Confidence Interval for Survival

Definition

A confidence interval for $\hat{S}(t)$ provides uncertainty bounds around the Kaplan-Meier estimate.

Pointwise Confidence Interval Methods

Let

• $z_{\alpha /2}$ : inverse cdf of Normal Distribution
• $\hat{S}(t)$ : Kaplan-Meier estimate
• $\widehat{\text{Var}}[\hat{S}(t)]$ : Greenwood’s Formula

Linear Scale (Naive)

$\hat{S}(t)\pm z_{\alpha /2}\sqrt{\widehat{\mathrm{Var}}[\hat{S}(t)]}$

Problem

Can produce values outside $[0,1]$impossible for probabilities.

Log-Log Transform (Recommended)

$\left[\hat{S}(t{)}^{\exp (\mp z_{\alpha /2}\hat{\sigma }(t))},\hat{S}(t{)}^{\exp (\pm z_{\alpha /2}\hat{\sigma }(t))}\right]$

where:

${\hat{\sigma }}^2(t)=\frac{\widehat{\mathrm{Var}}[\hat{S}(t)]}{[\hat{S}(t)\ln \hat{S}(t){]}^2}$

The log-log transform ensures the CI stays within $[0,1]$.

For 95% Confidence Interval

$z_{0.025}=1.96$

For Nelson-Aalen Estimator

Variance of $\hat{H}(t)$:

$\widehat{\mathrm{Var}}[\hat{H}(t)]={\sum }_{t_i\le t}\frac{d_i}{Y_i^2}$

Related

• Kaplan-Meier Estimator
• Greenwood’s Formula
• Nelson-Aalen Estimator