Logrank test

The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time. It is constructed by computing the observed and expected number of events in one of the groups at each observed event time and then adding these to obtain an overall summary across all-time points where there is an event.

Consider two groups of patients, e.g., treatment vs. control. Let ${\displaystyle 1,\ldots ,J}$ be the distinct times of observed events in either group. Let ${\displaystyle N_{1,j}}$ and ${\displaystyle N_{2,j}}$ be the number of subjects "at risk" (who have not yet had an event or been censored) at the start of period ${\displaystyle j}$ in the groups, respectively. Let ${\displaystyle O_{1,j}}$ and ${\displaystyle O_{2,j}}$ be the observed number of events in the groups at time ${\displaystyle j}$. Finally, define ${\displaystyle N_{j}=N_{1,j}+N_{2,j}}$ and ${\displaystyle O_{j}=O_{1,j}+O_{2,j}}$.

The null hypothesis is that the two groups have identical hazard functions, ${\displaystyle H_{0}:h_{1}(t)=h_{2}(t)}$. Hence, under ${\displaystyle H_{0}}$, for each group ${\displaystyle i=1,2}$, ${\displaystyle O_{i,j}}$ follows a hypergeometric distribution with parameters ${\displaystyle N_{j}}$, ${\displaystyle N_{i,j}}$, ${\displaystyle O_{j}}$. This distribution has expected value ${\displaystyle E_{i,j}=O_{j}{\frac {N_{i,j}}{N_{j}}}}$ and variance ${\displaystyle V_{i,j}=E_{i,j}\left({\frac {N_{j}-O_{j}}{N_{j}}}\right)\left({\frac {N_{j}-N_{i,j}}{N_{j}-1}}\right)}$.

For all ${\displaystyle j=1,\ldots ,J}$, the logrank statistic compares ${\displaystyle O_{i,j}}$ to its expectation ${\displaystyle E_{i,j}}$ under ${\displaystyle H_{0}}$. It is defined as

${\displaystyle Z_{i}={\frac {\sum _{j=1}^{J}(O_{i,j}-E_{i,j})}{\sqrt {\sum _{j=1}^{J}V_{i,j}}}}\ {\xrightarrow {d}}\ {\mathcal {N}}(0,1)}$      (for ${\displaystyle i=1}$ or ${\displaystyle 2}$)

By the central limit theorem, the distribution of each ${\displaystyle Z_{i}}$ converges to that of a standard normal distribution as ${\displaystyle J}$ approaches infinity and therefore can be approximated by the standard normal distribution for a sufficiently large ${\displaystyle J}$. An improved approximation can be obtained by equating this quantity to Pearson type I or II (beta) distributions with matching first four moments, as described in Appendix B of the Peto and Peto paper.[2]

If the two groups have the same survival function, the logrank statistic is approximately standard normal. A one-sided level ${\displaystyle \alpha }$ test will reject the null hypothesis if ${\displaystyle Z>z_{\alpha }}$ where ${\displaystyle z_{\alpha }}$ is the upper ${\displaystyle \alpha }$ quantile of the standard normal distribution. If the hazard ratio is ${\displaystyle \lambda }$, there are ${\displaystyle n}$ total subjects, ${\displaystyle d}$ is the probability a subject in either group will eventually have an event (so that ${\displaystyle nd}$ is the expected number of events at the time of the analysis), and the proportion of subjects randomized to each group is 50%, then the logrank statistic is approximately normal with mean ${\displaystyle (\log {\lambda })\,{\sqrt {\frac {n\,d}{4}}}}$ and variance 1.[4] For a one-sided level ${\displaystyle \alpha }$ test with power ${\displaystyle 1-\beta }$, the sample size required is ${\displaystyle n={\frac {4\,(z_{\alpha }+z_{\beta })^{2}}{d\log ^{2}{\lambda }}}}$ where ${\displaystyle z_{\alpha }}$ and ${\displaystyle z_{\beta }}$ are the quantiles of the standard normal distribution.

Suppose ${\displaystyle Z_{1}}$ and ${\displaystyle Z_{2}}$ are the logrank statistics at two different time points in the same study (${\displaystyle Z_{1}}$ earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio ${\displaystyle \lambda }$ and ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ are the probabilities that a subject will have an event at the two time points where ${\displaystyle d_{1}\leq d_{2}}$. ${\displaystyle Z_{1}}$ and ${\displaystyle Z_{2}}$ are approximately bivariate normal with means ${\displaystyle \log {\lambda }\,{\sqrt {\frac {n\,d_{1}}{4}}}}$ and ${\displaystyle \log {\lambda }\,{\sqrt {\frac {n\,d_{2}}{4}}}}$ and correlation ${\displaystyle {\sqrt {\frac {d_{1}}{d_{2}}}}}$. Calculations involving the joint distribution are needed to correctly maintain the error rate when the data are examined multiple times within a study by a Data Monitoring Committee.

• The logrank statistic can be derived as the score test for the Cox proportional hazards model comparing two groups. It is therefore asymptotically equivalent to the likelihood ratio test statistic based from that model.
• The logrank statistic is asymptotically equivalent to the likelihood ratio test statistic for any family of distributions with proportional hazard alternative. For example, if the data from the two samples have exponential distributions.
• If ${\displaystyle Z}$ is the logrank statistic, ${\displaystyle D}$ is the number of events observed, and ${\displaystyle {\hat {\lambda }}}$ is the estimate of the hazard ratio, then ${\displaystyle \log {\hat {\lambda }}\approx Z\,{\sqrt {4/D}}}$. This relationship is useful when two of the quantities are known (e.g. from a published article), but the third one is needed.
• The logrank statistic can be used when observations are censored. If censored observations are not present in the data then the Wilcoxon rank sum test is appropriate.
• The logrank statistic gives all calculations the same weight, regardless of the time at which an event occurs. The Peto logrank test statistic gives more weight to earlier events when there are a large number of observations.

The logrank test is based on the same assumptions as the Kaplan-Meier survival curve—namely, that censoring is unrelated to prognosis, the survival probabilities are the same for subjects recruited early and late in the study, and the events happened at the times specified. Deviations from these assumptions matter most if they are satisfied differently in the groups being compared, for example if censoring is more likely in one group than another.[5]

• Kaplan–Meier estimator
• Hazard ratio