Hazard Ratio, Median Ratio and Kaplan-Meier Curves

Reviewed and revised 26 August 2015

OVERVIEW

• Time-to-event curves analyzed by Cox proportional hazards regression are useful for analysing events occurring over time
• uses all available information, including patients who fail to follow up or reach the endpoint (censored data)
• These data are commonly depicted with a Kaplan-Meier curve
• if viewed as a race, the hazard ratio quantifies ‘the odds of winning the race’ whereas the median ratio quantifies the ‘margin of victory’ of the treatment

COX PROPORTIONAL REGRESSION MODEL

• used for time-to-event analysis (alternatives include log-rank and Wilcoxon two-sample test)
• provides an estimate of the hazard ratio and its confidence interval
• avoids bias from loss to follow up
• can incorporate information about subjects that may change over time (time-dependent covariates)
• avoids loss of clinically important information by only analysing data at one point in time (e.g. the end of a trial)

HAZARD RATIO

• a hazard rate is the rate at which a particular event happens
• the hazard ratio = treatment hazard rate/placebo hazard rate
• i.e. the ratio of the particular event taking place in treatment group compared to control group
• need to interpret hazard ratio alongside a measure of time
• used to reflect time survived to an event
• does not indicate how fast something occurs
• commonly used when presenting data from a clinical trial (not the same as a relative risk ratio)

Interpretation

• quantifies ‘the odds of winning the race’ not the margin of victory (see median ratio)
• hazard ratio of 1 = equal event rate between groups
• hazard ratio of 2 = twice as many patients in the active group will have the event compared to the control in the next unit of time
• hazard ratio of 0.5 = half as many patients in the active group are having the event compared to the control in the next unit of time

MEDIAN RATIO

• time-to-event curves can be constructed which allows the ratio of median times between treatment and placebo to be used to measure the magnitude of benefit to patients
• median ratio = placebo median time/treatment median time
• quantifies the ‘margin of victory’ of the treatment (see hazard ratio)

KAPLAN-MEIER CURVE

• length of time from study entry to disease end-point for a treatment and control group
• from this curve, we can derive:

-> median time (time at which 50% of cases resolve)
-> mean time (average resolution time)

• allows comparisons of patients throughout study and provides information on patients who may be lost to follow up

References and Links

Journal articles

• Spruance SL, Reid JE, Grace M, Samore M. Hazard ratio in clinical trials. Antimicrob Agents Chemother. 2004 Aug;48(8):2787-92. PMC478551.