Hazard Ratio

The Cox model assumes the hazard ratio between groups remains constant over time. This means the relative risk does not change as the study progresses.

• Violation detection: Use Schoenfeld residuals or log-minus-log survival plots
• Non-proportional hazards: If the HR changes over time (e.g., treatment effect diminishes), consider time-varying covariates or stratified models
• Clinical implication: A single HR of 0.75 means the treatment group has a 25% lower instantaneous risk of the event at every time point

The hazard ratio is a ratio of instantaneous event rates, not a ratio of median survival times or absolute risk.

• HR = 1: No difference between groups
• HR < 1: Treatment reduces the hazard (protective effect)
• HR > 1: Treatment increases the hazard (harmful effect)
• Example: HR = 0.60 means a 40% reduction in the instantaneous hazard rate, not a 40% reduction in the number of events

The precision of a hazard ratio estimate is captured by its 95% confidence interval, which reflects both sample size and event count.

• Narrow CI: High precision, typically from large sample sizes or many events
• Wide CI crossing 1.0: The result is not statistically significant at the 0.05 level
• Federated context: In distributed survival analysis, the effective sample size for HR estimation is driven by the total number of events across all sites, not the total number of patients

Hazard ratios can be computed from models with different covariate structures, and their interpretation differs fundamentally.

• Univariate HR: The crude association between a single predictor and the outcome, without accounting for confounders
• Adjusted HR: The effect of a predictor after controlling for other covariates in a multivariable Cox model
• Confounding control: In federated meta-analysis, adjusted HRs from each site can be pooled using inverse variance weighting, provided the same covariates are used across all institutions

Valid hazard ratio estimation requires that censoring is non-informative—the probability of being censored must be unrelated to the probability of experiencing the event.

• Right censoring: Patients lost to follow-up or event-free at study end
• Informative censoring: Occurs when patients drop out due to worsening condition, biasing the HR
• Sensitivity analysis: In federated settings, compare HRs under different censoring assumptions across sites to assess robustness

In federated survival analysis, site-specific hazard ratios are combined using meta-analytic techniques without sharing patient-level data.

• Fixed-effects model: Assumes a single true HR across all sites; weights studies by inverse variance
• Random-effects model: Allows the true HR to vary across sites; incorporates between-site heterogeneity using the DerSimonian-Laird estimator
• Heterogeneity metric: The I-squared statistic quantifies the percentage of total variation attributable to between-site differences rather than chance

The semi-parametric regression model used to estimate the hazard ratio. It models the hazard function as a baseline hazard multiplied by an exponential function of covariates. The key assumption is proportional hazards: the ratio of hazards between two groups is constant over time. In federated analytics, Cox models are fit locally and aggregated using inverse variance weighting to produce a global hazard ratio without sharing patient-level survival times.

A non-parametric statistic that estimates the survival function—the probability of surviving past a certain time point. It handles right-censored data (patients lost to follow-up) by recalculating the survival probability only at event times. The Kaplan-Meier curve visually complements the hazard ratio by showing the separation between treatment arms over time. In federated survival analysis, Kaplan-Meier curves are constructed by sharing only event-time and censoring indicators across sites.

Censoring occurs when the exact event time is unknown. Right censoring is most common: a patient drops out, is lost to follow-up, or the study ends before the event. Left censoring means the event occurred before study entry. Interval censoring means the event occurred between two observation points. The hazard ratio calculation in Cox regression assumes non-informative censoring—that censoring is unrelated to the probability of the event. Violations require sensitivity analyses.

A distributed framework for fitting time-to-event models across siled clinical datasets. Instead of centralizing longitudinal patient records, each institution computes local partial likelihoods and shares only aggregated statistics. The global hazard ratio is derived through meta-analysis engines that weight site contributions by precision. This architecture enables multi-center survival studies while maintaining HIPAA and GDPR compliance, as raw event times and covariates never leave the local firewall.

A graphical display used in meta-analysis to present hazard ratios with confidence intervals from individual study sites alongside the pooled summary effect. Each horizontal line represents a site's estimate; the diamond represents the combined hazard ratio. Forest plots enable rapid visual assessment of heterogeneity—whether treatment effects are consistent across institutions. In federated clinical analytics, the forest plot is the standard output for communicating multi-site survival results to clinical researchers.

An extension of survival analysis where patients are at risk of multiple mutually exclusive events. The standard Cox model treats competing events as independent censoring, which can bias the hazard ratio. The Fine-Gray subdistribution hazard model directly estimates the hazard ratio for a specific event while accounting for competing risks. In oncology federated studies, this distinguishes cancer-specific mortality from death due to other causes, providing a more clinically relevant risk estimate.