Hypothesis Testing Survival Cheatsheet

Notation

Symbol	Meaning
$t_i$	$i$-th ordered distinct event time (pooled)
$d_{ij}$	Events in group $j$ at time $t_i$
$Y_{ij}$	At risk in group $j$ just before $t_i$
$d_i={\sum }_j{d}_{ij}$	Total events at $t_i$
$Y_i={\sum }_j{Y}_{ij}$	Total at risk at $t_i$
$W(t_i)$	Weight function
$K$	Number of groups
$M$	Number of strata

Weight Functions

Test	$W(t_i)$	Characteristics
Log-Rank	$1$	Equal weight; most powerful under proportional hazards
Gehan / Breslow	$Y_i$	More weight to early times; generalization of Wilcoxon
Tarone-Ware	$\sqrt{Y_i}$	Intermediate between log-rank and Gehan

1-Sample Test

$H_0:h(t)=h_0(t),H_1:h(t)\ne h_0(t)$

$Z(\tau )={\sum }_{i=1}^{D}W(t_i)\frac{d_i}{Y(t_i)}-{\int }_0^{\tau }W(s)h_0(s)ds$

Under $H_0$: $\frac{Z(\tau {)}^2}{\mathrm{Var}[Z(\tau )]}\sim {\chi }_1^2$

With $W(t)=Y(t)$ (1-sample log-rank): $O(\tau )=\sum d_i,E(\tau )={\sum }_{j=1}^n[H_0(T_j)-H_0(L_j)]$

K-Sample Test

$H_0:h_1(t)=\dots =h_K(t)$

$Z_j(\tau )={\sum }_{i=1}^{D}W(t_i)\left[d_{ij}-Y_{ij}\frac{d_i}{Y_i}\right],j=1,\dots ,K$

${\hat{\sigma }}_{jj}={\sum }_{i=1}^{D}W(t_i{)}^2\frac{Y_{ij}}{Y_i}\left(1-\frac{Y_{ij}}{Y_i}\right)\left(\frac{Y_i-d_i}{Y_i-1}\right)d_i$

${\hat{\sigma }}_{jg}=-{\sum }_{i=1}^{D}W(t_i{)}^2\frac{Y_{ij}{Y}_{ig}}{Y_i^2}\left(\frac{Y_i-d_i}{Y_i-1}\right)d_i$

${\chi }^2=(Z_1,\dots ,Z_{K-1}){\hat{\Sigma }}^{-1}(Z_1,\dots ,Z_{K-1}{)}^T\sim {\chi }_{K-1}^2$

Two-Sample Special Case ($K=2$)

$Z=\frac{{\sum }_{i=1}^{D}W(t_i)\left[d_{i1}-Y_{i1}\frac{d_i}{Y_i}\right]}{\sqrt{{\sum }_{i=1}^{D}W(t_i{)}^2\frac{Y_{i1}}{Y_i}\left(1-\frac{Y_{i1}}{Y_i}\right)\left(\frac{Y_i-d_i}{Y_i-1}\right)d_i}}\sim N(0,1)$

Trend Test

For ordered groups with scores $a_1<a_2<\dots <a_K$:

$Z=\frac{{\sum }_{j=1}^K{a}_j{Z}_j(\tau )}{\sqrt{{\sum }_{j=1}^K{\sum }_{g=1}^K{a}_j{a}_g{\hat{\sigma }}_{jg}}}\sim N(0,1)$

Stratified Test

For $M$ strata, pool across strata:

$Z_{j\cdot }(\tau )={\sum }_{s=1}^M{Z}_{js}(\tau ),{\hat{\sigma }}_{jg\cdot }={\sum }_{s=1}^M{\hat{\sigma }}_{jgs}$

${\chi }^2=(Z_{1\cdot },\dots ,Z_{K-1\cdot }){\hat{\Sigma }}_{\cdot }^{-1}(Z_{1\cdot },\dots ,Z_{K-1\cdot }{)}^T\sim {\chi }_{K-1}^2$

Two-sample stratified: $Z=\frac{{\sum }_s{Z}_{1s}(\tau )}{\sqrt{{\sum }_s{\hat{\sigma }}_{11s}}}\sim N(0,1)$

Decision Rule Summary

Test	Statistic	Distribution	Reject $H_0$ if
1-sample (2-sided)	$Z^2/\mathrm{Var}(Z)$	${\chi }_1^2$	$>{\chi }_{1,\alpha }^2$
1-sample (1-sided)	$Z/\sqrt{\mathrm{Var}(Z)}$	$N(0,1)$	$\Vert Z\Vert >z_{\alpha }$
K-sample	${\chi }^2$	${\chi }_{K-1}^2$	$>{\chi }_{K-1,\alpha }^2$
Trend	$Z$	$N(0,1)$	$\Vert Z\Vert >z_{\alpha /2}$
Stratified	${\chi }^2$	${\chi }_{K-1}^2$	$>{\chi }_{K-1,\alpha }^2$

Related

• One-Sample Log-Rank Test
• Log-Rank Test
• Trend Test
• Stratified Test
• Non-Parametric Estimation Cheatsheet