.

Cox Proportional Hazards Model

Overview and Assumptions

• The Cox proportional hazards model is a semi-parametric survival analysis method used to investigate the association between the survival time of patients and one or more predictor variables (age, treatment group)
• The model assumes that the hazard function for an individual depends on the values of the covariates and the baseline hazard
  • The baseline hazard function is the hazard function when all covariates are equal to zero, and it is estimated non-parametrically
• The model assumes that the hazard ratio comparing any two specifications of predictors is constant over time, known as the proportional hazards assumption
• The Cox model estimates the effect of covariates on the hazard rate rather than directly modeling the survival time
• The model can handle both continuous (age, blood pressure) and categorical covariates (gender, treatment group), as well as time-dependent covariates

Estimation and Interpretation

• The partial likelihood function is used to estimate the coefficients in the Cox model, and the maximum partial likelihood estimates are obtained through iterative methods
• The coefficients in the Cox model represent the change in the log of the hazard ratio for a one-unit increase in the corresponding covariate, while holding other covariates constant
  • A positive coefficient indicates an increased hazard and shorter survival time, while a negative coefficient indicates a decreased hazard and longer survival time
• The exponentiated coefficients in the Cox model provide hazard ratios, which represent the multiplicative effect of the covariate on the hazard
  • A hazard ratio greater than 1 indicates an increased risk of the event (death, relapse), while a hazard ratio less than 1 indicates a decreased risk
• Confidence intervals for the hazard ratios can be used to assess the precision and statistical significance of the estimated effects

Interpreting Coefficients and Hazard Ratios

Hazard Ratios

• The exponentiated coefficients in the Cox model provide hazard ratios, which represent the multiplicative effect of the covariate on the hazard
  • For example, if the hazard ratio for a treatment group is 0.5, the hazard of the event in the treatment group is half that of the reference group
• A hazard ratio greater than 1 indicates an increased risk of the event, while a hazard ratio less than 1 indicates a decreased risk
• Confidence intervals for the hazard ratios can be used to assess the precision and statistical significance of the estimated effects
  • If the confidence interval for a hazard ratio includes 1, the effect of the covariate is not statistically significant at the chosen level (0.05)

Reference Levels and Interpretation

• The reference level for categorical covariates should be carefully chosen to facilitate meaningful interpretation of the hazard ratios
  • For example, when comparing treatment groups, the placebo or standard care group is often chosen as the reference level
• The interpretation of hazard ratios for continuous covariates depends on the scale of the covariate
  • For example, a hazard ratio of 1.05 for age (in years) indicates a 5% increase in the hazard for each one-year increase in age
• Interactions between covariates can be included in the model to assess whether the effect of one covariate depends on the level of another covariate
  • The interpretation of interaction terms requires careful consideration of the reference levels and the combined effects of the interacting covariates

Building and Assessing Cox Models

Model Building Strategies

• Model building in Cox regression involves selecting relevant covariates based on subject matter knowledge, statistical significance, and model fit
• Univariate Cox models can be used to assess the individual effects of covariates on the hazard, followed by multivariable model building
  • Covariates with p-values less than a specified threshold (0.1 or 0.2) in the univariate analyses may be considered for inclusion in the multivariable model
• Stepwise selection methods (forward, backward, or a combination) can be employed to identify a parsimonious set of covariates
  • Forward selection starts with an empty model and adds covariates one at a time based on a specified criterion (p-value, AIC)
  • Backward elimination starts with a full model and removes covariates one at a time based on a specified criterion

Model Assessment and Goodness-of-Fit

• The likelihood ratio test, Wald test, and score test can be used to assess the overall significance of the model and individual covariates
• The goodness-of-fit of a Cox model can be assessed using the Cox-Snell residuals, which should follow a standard exponential distribution if the model fits well
  • Plotting the Nelson-Aalen cumulative hazard estimate against the Cox-Snell residuals can help visualize the model fit
  • Deviations from a straight line with a slope of 1 indicate a lack of fit
• Other methods for assessing model fit include the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), which balance model complexity and fit
  • Lower values of AIC and BIC indicate better model fit, while penalizing for the number of parameters in the model

Proportional Hazards Assumption vs Violations

Assessing the Proportional Hazards Assumption

• The proportional hazards assumption states that the hazard ratio comparing any two specifications of predictors is constant over time
• Graphical methods, such as plotting the scaled Schoenfeld residuals against time for each covariate, can help assess the proportional hazards assumption
  • A non-random pattern or trend in the residuals suggests a violation of the assumption
  • Parallel lines indicate that the proportional hazards assumption is met
• Formal statistical tests, such as the Grambsch-Therneau test, can be used to test the proportional hazards assumption for individual covariates and the overall model
  • A significant p-value indicates a violation of the proportional hazards assumption

Addressing Non-Proportional Hazards

• If the proportional hazards assumption is violated, alternative modeling strategies can be employed:
  • Stratified Cox models allow for different baseline hazard functions for each stratum of a categorical covariate (disease stage) while assuming proportional hazards within each stratum
  • Time-dependent covariates can be included in the model to allow for non-proportional hazards, where the effect of a covariate changes over time
    • For example, the effect of a treatment may be strong initially but diminish over time
  • Piecewise constant models or flexible parametric models can be used to model non-proportional hazards by allowing the baseline hazard to vary over different time intervals
• Model diagnostics, such as examining the martingale and deviance residuals, can help identify influential observations and assess the functional form of continuous covariates
  • Martingale residuals can be plotted against each covariate to assess the functional form and identify potential outliers
  • Deviance residuals can be used to identify observations that are not well fit by the model