Kaplan-Meier Estimator

The estimator calculates survival probability as a product of conditional probabilities over time. At each distinct event time t, the survival probability is updated by multiplying the current estimate by the proportion of subjects surviving past t among those at risk just before t.

• Formula: Ŝ(t) = ∏ (1 - dᵢ/nᵢ) for all tᵢ ≤ t
• dᵢ: number of events at time tᵢ
• nᵢ: number of individuals at risk just before tᵢ
• The curve only drops at observed event times, remaining flat between events.

A defining strength of the Kaplan-Meier estimator is its ability to incorporate censored observations—subjects lost to follow-up, who withdraw, or who survive past the study end without experiencing the event.

• Censored subjects contribute information up to their last known follow-up time
• They are removed from the risk set after their censoring time
• The estimator assumes non-informative censoring: censoring is unrelated to the event risk
• This prevents overestimation of event probability compared to naive methods that ignore censoring

The output is a step function plotted over time, visually representing the estimated survival probability. The curve starts at 1.0 (100% survival) at time zero and descends toward zero as events accumulate.

• Drops occur only at observed event times
• Tick marks on the curve typically indicate censored observations
• The curve provides an intuitive visual comparison between treatment groups
• Median survival time is read directly from the curve at the 0.5 probability mark

While the Kaplan-Meier estimator generates survival curves, the log-rank test is the standard statistical method for comparing two or more curves to determine if observed differences are statistically significant.

• Tests the null hypothesis that there is no difference between groups
• Weighs all time points equally, giving more weight to later events
• Assumes proportional hazards between groups
• Often paired with Kaplan-Meier visualizations in clinical trial publications

In federated clinical analytics, the Kaplan-Meier estimator can be computed across multiple institutions without centralizing patient-level data. Each site calculates local event and risk-set counts, sharing only aggregated statistics.

• Local sites compute dᵢ and nᵢ at each event time
• A secure aggregation protocol combines these counts into a global survival curve
• Preserves patient privacy while enabling multi-site survival analysis
• Integrates with federated cohort discovery for defining analysis populations

Valid interpretation requires understanding the estimator's underlying assumptions. Violations can lead to biased survival estimates and incorrect clinical conclusions.

• Non-informative censoring: Censoring must be independent of the event risk
• Homogeneity: All subjects should have the same survival prospects at a given time
• Event definition: The event of interest must be clearly and consistently defined
• Does not adjust for confounding variables—use the Cox model for covariate adjustment

A fundamental condition in survival analysis where the exact event time is unknown. Right censoring occurs when a subject leaves the study or the study ends before the event. Left censoring happens when the event occurred before observation began. Interval censoring arises when the event is known only to have occurred within a time window. The Kaplan-Meier estimator explicitly accounts for right-censored observations by adjusting the risk set at each time point, preventing bias that would occur if censored subjects were simply excluded.

A semi-parametric regression model that extends survival analysis to assess the effect of multiple covariates on the hazard rate simultaneously. Unlike the Kaplan-Meier estimator, which estimates a single survival curve per group, the Cox model quantifies how factors like treatment type, age, or biomarker levels influence risk. The key assumption is proportional hazards: the hazard ratio between groups remains constant over time. Output is expressed as a hazard ratio, where a value above 1 indicates increased risk.

A measure of relative risk derived from survival models, representing the ratio of hazard rates between two groups over the entire study period. Interpretation:

• HR = 1: No difference between groups
• HR < 1: Reduced risk in the treatment group
• HR > 1: Increased risk in the treatment group In federated survival analysis, hazard ratios are computed without centralizing patient-level data, using only aggregated statistics shared across institutions.

A non-parametric hypothesis test that compares the survival distributions of two or more groups. It tests the null hypothesis that there is no difference between groups in the probability of an event at any time point. The test statistic is calculated by comparing observed versus expected events across all time points, weighted equally. It is most sensitive to differences that are consistent over time and is commonly paired with Kaplan-Meier curves to determine if observed separations are statistically significant.

The mathematical function S(t) that gives the probability that a subject survives beyond time t. Properties:

• S(0) = 1 (all subjects alive at baseline)
• S(t) is non-increasing
• S(∞) = 0 (eventually all subjects experience the event) The Kaplan-Meier estimator provides a step-function approximation of S(t) that drops only at observed event times, remaining flat during intervals with only censored observations.