Why the Hazard Function and Survival Function Measure Different Aspects of Time‑to‑Event Data

What Are the Hazard and Survival Functions?

Formally:

\[ S(t) = P(T > t) \]

\[ h(t) = \lim_{\Delta t \to 0} \frac{P(t \le T < t+\Delta t \mid T \ge t)}{\Delta t} \]

The survival function is cumulative; the hazard function is instantaneous.

Why the Hazard and Survival Functions Measure Different Concepts

1. Survival is a probability; hazard is a rate

Survival answers: “What is the chance I make it past time t?” Hazard answers: “Given I made it to t, how risky is the next instant?”

2. Survival accumulates risk over time

It integrates all past hazards.

3. Hazard describes the shape of risk

Hazard can increase, decrease, or be constant — even when survival is monotone.

4. Hazard can be high even when survival is high

Early in time, survival is near 1, but hazard may already be rising.

5. Survival cannot increase; hazard can

Survival is a non‑increasing function. Hazard can fluctuate dramatically.

How the Hazard and Survival Functions Are Connected

Step 1: Define the cumulative hazard

\[ H(t) = \int_0^t h(u)\,du \]

Step 2: Survival is the exponential of negative cumulative hazard

\[ S(t) = e^{-H(t)} \]

Step 3: Hazard is the derivative of cumulative hazard

\[ h(t) = H'(t) \]

Step 4: Hazard shapes survival

A high hazard early → survival drops quickly. A low hazard early → survival stays high.

Step 5: Hazard ratios compare hazards, not survival

This is why Cox models estimate hazard ratios, not survival differences.

Numerical Example

Suppose the hazard is constant:

\[ h(t) = 0.1 \]

Then:

\[ H(t) = 0.1t \]

\[ S(t) = e^{-0.1t} \]

Interpretation:

• Hazard stays constant at 0.1
• Survival decreases exponentially
• At t = 10, survival = \(e^{-1} \approx 0.37\)

This example shows how a simple hazard pattern generates a predictable survival curve.