## What Is Survival Analysis?
Survival analysis models the time until an event occurs. Originally developed for medical contexts (time until patient death), it applies naturally to sports events that occur as a function of time.
## Applications in Sports Betting
**Time to first goal:**
"What is the probability that the first goal occurs before the 30-minute mark?"
Survival analysis models the hazard rate (probability of a goal in the next minute, given no goal has yet occurred) as a function of time and match context.
**Manager survival:**
"How long until a manager is sacked?" Teams in poor form have higher sacking hazard rates. This powers outright "first manager to leave" markets.
**Player injury return:**
"When will this player return from injury?" Injury duration models use survival analysis to estimate return probabilities by date.
## The Key Concepts
**Survival function S(t):** P(event occurs after time t) — the probability of surviving (no event) past time t.
**Hazard function h(t):** The instantaneous rate of event occurrence at time t, given survival to t.
For football goals: the hazard rate is not constant — it peaks around the 75–90 minute mark as teams tire and matches are decided.
## The Cox Proportional Hazards Model
The Cox model estimates how covariates (team quality, score state) modify the hazard rate:
h(t|X) = h₀(t) × exp(β₁X₁ + β₂X₂ + ...)
Where h₀(t) is the baseline hazard and the exponential term modifies it by team-specific factors.
This produces: "A high-quality home team with a score advantage has a hazard rate for conceding 30% lower than the baseline."
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