Log-Rank Test

Tests whether hazard function of different groups are equal.

Definition

Let:

• $t_1<t_2<\dots <t_D$ : ordered distinct event times (across all groups pooled)
• $D$ : number of distinct event times
• $K$ : number of groups (populations) being compared
• $d_{ij}$ : number of events in group $j$ at time $t_i$
• $Y_{ij}$ : number at risk in group $j$ just before $t_i$
• $d_i={\sum }_{j=1}^K{d}_{ij}$ : total events at time $t_i$ (pooled)
• $Y_i={\sum }_{j=1}^K{Y}_{ij}$ : total at risk at time $t_i$ (pooled)
• $W(t_i)$ : weight function applied at time $t_i$ (see Weight Functions)

Hypotheses

$$\begin{array}{rl}{H}_0& :h_1(t)=h_2(t)=\dots =h_K(t)\forall t\le \tau \\ & \\ H_1& :\text{not all }{h}_j(t)\text{ are equal for some }t\le \tau \end{array}$$

where $\tau$ is the end of study time (maximum follow-up).

Under $H_0$, the expected hazard in each group is $E[h_j(t)]=\frac{d_i}{Y_i}$

Test Statistic

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

Variance and Covariance

$$\begin{array}{rl}{\hat{\sigma }}_{jj}& =\sum _{i=1}^{D}W(t_i{)}^2\frac{Y_{ij}}{Y_i}\left(1-\frac{Y_{ij}}{Y_i}\right)\left(\frac{Y_i-d_i}{Y_i-1}\right)d_i\\ {\hat{\sigma }}_{jg}& =-\sum _{i=1}^{D}W(t_i{)}^2\frac{Y_{ij}{Y}_{ig}}{Y_i^2}\left(\frac{Y_i-d_i}{Y_i-1}\right)d_i,g\ne j\end{array}$$

Overall Test

Let $\hat{\Sigma }$ be the $(K-1)\times (K-1)$ estimated covariance matrix with entries ${\hat{\sigma }}_{jj}$ on the diagonal and ${\hat{\sigma }}_{jg}$ off-diagonal. Then:

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

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

Weight Functions

Test Name	$W(t_i)$	Description
Log-Rank	$1$	Equal weight to all event times
Gehan / Breslow	$Y_i$	Generalization of Mann-Whitney-Wilcoxon / Kruskal-Wallis
Tarone-Ware	$\sqrt{Y_i}$	Intermediate weighting between log-rank and Gehan

Interpretation

The log-rank test ($W=1$) is most powerful when hazard ratios are constant over time. Gehan’s test ($W=Y_i$) gives more weight to early event times. Tarone-Ware ($W=\sqrt{Y_i}$) balances the two.

Example: Leukemia Remission Data

Case: Compare survival between two leukemia treatment groups — Group 1 (6-MP treatment) vs Group 2 (placebo).

# Group 1 (6-MP treatment)
time1  <- c(6, 6, 6, 7, 10, 13, 16, 22, 23, 6, 9, 10, 11, 17, 19, 20, 25, 32, 32, 34, 35)
status1 <- c(1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
 
# Group 2 (placebo)
time2  <- c(1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23)
status2 <- c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Step 1 — KM curves:

library(survival)
 
fit1 <- survfit(Surv(time1, status1) ~ 1)
fit2 <- survfit(Surv(time2, status2) ~ 1)
 
plot(fit1, conf.int="none", col="blue",
     xlab="Time (weeks)", ylab="Survival Probability")
lines(fit2, conf.int="none", col="red")
legend(19, 1, c("Treatment", "Placebo"), col=c("blue","red"), lty=1)

Step 2 — Log-rank test:

time      <- c(time1, time2)
status    <- c(status1, status2)
treatment <- c(rep(1, length(time1)), rep(2, length(time2)))
 
fit <- survdiff(Surv(time, status) ~ treatment)
fit
# Output: N, Observed, Expected, (O-E)^2/E, (O-E)^2/V, Chisq, p-value

Interpretation: The log-rank test answers: “Do the two survival curves differ beyond random chance?” The KM plot provides visual comparison; the log-rank test provides statistical confirmation. A significant p-value means the treatment effect is statistically significant.

Interpretation

For survival analysis, the log-rank test ($W=1$) is most powerful under the proportional hazards assumption. If PH is violated, consider alternative weights (Gehan/Breslow) or methods.

Related

• One-Sample Log-Rank Test
• Trend Test
• Stratified Test
• Kaplan-Meier Estimator