How to Compute Survival and Hazard Functions: What They Measure and How They Relate

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Survival vs Hazard Functions
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Survival and hazard functions are core concepts in biostatistics tutoring, especially in survival analysis, epidemiology, and medical statistics. Students often memorize formulas without understanding how these functions relate. This page explains the meaning, intuition, and mathematical connection between survival and hazard functions.

The survival function gives the probability of surviving past time t. The hazard function gives the instantaneous risk of failure at time t, given survival until t.

\(S(t) = P(T > t)\) \(h(t) = \frac{f(t)}{S(t)}\)

Why do we need both functions? Because the survival function describes the overall probability of surviving, while the hazard function describes the instantaneous risk at each moment. Together, they provide a complete picture of time‑to‑event behavior. The hazard function is especially useful in medical studies where risk changes over time.

  1. Start with a time‑to‑event variable. Let \(T\) be a non‑negative random variable representing survival time.
  2. Define the survival function. \(S(t) = P(T > t)\).
  3. Define the density. \(f(t) = \frac{d}{dt}(1 – S(t)) = -S'(t)\).
  4. Form the hazard function. \(h(t) = \frac{f(t)}{S(t)}\).
  5. Connect hazard to survival. \(S(t) = \exp\left(-\int_0^t h(u)\,du\right)\).
  6. Interpret the relationship. The hazard function determines the shape of the survival curve.

Suppose the hazard function is constant: \(h(t) = 0.2\). Then:

\(S(t) = e^{-0.2t}\)

At \(t = 5\):

\(S(5) = e^{-1} \approx 0.3679\)

This means there is a 36.8% chance of surviving past time 5.

  • Thinking the hazard is a probability (it is not).
  • Confusing density with hazard.
  • Assuming high hazard means low survival (depends on cumulative hazard).
  • Forgetting that survival is the exponential of negative cumulative hazard.

Survival and hazard functions are essential for Kaplan–Meier curves, Cox models, parametric survival models, and medical risk analysis. They appear in nearly every graduate biostatistics exam and form the backbone of survival analysis.

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