Greenwood's Formula

Definition

Greenwood’s Formula estimates the variance of the Kaplan-Meier estimator:

$\widehat{\mathrm{Var}}\text{[}\hat{S}(t)]=\text{[}\hat{S}(t){]}^2\sum \_t\_i\le t\frac{d\_i}{Y\_i(Y\_i-d\_i)}$

The standard error is:

$\widehat{\mathrm{SE}}\text{[}\hat{S}(t)]=\sqrt{\widehat{\mathrm{Var}}\text{[}\hat{S}(t)]}$

where:

• $\hat{S}(t)$ =Kaplan-Meier estimate at time $t$
• $d\_i$ = number of events at time $t\_i$
• $Y\_i$ = number at risk just before $t\_i$

About

This is NOT the normal formula for sample proportion variance.

Greenwood’s formula accounts for the fact that $\hat{S}(t)$ is a product of conditional probabilities, not a simple average. The variance accumulates multiplicatively through the product-limit estimation.

NOTE

Terms where $d\_i=Y\_i$ (everyone at risk fails) are excluded to avoid division by zero.

Related

• Kaplan-Meier Estimator
• Survival Function