Estimating Hazard Ration (HR) and Confidence Interval from Other Information

Scenario 1: Approximately calculate HR and confidence interval based on median survival time and number of event.[1, 2]
Median survival time of group 1
Number of events in group 1
Median survival time of group 2
Number of events in group 2
Confidence level

HR
Lower limit Upper limit

  • The ratio of median survival times is approximately equivalent to the HR value. Limitations can be found in reference [1].
  • This approximation assumes that the survival curve follows an exponential distribution.

Scenario 2: Approximately calculate confidence interval of HR based on HR value and log-rank's p-value[3,4]
HR
P value from Log-rank test
Confidence Level

Standard error of Log(HR)
Lower Limit Upper Limit

Scenario 3: Approximately calculate confidence interval of HR based on median survival time and log-rank's p-value.[1,2,3,4]
Median survival time of group 1
Median survival time of group 2
P value from Log-rank test
Confidence Level

HR
Lower Limit Upper Limit

Scenario 4: Approximately calculate HR value based on number of event and follow-up time.[5]
Number of events in group 1 Follow-up person-years for group 1
Number of events in group 2 Follow-up person-years for group 2

HR

  • Incidence rate = Number of events / Follow-up person-years
  • The incidence rate ratio is approximately equivalent to the HR value. Limitations are detailed in reference [5]
  • Follow-up person-years: The total observation time for all subjects in the study. If not reported, it is assumed that the person-years for subjects with events and dropouts are half of the total trial duration, while the rest are considered equal to the total trial duration. This assumption is based on the method from reference [1].

Reference:

[1] Cortés J, González JA, Campbell MJ, Cobo E. A hazard ratio was estimated by a ratio of median survival times, but with considerable uncertainty. J Clin Epidemiol. 2014 Oct;67(10):1172-7.
[2] https://www.graphpad.com/support/faq/the-confidence-interval-for-the-ratio-of-median-survival-times/
[3] https://www.bmj.com/content/suppl/2007/01/22/bmj.39063.689375.55.dc1/fdetails.pdf
[4] Gillies CL, Abrams KR, Lambert PC, Cooper NJ, Sutton AJ, Hsu RT, Khunti K. Pharmacological and lifestyle interventions to prevent or delay type 2 diabetes in people with impaired glucose tolerance: systematic review and meta-analysis. BMJ. 2007 Feb 10;334(7588):299.
[5] Bender, R., Beckmann, L. Limitations of the incidence density ratio as approximation of the hazard ratio. Trials 20, 485 (2019). https://doi.org/10.1186/s13063-019-3590-2

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