*New York Times*, LCZ696 “has shown a striking efficacy in prolonging the lives of people with heart failure and could replace what has been the bedrock treatment for more than 20 years.” [1] Specifically, more than 8,400 patients in 47 countries enrolled in a randomized, double-blind experiment in which they received either LCZ696 or an ACE inhibitor called enalapril (in addition to whatever else their doctors prescribed).

The trial was halted after a median follow-up time of 27 months “ because the boundary for an overwhelming benefit with LCZ696 had been crossed.” [2] “By that point, 21.8 percent of those who received LCZ696 had died from a cardiovascular cause or had been hospitalized for worsening heart failure. That figure was 26.5 percent for those receiving enalapril. That represents a 20 percent relative reduction in risk using a statistical measure called the hazard ratio.” [1]

This is good news for patients (if the drug receives regulatory approval and performs as expected in practice). But the account in the

*Times*poses a small statistical puzzle. How does the difference between 21.8 and 26.5 percentage points translate into “a 20 percent relative reduction in risk”? The average risk across patients dropped by 26.5 – 21.8 = 4.7 percentage points. This absolute reduction is appreciable, but 4.7 percentage points is not 20% of the original 26.5 percent risk of hospitalizations or deaths in the control group (4.7 / 26.5 = 17.7%). What accounts for the discrepancy?

The answer lies in the details of a technique known in biostatistics as survival analysis. The statistical technique is not limited to the analysis of death rates. It can be applied to all sorts of situations involving different times to some outcome. The outcome can be the overruling of a Supreme Court case, the firing of a worker, or the exoneration of a prison inmate sentenced to die, to pick a few examples from forensic statistics.

So what does the 20% “relative reduction in risk” cited in the

*Times*article mean? Well, a hazard function is the probability that if you survive to a given time

*t*(the event in question has not already occurred), you will survive in the next instant. A hazard ratio is the ratio of the hazard in the treatment group to the hazard in the control group at

*t*. The heart failure study used an estimation procedure known as proportional hazards regression, which assumes that the hazard in one group is a constant proportion of the hazard in the other group. Under this assumption, in a clinical trial where death is the endpoint, the hazard ratio indicates the relative likelihood of death in treated versus control subjects at any given point in time.

Thus, unlike the ordinary relative risk discussed in many court opinions, the “hazard ratio” is not simply the proportion with a disease in an exposed group divided by the proportion in an unexposed group. In the LCZ696 study, the hazard ratio was 0.80, meaning that the probability that a randomly selected patient taking LCZ696 would die from or be hospitalized for heart failure the next day is 80% of the probability for a randomly selected patient taking enalapril. To put it another way, the probability of hospitalization or death tomorrow from heart failure drops by 20% when LCZ696 is substituted for enalapril.

Yet a third formulation is that the odds that a randomly selected patient treated with LCZ696 will be hospitalized or die sooner than a randomly selected control patient are 0.8 (to 1) — that's 4 to 5, corresponding to a probability of 4/9 = 44%. [3]

How long either patient can expect to live and avoid hospitalization from heart failure is another story. As one article on hazard ratios explains, “[t]he difference between hazard-based and time-based measures is analogous to the odds of winning a race and the margin of victory.” [3] By itself, the hazard ratio picks the winning horse (probably), but it does not give the number of lengths for its expected success.

**References**

- Andrew Pollack, New Novartis Drug Effective in Treating Heart Failure, N.Y. Times, Aug.31, 2014, at A4
- John J.V. McMurray et al., Angiotensin–Neprilysin Inhibition versus Enalapril in Heart Failure, New Engl. J. Med., Aug. 30, 2014
- Spotswood L. Spruance et al., Hazard Ratio in Clinical Trials, 48 Antimicrobial Agents and Chemotherapy 2787 (2004)