Survival Analysis

Survival Analysis: Intro - Time's Ticking Tale

• Survival Analysis: Statistical methods for time-to-event data.
• Time-to-event data: Measures time from a defined start to an event.
  • Event: Death, disease recurrence, recovery, or other outcome.
  • Survival Time: Duration until the event occurs.
• Censoring: Incomplete information on survival time.
  • Right Censoring: Event not observed by study end or patient lost to follow-up.

    ⭐ Right censoring is the most common type of censoring in medical studies, where the event has not occurred by the end of the study or the patient is lost to follow-up.

  • Left Censoring: Event occurred before observation began.
  • Interval Censoring: Event occurred within a known time interval.
• Applications: Assessing drug efficacy, patient prognosis, comparing treatment effectiveness_

Survival & Hazard Functions - Life's Ebb & Flow

• Survival Function $S(t)$:
  • Definition: $S(t) = P(T > t)$, probability of surviving beyond time $t$.
  • Properties: Non-increasing; starts at 1 (time 0), ends at/near 0.
  • Interpretation: Proportion surviving at least up to time $t$.
• Hazard Function $h(t)$ or $\lambda(t)$:
  • Definition: $h(t) = \lim_{\Delta t \to 0} \frac{P(t \le T < t+\Delta t | T \ge t)}{\Delta t}$.
  • Interpretation: Instantaneous risk of event at time $t$, given survival up to $t$.
  • 📌 Mnemonic: Hazard = 'H'ealth 'H'arming rate.
• Relationship:
  • $S(t) = \exp(-\int_0^t h(u)du)$
  • $h(t) = -\frac{d}{dt} \ln(S(t)) = \frac{f(t)}{S(t)}$ ($f(t)$ = event density).

and various Hazard Function h(t) shapes over time)

⭐ The hazard function represents the instantaneous potential per unit time for the event to occur, given that the individual has survived up to time t.

Kaplan-Meier Curves - Stepwise Survival Story

• Kaplan-Meier (KM) Estimator: Non-parametric method to estimate survival function $S(t)$ from time-to-event data, especially with censored observations.
• Interpreting KM Curve:
  • Step function: survival probability is constant between events.
  • Drops occur only at event times (e.g., death).
  • Censored observations (lost to follow-up, study ends) marked by ticks; do not cause drops.
• Median Survival Time: Time point where survival probability $S(t) = \textbf{0.5}$.
• Advantages: Handles censored data effectively; intuitive visual.
• Limitations: Primarily descriptive; difficult to adjust for covariates.

⭐ The Kaplan-Meier curve is a step function that changes (drops) only at the time of an event; censored observations are indicated but do not cause a drop in the curve.

Comparing & Modeling Survival - Tests & Predictions

• Log-Rank Test:
  • Purpose: Compares survival distributions (≥2 groups).
  • $H_0$: No difference in survival.
  • p < 0.05: Significant difference.
• Cox Proportional Hazards Model (Multivariable):
  • Purpose: Multivariable; assesses covariate effects on hazard rate.
  • Hazard Ratio (HR):
    • HR = 1.0: No effect.
    • HR > 1.0: ↑ Hazard (HR=2.0 → double hazard).
    • HR < 1.0: ↓ Hazard (protective).
  • Formula: $h(t, X) = h_0(t) \exp(\beta_1 X_1 + ... + \beta_p X_p)$.
  • Key Assumption: Proportional hazards (HR constant over time).
    • Check: Log-log plots.

Flowchart: Log-Rank vs. Cox Model

⭐ A Hazard Ratio (HR) from a Cox model of 2.0 for an exposure group compared to an unexposed group means the exposed group has twice the hazard (risk of event) at any given time, assuming proportional hazards.

High‑Yield Points - ⚡ Biggest Takeaways

• Kaplan-Meier curves show survival probability; steps down at each event.
• Log-rank test compares survival curves between groups.
• Cox Proportional Hazards model evaluates multiple predictors, yielding Hazard Ratios (HR).
• HR > 1 means ↑ risk; HR < 1 means ↓ risk of event.
• Censoring (e.g., right-censoring) handles incomplete follow-up data.
• Median survival time: Time at which 50% survive; robust to outliers.
• Survival function S(t): Probability of surviving beyond time t.