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 at time $t\_i$
• $Y\_i$ = 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. $S(t\_2)=S(t\_1)\times \left(1-\frac{d\_2}{Y\_2}\right)$

Related

• Nelson-Aalen Estimator
• Survival Function