One-Sample Log-Rank Test

Definition

Let:

• $n$ : sample size (with censored observations) from a population
• $h(t)$ : true hazard function
• $h_0(t)$ : known/hypothesized hazard function (reference population)
• $\tau$ : maximum follow-up time
• $t_1<t_2<\dots <t_D$ : ordered distinct event times

Test Hypotheses: $H_0:h(t)=h_0(t)\forall t\le \tau \text{vs}{H}_1:h(t)\ne h_0(t)\text{for some }t\le \tau$

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

where:

• $d_i$ : number of events at time $t_i$
• $Y(t_i)$ : number at risk just before $t_i$
• $W(t)$ : weight function, with $W(t)=0$ when $Y(t)=0$

Under $H_0$: $\mathrm{Var}[Z(\tau )]={\int }_0^{\tau }{W}^2(s)\frac{h_0(s)}{Y(s)}ds$

For large $n$: $\frac{Z(\tau {)}^2}{\mathrm{Var}[Z(\tau )]}\sim {\chi }_1^2$

For one-sided alternative $h(t)>h_0(t)$: $\frac{Z(\tau )}{\sqrt{\mathrm{Var}[Z(\tau )]}}\sim N(0,1)$

1-Sample Log-Rank Test ($W(t)=Y(t)$):

• $O(\tau )={\sum }_{i=1}^D{d}_i$ : total observed events up to $\tau$
• $E(\tau )=\mathrm{Var}[Z(\tau )]={\sum }_{j=1}^n\left[H_0(T_j)-H_0(L_j)\right]$
• $L_j$ : entry time of subject $j$ into the study
• $T_j$ : exit time (event or censoring) of subject $j$

Interpretation

Compare whether the observed number of events $O(\tau )$ matches what is expected under a known hazard $h_0(t)$. If $Z(\tau )$ is large (positive), the sample hazard exceeds the reference hazard.

Related

• Log-Rank Test
• Nelson-Aalen Estimator
• Cumulative Hazard