Kaplan–Meier Survival Analysis

Kaplan–Meier Survival Analysis

1. Definition, scope & why survival data are special

• Survival analysis is the branch of statistics concerned with the time between entry to a study and a subsequent event ("time-to-event" or TTE data) (Swinscow & Campbell 10e Ch.12, p.126).
  • Originally developed for time from treatment until death — hence the name "survival" — but the methods apply to any well-defined endpoint, not only mortality (Swinscow & Campbell 10e Ch.12, p.126).
  • The two pieces of information for each subject are (i) a time and (ii) a status indicator stating whether the event occurred or the observation was censored at that time (Swinscow & Campbell 10e Ch.12, p.127).
• The variable analysed is the survival time t (a duration), with an associated event indicator d (1 = event occurred; 0 = censored) (Swinscow & Campbell 10e Ch.12, pp.126–7).
• Examples of survival/time-to-event endpoints beyond death (Swinscow & Campbell 10e Ch.12, p.126):
  • Time to discontinuation of a contraceptive.
  • Maximum dose of bronchoconstrictor required to reduce a patient's lung function to 80% of baseline.
  • Time taken to exercise to maximum tolerance.
  • Time a transdermal patch can be left in place.
  • Time for a leg fracture to heal.
• Two key problems make ordinary statistics (t-test, mean ± SD) inappropriate when the outcome is the time between one event and another (Swinscow & Campbell 10e Ch.12, p.126):
  • (1) Non-Normality — survival times are unlikely to be Normally distributed; they are typically right-skewed (a long tail of long survivors), so the arithmetic mean and SD are misleading and Normal-theory tests do not hold (Swinscow & Campbell 10e Ch.12, p.126).
  • (2) Incomplete follow-up / censoring — we cannot afford to wait until the event has happened to every subject (e.g. until all are dead). At analysis some subjects are still event-free. For these the only information is that they were event-free up to their last contact (Swinscow & Campbell 10e Ch.12, p.126).
• Because of skew + censoring, survival data demand specialised non-parametric estimators (Kaplan–Meier), specialised hypothesis tests (log-rank), and specialised regression (Cox) rather than means, t-tests, or ordinary linear/logistic regression (Swinscow & Campbell 10e Ch.12, pp.126, 131).
• Pharmacology / clinical-trial relevance — survival (TTE) endpoints are the workhorse of oncology and cardiovascular outcome trials: overall survival (OS), progression-free survival (PFS), disease-free survival, time-to-treatment-failure, time-to-first-MACE. The Kaplan–Meier curve + log-rank p + hazard ratio (with 95% CI) is the standard summary triplet for a two-arm trial (Swinscow & Campbell 10e Ch.12, pp.129–133).