Hazard Ratios: an overview
As you read in the Medpage summary of treatments for Diabetes, hazard ratios are everywhere in the medical literature. But what are they and where do they come from? The goal of this handout is to explain the origins of the methods used to understand why hazard ratios are so common, the methods of calculating hazard ratios, and what the hazard ratio means.
Hazard ratios are a specific type of relative risk that are calculated using a statistical technique known as Survival Analysis. Survival analysis was created by people doing time-to-death analyses for one or more groups, which is where the name “Survival” comes from. However, survival analysis' usefulness is not limited to studies in which subjects die. In fact, survival analysis can be applied to any situation where the subjects will have different times to some event or outcome of interest. The outcome can be a positive event (age until first child born, recovery from an infection) or a negative event (heart attack, cancer recurrence). Survival analysis accounts for the reality that some subjects may drop out of the study before the event of interest happens, or that the study may end before all of the subjects experience the event. Survival analysis is often used in clinical trials, for instance, to assess the effectiveness of a new drug or treatment, and hazard ratios are often the statistic of choice in reporting the results of clinical trials.
Survival analysis keeps track of how many subjects have NOT experienced the event at a given time or during a given time interval. The data is then plotted over the entire time of the study, and the results are graphed as a decreasing curve. The curve decreases since as time increases, so does the number who have experienced the event. For an example of survival curves, see figure 1 (page 2788) in the article “Hazard Ratio in Clinical Trials” by S.L. Spruance, J.E. Reid, M. Grace, and M. Samore. The accompanying spreadsheet will familiarize you with how to construct a survival curve.
In a clinical trial, we often have two groups: one that receives a treatment of some sort (the treatment group) and one that received a placebo (the control group). Each group has lots of subjects, and the event of interest can happen at any time for a given subject. Most trials are designed so that there is no difference between the two groups other than the treatment. Therefore, we'd expect that the two groups would have the same rate if the treatment had no effect. The rates of the two groups can be therefore used to assess whether the treatment had an effect. The ratio of the two rates, the treatment hazard rate divided by the placebo hazard rate, is called the hazard ratio, and this is the statistics used for further analysis
A hazard ratio of one means that there is no difference between the two groups. A hazard ratio greater than one indicates that desired event is happening faster for the treatment group than for the control group. A hazard ratio less than one indicates that the event of interest is happening slower for the treatment group than for the control group. Note that these ratios are comparisons between the two groups and give no indication of how long it takes for the average subject in either group.
Let's look at an example. Let's say we want to test a new treatment for a disease called sneezeathon. The main symptom of this disease is that, whenever the infected person is awake, he or she will sneeze about once every 15 minutes. 50% of people recover after about 9 days, with most people recovering somewhere between 5 and 15 days. Recovery is measured by not having sneezed within 3 hours.
A new treatment is proposed and we're hoping that the new treatment will reduce the time to recovery. That is, we're hoping that recovery will be faster for the treatment group than for the control group. In terms of the hazard ratios, we expect the hazard ratio to be significantly larger than one (a statistical analysis would determine how large the value needs to be to be significantly larger). Let's say that we run our experiment and, after doing the survival analysis, our hazard ratio turns out to be 3. This means that at any given time, a treatment patient who has not yet recovered from sneezeathon is three times as likely to recover during the next (very small) time tinterval than a patient who is in the control group. If the patients still had sneezeathon at 5 days, the patients who were given the treatment are approximately 3 times as likely to recover by day 6 than those who weren't treated. Similarly, for those patients that still had sneezeathon at 15 days after infection, those given the treatment were also approximately three times as likely to recover in the next day than those who were in the control group.
So, in the above example, we see that those who were given the treatment were approximately 3 times more likely than the non-treated patients to get better in the next instant at any given point in time. This information says nothing about how fast the treated patients actually got well, only that they were about 3 times as likely to get better on any given day. How fast the treated patients recover depends on the rate at which the untreated patients recover. Therefore, the hazard ratio is a proportional measure. Parts A and B of figure 2 of the above paper (page 2789) shows us that very different survival curves can have the same hazard ratio. They have the same hazard ratio since the ratio is only a measure of the treatment divided by the control.
Sometimes it's easier to understand hazard ratios when they are translated into odds or probabilities. In other words, you can get the odds that the treatment group will experience the event before the control group from the hazard ratio. If the odds are 50% (i.e. 1:1), then the groups are the same. Keep in mind that probabilities range from zero to one, and are often represented as a percentage (0 % to 100%). The formula for translating a hazard ratio to a probability is:
probability = (hazard ratio) / (1 + hazard ratio).
Applying this formula to our sneezeathon example above, we see that the probability that at treated patient will heal before an untreated patient is:
probability = (3)/(1+3) = ¾ = 0.75
So there is a 75% chance that the the treated patient will heal before the control patients. For more on the practical applications of hazard ratios, please see the section “Questions Asked By Patients” on page 2791 of the paper “Hazard Ratio in Clinical Trials” by S.L. Spruance et al.
Hazard ratios are often used in medical studies because they describe nicely the results of clinical trials. Understanding that these are relative measures of treatment versus control and how to translate it into odds is essential to applying the results.