Kaplan-Meier Curves

Overview and Fundamentals

• The KM method was introduced in 1958 by Edward Kaplan and Paul Meier and it is a nonparametric estimator used in survival analysis.^1-3
  • It is used to estimate how long it takes for a specific event to occur.^1,2
  • It can represent:
  • Tumor recurrence/progression¹
  • Drug response or withdrawal^1,2
• It addresses how to handle incomplete follow-up data.¹
  • It allows for estimation of survival probabilities even when subjects do not experience the event.^1-3
• In anesthesiology, KM analysis could be used to study:
  • Time to extubation following propofol vs sevoflurane anesthesia
  • Time to onset of postoperative nausea and vomiting (PONV)
• The goal is to quantify the probability of remaining event free (i.e., extubation not achieved or no PONV) across a period.

Time-to-Event Data

• An event captures whether and when an event occurs.⁴
  • Each participant contributes a “serial time” variable composed of:
    • A start point when the treatment is initiated^1,2
    • An end point when an event occurs or censoring^1,2
• Serial vs calendar time:
  • Serial time aligns participants by follow-up duration not by calendar date.¹
  • This allows meaningful comparison despite staggered enrollment.¹
• The KM estimator incorporates all subjects up to their event or censoring time, maintaining accuracy in longitudinal outcome measurement.²

Censoring and the KM Estimator

Censoring

• Occurs when a subject’s total time to an event is not fully observed and it represents incomplete information.^2-4
• There are several reasons for censoring:
  • Participants are lost to follow-up.¹
  • Study ends before the event occurs.¹
  • Subject remains event-free at study completion.¹
• On KM curves, censored subjects appear as tick marks.^1,3
  • Tick marks do not cause downward steps, preserving survival probability integrity.^1,3

Types of Censoring

• Right censoring:
  • The event has not yet occurred at last observation.^2-5
    • Type I: Study ends before all the events occur.^2,5
    • Type II: Study ends once a set number of events have occurred.^4,5
• Left censoring:
  • The event occurred before study entry; the subject entered was already at risk.^4,5
• Interval censoring:
  • The event occurred between two known times, but the exact moment is unknown.^4,5

Effect on Analysis

• Censored participants are removed from the risk set for subsequent time intervals.^1,2
• If censoring coincides with an event, it is conventionally treated as occurring immediately after the event to avoid underestimation.¹

The KM Estimator

• Also called the product limit estimator.²
• Calculates the conditional probability of survival at each observed event time.²
• At each event time, the probability of surviving that interval is:²
  • Survival probability = (number living at start – number died) / number living at start²
• Cumulative survival up to any point is obtained by multiplying the survival probabilities of all preceding intervals.²
• The resulting curve is a step function, remaining flat between events and dropping vertically at each event.^2-4
• The median survival time corresponds to the point where cumulative survival equals 0.5.^1-4

Interpretation, Assumptions, and Clinical Implications

Interpretation of KM Curves

• X-axis represents time^1-4,6
• Y-axis represents the proportion of patients who have not experienced the events.^1-4,6
• The step-shaped curve reflects event occurrences and censoring.^1-4,6
  • Horizontal segments represent intervals with no events.^1-4,6
  • Vertical drop represents one or more events.^1-4,6
  • Tick marks are censored observations.^1,3
• Differences between groups are commonly evaluated using the log-rank test.^1-4,6

Assumptions of the KM Method

• Each participant’s outcome is independent of others.⁴
• Censoring is unrelated to the likelihood of the event.⁴
• Event risk is consistent for participants regardless of when they enter the study.^2,4
• Events can occur at any time therefore, frequent follow-ups improve accuracy.⁴
• Violating these assumptions can lead to bias and reduced validity.⁴

Clinical and Research Implications

• Advantages
  • It accurately incorporates censored data which allows inclusion of patients that are lost to follow-up or event free at the completion of the study.⁷
  • It is a visually intuitive approach that clearly represents survival probability over time.⁷
  • The method allows straightforward comparison between treatment groups.⁷
• Disadvantages
  • It can overestimate survival probabilities in the presence of competing risks (i.e., deaths from unrelated causes).⁷
  • Missing data can introduce bias and reduce validity of results.^3,7
  • The reliability of the curve decreases near the end of follow-up when the number of patients at risk becomes small, increasing uncertainty.^3,7
• Understanding the assumptions, construction and limitations of KM analysis is essential for valid interpretation of survival data and evidence-based clinical decision-making.