This JAMA Guide to Statistics and Methods explains the meaning underlying the proportional hazards (PH) assumption underlying Cox regression and survival analyses, and proposes that reports of survival differences might replace statistical tests of the PH assumption because they are more meaningful.
The Cox proportional hazards model, introduced in 1972,1 has become the default approach for survival analysis in randomized trials. The Cox model estimates the ratio of the hazard of the event or outcome of interest (eg, death) between 2 treatment groups. Informally, the hazard at any given time is the probability of experiencing the event of interest in the next interval among individuals who had not yet experienced the event by the start of the interval. Because the Cox model requires the hazards in both groups to be proportional, researchers are often asked to “test” whether hazards are proportional.
WHAT DOES IT MEAN THAT HAZARDS ARE PROPORTIONAL?
The hazards are proportional if the hazard ratio remains constant from day 1 of the study until the end of follow-up. In practice, this does not occur for most medical interventions. Three articles previously published in JAMA illustrate different scenarios regarding proportional hazards.
Scenario 1—No Immediate Effect
The Air Force/Texas Coronary Atherosclerosis Prevention Study2 randomly assigned patients with atherosclerotic cardiovascular disease to either statin therapy or placebo. The hazard ratio of a major adverse cardiovascular event was 0.63 (95% CI, 0.50-0.79) for statin vs placebo. However, the cumulative incidences of major adverse cardiovascular event in the statin and placebo groups were almost identical during the first 6 months of follow-up and diverged thereafter. That is, the overall hazard ratio of 0.63 was a weighted average of the time-varying hazard ratios, which were close to 1 in the first months of follow-up and declined later.
Scenario 2—Immediate and Delayed Effects in Opposite Directions
The Norwegian Colorectal Cancer Prevention Trial3 randomly assigned individuals aged 50 to 64 years to flexible sigmoidoscopy screening or no screening. The hazard ratio of colorectal cancer was 0.80 (95% CI, 0.70-0.92) for screening vs no screening. However, the cumulative incidence was greater in the screening group until about 5 years of follow-up and lower after that time. That is, the hazard ratio of 0.80 was a weighted average of the time-varying hazard ratios, which were greater than 1 in the early follow-up and less than 1 in the later follow-up.
Scenario 3—Variations in Disease Susceptibility
A Women’s Health Initiative study4 randomly assigned postmenopausal women to either estrogen plus progestin hormone therapy or placebo. The hazard ratio of coronary heart disease was 1.24 (95% CI, 1.00-1.54) for hormone therapy vs placebo. However, the hazard ratio was 1.8 during the first year and 0.70 after 5 years of follow-up. The overall hazard ...