Cox Proportional Hazards Model

The defining constraint of the model: the hazard ratio between any two individuals is constant over time. This means the effect of a covariate—like a treatment group—is multiplicative and does not change as time progresses.

• Diagnostic Tool: Schoenfeld residuals are plotted against time to test this assumption.
• Violation Consequence: If the effect of a drug wears off (non-proportionality), the model's single summary hazard ratio is misleading.
• Remedy: Stratification or time-varying covariates can be introduced to handle variables that violate the assumption.

The Cox model is semi-parametric because it makes no assumption about the shape of the baseline hazard function, h₀(t).

• Non-Parametric Part: The baseline hazard is left unspecified, allowing it to take any form—increasing, decreasing, or constant.
• Parametric Part: The covariate effects are estimated parametrically using a log-linear model.
• Advantage: This flexibility allows the model to fit a wide variety of survival patterns without forcing the data into a specific distribution like Weibull or Exponential.

Instead of a full likelihood, Cox proposed a partial likelihood that discards the baseline hazard component. This ingenious method estimates the regression coefficients (β) by considering only the ordering of event times.

• Risk Sets: At each event time, the likelihood compares the covariate values of the subject who experienced the event against all subjects still at risk.
• Tied Handling: Methods like Efron or Breslow approximations are used when multiple events occur at the exact same time.
• Output: Maximizing this partial likelihood yields coefficient estimates that are consistent and asymptotically normal.

The core output is the hazard ratio (HR), calculated as exp(β). It represents the instantaneous relative risk associated with a one-unit increase in a covariate.

• HR > 1: Indicates an increased hazard (worse prognosis). For example, HR=2.0 means the event rate doubles.
• HR < 1: Indicates a decreased hazard (protective effect).
• Continuous Variables: For a 10-year age increase, the HR is exp(10 * β_age).
• Confidence Intervals: 95% CIs for the HR are calculated to assess statistical significance; if the interval crosses 1.0, the effect is not significant.

The Cox model elegantly handles right-censoring, the most common type in clinical studies where a patient leaves the study or the study ends before the event occurs.

• Mechanism: It assumes non-informative censoring—that the censoring mechanism is independent of the event risk.
• Contribution: Censored subjects contribute information to the risk set up until their censoring time, after which they are removed from the denominator.
• Power: By retaining partial information from censored patients, the model avoids the bias that would result from simply excluding them.

The standard model assumes covariates are fixed at baseline, but an extension allows for time-dependent covariates that change value during follow-up.

• Data Structure: Requires a counting process format where each subject can have multiple rows of data representing distinct time intervals.
• Clinical Example: Modeling the effect of a biomarker that is measured repeatedly, or a treatment that is initiated after a specific event.
• Interpretation: The hazard at time t depends on the current value of the covariate, not just the baseline value.

The primary effect measure derived from the Cox model. It represents the relative risk of an event occurring in one group versus a control group over the entire study duration. A hazard ratio of 1.0 indicates no difference, < 1.0 indicates a reduced hazard, and > 1.0 indicates an increased hazard. In federated survival analysis, the hazard ratio is computed without centralizing patient-level data.

A fundamental condition in survival analysis where the exact time of the event is unknown. Common types include:

• Right censoring: The subject is lost to follow-up, withdraws, or the study ends before the event occurs.
• Left censoring: The event occurred before the study observation period began.
• Interval censoring: The event is known only to have occurred within a specific time window. Proper handling of censored data is critical for unbiased hazard estimation.

A non-parametric statistic used to estimate the survival function from lifetime data. It calculates the probability of surviving past specific time points while properly accounting for censored observations. The resulting step-function curve visually represents survival probabilities over time. While the Cox model estimates covariate effects, the Kaplan-Meier curve provides the baseline survival visualization.

A distributed statistical framework that extends the Cox model to multi-institutional settings. It enables the calculation of hazard ratios and survival curves across siled datasets without centralizing longitudinal patient records. Key approaches include:

• Meta-analysis: Combining locally computed hazard ratios using inverse variance weighting.
• Distributed optimization: Iteratively updating a global model by sharing only aggregated gradient statistics.
• Secure aggregation: Using cryptographic protocols to protect intermediate calculations.

The foundational assumption of the Cox model stating that the ratio of hazards between any two groups remains constant over time. This means the effect of a covariate is multiplicative and does not change with time. Violations can be detected using:

• Schoenfeld residuals: Testing for correlation between residuals and time.
• Log-log plots: Checking if survival curves for different groups remain parallel. When violated, extensions like time-varying covariates or stratified Cox models are required.

A computational system that statistically combines the results of independent studies to produce a single, more precise estimate of treatment effect. In federated Cox analysis, each institution computes a local log hazard ratio and its standard error. The meta-analysis engine then pools these using inverse variance weighting, giving greater influence to sites with larger sample sizes or more events.