Hazard Functions: Why Time-to-Event Beats Traditional Risk Measures

🚨 Why Hazard? The Limits of Traditional Risk Measures

Most measures (Risk, Odds, Rate) ask:

“Did the event happen?”

But Time-to-Event asks:

“When did it happen — if at all?”

In long-term studies, many patients may:

• Be followed for different lengths

• Drop out or die from other causes (censoring)

• Survive beyond study period

That’s where hazard functions shine — they handle variable follow-up, censoring, and time-dependent risk.

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🧾 Section 1: Definitions You Must Master

Term	Meaning
Event	The outcome of interest (e.g., death, relapse)
Time-to-event	Time from baseline to event (or censoring)
Censoring	The event is unknown because patient left or study ended
Survival function S(t)	Probability of surviving beyond time t: P(T > t)
Hazard function h(t)	Instantaneous rate of event at time t, given survival until t
Failure function F(t)	Probability that event has occurred by time t: 1 − S(t)

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⚙️ Section 2: Formal Definition of Hazard

Intuitive First:

“Among those who survived up to time t, what’s the instantaneous risk of failure right after t ?”

Mathematical Form:

This is the instantaneous incidence rate at time t .

Contrast this with:

• Incidence rate → average over time

• Hazard → moment-by-moment risk
  

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🔍 Section 3: Hazard vs Rate vs Risk

Concept	Unit	Accounts for Time?	Deals with Censoring?	Best For
Risk	Probability (0–1)	❌	❌	Short-term fixed cohorts
Incidence Rate	Events per time	✅	❌	Longitudinal with fixed follow-up
Hazard	Rate at time ttt	✅✅	✅	Dynamic, censored, long follow-up

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📈 Section 4: Shapes of Hazard Functions

• Constant Hazard → exponential survival (e.g., radioactive decay)

• Increasing Hazard → aging-related mortality (e.g., cancer)

• Decreasing Hazard → surgical recovery (e.g., stroke rehab)

This shape tells the story of risk over time — essential for modeling.

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⏳ Section 5: Kaplan-Meier & Survival Function

To visualize survival:

Kaplan-Meier is a non-parametric estimator of the survival function.

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📊 Section 6: Cox Proportional Hazards Model

Used to model hazard without assuming baseline hazard shape:

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🧪 Section 7: Time-to-Event Analysis Outputs

You may see:

• Median survival time:t when S(t) = 0.5

• Disease-free survival (DFS): No relapse or event

• Overall survival (OS): Any-cause mortality

• 3-year survival: S(3)

  
  

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🔚 Section 8: Key Takeaways

Hazard is the time-specific, instantaneous risk of event — not just “if” but “when.”

• It captures the speed of deterioration over time.

• Survival function tells us probability of being alive beyond t.

• Use Cox models to adjust for covariates without assuming rate shapes.

• Hazard Ratio ≠ Risk Ratio. HR reflects ongoing risk; RR is fixed in time.