Log-Rank Test

A statistical test used to determine if two conditions produce differing survival curves
It works by applying a Chi-Squared Test across each time point of the survival curve

${\chi }^2=\frac{({\sum }_{i=1}^k{O}_{1i}-{\sum }_{i=1}^k{E}_{1i}{)}^2}{{\sum }_{i=1}^k{V}_i}$

• The Log-Rank statistic (${\chi }^2$) follows a Chi-squared distribution with 1 degree of freedom (for two groups)
• $k$: The total number of unique time points where at least one event occurred
• $O_{1i}$: The observed number of events in Group 1 at time $t_i$
• $E_{1i}$: The expected number of events in Group 1 at time $t_i$ under the null hypothesis
• $V_i$: The variance of the differences at time $t_i$

$E_{1i}=\frac{n_{1i}}{N_i}\times O_i$

• At each event time $t_i$, we calculate how many deaths we would expect in Group 1 if the risk were identical across both groups
• $n_{1i}$: Number of individuals at risk in Group 1 just before time $t_i$
• $N_i$: Total number of individuals at risk across both groups just before time $t_i$ ($n_{1i}+n_{2i}$)
• $O_i$: Total number of events (deaths) observed in both groups at time $t_i$

$V_i=\frac{O_i(n_{1i}/N_i)(1-n_{1i}/N_i)(N_i-O_i)}{N_i-1}$

• Hypergeometric Variance
• Because the Log-Rank test is based on a hypergeometric distribution (sampling without replacement from the “at risk” pool), the variance is more complex than a standard Poisson variance
• $O_i$: Total events at time $t_i$
• $n_{1i}$: Number at risk in Group 1
• $N_i$: Total number at risk (Group 1 + Group 2)
• $(N_i-O_i)/(N_i-1)$: The finite population correction factor, which distinguishes this from simple binomial variance