Commentary: The Hazards of Survival Comparisons

Abstract

A standard way to present time-to-event analyses—such as comparisons of two survival distributions—is to use a hazardratio(HR).Thisisaweightedaverage(overtime)ofthe corresponding hazard functions. The term “hazard function” stems from reliability theory, where it is also called failure rate. Actuaries use the term “force of mortality” for the same concept. The annual premium that you pay for life insurance is a bet that you’re going to die in the next year. Appropriate odds on the bet (padded somewhat so the insurance company can make money) are based on the proportion of peoplelikeyouwhoarealiveatyourageandwhowilldieinthe next year. That’s your hazard of dying, and it depends on your age. Next year it will be different. Picture a curve (or function) showing the hazard of dying within the next small interval of time, for each time from birth to age 100. The same kind of curve applies for the occurrence of an event (recurrence, death, or whatever) in a clinical trial for patients in a particular treatment group. An estimate of the population hazard over the next year, say,isthenumberofeventsinthatyeardividedbythenumber of patients at risk for the event at the beginning of the year. Time 0 is the time of randomization, for example. Comparing two treatment groups means comparing the two curves. One such comparison is the ratio of the two curves, the hazard ratio function (HRF). But that’s still a curve, and it may take on a different value for each time point. The HRF may be bigger than 1.0 sometimes (at times when the denominator treatment is looking better) and at other times it may be smaller than 1.0 (when the numerator treatment is lookingbetter).Ifso,whichtreatmentisbetteroverall?And by how much?