Log-rank test

The Log-rank test is a statistical test used to compare the survival distributions of two or more groups. It is widely used in biostatistics and epidemiology to analyze randomized controlled trials and observational studies in which the endpoint is time until an event occurs, commonly referred to as "failure time data."

Overview

The Log-rank test, also known as the Mantel-Cox test, focuses on comparing the survival curves of different groups. It is a non-parametric test that is based on the ranks of the data rather than their numerical values, making it particularly useful when the data do not conform to a normal distribution. The test is most effective when the assumption that the hazard ratios are proportional over time (proportional hazards assumption) is reasonable.

Methodology

The test statistic for the Log-rank test is calculated by comparing observed events in each group at each observed event time to what would be expected under the null hypothesis of no difference between the groups. If there are \( k \) groups being compared, the test statistic approximately follows a chi-square distribution with \( k-1 \) degrees of freedom under the null hypothesis.

The formula for the test statistic is: \[ Q = \sum \frac{(O_i - E_i)^2}{E_i} \] where \( O_i \) is the observed number of events in group \( i \) and \( E_i \) is the expected number of events in group \( i \), calculated under the null hypothesis.

Applications

The Log-rank test is commonly used in the analysis of clinical trials where the interest lies in comparing the efficacy of different treatments over time. It is also used in other fields such as engineering and sociology, where the timing of events is crucial.

Limitations

While the Log-rank test is a powerful tool for analyzing survival data, it does have limitations. It is most suitable when the proportional hazards assumption holds. If this assumption is violated, other methods such as the Cox proportional hazards model may be more appropriate. Additionally, the test may have reduced power in situations with small sample sizes or when the hazard rates cross.

See Also

• Kaplan-Meier estimator
• Cox proportional hazards model
• Survival function
• Hazard ratio