Survival Curve Estimation | Survival Analysis

Survival analysis examines the time to a prespecified event such as death, recurrence, or recovery. The Kaplan–Meier method estimates the survival function while retaining information from participants who remain event-free when follow-up ends and are therefore right-censored.

Population Survival Function and Sample Survival Curve#

population survival function

S(t)=P(T>t)=1-F(t)

sample proportion without censoring

\hat S(t)=\frac{\#\{T_i>t\}}n

S(t) is the probability of remaining event-free beyond time t measured from the defined origin. It typically starts at 1 and cannot increase over time. With censoring, a simple proportion cannot correctly account for differences in follow-up duration.

Events, Right Censoring, and the Time Origin#

Define a consistent time origin for all participants
Prespecify what constitutes an event
Distinguish the event time from the last known event-free time
Treat participants without a confirmed event at loss to follow-up or study end as right-censored
State clearly how the event indicator is coded

Why Kaplan–Meier Multiplies Conditional Probabilities#

Let the distinct event times be t₁<t₂<…, with nⱼ participants at risk immediately before tⱼ and dⱼ events at tⱼ. Survival through each time is the product of the successive conditional survival probabilities.

conditional survival proportion

P(T>t_j\mid T\ge t_j)=\frac{n_j-d_j}{n_j}

product-limit estimator

\hat S(t)=\prod_{t_j\le t}\left(\frac{n_j-d_j}{n_j}\right)

The curve steps downward at event times and remains at the same height at censoring times. Tied events are handled together through dⱼ.

Step-by-Step Example with 10 Participants#

Time	At risk nⱼ	Events dⱼ	Conditional survival	Ŝ(t)
2	10	1	9/10	0.900
6	9	1	8/9	0.800
7	8	2	6/8	0.600
8	5	1	4/5	0.480
9	4	1	3/4	0.360
12	2	1	1/2	0.180

t=7

\hat S(7)=\frac9{10}\frac8{9}\frac6{8}=0.600

t=12

\hat S(12)=0.360\times\frac12=0.180

Greenwood’s Formula: Uncertainty in Survival Estimates#

variance

\widehat{\operatorname{Var}}[\hat S(t)]=\hat S(t)^2\sum_{t_j\le t}\frac{d_j}{n_j(n_j-d_j)}

standard error

SE[\hat S(t)]=\hat S(t)\sqrt{\sum_{t_j\le t}\frac{d_j}{n_j(n_j-d_j)}}

As the risk set shrinks later in follow-up, each event contributes more uncertainty. Because the tail of the curve may be supported by very few participants, it should not be overinterpreted.

Wald and Log–Log Confidence Intervals#

\hat S(t)\pm z_{1-\alpha/2}SE[\hat S(t)]

A conventional Wald interval can be less than 0 or greater than 1. Although truncation is easy, coverage can be poor, so log–log transformation is often used to keep the probability range natural.

transformation

g(t)=\ln[-\ln\hat S(t)]

SE after transformation

SE[g(t)]=\sqrt{\frac1{[\ln\hat S(t)]^2}\sum_{t_j\le t}\frac{d_j}{n_j(n_j-d_j)}}

interval on the original scale

\hat S(t)^{\exp[zSE(g)]}<S(t)<\hat S(t)^{\exp[-zSE(g)]}

Relationship to the Hazard Function#

The survival function gives the probability of remaining event-free through time t. The hazard h(t) is the instantaneous event rate at t among individuals who have survived to that time; it is a rate, not a probability.

hazard

h(t)=\lim_{\Delta t\to0}\frac{P(t\le T<t+\Delta t\mid T\ge t)}{\Delta t}

relationship to the density

h(t)=\frac{f(t)}{S(t)}=-\frac{S'(t)}{S(t)}

cumulative hazard

H(t)=\int_0^t h(u)du=-\ln S(t)

inverse relationship

S(t)=e^{-H(t)}

Checks before Kaplan–Meier Estimation#

Use consistent definitions for the time origin, event, and final follow-up time
Distinguish event indicators from right-censoring correctly
Assess independence between participants
Consider whether censoring is conditionally noninformative
Report numbers at risk, event counts, censoring marks, Ŝ(t), and confidence intervals
Do not overinterpret the sparse tail of the curve