Survival Analysis Reference Cheat Sheet

Key Facts

• The survival function is $S(t)=P(T>t)$, the probability that the event time $T$ is greater than time $t$.
• The cumulative distribution function is $F(t)=P(T\le t)=1-S(t)$ for an event time $T$.
• The hazard function is $h(t)=\lim_{\Delta t\to 0}\frac{P(t\le T<t+\Delta t\mid T\ge t)}{\Delta t}$, the instantaneous event rate among those still at risk.
• The cumulative hazard is $H(t)=\int_0^t h(u)\,du$, and it relates to survival by $S(t)=e^{-H(t)}$.
• The Kaplan-Meier estimator is $\hat{S}(t)=\prod_{t_i\le t}\left(1-\frac{d_i}{n_i}\right)$, where $d_i$ events occur among $n_i$ subjects at risk at time $t_i$.
• Right censoring occurs when a subject is known to survive beyond a time $c$, so the observed data are $Y=\min(T,c)$ and $\delta=I(T\le c)$.
• The log-rank test compares groups using observed and expected event counts, often summarized by $\chi^2=\frac{(O_1-E_1)^2}{V_1}$ for two groups.
• The Cox proportional hazards model is $h(t\mid x)=h_0(t)e^{\beta^Tx}$, and a one-unit increase in $x_j$ multiplies the hazard by $e^{\beta_j}$.

Vocabulary

Event time

The event time $T$ is the time from a defined starting point until the event of interest occurs.

Censoring

Censoring occurs when the exact event time is unknown but partial information about $T$ is still available.

Survival function

The survival function $S(t)$ gives the probability that a subject has not experienced the event by time $t$.

Hazard function

The hazard function $h(t)$ describes the instantaneous risk of the event at time $t$ among subjects still at risk.

Kaplan-Meier estimator

The Kaplan-Meier estimator $\hat{S}(t)$ is a step function estimate of survival that updates only at observed event times.

Hazard ratio

A hazard ratio compares hazards between groups or covariate levels, with $HR=e^{\beta}$ in a Cox model.