Intended for healthcare professionals

Endgames Statistical Question

# The log rank test

BMJ 2010; 341 (Published 21 July 2010) Cite this as: BMJ 2010;341:c3773
1. Philip Sedgwick, senior lecturer in medical statistics
1. 1Centre for Medical and Healthcare Education, St George’s, University of London, Tooting, London
1. p.sedgwick{at}sgul.ac.uk

The previous two statistical questions described survival (time to event) data.1 2 The example used was a randomised controlled trial that evaluated the effectiveness of an integrated care programme compared with usual care in facilitating the return to work of patients with chronic low back pain.3 The integrated care programme was a combined patient and workplace directed intervention.

Trial participants were adults aged between 18 and 65 years who had experienced low back pain for more than 12 weeks, were in paid work, and were absent or partially absent from work. The primary outcome was duration of time off work—that is, from randomisation until a fully sustained return to work.

Participants were followed for 12 months. The survival (time to event) data for the two treatment groups were compared statistically using the log rank test (P=0.003). Days until a fully sustainable return to work for the group on the integrated care programme and the usual care group were plotted as Kaplan-Meier survival curves.

Kaplan-Meier survival curves showing the time until a fully sustained return to work for patients in the integrated care programme and those receiving usual care. Censored observations are displayed by crosses on the curves

Which of the following statements, if any, are true?

• a) The log rank test facilitates testing of the null hypothesis—in this case, that there are no differences in survival times in the population between the integrated care programme and usual care

• b) Censored observations were excluded before the interventions were compared statistically

• c) A statistically significant difference in survival times existed between interventions at the 5% level of significance

• d) The log rank test provides an estimate of the magnitude of the difference in survival times between interventions

Answers a and c are true; b and d are false.

The log rank test is a statistical test used to compare the survival times between two treatment groups. “Survival time” is the time it takes an individual to reach an end point. For the above study, the end point was a fully sustained return to work following work absenteeism owing to chronic low back pain. As with traditional hypothesis testing, the log rank test starts at the position of equipoise.4 The null hypothesis states that there are no differences in survival times between people on the integrated care programme and those who receive standard care in the population where the sample was obtained (a is true). The alternative hypothesis is two sided and states that a difference exists—that in the population where the samples were obtained, the survival times for people receiving the two interventions are not equal.

The log rank test assesses whether there are any differences between the two interventions at any point in time during the 12 month study period. All survival times are included in the analysis, including both exact and censored ones (b is false). Censored survival times provide useful information up to the point of censoring and are, therefore, included in the calculation of survival curves and in statistical comparison of treatment groups.

The traditional level of significance for statistical hypothesis testing is 0.05 (that is, 5%), which is termed the critical level of significance.5 The resulting P value for the log rank test was 0.003. Given that this value is smaller than 0.05, the result was statistically significant at the 5% level of significance (c is true). We thus reject the null hypothesis in favour of the alternative, and conclude that the survival times for the two interventions are not equal. The figure shows that the integrated care group as a whole had shorter survival times than the usual care group.

The log rank test is purely a test of significance. It cannot provide an estimate of the magnitude of difference in survival times between interventions (d is false). The hazard ratio is probably the most commonly used method of providing an estimate of the magnitude of the difference in survival times between two interventions.

## Notes

Cite this as: BMJ 2010;341:c3773

## Footnotes

• Competing interests: None declared.

## References

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