Survival analysis is a branch of statistics focused on analyzing time-to-event data, where the outcome is the time until a specific event occurs (death, failure, churn, recovery). Originating in biomedical research to model patient lifespans, survival analysis has expanded across clinical trials, reliability engineering, customer retention analytics, and social sciences. The key feature that distinguishes survival analysis from standard regression is its ability to handle censored observations—cases where the event has not yet occurred by the end of the study period. Unlike traditional methods that discard incomplete data, survival analysis incorporates partial information through specialized estimators and models, providing unbiased estimates even when exact event times are unknown for some subjects. The central insight is that knowing someone hasn't experienced the event by time t is itself valuable information.
What This Cheat Sheet Covers
This topic spans 26 focused tables and 125 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Core Concepts
| Concept | Example | Description |
|---|---|---|
S(t) = P(T > t) | • Probability that the event has not occurred by time t• decreases from 1 at t=0 toward 0 as time increases. | |
h(t) = \lim_{\Delta t \to 0} \frac{P(t \leq T < t + \Delta t \mid T \geq t)}{\Delta t} | • Instantaneous rate of event occurrence at time t given survival up to t• describes risk intensity at each moment. | |
H(t) = \int_0^t h(u) du | • Total accumulated hazard from time 0 to t• related to survival via S(t) = e^{-H(t)}. |