Kaplan-Meier Estimator

The estimator calculates the survival probability at time t as the cumulative product of conditional survival probabilities. At each distinct event time tᵢ, the probability of surviving past that point is (1 - dᵢ/nᵢ), where dᵢ is the number of events and nᵢ is the number at risk just before tᵢ.

• Formula: Ŝ(t) = ∏ (1 - dᵢ/nᵢ) for all tᵢ ≤ t
• The curve only drops at observed event times, not at censoring times.
• This non-parametric approach makes no assumption about the underlying hazard distribution.

The statistical processes defining why event times are incomplete. The Kaplan-Meier estimator is specifically designed to handle right-censoring, where a subject leaves the study before an event occurs or survives past the study end date.

• Right-censoring: The most common type; the event time is only known to be greater than the observed time.
• Left-censoring: The event occurred before observation began.
• Interval-censoring: The event is known to have occurred within a specific time window.
• Non-informative censoring: The critical assumption that censoring is independent of the event risk.

The standard non-parametric hypothesis test used to compare the survival distributions of two or more groups visualized by Kaplan-Meier curves. It tests the null hypothesis that there is no difference between the populations.

• Assigns equal weight to failures at all time points, making it most sensitive to proportional hazards alternatives.
• Calculated by comparing observed versus expected event counts in each group at each distinct event time.
• The Mantel-Cox version is the most widely implemented variant in statistical software.

A semiparametric regression model that extends Kaplan-Meier analysis by quantifying the effect of multiple covariates on the hazard rate. While Kaplan-Meier estimates unadjusted survival, the Cox model provides adjusted hazard ratios.

• Does not require specifying the baseline hazard function, making it highly flexible.
• Relies on the proportional hazards assumption: the effect of a covariate is constant over time.
• Often used in conjunction with Kaplan-Meier plots to visually validate model assumptions.

A clinically interpretable summary measure derived directly from the area under the Kaplan-Meier curve up to a specified time point τ. It represents the average event-free time experienced by patients.

• Provides a robust alternative to the hazard ratio when the proportional hazards assumption is violated.
• Reported in absolute time units (e.g., months), making it easier for clinicians to communicate treatment benefit.
• The difference in RMST between two arms quantifies the average gain or loss in event-free time.

A survival analysis framework required when a subject can experience a different event that precludes the primary event of interest. Standard Kaplan-Meier estimates overestimate the incidence in this setting.

• Uses the Cumulative Incidence Function (CIF) instead of 1-KM to estimate the probability of a specific event.
• Example: In an oncology trial, death from a car accident is a competing risk that prevents observing cancer recurrence.
• The Fine-Gray subdistribution hazard model is the corresponding regression method for CIF data.

The primary diagnostic tool used to test the proportional hazards assumption required by the Cox model, often applied after visualizing Kaplan-Meier curves. These residuals check if covariate effects are constant over time.

• A non-zero slope in a plot of scaled Schoenfeld residuals against time suggests a violation.
• The Grambsch-Therneau test provides a formal statistical p-value for the correlation.
• If the assumption fails, alternatives like time-varying covariates or RMST analysis are warranted.