Kaplan-Meier Estimator

Definition

Kaplan-Meier Estimator for survival function:

$\hat{S}(t)={\prod }_{t_i\le t}\left(1-\frac{d_i}{Y_i}\right)$

where:

• $d_i$ = number of events occurred at time $t_i$ (not censored)
• $Y_i$ = number at risk just before $t_i$

About Kaplan-Meier Approach

The Kaplan-Meier estimator builds the survival curve step-by-step, calculating the conditional probability of surviving past each observed event time.

Timeline and Definitions:

• $n=Y_0$: Total number of subjects at the start
• $Y_i$: Number at risk just before time $t_i$
• $d_i$: Number of events at time $t_i$
• $c_i$: Number censored between $t_i$ and $t_{i+1}$

Step-by-step Intuition:

1. At time $t_0$: Everyone is alive. $S(t_0)=\mathrm{Pr}(T>t_0)=1$

2. At time $t_1$: $Y_1$ people at risk. $d_1$ events. $S(t_1)=S(t_0)\times \left(1-\frac{d_1}{Y_1}\right)$

3. At time $t_2$: $Y_2=n-d_1-c_1$ people at risk. $S(t_2)=S(t_1)\times \left(1-\frac{d_2}{Y_2}\right)$

Example

Suppose a study follows 5 subjects with the following data:

Time $t_i$	$Y_i$ (at risk)	$d_i$ (events)	$1-d_i/Y_i$	$\hat{S}(t)$
0	5	0	1.000	1.000
3	5	1	$1-1/5=0.800$	$1.000\times 0.800=0.800$
5	4	1	$1-1/4=0.750$	$0.800\times 0.750=0.600$
8	3	1	$1-1/3=0.667$	$0.600\times 0.667=0.400$

Interpretation: After $t=3$, 80% of subjects survive. By $t=8$, only 40% remain. Each step down corresponds to an event time; the curve stays flat between events.

Interpretation

The Kaplan-Meier curve is a step function: it drops only at event times and stays constant between them. Censored observations reduce $Y_i$ for subsequent steps but don’t cause a drop themselves.

Related

• Nelson-Aalen Estimator
• Survival Function