Non-Parametric Estimation Cheatsheet

Notation

Symbol	Meaning
$t_i$	$i$-th ordered distinct event time
$d_i$	Number of events (failures) at time $t_i$
$Y_i$	Number at risk just before time $t_i$
$c_i$	Number censored between $t_i$ and $t_{i+1}$
$n=Y_0$	Total subjects at start

Estimators

Kaplan-Meier (Survival Function)

$$\begin{array}{rl}\hat{S}(t)& =\prod _{t_i\le t}\left(1-\frac{d_i}{Y_i}\right)\\ \widehat{\mathrm{Var}}[\hat{S}(t)]& =[\hat{S}(t){]}^2\sum _{t_i\le t}\frac{d_i}{Y_i(Y_i-d_i)}\\ \widehat{\mathrm{SE}}[\hat{S}(t)]& =\sqrt{\widehat{\mathrm{Var}}[\hat{S}(t)]}\end{array}$$

Nelson-Aalen (Cumulative Hazard)

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

Confidence Intervals for $\hat{S}(t)$

Linear Scale (Naive)

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

WARNING

Can produce values outside $[0,1]$. Not recommended.

Log-Log Transform (Recommended)

$$\begin{array}{rl}& \left[\hat{S}(t{)}^{\exp \left(\mp z_{\alpha /2}\hat{\sigma }(t)\right)},\hat{S}(t{)}^{\exp \left(\pm z_{\alpha /2}\hat{\sigma }(t)\right)}\right]\\ & {\hat{\sigma }}^2(t)=\frac{\widehat{\mathrm{Var}}[\hat{S}(t)]}{[\hat{S}(t)\ln \hat{S}(t){]}^2}\end{array}$$

Common $z_{\alpha /2}$ values

Confidence Level	$z_{\alpha /2}$
90%	1.645
95%	1.960
99%	2.576

Key Relationships

$${\hat{S}}_{KM}(t)\approx \exp (-{\hat{H}}_{NA}(t))$$

$Y_{i+1}=Y_i-d_i-c_i$

To Estimate	Use	Variance
$S(t)$	Kaplan-Meier	Greenwood’s formula
$H(t)$	Nelson-Aalen	$\sum \frac{d_i}{Y_i^2}$
$h(t)$	$d_i/Y_i$ at each event time	—

Related

• Kaplan-Meier Estimator
• Nelson-Aalen Estimator
• Greenwood’s Formula
• Confidence Interval for Survival